,,the ,,macmillan ,,company ,,new ,,york 4 ,,boston 4 ,,chicago 4 ,,dallas ,,atlanta 4 ,,san ,,francisco ,,macmillan ,,co41 ,,limited ,,london 4 ,,bombay 4 ,,calcutta ,,melbourne ,,the ,,macmillan ,,company ,,of ,,canada1 ,,limited ,,toronto ,,the ,,psychology ,,of ,,arithmetic ,,by ,,edward ,l4 ,,thorndike ,,teachers ,,college1 ,,columbia ,,university ,new ,york ,,the ,,macmillan ,,company #aibi ,all rights reserved ,,copyright1 #aibb1 ,,by ,,the ,,macmillan ,,company4 ,set up and electrotyped4 ,published ,january1 #aibb4 ,reprinted ,october1 #aibd2 ,may1 #aibf2 ,august1 #aibg2 ,october1 #aibi4 4 ,,printed ,,in ,,the ,,united ,,states ,,of ,,america 4 ,,preface ,within recent years there have been three lines of advance in psychology which are of notable significance for teaching4 ,the first is the new point of view concerning the general process of learning4 ,we now understand that learning is essentially the formation of connections or bonds between situations and responses1 that the satisfyingness of the result is the chief force that forms them1 and that habit rules in the realm of thought as truly and as fully as in the realm of action4 ,the second is the great increase in knowledge of the amount1 rate1 and conditions of improvement in those organized groups or hierarchies of habits which we call abilities1 such as ability to add or ability to read4 ,practice and improvement are no longer vague generalities1 but concern changes which are definable and measurable by standard tests and scales4 ,the third is the better understanding of the so-called higher processes of analysis1 abstraction1 the formation of general notions1 and reasoning4 ,the older view of a mental chemistry whereby sensations were compounded into percepts1 percepts were duplicated by images1 percepts and images were amalgamated into abstractions and concepts1 and these were manipulated by reasoning1 has given way to the understanding of the laws of response to elements or aspects of situations and to many situations or elements thereof in combination4 ,james' view of reasoning as selection of essentials and thinking things together in a revised and clarified form has important applications in the teaching of all the school subjects4 ,this book presents the applications of this newer dynamic psychology to the teaching of arithmetic4 ,its contents are substantially what have been included in a course of lectures on the psychology of the elementary school subjects given by the author for some years to students of elementary education at ,teachers ,college4 ,many of these former students1 now in supervisory charge of elementary schools1 have urged that these lectures be made available to teachers in general4 ,so they are now published in spite of the author's desire to clarify and reinforce certain matters by further researches4 ,a word of explanation is necessary concerning the exercises and problems cited to illustrate various matters1 especially erroneous pedagogy4 ,these are all genuine1 having their source in actual textbooks1 courses of study1 state examinations1 and the like4 ,to avoid any possibility of invidious comparisons they are not quotations1 but equivalent problems such as represent accurately the spirit and intent of the originals4 ,i take pleasure in acknowledging the courtesy of ,mr4 ,s4 ,a4 ,courtis1 ,ginn and ,company1 ,d4 ,c4 ,heath and ,company1 ,the ,macmillan ,company1 ,the ,oxford ,university ,press1 ,rand1 ,mc,nally and ,company1 ,dr4 ,c4 ,w4 ,stone1 ,the ,teachers ,college ,bureau of ,publications1 and ,the ,world ,book ,company1 in permitting various quotations4 ,,edward ,l4 ,,thorndike4 ,,teachers ,,college ,,columbia ,,university ,april #a1 #aibj ,,contents ,,chapter ,,page ,,introduction3 ,,the ,,psychology ,,of ,,the ,,elementary ,,school ,,subjects xi ,i4 ,,the ,,nature ,,of ,,arithmetical ,,abilities #a ,knowledge of the ,meanings of ,numbers ,arithmetical ,language ,problem ,solving ,arithmetical ,reasoning ,summary ,the ,sociology of ,arithmetic ,,ii4 ,,the ,,measurement ,,of ,,arithmetical ,,abilities #bg ,a ,sample ,measurement of an ,arithmetical ,ability ,ability to ,add ,integers ,measurements of ,ability in ,computation ,measurements of ,ability in ,applied ,arithmetic3 the ,solution of ,problems ,,iii4 ,,the ,,constitution ,,of ,,arithmetical ,,abilities #ea ,the ,elementary ,functions of ,arithmetical ,learning ,knowledge of the ,meaning of a ,fraction ,learning the ,processes of ,computation ,,iv4 ,,the ,,constitution ,,of ,,arithmetical ,,abilities 7continued7 #gj ,the ,selection of the ,bonds to ,be ,formed ,the ,importance of ,habit ,formation ,desirable ,bonds ,now ,often ,neglected ,wasteful and ,harmful ,bonds ,guiding ,principles ,v4 ,,the ,,psychology ,,of ,,drill ,,in ,,arithmetic3 ,,the ,,strength ,,of ,,bonds #ajb ,the ,need of ,stronger ,elementary ,bonds ,early ,mastery ,the ,strength of ,bonds for ,temporary ,service ,the ,strength of ,bonds with ,technical ,facts and ,terms ,the ,strength of ,bonds ,concerning the ,reasons for ,arithmetical ,processes ,propaedeutic ,bonds ,,vi4 ,,the ,,psychology ,,of ,,drill ,,in ,,arithmetic3 ,,the ,,amount ,,of ,,practice ,,and ,,the ,,organization ,,of ,,abilities #abb ,the ,amount of ,practice ,under-learning and ,over-learning ,the ,organization of ,abilities ,,vii4 ,,the ,,sequence ,,of ,,topics3 ,,the ,,order ,,of ,,formation ,,of ,,bonds #ada ,conventional versus ,effective ,orders ,decreasing ,interference and ,increasing ,facilitation ,interest ,general ,principles ,,viii4 ,,the ,,distribution ,,of ,,practice #aef ,the ,problem ,sample ,distributions ,possible ,improvements ,,ix4 ,,the ,,psychology ,,of ,,thinking3 ,,abstract ,,ideas ,,and ,,general ,,notions ,,in ,,arithmetic #afi ,responses to ,elements and ,classes ,facilitating the ,analysis of ,elements ,systematic and ,opportunistic ,stimuli to ,analysis ,adaptations to ,elementary-school ,pupils ,x4 ,,the ,,psychology ,,of ,,thinking3 ,,reasoning ,,in ,,arithmetic #ahe ,the ,essentials of ,arithmetical ,reasoning ,reasoning as the ,cooeperation of ,organized ,habits ,,xi4 ,,original ,,tendencies ,,and ,,acquisitions ,,before ,,school #aie ,the ,utilization of ,instinctive ,interests ,the ,order of ,development of ,original ,tendencies ,inventories of ,arithmetical ,knowledge and ,skill ,the ,perception of ,number and ,quantity ,the ,early ,awareness of ,number ,,xii4 ,,interest ,,in ,,arithmetic #bji ,censuses of ,pupils' ,interests ,relieving ,eye ,strain ,significance for ,related ,activities ,intrinsic ,interest in ,arithmetical ,learning ,,xiii4 ,,the ,,conditions ,,of ,,learning #bbg ,external ,conditions ,the ,hygiene of the ,eyes in ,arithmetic ,the ,use of ,concrete ,objects in ,arithmetic ,oral1 ,mental1 and ,written ,arithmetic ,,xiv4 ,,the ,,conditions ,,of ,,learning3 ,,the ,,problem ,,attitude #bff ,illustrative ,cases ,general ,principles ,difficulty and ,success as ,stimuli ,false ,inferences ,,xv4 ,,individual ,,differences #bhe ,nature and ,amount ,differences within ,one ,class ,the ,causes of ,individual ,differences ,the ,interrelations of ,individual ,differences ,,bibliography ,,of ,,references #cjb ,,index #caa ,,general ,,introduction ,,the ,,psychology ,,of ,,the ,,elementary ,,school ,,subjects ,the psychology of the elementary school subjects is concerned with the connections whereby a child is able to respond to the sight of printed words by thoughts of their meanings1 to the thought of six and eight by thinking fourteen1 to certain sorts of stories1 poems1 songs1 and pictures by appreciation thereof1 to certain situations by acts of skill1 to certain others by acts of courtesy and justice1 and so on and on through the series of situations and responses which are provided by the systematic training of the school subjects and the less systematic training of school life during their study4 ,the aims of elementary education1 when fully defined1 will be found to be the production of changes in human nature represented by an almost countless list of connections or bonds whereby the pupil thinks or feels or acts in certain ways in response to the situations the school has organized and is influenced to think and feel and act similarly to similar situations when life outside of school confronts him with them4 ,we are not at present able to define the work of the elementary school in detail as the formation of such and such bonds between certain detached situations and certain specified responses4 ,as elsewhere in human learning1 we are at present forced to think somewhat vaguely in terms of mental functions1 like ability to read the vernacular1 ability to spell common words1 ability to add1 subtract1 multiply1 and divide with integers1 knowledge of the history of the ,united ,states1 honesty in examinations1 and appreciation of good music1 defined by some general results obtained rather than by the elementary bonds which constitute them4 ,the psychology of the school subjects begins where our common sense knowledge of these functions leaves off and tries to define the knowledge1 interest1 power1 skill1 or ideal in question more adequately1 to measure improvement in it1 to analyze it into its constituent bonds1 to decide what bonds need to be formed and in what order as means to the most economical attainment of the desired improvement1 to survey the original tendencies and the tendencies already acquired before entrance to school which help or hinder progress in the elementary school subjects1 to examine the motives that are or may be used to make the desired connections satisfying1 to examine any other special conditions of improvement1 and to note any facts concerning individual differences that are of special importance to the conduct of elementary school work4 ,put in terms of problems1 the task of the psychology of the elementary school subjects is1 in each case3-- 7#a7 ,what is the function8 ,for example1 just what is ability to read8 ,just what does the understanding of decimal notation mean8 ,just what are the moral effects to be sought from the teaching of literature8 7#b7 ,how are degrees of ability or attainment1 and degrees of progress or improvement in the function or a part of the function measured8 ,for example1 how can we determine how well a pupil should write1 or how hard words we expect him to spell1 or what good taste we expect him to show8 ,how can we define to ourselves what knowledge of the meaning of a fraction we shall try to secure in grade #d8 7#c7 ,what can be done toward reducing the function to terms of particular situation-response connections1 whose formation can be more surely and easily controlled8 ,for example1 how far does ability to spell involve the formation one by one of bonds between the thought of almost every word in the language and the thought of that word's letters in their correct order2 and how far does1 say1 the bond leading from the situation of the sound of ceive in receive and deceive to their correct spelling insure the correct spelling of that part of perceive8 ,does ability to add involve special bonds leading from #bg and #d to #ca1 from #bg and #e to #cb1 and #bg and #f to #cc2 or will the bonds leading from #g and #d to #aa1 #g and #e to #ab and #g and #f to #ac 7each plus a simple inference7 serve as well8 ,what are the situations and responses that represent in actual behavior the quality that we call school patriotism8 7#d7 ,in almost every case a certain desired change of knowledge or skill or power can be attained by any one of several sets of bonds4 ,which of them is the best8 ,what are the advantages of each8 ,for example1 learning to add may include the bonds #j and #j are #j1 #j and #a are #a1 #j and #b are #b1 #a and #j are #a1 #b and #j are #b1 etc42 or these may be all left unformed1 the pupil being taught the habits of entering #j as the sum of a column that is composed of zeros and otherwise neglecting #j in addition4 ,are the rules of usage worth teaching as a means toward correct speech1 or is the time better spent in detailed practice in correct speech itself8 7#e7 ,a bond to be formed may be formed in any one of many degrees of strength4 ,which of these is1 at any given stage of learning the subject1 the most desirable1 all things considered8 ,for example1 shall the dates of all the early settlements of ,north ,america be learned so that the exact year will be remembered for ten years1 or so that the exact date will be remembered for ten minutes and the date with an error plus or minus of ten years will be remembered for a year or two8 ,shall the tables of inches1 feet1 and yards1 and pints1 quarts1 and gallons be learned at their first appearance so as to be remembered for a year1 or shall they be learned only well enough to be usable in the work of that week1 which in turn fixes them to last for a month or so8 ,should a pupil in the first year of study of ,french have such perfect connections between the sounds of ,french words and their meanings that he can understand simple sentences containing them spoken at an ordinary rate of speaking8 ,or is slow speech permissible1 and even imperative1 on the part of the teacher1 with gradual increase of rate8 7#f7 ,in almost every case1 any set of bonds may produce the desired change when presented in any one of several orders4 ,which is the best order8 ,what are the advantages of each8 ,certain systems for teaching handwriting perfect the elementary movements one at a time and then teach their combination in words and sentences4 ,others begin and continue with the complex movement-series that actual words require4 ,what do the latter lose and gain8 ,the bonds constituting knowledge of the metric system are now formed late in the pupil's course4 ,would it be better if they were formed early as a means of facilitating knowledge of decimal fractions8 7#g7 ,what are the original tendencies and pre-school acquisitions upon which the connection-forming of the elementary school may be based or which it has to counteract8 ,for example1 if a pupil knows the meaning of a heard word1 he may read it understandingly from getting its sound1 as by phonic reconstruction4 ,what words does the average beginner so know8 ,what are the individual differences in this respect8 ,what do the instincts of gregariousness1 attention-getting1 approval1 and helpfulness recommend concerning group-work versus individual-work1 and concerning the size of a group that is most desirable8 ,the original tendency of the eyes is certainly not to move along a line from left to right of a page1 then back in one sweep and along the next line4 ,what is their original tendency when confronted with the printed page1 and what must we do with it in teaching reading8 7#h7 ,what armament of satisfiers and annoyers1 of positive and negative interests and motives1 stands ready for use in the formation of the intrinsically uninteresting connections between black marks and meanings1 numerical exercises and their answers1 words and their spelling1 and the like8 ,school practice has tried1 more or less at random1 incentives and deterrents from quasi-physical pain to the most sentimental fondling1 from sheer cajolery to philosophical argument1 from appeals to assumed savage and primitive traits to appeals to the interest in automobiles1 flying-machines1 and wireless telegraphy4 ,can not psychology give some rules for guidance1 or at least limit experimentation to its more hopeful fields8 7#i7 ,the general conditions of efficient learning are described in manuals of educational psychology4 ,how do these apply in the case of each task of the elementary school8 ,for example1 the arrangement of school drills in addition and in short division in the form of practice experiments has been found very effective in producing interest in the work and in improvement at it4 ,in what other arithmetical functions may we expect the same8 7#aj7 ,beside the general principles concerning the nature and causation of individual differences1 there must obviously be1 in existence or obtainable as a possible result of proper investigation1 a great fund of knowledge of special differences relevant to the learning of reading1 spelling1 geography1 arithmetic1 and the like4 ,what are the facts as far as known8 ,what are the means of learning more of them8 ,courtis finds that a child may be specially strong in addition and yet be specially weak in subtraction in comparison with others of his age and grade4 ,it even seems that such subtle and intricate tendencies are inherited4 ,how far is such specialization the rule8 ,is it1 for example1 the case that a child may have a special gift for spelling certain sorts of words1 for drawing faces rather than flowers1 for learning ancient history rather than modern8 99 99 99 99 99 ,such are our problems3 this volume discusses them in the case of arithmetic4 ,the student who wishes to relate the discussion to the general pedagogy of arithmetic may profitably read1 in connection with this volume3 ,the ,teaching of ,elementary ,mathematics1 by ,d4 ,e4 ,smith '#ja1 ,the ,teaching of ,primary ,arithmetic1 by ,h4 ,suzzallo '#aa1 ,how to ,teach ,arithmetic1 by ,j4 ,c4 ,brown and ,l4 ,d4 ,coffman '#ad1 ,the ,teaching of ,arithmetic1 by ,paul ,klapper '#af1 and ,the ,new ,methods in ,arithmetic1 by the author '#ba4 ,,the ,,psychology ,,of ,,arithmetic ,,chapter ,i ,,the ,,nature ,,of ,,arithmetical ,,abilities ,according to common sense1 the task of the elementary school is to teach3--7#a7 the meanings of numbers1 7#b7 the nature of our system of decimal notation1 7#c7 the meanings of addition1 subtraction1 multiplication1 and division1 and 7#d7 the nature and relations of certain common measures2 to secure 7#e7 the ability to add1 subtract1 multiply1 and divide with integers1 common and decimal fractions1 and denominate numbers1 7#f7 the ability to apply the knowledge and power represented by 7#a7 to 7#e7 in solving problems1 and 7#g7 certain specific abilities to solve problems concerning percentage1 interest1 and other common occurrences in business life4 ,this statement of the functions to be developed and improved is sound and useful so far as it goes1 but it does not go far enough to make the task entirely clear4 ,if teachers had nothing but the statement above as a guide to what changes they were to make in their pupils1 they would often leave out important features of arithmetical training1 and put in forms of training that a wise educational plan would not tolerate4 ,it is also the case that different leaders in arithmetical teaching1 though they might all subscribe to the general statement of the previous paragraph1 certainly do not in practice have identical notions of what arithmetic should be for the elementary school pupil4 ,the ordinary view of the nature of arithmetical learning is obscure or inadequate in four respects4 ,it does not define what 'knowledge of the meanings of numbers' is2 it does not take account of the very large amount of teaching of language which is done and should be done as a part of the teaching of arithmetic2 it does not distinguish between the ability to meet certain quantitative problems as life offers them and the ability to meet the problems provided by textbooks and courses of study2 it leaves 'the ability to apply arithmetical knowledge and power' as a rather mystical general faculty to be improved by some educational magic4 ,the four necessary amendments may be discussed briefly4 ,,knowledge ,,of ,,the ,,meanings ,,of ,,numbers ,knowledge of the meanings of the numbers from one to ten may mean knowledge that 'one' means a single thing of the sort named1 that two means one more than one1 that three means one more than two1 and so on4 ,this we may call the series meaning4 ,to know the meaning of 'six' in this sense is to know that it is one more than five and one less than seven--that it is between five and seven in the number series4 ,or we may mean by knowledge of the meanings of numbers1 knowledge that two fits a collection of two units1 that three fits a collection of three units1 and so on1 each number being a name for a certain sized collection of discrete things1 such as apples1 pennies1 boys1 balls1 fingers1 and the other customary objects of enumeration in the primary school4 ,this we may call the collection meaning4 ,to know the meaning of six in this sense is to be able to name correctly any collection of six separate1 easily distinguishable individual objects4 ,in the third place1 knowledge of the numbers from one to ten may mean knowledge that two is twice whatever is called one1 that three is three times whatever is one1 and so on4 ,this is1 of course1 the ratio meaning4 ,to know the meaning of six in this sense is to know that if is one1 a line half a foot long is six1 that if is one1 is about six1 while if is one1 is about six1 and the like4 ,in the fourth place1 the meaning of a number may be a smaller or larger fraction of its implications--its numerical relations1 facts about it4 ,to know six in this sense is to know that it is more than five or four1 less than seven or eight1 twice three1 three times two1 the sum of five and one1 or of four and two1 or of three and three1 two less than eight--that with four it makes ten1 that it is half of twelve1 and the like4 ,this we may call the 'nucleus of facts' or relational meaning of a number4 ,ordinary school practice has commonly accepted the second meaning as that which it is the task of the school to teach beginners1 but each of the other meanings has been alleged to be the essential one--the series idea by ,phillips '#ig1 the ratio idea by ,mc,lellan and ,dewey '#ie and ,speer '#ig1 and the relational idea by ,grube and his followers4 ,this diversity of views concerning what the function is that is to be improved in the case of learning the meanings of the numbers one to ten is not a trifling matter of definition1 but produces very great differences in school practice4 ,consider1 for example1 the predominant value assigned to counting by ,phillips in the passage quoted below1 and the samples of the sort of work at which children were kept employed for months by too ardent followers of ,speer and ,grube4 ,,the ,,series ,,idea ,,overemphasized ,this is essentially the counting period1 and any words that can be arranged into a series furnish all that is necessary4 ,counting is fundamental1 and counting that is spontaneous1 free from sensible observation1 and from the strain of reason4 ,a study of these original methods shows that multiplication was developed out of counting1 and not from addition as nearly all textbooks treat it4 ,multiplication is counting4 ,when children count by #d's1 etc41 they accent the same as counting gymnastics or music4 ,when a child now counts on its fingers it simply reproduces a stage in the growth of the civilization of all nations4 ,i would emphasize again that during the counting period there is a somewhat spontaneous development of the number series-idea which ,preyer has discussed in his ,arithmogenesis2 that an immense momentum is given by a systematic series of names2 and that these names are generally first learned and applied to objects later4 ,a lady teacher told me that the ,superintendent did not wish the teachers to allow the children to count on their fingers1 but she failed to see why counting with horse-chestnuts was any better4 ,her children could hardly avoid using their fingers in counting other objects yet they followed the series to #ajj without hesitation or reference to their fingers4 ,this spontaneous counting period1 or naming and following the series1 should precede its application to objects4 ,d4,e4 ,phillips1 '#ig1 p4 #bch4 ,,the ,,ratio ,,idea ,,overemphasized ,illustration3 ,,fig4 #a4 ,ratios4--#a4 ,select solids having the relation1 or ratio1 of a1 b1 c1 d1 o1 e4 #b4 ,name the solids1 a1 b1 c1 d1 o1 e4 ,the means of expressing must be as freely supplied as the means of discovery4 ,the pupil is not expected to invent terms4 #c4 ,tell all you can about the relation of these units4 #d4 ,unite units and tell what the sum equals4 #e4 ,make statements like this3 o less e equals b4 #f4 c can be separated into how many d's8 into how many b's8 #g4 c can be separated into how many b's8 ,what is the name of the largest unit that can be found in both c and d an exact number of times8 #h4 ,each of the other units equals what part of c8 #i4 ,if b is #a1 what is each of the other units8 #aj4 ,if a is #a1 what is each of the other units8 #aa4 ,if b is #a1 how many #a's are there in each of the other units8 #ab4 ,if d is #a1 how many #a's and parts of #a in each of the other units8 #ac4 #b is the relation of what units8 #ad4 #c is the relation of what units8 #ae4 #a_/b is the relation of what units8 #af4 #b_/c is the relation of what units8 #ag4 ,which units have the relation #c_/b8 #ah4 ,which unit is #c times as large as #a_/b of b8 #ai4 c equals #f times #a_/c of what unit8 #bj4 #a_/c of what unit equals #a_/f of c8 #ba4 ,what equals #a_/b of c8 d equals how many sixths of c8 #bb4 o equals #e times #a_/c of what unit8 #bc4 #a_/c of what unit equals #a_/e of o8 #bd4 #b_/c of d equals what unit8 b equals how many thirds of d8 #be4 #b is the ratio of d to #a_/c of what unit8 #c is the ratio of d to #a_/b of what unit8 #bf4 d equals #c_/d of what unit8 #c_/d is the ratio of what units8 ,speer1 '#ig1 p4 #i;f4 ,,the ,,relational ,,idea ,,overemphasized ,an inspection of books of the eighties which followed the ,grube method 7for example1 the ,new ,elementary ,arithmetic by ,e4,e4 ,white '#hc7 will show undue emphasis on the relational ideas4 ,there will be over a hundred and fifty successive tasks all1 or nearly all1 on #g and -#g4 ,there will be much written work of the sort shown below3 ,add3 #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #d #a #b -- -- -- which must have sorely tried the eyes of all concerned4 ,pupils are taught to give the analysis and synthesis of each of the nine digits4 ,yet the author states that he does not carry the principle of the ,grube method to the extreme of useless repetition and mechanism4 ,it should be obvious that all four meanings have claims upon the attention of the elementary school4 ,four is the thing between three and five in the number series2 it is the name for a certain sized collection of discrete objects2 it is also the name for a continuous magnitude equal to four units--for four quarts of milk in a gallon pail as truly as for four separate quart-pails of milk2 it is also1 if we know it well1 the thing got by adding one to three or subtracting six from ten or taking two two's or half of eight4 ,to know the meaning of a number means to know somewhat about it in all of these respects4 ,the difficulty has been the narrow vision of the extremists4 ,a child must not be left interminably counting2 in fact the one-more-ness of the number series can almost be had as a by-product4 ,a child must not be restricted to exercises with collections objectified as in ,fig4 #b or stated in words as so many apples1 oranges1 hats1 pens1 etc41 when work with measurement of continuous quantities with varying units--inches1 feet1 yards1 glassfuls1 pints1 quarts1 seconds1 minutes1 hours1 and the like--is so easy and so significant4 ,on the other hand1 the elaboration of artificial problems with fictitious units of measure just to have relative magnitudes as in the exercises on page #e is a wasteful sacrifice4 ,similarly1 special drills emphasizing the fact that eighteen is eleven and seven1 twelve and six1 three less than twenty-one1 and the like1 are simply idolatrous2 these facts about eighteen1 so far as they are needed1 are better learned in the course of actual column-addition and -subtraction4 ,illustration3 ,,fig4 #b4 ,,arithmetical ,,language ,the second improvement to be made in the ordinary notion of what the functions to be improved are in the case of arithmetic is to include among these functions the knowledge of certain words4 ,the understanding of such words as both1 all1 in all1 together1 less1 difference1 sum1 whole1 part1 equal1 buy1 sell1 have left1 measure1 is contained in1 and the like1 is necessary in arithmetic as truly as is the understanding of numbers themselves4 ,it must be provided for by the school2 for pre-school and extra-school training does not furnish it1 or furnishes it too late4 ,it can be provided for much better in connection with the teaching of arithmetic than in connection with the teaching of ,english4 ,it has not been provided for4 ,an examination of the first fifty pages of eight recent textbooks for beginners in arithmetic reveals very slight attention to this matter at the best and no attention at all in some cases4 ,three of the books do not even use the word sum1 and one uses it only once in the fifty pages4 ,in all the four hundred pages the word difference occurs only twenty times4 ,when the words are used1 no great ingenuity or care appears in the means of making sure that their meanings are understood4 ,the chief reason why it has not been provided for is precisely that the common notion of what the functions are that arithmetic is to develop has left out of account this function of intelligent response to quantitative terms1 other than the names of the numbers and processes4 ,knowledge of language over a much wider range is a necessary element in arithmetical ability in so far as the latter includes ability to solve verbally stated problems4 ,as arithmetic is now taught1 it does include that ability1 and a large part of the time of wise teaching is given to improving the function 'knowing what a problem states and what it asks for4' ,since1 however1 this understanding of verbally stated problems may not be an absolutely necessary element of arithmetic1 it is best to defer its consideration until we have seen what the general function of problem-solving is4 ,,problem-,,solving ,the third respect in which the function1 'ability in arithmetic1' needs clearer definition1 is this 'problem-solving4' ,the aim of the elementary school is to provide for correct and economical response to genuine problems1 such as knowing the total due for certain real quantities at certain real prices1 knowing the correct change to give or get1 keeping household accounts1 calculating wages due1 computing areas1 percentages1 and discounts1 estimating quantities needed of certain materials to make certain household or shop products1 and the like4 ,life brings these problems usually either with a real situation 7as when one buys and counts the cost and his change71 or with a situation that one imagines or describes to himself 7as when one figures out how much money he must save per week to be able to buy a forty-dollar bicycle before a certain date74 ,sometimes1 however1 the problem is described in words to the person who must solve it by another person 7as when a life insurance agent says1 ',you pay only #be cents a week from now till--and you get #bej then'2 or when an employer says1 ',your wages would be #i dollars a week1 with luncheon furnished and bonuses of such and such amounts'74 ,sometimes also the problem is described in printed or written words to the person who must solve it 7as in an advertisement or in the letter of a customer asking for an estimate on this or that74 ,the problem may be in part real1 in part imagined or described to oneself1 and in part described to one orally or in printed or written words 7as when the proposed articles for purchase lie before one1 the amount of money one has in the bank is imagined1 the shopkeeper offers #aj percent discount1 and the printed price list is there to be read74 ,to fit pupils to solve these real1 personally imagined1 or self-described problems1 and 'described-by-another' problems1 schools have relied almost exclusively on training with problems of the last sort only4 ,the following page taken almost at random from one of the best recent textbooks could be paralleled by thousands of others2 and the oral problems put by teachers have1 as a rule1 no real situation supporting them4 #a4 ,at #gj cents per #ajj pounds1 what will be the amount of duty on an invoice of #cfbb steel rails1 each rail being #bg feet long and weighing #fj pounds to the yard8 #b4 ,a man had property valued at #fejj4 ,what will be his taxes at the rate of #aj4hj per #ajjj8 #c4 ,multiply seventy thousand fourteen hundred-thousandths by one hundred nine millionths1 and divide the product by five hundred forty-five4 #d4 ,what number multiplied by #dc-c_/d will produce #bfe-e_/h8 #e4 ,what decimal of a bushel is #c quarts8 #f4 ,a man sells #e_/h of an acre of land for #ic4ge4 ,what would be the value of his farm of #aej-c_/d acres at the same rate8 #g4 ,a coal dealer buys #cge tons coal at #d4be per ton of #bbdj pounds4 ,he sells it at #d4ej per ton of #bjjj pounds4 ,what is his profit8 #h4 ,bought #fj yards of cloth at the rate of #b yards for #e1 and #hj yards more at the rate of #d yards for #i4 ,i immediately sold the whole of it at the rate of #e yards for #ab4 ,how much did ,i gain8 #i4 ,a man purchased #dj bushels of apples at #a4ej per bushel4 ,twenty-five hundredths of them were damaged1 and he sold them at #bj cents per peck4 ,he sold the remainder at #ej cents per peck4 ,how much did he gain or lose8 #aj4 ,if oranges are #cg-a_/b cents per dozen1 how many boxes1 each containing #dhj1 can be bought for #fj8 #aa4 ,a man can do a piece of work in #ah-c_/d days4 ,what part of it can he do in #f-b_/c days8 #ab4 ,how old to-day is a boy that was born ,oct4 #bi1 #ahif8 ,walsh1 '#jf1 ,part ,i1 p4 #afe4 ,as a result1 teachers and textbook writers have come to think of the functions of solving arithmetical problems as identical with the function of solving the described problems which they give in school in books1 examination papers1 and the like4 ,if they do not think explicitly that this is so1 they still act in training and in testing pupils as if it were so4 ,it is not4 ,problems should be solved in school to the end that pupils may solve the problems which life offers4 ,to know what change one should receive after a given real purchase1 to keep one's accounts accurately1 to adapt a recipe for six so as to make enough of the article for four persons1 to estimate the amount of seed required for a plot of a given size from the statement of the amount required per acre1 to make with surety the applications that the household1 small stores1 and ordinary trades require--such is the ability that the elementary school should develop4 ,other things being equal1 the school should set problems in arithmetic which life then and later will set1 should favor the situations which life itself offers and the responses which life itself demands4 ,other things are not always equal4 ,the same amount of time and effort will often be more productive toward the final end if directed during school to 'made-up' problems4 ,the keeping of personal financial accounts as a school exercise is usually impracticable1 partly because some of the children have no earnings or allowance--no accounts to keep1 and partly because the task of supervising work when each child has a different problem is too great for the teacher4 ,the use of real household and shop problems will be easy only when the school program includes the household arts and industrial education1 and when these subjects themselves are taught so as to improve the functions used by real life4 ,very often the most efficient course is to make sure that arithmetical procedures are applied to the real and personally initiated problems which they fit1 by having a certain number of such problems arise and be solved2 then to make sure that the similarity between these real problems and certain described problems of the textbook or teacher's giving is appreciated2 and then to give the needed drill work with described problems4 ,in many cases the school practice is fairly well justified in assuming that solving described problems will prepare the pupil to solve the corresponding real problems actually much better than the same amount of time spent on the real problems themselves4 ,all this is true1 yet the general principle remains that1 other things being equal1 the school should favor real situations1 should present issues as life will present them4 ,where other things make the use of verbally described problems of the ordinary type desirable1 these should be chosen so as to give a maximum of preparation for the real applications of arithmetic in life4 ,we should not1 for example1 carelessly use any problem that comes to mind in applying a certain principle1 but should stop to consider just what the situations of life really require and show clearly the application of that principle4 ,for example1 contrast these two problems applying cancellation3-- ,a4 ,a man sold #bd lambs at #ah apiece on each of six days1 and bought #h pounds of metal with the proceeds4 ,how much did he pay per ounce for the metal8 ,b4 ,how tall must a rectangular tank #af long by #h wide be to hold as much as a rectangular tank #bd by #ah by #f8 ,the first problem not only presents a situation that would rarely or never occur1 but also takes a way to find the answer that would not1 in that situation1 be taken since the price set by another would determine the amount4 ,much thought and ingenuity should in the future be expended in eliminating problems whose solution does not improve the real function to be improved by applied arithmetic1 or improves it at too great cost1 and in devising problems which prepare directly for life's demands and still can fit into a curriculum that can be administered by one teacher in charge of thirty or forty pupils1 under the limitations of school life4 ,the following illustrations will to some extent show concretely what the ability to apply the knowledge and power represented by abstract or pure arithmetic--the so-called fundamentals--in solving problems should mean and what it should not mean4 ,samples of ,desirable ,applications of ,arithmetic in ,problems where the ,situation is ,actually ,present to ,sense in ,whole or in ,part ,keeping the scores and deciding which side beat and by how much in appropriate classroom games1 spelling matches1 and the like4 ,computing costs1 making and inspecting change1 taking inventories1 and the like with a real or play store4 ,mapping the school garden1 dividing it into allotments1 planning for the purchase of seeds1 and the like4 ,measuring one's own achievement and progress in tests of word-knowledge1 spelling1 addition1 subtraction1 speed of writing1 and the like4 ,measuring the rate of improvement per hour of practice or per week of school life1 and the like4 ,estimating costs of food cooked in the school kitchen1 articles made in the school shops1 and the like4 ,computing the cost of telegrams1 postage1 expressage1 for a real message or package1 from the published tariffs4 ,computing costs from mail order catalogues and the like4 ,samples of ,desirable ,applications of ,arithmetic where the ,situation is ,not ,present to ,sense ,the samples given here all concern the subtraction of fractions4 ,samples concerning any other arithmetical principle may be found in the appropriate pages of any text which contains problem-material selected with consideration of life's needs4 ,a #a4 ,dora is making jelly4 ,the recipe calls for #bd cups of sugar and she has only #ba-a_/b4 ,she has no time to go to the store so she has to borrow the sugar from a neighbor4 ,how much must she get8 ,subtract #bd ,think #a_/b and #a_/b #a4 ,write #a_/b4 #ba-a_/b ,think #b and #b #d4 ,write the #b4 -------- #b-a_/b #b4 ,a box full of soap weighs #bi-a_/b lb4 ,the empty box weighs #c-a_/b lb4 ,how much does the soap alone weigh8 #c4 ,on ,july #a1 ,mr4 ,lewis bought a #ej-lb4 bag of ice-cream salt4 ,on ,july #ae there were just #aa-a_/b lb4 left4 ,how much had he used in the two weeks8 #d4 ,grace promised to pick #cj qt4 blueberries for her mother4 ,so far she has picked #ah-a_/b qt4 ,how many more quarts must she pick8 ,b ,this table of numbers tells ,weight of ,mary ,adams what ,nell's baby sister ,mary ,when born #g-c_/h lb4 weighed every two months from #b months old #aa-a_/d lb4 the time she was born till she #d months old #ad-a_/h lb4 was a year old4 #f months old #ae-c_/d lb4 #h months old #ag-e_/h lb4 #aj months old #ai-a_/b lb4 #ab months old #ba-c_/h lb4 #a4 ,how much did the ,adams baby gain in the first two months8 #b4 ,how much did the ,adams baby gain in the second two months8 #c4 ,in the third two months8 #d4 ,in the fourth two months8 #e4 ,from the time it was #h months old till it was #aj months old8 #f4 ,in the last two months8 #g4 ,from the time it was born till it was #f months old8 ,c #a4 ,helen's exact average for ,december was #hg-a_/c4 ,kate's was #hd-a_/b4 ,how much higher was ,helen's than ,kate's8 #hg-a_/c ,how do you think of #a_/b and #a_/c8 #hd-a_/b ,how do you think of #a-b_/f8 ------ ,how do you change the #d8 #b4 ,find the exact average for each girl in the following list4 ,write the answers clearly so that you can see them easily4 ,you will use them in solving problems #c1 #d1 #e1 #f1 #g1 and #h4 ,alice ,dora ,emma ,grace ,louise ,mary ,nell ,rebecca ,reading #ia #hg #hc #ha #gi #gg #gf #gc ,language #hh #gh #hb #gi #gc #gh #gc #ge ,arithmetic #hi #he #gi #ge #hd #hg #hi #hj ,spelling #ij #gi #ge #hj #hb #ia #fh #ha ,geography #ia #hg #hc #ge #gh #he #gc #gi ,writing #ij #hh #ge #gb #ic #ib #ie #gh #c4 ,which girl had the highest average8 #d4 ,how much higher was her average than the next highest8 #e4 ,how much difference was there between the highest and the lowest girl8 #f4 ,was ,emma's average higher or lower than ,louise's8 ,how much8 #g4 ,how much difference was there between ,alice's average and ,dora's8 #h4 ,how much difference was there between ,mary's average and ,nell's8 #i4 ,write five other problems about these averages1 and solve each of them4 ,samples of ,undesirable ,applications of ,arithmetic#a ,will has ,,xxi marbles1 ,,xii jackstones1 ,,xxxvi pieces of string4 ,how many things had he8 ,george's kite rose ,,cdxxxv feet and ,tom's went ,,lxiii feet higher4 ,how high did ,tom's kite rise8 ,if from ,,dciv we take ,,cciv the result will be a number ,,iv times as large as the number of dollars ,mr4 ,dane paid for his horse4 ,how much did he pay for his horse8 ,hannah has #e_/h of a dollar1 ,susie #g_/be1 ,nellie #c_/d1 ,norah #ac_/af4 ,how much money have they all together8 ,a man saves #c-ag_/hj dollars a week4 ,how much does he save in a year8 ,a tree fell and was broken into #d pieces1 #ac-a_/f feet1 #aj-c_/g feet1 #h-a_/b feet1 and #g-af_/ba feet long4 ,how tall was the tree8 ,annie's father gave her #bj apples to divide among her friends4 ,she gave each one #b-b_/i apples apiece4 ,how many playmates had she8 ,john had #ag-b_/e apples4 ,he divided his whole apples into fifths4 ,how many pieces had he in all8 ,a landlady has #c-c_/g pies to be divided among her #h boarders4 ,how much will each boarder receive8 ,there are twenty columns of spelling words in ,mary's lesson and #af words in each column4 ,how many words are in her lesson8 ,there are #i nuts in a pint4 ,how many pints in a pile of #e1hhh1fgc nuts8 ,the ,adams school contains eight rooms2 each room contains #dh pupils2 if each pupil has eight cents1 how much have they together8 ,a pile of wood in the form of a cube contains #ae-a_/b cords4 ,what are the dimensions to the nearest inch8 ,a man #f ft4 high weighs #age lb4 ,how tall is his wife who is of similar build1 and weighs #abe lb48 ,a stick of timber is in the shape of the frustum of a square pyramid1 the lower base being #bb in4 square and the upper #ad in4 square4 ,how many cubic feet in the log1 if it is #bb ft4 long8 ,mr4 ,ames1 being asked his age1 replied3 ,if you cube one half of my age and add #da1dgb to the result1 the sum will be one half the cube of my age4 ,how old am ,i8 #a ,the following and later problems are taken from actual textbooks or courses of study or state examinations2 to avoid invidious comparisons1 they are not exact quotations1 but are equivalents in principle and form1 as stated in the preface4 ,these samples1 just given1 of the kind of problem-solving that should not be emphasized in school training refer in some cases to books of forty years back1 but the following represent the results of a collection made in #aiaj from books then in excellent repute4 ,it required only about an hour to collect them2 and ,i am confident that a thousand such problems describing situations that the pupil will never encounter in real life1 or putting questions that he will never be asked in real life1 could easily be found in any ten textbooks of the decade from #aijj to #aiaj4 ,if there are #bej kernels of corn on one ear1 how many are there on #bd ears of corn the same size8 ,maud is four times as old as her sister1 who is #d years old4 ,what is the sum of their ages8 ,if the first century began with the year #a1 with what year does it end8 ,every spider has #h compound eyes4 ,how many eyes have #ba spiders8 ,a nail #d inches long is driven through a board so that it projects #a4fie inches on one side and #a4dbh on the other4 ,how thick is the board8 ,find the perimeter of an envelope #e in4 by #c-a_/d in4 ,how many minutes in #e_/i of #i_/d of an hour8 ,mrs4 ,knox is #c_/d as old as ,mr4 ,knox1 who is #dh years old4 ,their son ,edward is #d_/i as old as his mother4 ,how old is ,edward8 ,suppose a pie to be exactly round and #aj-a_/b miles in diameter4 ,if it were cut into #f equal pieces1 how long would the curved edge of each piece be8 #h-a_/c of a class of #cf boys were absent on a rainy day4 #cc-a_/c of those present went out of the room to the school yard4 ,how many were left in the room8 ,just after a ton of hay was weighed in market1 a horse ate one pound of it4 ,what was the ratio of what he ate to what was left8 ,if a fan having #ae rays opens out so that the outer rays form a straight line1 how many degrees are there between any two adjacent rays8 ,one half of the distance between ,st4 ,louis and ,new ,orleans is #bhj miles more than #a_/aj of the distance2 what is the distance between these places8 ,if the pressure of the atmosphere is #ad4g lb4 per square inch what is the pressure on the top of a table #a-a_/d yd4 long and #b_/c yd4 wide8 #ac_/bh of the total acreage of barley in #aijj was #ajj1jjj acres2 what was the total acreage8 ,what is the least number of bananas that a mother can exactly divide between her #b sons1 or among her #d daughters1 or among all her children8 ,if ,alice were two years older than four times her actual age she would be as old as her aunt1 who is #ch years old4 ,how old is ,alice8 ,three men walk around a circular island1 the circumference of which is #cfj miles4 ,a walks #ae miles a day1 ,b #ah miles a day1 and ,c #bd miles a day4 ,if they start together and walk in the same direction1 how many days will elapse before they will be together again8 ,with only thirty or forty dollars a year to spend on a pupil's education1 of which perhaps eight dollars are spent on improving his arithmetical abilities1 the immediate guidance of his responses to real situations and personally initiated problems has to be supplemented largely by guidance of his responses to problems described in words1 diagrams1 pictures1 and the like4 ,of these latter1 words will be used most often4 ,as a consequence the understanding of the words used in these descriptions becomes a part of the ability required in arithmetic4 ,such word knowledge is also required in so far as the problems to be solved in real life are at times described1 as in advertisements1 business letters1 and the like4 ,this is recognized by everybody in the case of words like remainder1 profit1 loss1 gain1 interest1 cubic capacity1 gross1 net1 and discount1 but holds equally of let1 suppose1 balance1 average1 total1 borrowed1 retained1 and many such semi-technical words1 and may hold also of hundreds of other words unless the textbook and teacher are careful to use only words and sentence structures which daily life and the class work in ,english have made well known to the pupils4 ,to apply arithmetic to a problem a pupil must understand what the problem is2 problem-solving depends on problem-reading4 ,in actual school practice training in problem-reading will be less and less necessary as we get rid of problems to be solved simply for the sake of solving them1 unnecessarily unreal problems1 and clumsy descriptions1 but it will remain to some extent as an important joint task for the 'arithmetic' and 'reading' of the elementary school4 ,,arithmetical ,,reasoning ,the last respect in which the nature of arithmetical abilities requires definition concerns arithmetical reasoning4 ,an adequate treatment of the reasoning that may be expected of pupils in the elementary school and of the most efficient ways to encourage and improve it cannot be given until we have studied the formation of habits4 ,for reasoning is essentially the organization and control of habits of thought4 ,certain matters may1 however1 be decided here4 ,the first concerns the use of computation and problems merely for discipline1--that is1 the emphasis on training in reasoning regardless of whether the problem is otherwise worth reasoning about4 ,it used to be thought that the mind was a set of faculties or abilities or powers which grew strong and competent by being exercised in a certain way1 no matter on what they were exercised4 ,problems that could not occur in life1 and were entirely devoid of any worthy interest1 save the intellectual interest in solving them1 were supposed to be nearly or quite as useful in training the mind to reason as the genuine problems of the home1 shop1 or trade4 ,anything that gave the mind a chance to reason would do2 and pupils labored to find when the minute hand and hour hand would be together1 or how many sheep a shepherd had if half of what he had plus ten was one third of twice what he had6 ,we now know that the training depends largely on the particular data used1 so that efficient discipline in reasoning requires that the pupil reason about matters of real importance4 ,there is no magic essence or faculty of reasoning that works in general and irrespective of the particular facts and relations reasoned about4 ,so we should try to find problems which not only stimulate the pupil to reason1 but also direct his reasoning in useful channels and reward it by results that are of real significance4 ,we should replace the purely disciplinary problems by problems that are also valuable as special training for important particular situations of life4 ,reasoning sought for reasoning's sake alone is too wasteful an expenditure of time and is also likely to be inferior as reasoning4 ,the second matter concerns the relative merits of 'catch' problems1 where the pupil has to go against some customary habit of thinking1 and what we may call 'routine' problems1 where the regular ways of thinking that have served him in the past will1 except for some blunder1 guide him rightly4 ,consider1 for example1 these four problems3 #a4 ,a man bought ten dozen eggs for #b4ej and sold them for #cj cents a dozen4 ,how many cents did he lose8 #b4 ,i went into ,smith's store at #i ,a4,m4 and remained until #aj ,a4,m4 ,i bought six yards of gingham at #dj cents a yard and three yards of muslin at #bj cents a yard and gave a #e4jj bill4 ,how long was ,i in the store8 #c4 ,what must you divide #dh by to get half of twice #f8 #d4 ,what must you add to #ai to get #cj8 ,the 'catch' problem is now in disrepute1 the wise teacher feeling by a sort of intuition that to willfully require a pupil to reason to a result sharply contrary to that to which previous habits lead him is risky4 ,the four illustrations just given show1 however1 that mere 'catchiness' or 'contra-previous-habit-ness' in a problem is not enough to condemn it4 ,the fourth problem is a catch problem1 but so useful a one that it has been adopted in many modern books as a routine drill6 ,the first problem1 on the contrary1 all1 save those who demand no higher criterion for a problem than that it make the pupil 'think1' would reject4 ,it demands the reversal of fixed habits to no valid purpose2 for in life the question in such case would never 7or almost never7 be ',how many cents did he lose8' but ',what was the result8' or simply ',what of it8' ,this problem weakens without excuse the child's confidence in the training he has had4 ,problems like 7#b7 are given by teachers of excellent reputation1 but probably do more harm than good4 ,if a pupil should interrupt his teacher during the recitation in arithmetic by saying1 ,i got up at #g o'clock to multiply #i by #b-c_/d and got #bd-c_/d for my answer2 was that the right time to get up8 the teacher would not thank fortune for the stimulus to thought but would think the child a fool4 ,such catch questions may be fairly useful as an object lesson on the value of search for the essential element in a situation if a great variety of them are given one after another with routine problems intermixed and with warning of the general nature of the exercise at the beginning4 ,even so1 it should be remembered that reasoning should be chiefly a force organizing habits1 not opposing them2 and also that there are enough bad habits to be opposed to give all necessary training4 ,fabricated puzzle situations wherein a peculiar hidden element of the situation makes the good habits called up by the situation misleading are useful therefore rather as a relief and amusing variation in arithmetical work than as stimuli to thought4 ,problems like the third quoted above we might call puzzling rather than 'catch' problems4 ,they have value as drills in analysis of a situation into its elements that will amuse the gifted children1 and as tests of certain abilities4 ,they also require that of many confusing habits1 the right one be chosen1 rather than that ordinary habits be set aside by some hidden element in the situation4 ,not enough is known about their effect to enable us to decide whether or not the elementary school should include special facility with them as one of the arithmetical functions that it specially trains4 ,the fourth 'catch' quoted above1 which all would admit is a good problem1 is good because it opposes a good habit for the sake of another good habit1 forces the analysis of an element whose analysis life very much requires1 and does it with no obvious waste4 ,it is not safe to leave a child with the one habit of responding to 'add1 #ai1 #cj' by #di1 for in life the 'have #ai1 must get 4444 to have #cj' situation is very frequent and important4 ,on the whole1 the ordinary problems which ordinary life proffers seem to be the sort that should be reasoned out1 though the elementary school may include the less noxious forms of pure mental gymnastics for those pupils who like them4 ,,summary ,these discussions of the meanings of numbers1 the linguistic demands of arithmetic1 the distinction between scholastic and real applications of arithmetic1 and the possible restrictions of training in reasoning1--may serve as illustrations of the significance of the question1 ,what are the functions that the elementary school tries to improve in its teaching of arithmetic8 ,other matters might well be considered in this connection1 but the main outline of the work of the elementary school is now fairly clear4 ,the arithmetical functions or abilities which it seeks to improve are1 we may say3-- 7#a7 ,working knowledge of the meanings of numbers as names for certain sized collections1 for certain relative magnitudes1 the magnitude of unity being known1 and for certain centers or nuclei of relations to other numbers4 7#b7 ,working knowledge of the system of decimal notation4 7#c7 ,working knowledge of the meanings of addition1 subtraction1 multiplication1 and division4 7#d7 ,working knowledge of the nature and relations of certain common measures4 7#e7 ,working ability to add1 subtract1 multiply1 and divide with integers1 common and decimal fractions1 and denominate numbers1 all being real positive numbers4 7#f7 ,working knowledge of words1 symbols1 diagrams1 and the like as required by life's simpler arithmetical demands or by economical preparation therefor4 7#g7 ,the ability to apply all the above as required by life's simpler arithmetical demands or by economical preparation therefor1 including 7#g a7 certain specific abilities to solve problems concerning areas of rectangles1 volumes of rectangular solids1 percents1 interest1 and certain other common occurrences in household1 factory1 and business life4 ,,the ,,sociology ,,of ,,arithmetic ,the phrase 'life's simpler arithmetical demands' is necessarily left vague4 ,just what use is being made of arithmetic in this country in #aibj by each person therein1 we know only very roughly4 ,what may be called a 'sociology' of arithmetic is very much needed to investigate this matter4 ,for rare or difficult demands the elementary school should not prepare2 there are too many other desirable abilities that it should improve4 ,a most interesting beginning at such an inventory of the actual uses of arithmetic has been made by ,wilson '#ai and ,mitchell4#b ,although their studies need to be much extended and checked by other methods of inquiry1 two main facts seem fairly certain4 ,first1 the great majority of people in the great majority of their doings use only very elementary arithmetical processes4 ,in #agcg cases of addition reported by ,wilson1 seven eighths were of five numbers or less4 ,over half of the multipliers reported were one-figure numbers4 ,over #ie per cent of the fractions operated with were included in this list3 #a_/b #a_/d #c_/d #a_/c #b_/c #a_/h #c_/h #a_/e #b_/e #d_/e4 ,three fourths of all the cases reported were simple one-step computations with integers or ,united ,states money4 ,second1 they often use these very elementary processes1 not because such are the quickest and most convenient1 but because they have lost1 or maybe never had1 mastery of the more advanced processes which would do the work better4 ,the #e and #aj cent stores1 the counter with ,anything on this counter for #be;c1 and the arrangements for payments on the installment plan are familiar instances of human avoidance of arithmetic4 ,wilson found very slight use of decimals2 and ,mitchell found men computing with #diths as common fractions when the use of decimals would have been more efficient4 ,if given #abj seconds to do a test like that shown below1 leading lawyers1 physicians1 manufacturers1 and business men and their wives will1 according to my experience1 get only about half the work right4 ,many women1 finding on their meat bill #g-c_/h lb4 roast beef #b4cf1 will spend time and money to telephone the butcher asking how much roast beef was per pound1 because they have no sure power in dividing by a mixed number4 #b ,the work of ,mitchell has not been published1 but the author has had the privilege of examining it4 ,test ,perform the operations indicated4 ,express all fractions in answers in lowest terms4 ,add3 #c_/d #a_/f 4#be #d yr4 #f mo4 #a yr4 #b mo4 #f yr4 #i mo4 #c yr4 #f mo4 #d yr4 #e mo4 ----------- ,subtract3 #h4f - #f4jejjg #g_/h - #b_/c #e-g_/af - #b-c_/af ,multiply3 #bi ft4 #f in4 #g x #h x #d-a_/b #h ------------ ,divide3 #d-a_/b _/ #g ,it seems probable that the school training in arithmetic of the past has not given enough attention to perfecting the more elementary abilities4 ,and we shall later find further evidence of this4 ,on the other hand1 the fact that people in general do not at present use a process may not mean that they ought not to use it4 ,life's simpler arithmetical demands certainly do not include matters like the rules for finding cube root or true discount1 which no sensible person uses4 ,they should not include matters like computing the lateral surface or volume of pyramids and cones1 or knowing the customs of plasterers and paper hangers1 which are used only by highly specialized trades4 ,they should not include matters like interest on call loans1 usury1 exact interest1 and the rediscounting of notes1 which concern only brokers1 bank clerks1 and rich men4 ,they should not include the technique of customs which are vanishing from efficient practice1 such as simple interest on amount for times longer than a year1 days of grace1 or extremes and means in proportions4 ,they should not include any elaborate practice with very large numbers1 or decimals beyond thousandths1 or the addition and subtraction of fractions which not one person in a hundred has to add or subtract oftener than once a year4 ,when we have an adequate sociology of arithmetic1 stating accurately just who should use each arithmetical ability and how often1 we shall be able to define the task of the elementary school in this respect4 ,for the present1 we may proceed by common sense1 guided by two limiting rules4 ,the first is1--,it is no more desirable for the elementary school to teach all the facts of arithmetic than to teach all the words in the ,english language1 or all the topography of the globe1 or all the details of human physiology4 ,the second is1--,it is not desirable to eliminate any element of arithmetical training until you have something better to put in its place4 ,,chapter ,,ii ,,the ,,measurement ,,of ,,arithmetical ,,abilities ,one of the best ways to clear up notions of what the functions are which schools should develop and improve is to get measures of them4 ,if any given knowledge or skill or power or ideal exists1 it exists in some amount4 ,a series of amounts of it1 varying from less to more1 defines the ability itself in a way that no general verbal description can do4 ,thus1 a series of weights1 #a lb41 #b lb41 #c lb41 #d lb41 etc41 helps to tell us what we mean by weight4 ,by finding a series of words like only1 smoke1 another1 pretty1 answer1 tailor1 circus1 telephone1 saucy1 and beginning1 which are spelled correctly by known and decreasing percentages of children of the same age1 or of the same school grade1 we know better what we mean by 'spelling-difficulty4' ,indeed1 until we can measure the efficiency and improvement of a function1 we are likely to be vague and loose in our ideas of what the function is4 ,a ,,sample ,,measurement ,,of ,,an ,,arithmetical ,,ability3 ,,the ,,ability ,,to ,,add ,,integers ,consider first1 as a sample1 the measurement of ability to add integers4 ,the following were the examples used in the measurements made by ,stone '#jh3 #eif #dfie #dbh #hgb #bcge #id #gidh #djeb #ge #fghf #fced #cjd #efg #bfj #fde #heh #ejda #ihd #iddg #aedc #hig #gdii ---- --- ---- ,the scoring was as follows3 ,credit of #a for each column added correctly4 ,stone combined measures of other abilities with this in a total score for amount done correctly in #ab minutes4 ,stone also scored the correctness of the additions in certain work in multiplication4 ,courtis uses a sheet of twenty-four tasks or 'examples1' each consisting of the addition of nine three-place numbers as shown below4 ,eight minutes is allowed4 ,he scores the amount done by the number of examples1 and also scores the number of examples done correctly1 but does not suggest any combination of these two into a general-efficiency score4 #ibg #cgi #gef #hcg #ibd #aaj #hed #ife #cdd --- ,the author long ago proposed that pupils be measured also with series like a to g shown below1 in which the difficulty increases step by step4 a4 #c #b #b #c #b #b #a #b #b #c #a #b #d #e #e #a #d #b #c #c #c #b #b #b - - - - - - - - b4 #ba #cb #ab #bd #cd #cd #bb #ab #bc #ab #eb #ca #cc #ab #bc #ac #bd #be #ae #ad #cb #bc #dc #fa -- -- -- -- -- -- -- -- c4 #bb #c #d #ce #cb #hc #bb #c #c #ca #c #b #cc #aa #c #ba #ch #de #eb #eb #b #d #cc #fd -- -- -- -- -- -- -- -- d4 #cj #bj #aj #bb #aj #bj #eb #ab #bj #ej #dj #dc #cj #d #f #bb #dj #ag #bd #ac #dj #bc #cj #dd -- -- -- -- -- -- -- -- e4 #d #e #bj #ab #ab #bj #aj #bj #cj #c #dj #d #aa #bj #bj #aj #cj #bj #d #a #bc #g #b #bj #b #dj #bc #dj #aa #aj #cj #bj #bj #aj #aa #bj #bb #cj #be -- -- -- -- -- -- -- -- f4 #ai #i #i #ad #b #ai #bd #i #d #ac #i #ad #ac #ab #ac #ac #i #ad #ag #bc #ac #ae #ae #cd #ab #be #bf #bi #ah #ai #be #bh #ah #ci -- -- -- -- -- -- -- -- g4 #ac #ac #i #ad #ab #i #i #ac #ab #i #ad #bd #bc #ai #ai #bi #i #i #ac #ba #bh #bf #bf #ad #h #h #bi #bc #bi #af #ae #ai #ag #ai #ai #bb -- -- -- -- -- -- -- -- ,woody '#af has constructed his well-known tests on this principle1 though he uses only one example at each step of difficulty instead of eight or ten as suggested above4 ,his test1 so far as addition of integers goes1 is3-- ,,series ,a4 ,,addition ,,scale 7in part7 ,by ,clifford ,woody 7#a7 7#b7 7#c7 7#d7 7#e7 7#f7 7#g7 7#h7 7#i7 #b #b #ag #ec #gb #fj #c #a #b #e #a #bj #c #d #b #de #bf #cg #aj -- #c -- -- -- -- #b -- #cj #be -- 7#aj7 7#aa7 7#ab7 7#ac7 7#ad7 7#ae7 7#af7 7#ag7 7#ah7 #ba #cb #dc #bc #be #db #ajj #i #aii #befc #cc #ei #a #be #cc #bd #aid #achg #ce #ag #b #af #de #ab #bie #died -- -- #ac -- #bja #ae #aef #bjfe -- #df #ai --- ---- --- -- 7#ai7 7#bj7 7#ba7 7#bb7 4#ge #ab4ej #h4jj #edg #a4be #af4ge #e4ge #aig 4#di #ae4ge #b4cc #fhe ----- ------ #d4af #fgh 4#id #def #f4cb #cic ----- #ebe #bdj #aeb --- ,in his original report1 ,woody gives no scheme for scoring an individual1 wisely assuming that1 with so few samples at each degree of difficulty1 a pupil's score would be too unreliable for individual diagnosis4 ,the test is reliable for a class2 and for a class ,woody used the degree of difficulty such that a stated fraction of the class can do the work correctly1 if twenty minutes is allowed for the thirty-eight examples of the entire test4 ,the measurement of even so simple a matter as the efficiency of a pupil's responses to these tests in adding integers is really rather complex4 ,there is first of all the problem of combining speed and accuracy into some single estimate4 ,stone gives no credit for a column unless it is correctly added4 ,courtis evades the difficulty by reporting both number done and number correct4 ,the author's scheme1 which gives specified weights to speed and accuracy at each step of the series1 involves a rather intricate computation4 ,this difficulty of equating speed and accuracy in adding means precisely that we have inadequate notions of what the ability is that the elementary school should improve4 ,until1 for example1 we have decided whether1 for a given group of pupils1 fifteen ,courtis attempts with ten right1 is or is not a better achievement than ten ,courtis attempts with nine right1 we have not decided just what the business of the teacher of addition is1 in the case of that group of pupils4 ,there is also the difficulty of comparing results when short and long columns are used4 ,correctness with a short column1 say of five figures1 testifies to knowledge of the process and to the power to do four successive single additions without error4 ,correctness with a long column1 say of ten digits1 testifies to knowledge of the process and to the power to do nine successive single additions without error4 ,now if a pupil's precision was such that on the average he made one mistake in eight single additions1 he would get about half of his five-digit columns right and almost none of his ten-digit columns right4 7,he would do this1 that is1 if he added in the customary way4 ,if he were taught to check results by repeated addition1 by adding in half-columns and the like1 his percentages of accurate answers might be greatly increased in both cases and be made approximately equal47 ,length of column in a test of addition under ordinary conditions thus automatically overweights precision in the single additions as compared with knowledge of the process1 and ability at carrying4 ,further1 in the case of a column of whatever size1 the result as ordinarily scored does not distinguish between one1 two1 three1 or more 7up to the limit7 errors in the single additions4 ,yet1 obviously1 a pupil who1 adding with ten-digit columns1 has half of his answer-figures wrong1 probably often makes two or more errors within a column1 whereas a pupil who has only one column-answer in ten wrong1 probably almost never makes more than one error within a column4 ,a short-column test is then advisable as a means of interpreting the results of a long-column test4 ,finally1 the choice of a short-column or of a long-column test is indicative of the measurer's notion of the kind of efficiency the world properly demands of the school4 ,twenty years ago the author would have been readier to accept a long-column test than he now is4 ,in the world at large1 long-column addition is being more and more done by machine1 though it persists still in great frequency in the bookkeeping of weekly and monthly accounts in local groceries1 butcher shops1 and the like4 ,the search for a measure of ability to add thus puts the problem of speed versus precision1 and of short-column versus long-column additions clearly before us4 ,the latter problem has hardly been realized at all by the ordinary definitions of ability to add4 ,it may be said further that the measurement of ability to add gives the scientific student a shock by the lack of precision found everywhere in schools4 ,of what value is it to a graduate of the elementary school to be able to add with examples like those of the ,courtis test1 getting only eight out of ten right8 ,nobody would pay a computer for that ability4 ,the pupil could not keep his own accounts with it4 ,the supposed disciplinary value of habits of precision runs the risk of turning negative in such a case4 ,it appears1 at least to the author1 imperative that checking should be taught and required until a pupil can add single columns of ten digits with not over one wrong answer in twenty columns4 ,speed is useful1 especially indirectly as an indication of control of the separate higher-decade additions1 but the social demand for addition below a certain standard of precision is nil1 and its disciplinary value is nil or negative4 ,this will be made a matter of further study later4 ,,measurements ,,of ,,abilities ,,in ,,computation ,measurements of these abilities may be of two sorts--7#a7 of the speed and accuracy shown in doing one same sort of task1 as illustrated by the ,courtis test for addition shown on page #bh2 and 7#b7 of how hard a task can be done perfectly 7or with some specified precision7 within a certain assigned time or less1 as illustrated by the author's rough test for addition shown on pages #bh and #bi1 and by the ,woody tests1 when extended to include alternative forms4 ,the ,courtis tests1 originated as an improvement on the ,stone tests and since elaborated by the persistent devotion of their author1 are a standard instrument of the first sort for measuring the so-called 'fundamental' arithmetical abilities with integers4 ,they are shown on this and the following page4 ,tests of the second sort are the ,woody tests1 which include operations with integers1 common and decimal fractions1 and denominate numbers1 the ,ballou test for common fractions '#af1 and the ,ladder exercises of the ,thorndike arithmetics4 ,some of these are shown on pages #cf to #da4 ,courtis ,test ,arithmetic4 ,test ,no4 #a4 ,addition ,series ,b ,you will be given eight minutes to find the answers to as many of these addition examples as possible4 ,write the answers on this paper directly underneath the examples4 ,you are not expected to be able to do them all4 ,you will be marked for both speed and accuracy1 but it is more important to have your answers right than to try a great many examples4 #ibg #big #acf #dhf #chd #agf #bgg #hcg #cgi #ibe #cdj #gfe #dgg #ghc #dde #hhb #gef #dgc #ihh #ebd #hha #fig #fhb #iei #hcg #ihc #chf #adj #bff #bjj #eid #fjc #ibd #cae #cec #hab #fgi #cff #dha #aah #aaj #ffa #ijd #dff #bda #hea #ggh #gha #hed #gid #edg #cee #gif #ece #hdi #gef #ife #agg #aib #hcd #hej #cbc #aeg #bbb #cdd #abd #dci #efg #gcc #bbi #iec #ebe --- --- --- --- --- --- --- --- and sixteen more addition examples of nine three-place numbers4 ,courtis ,test ,arithmetic4 ,test ,no4 #b4 ,subtraction ,series ,b ,you will be given four minutes to find the answers to as many of these subtraction examples as possible4 ,write the answers on this paper directly underneath the examples4 ,you are not expected to be able to do them all4 ,you will be marked for both speed and accuracy1 but it is more important to have your answers right than to try a great many examples4 #ajggiedia #gejhhhbd #iaejjjec #hgiciihc #ggaigjbi #egdjfcid #aiijaefc #gbbjgcaf --------- -------- -------- -------- and twenty more tasks of the same sort4 ,courtis ,test ,arithmetic4 ,test ,no4 #c4 ,multiplication ,series ,b ,you will be given six minutes to work as many of these multiplication examples as possible4 ,you are not expected to be able to do them all4 ,do your work directly on this paper2 use no other4 ,you will be marked for both speed and accuracy1 but it is more important to get correct answers than to try a large number of examples4 #hbdf #ghdc #dhcg #cdgh #fdhb #bi #gjb #hc #ae #df ---- ---- ---- ---- ---- and twenty more multiplication examples of the same sort4 ,courtis ,test ,arithmetic4 ,test ,no4 #d4 ,division ,series ,b ,you will be given eight minutes to work as many of these division examples as possible4 ,you are not expected to be able to do them all4 ,do your work directly on this paper2 use no other4 ,you will be marked for both speed and accuracy1 but it is more important to get correct answers than to try a large number of examples4 #be7fgge #id7heceb #cg7iiij #hf7hjjff and twenty more division examples of the same sort4 ,,series ,b4 ,,multiplication ,,scale ,by ,clifford ,woody 7#a7 7#c7 7#d7 7#e7 #c x #g #b x #c #d x #h #bc #c -- 7#h7 7#i7 7#aa7 7#ab7 #ej #bed #ajcf #ejif #c #f #h #f -- --- ---- ---- 7#ac7 7#af7 7#ah7 7#bj7 #hged #ghih #bd #bhg #h #i #bcd 4#je ---- ---- --- --- 7#bd7 7#bf7 7#bg7 7#bi7 #af #igdb #f4be #a_/h x #b #b-e_/h #ei #c4b --- ---- ---- 7#cc7 7#ce7 7#cg7 7#ch7 #b-a_/b x #c-a_/b #ihg-c_/d #b-a_/d x #d-a_/b x #a-a_/b 4#jifc-a_/h #be 4#jhd ---- ----- ,,series ,b4 ,,division ,,scale ,by ,clifford ,woody 7#a7 7#b7 7#g7 7#h7 #c7f #i7bg #d _/ #b #i7j 7#aa7 7#ad7 7#ae7 7#ag7 #b7ac #h7ehef #a_/d of #abh #ej _/ #g 7#ai7 7#bc7 7#bg7 7#bh7 #bdh _/ #g #bc7dfi #g_/h of #fbd 4#jjc74jicf 7#cj7 7#cd7 7#cf7 #c_/d _/ #e #fb4ej _/ #a-a_/d #i7fi lbs4 #i oz4 ,ballou ,test ,addition of ,fractions ,test #a ,test #b 7#a7 #a_/d 7#b7 #c_/ad 7#a7 #a_/c 7#b7 #b_/g #a_/d #a_/ad #a_/f #c_/ad --- ---- --- ---- ,test #c ,test #d 7#a7 #c_/e 7#b7 #e_/f 7#a7 #a_/g 7#b7 #g_/i #aa_/ae #a_/b #i_/aj #a_/d ----- --- ---- --- ,test #e ,test #f 7#a7 #a_/aj 7#b7 #d_/i 7#a7 #a_/f 7#b7 #e_/f #a_/f #e_/ab #i_/aj #c_/h ---- ---- ---- --- ,an ,addition ,ladder ,thorndike1 '#ag1 ,,iii1 #e ,begin at the bottom of the ladder4 ,see if you can climb to the top without making a mistake4 ,be sure to copy the numbers correctly4 ,step #f4 a4 ,add #a-a_/c yd41 #g_/h yd41 #a-a_/d yd41 #c_/d yd41 #g_/h yd41 and #a-a_/b yd4 b4 ,add #fb-a_/b;c1 #ff-b_/c;c1 #ef-a_/d;c1 #fj;c1 and #fb-a_/b;c4 c4 ,add #a-e_/af1 #a-i_/cb1 #a-c_/h1 #a-aa_/cb1 and #a-g_/af4 d4 ,add #a-a_/c yd41 #a-a_/d yd41 #a-a_/b yd41 #b yd41 #c_/d yd41 and #b_/c yd4 ,step #e4 a4 ,add #d ft4 #f-a_/b in41 #ec-a_/d in41 #e ft4 #a_/b in41 #ef-c_/d in41 and #e ft4 b4 ,add #g lb41 #f lb4 #aa oz41 #g-a_/b lb41 #f lb4 #d-a_/b oz41 and #h-a_/b lb4 c4 ,add #a hr4 #f min4 #bj sec41 #eh min4 #ae sec41 #a hr4 #d min41 and #ee min4 d4 ,add #g dollars1 #ac half dollars1 #ba quarters1 #ag dimes1 and #ai nickels4 ,step #d4 a4 ,add 4#je-a_/b1 4#jf1 4#jd-c_/d1 4#jb-c_/d1 and 4#je-a_/d4 b4 ,add 4#cc-a_/c1 4#ab-a_/b1 4#ah1 4#af-b_/c1 4#jh-a_/c and 4#ae4 c4 ,add 4#jh-a_/c1 4#jf-a_/d1 4#ba1 4#jc-c_/d1 and 4#af-b_/c4 d4 ,add 4#fb1 4#fd-a_/b1 4#ff-b_/c1 4#aj-a_/d1 and 4#fh4 ,step #c4 a4 ,add #g-a_/d1 #f-a_/b1 #h-c_/h1 #e-c_/d1 #i-e_/h and #c-g_/h4 b4 ,add #d-e_/h1 #ab1 #g-a_/b1 #h-c_/d1 #f and #e-a_/d4 c4 ,add #i-c_/d1 #e-g_/h1 #d-a_/h1 #f-a_/b1 #g1 #c-e_/h4 d4 ,add #ab1 #h-a_/b1 #g-a_/c1 #e1 #f-b_/c1 and #i-a_/b4 ,step #b4 a4 ,add #ab4jd1 4#if1 #d4g1 #i4fbe1 #c4be1 and #bj4 b4 ,add 4#eh1 #f4jc1 4#jgi1 #d4bjf1 #b4ge1 and #aj4d4 c4 ,add #eb1 #bi4h1 #da4jg1 #a4iac1 #b4f1 and #aaj4 d4 ,add #bi4g1 #cae1 #bf4ge1 #ai4jjd1 #h4gic1 and #bj4je4 ,step #a4 a4 ,add #aj-c_/e1 #aa-a_/e1 #aj-d_/e1 #aa1 #aa-b_/e1 #aj-c_/e1 and #aa4 b4 ,add #g-c_/h1 #f-e_/h1 #h1 #i-a_/h1 #g-g_/h1 #e-c_/h1 and #h-a_/h4 c4 ,add #ba-a_/b1 #ah-c_/d1 #ca-a_/b1 #ai-a_/d1 #ag-a_/d1 #bb1 and #af-a_/b4 d4 ,add #ad-e_/ab1 #ab-g_/ab1 #i-aa_/ab1 #f-a_/ab1 and #e4 ,a ,subtraction ,ladder ,thorndike1 '#ag1 ,,iii1 #aa ,step #i4 a4 #b4af mi4 - #a-c_/d mi4 b4 #e4gb ft4 - #e ft4 #c in4 c4 #b min4 #aj-a_/b sec4 - #ic4d sec4 d4 #cj4bh ,a4 - #aj-a_/e ,a4 e4 #aj gal4 #b-a_/b qt4 - #d4fbc gal4 ,step #h4 a b c d e #be-g_/ab #aj-a_/d #i-e_/af #e-g_/af #d-b_/c #ab-c_/d #g-a_/c #f-c_/h #b-c_/d #a-c_/d ------- ------ ------ ------ ----- ,step #g4 a b c d e #bh-c_/d #dj-a_/b #aj-a_/d #bd-a_/c #cg-a_/b #af-a_/h #ad-c_/h #f-a_/b #aa-a_/b #ad-c_/d ------ ------ ------ ------ ------ ,step #f4 a b c d e #aj-a_/c #g-a_/d #ae-a_/h #ab-a_/e #d-a_/af #d-b_/c #b-c_/d #f-c_/h #aa-d_/e #b-g_/af ------ ----- ------ ------ ------ ,step #e4 a b c d e #eh-d_/e #ff-b_/c #bh-g_/h #fb-a_/b #i-g_/ab #eb-a_/e #cc-a_/c #g-e_/h #cg-a_/b #d-e_/ab ------ ------ ------ ------ ------ ,step #d4 a4 #d hr4 - #b hr4 #ag min4 b4 #d lb4 #g oz4 - #b lb4 #aa oz4 c4 #a lb4 #e oz4 - #ac oz4 d4 #g ft4 - #b ft4 #h in4 e4 #a bu4 - #a pk4 ,step #c4 a b c d e #ib mi4 #fgce mi4 #c - #hi;c #bh4d mi4 #ejh4dj #hd4ae mi4 #ffhi mi4 #ah4jd mi4 #bjh4fb --------- -------- -------- --------- ------- ,step #b4 a b c d e #be4jj #ajj4jj #gej4jj #fabd sq4 mi4 #ghdf sq4 mi4 #i4cf #ga4bh #gcf4ej #bdid sq4 mi4 #bghi sq4 mi4 ------ ------- ------- ------------ ------------ ,step #a4 a b c d e #ah4fd #be4ci #ef4gj #hai4d mi4 #fg4ee mi4 #g4dj #ac4cg #de4fj #bji4b mi4 #cf4ad mi4 ------ ------ ------ --------- --------- ,an ,average ,ladder ,thorndike1 '#ag1 ,,iii1 #acb ,find the average of the quantities on each line4 ,begin with ,step #a4 ,climb to the top without making a mistake4 ,be sure to copy the numbers correctly4 ,extend the division to two decimal places if necessary4 ,step #f4 a4 #b-b_/c1 #a-g_/h1 #b-c_/d1 #d-a_/d1 #c-e_/h1 #c-a_/b b4 #fb-a_/b;c1 #ff-b_/c;c1 #dj;c1 #hc-a_/c;c1 #a4ge1 #b4be c4 #c-aa_/af1 #c-i_/cb1 #c-c_/h1 #c-ag_/cb1 #c-g_/af d4 4#ag1 #ai1 4#af-b_/c1 4#ae-a_/b1 4#bc-a_/d1 4#ah ,step #e4 a4 #e ft4 #c-a_/b in41 #fa-a_/d in41 #eh-c_/d in41 #d ft4 #aa in4 b4 #f lb4 #i oz41 #f lb4 #aa oz41 #g-a_/d lb41 #g-c_/h lb4 c4 #a hr4 #d min4 #dj sec41 #eh min4 #ce sec41 #a-a_/d hr4 d4 #b4h miles1 #c-a_/b miles1 #b4gb miles ,step #d4 a4 4#jc-a_/b1 4#jf1 4#jd-c_/d1 4#je-a_/b1 4#je-a_/d b4 4#jdc1 4#jde1 4#jdi1 4#jdg1 4#jdf1 4#jde c4 #b4bj1 4#hg-a_/b1 #a4ah1 4#ic-c_/d1 #a4bibe1 4#hj d4 4#ad-a_/b1 4#ab-a_/b1 4#cc-a_/c1 4#af-b_/c1 4#ae1 4#ag ,step #c4 a4 #e-a_/d1 #d-a_/b1 #h-c_/h1 #g-c_/d1 #f-e_/h1 #i-c_/h b4 #i-e_/h1 #ab1 #h-a_/b1 #h-c_/d1 #f1 #e-a_/d1 #i c4 #i-c_/h1 #e-c_/d1 #d-a_/h1 #g-a_/b1 #f d4 #aa1 #i-a_/b1 #aj-a_/c1 #ac1 #af-b_/c1 #i-a_/b ,step #b4 a4 #ac4je1 4#ig1 #d4h1 #aj4fbe1 #c4cg b4 #a4dh1 #g4jb1 4#ic1 #e4cjg1 #d4a1 #g1 #aj4d c4 #fh1 #ga4d1 #ei4h1 #aab1 #if4a1 #gi4h d4 #b4jgi1 #c4ijh1 #d4afe1 #b4gd ,step #a4 a4 #d1 #i-a_/b1 #f1 #e1 #g-a_/b1 #h1 #aj1 #i b4 #f1 #e1 #c4i1 #g4a1 #h c4 #ajhf1 #aada1 #ajei1 #acjb1 #abhd d4 #ajj4hb1 #bjf4di1 #cag4be1 #bdd4gc ,as such tests are widened to cover the whole task of the elementary school in respect to arithmetic1 and accepted by competent authorities as adequate measures of achievement in computing1 they will give1 as has been said1 a working definition of the task4 ,the reader will observe1 for example1 that work such as the following1 though still found in many textbooks and classrooms1 does not1 in general1 appear in the modern tests and scales4 ,reduce the following improper fractions to mixed numbers3-- #ai_/ac #dc_/ba #agf_/be #aih_/ad ,reduce to integral or mixed numbers3-- #facha_/cg #bacd_/fg #dac_/dac #fig_/bbe ,simplify3-- #c_/d of #h_/i of #c_/e of #ae_/bb ,reduce to lowest terms3-- #ceg_/ebg #bfd_/cab #dib_/ggi #dah_/hgd #hed_/agfi #cj_/gce #dd_/bdb #gg_/hdg #ah_/bdc #if_/bbd ,find differences3-- #f-b_/g #h-e_/aa #h-d_/ac #e-a_/d #g-a_/h #c-a_/ad #e-a_/g #c-g_/ac #b-aa_/ad #b-a_/g ------ ------ ------ ------- ------ ,square3-- #b_/c #d_/e #e_/g #f_/i #aj_/aa #ab_/ac #b_/g #ae_/af #ai_/bj #ag_/ah #be_/cj #da_/ec ,multiply3-- #b_/aa x #cc #cb x #c_/ad #ci x #b_/ac #fj x #aa_/bh #gg x #d_/aa #fc x #b_/bg #ed x #h_/de #fe x #c_/ac #cdd-af_/ba #dcb-b_/g ,,measurements ,,of ,,ability ,,in ,,applied ,,arithmetic3 ,,the ,,solution ,,of ,,problems ,stone '#jh measured achievement with the following problems1 fifteen minutes being the time allowed4 ,solve as many of the following problems as you have time for2 work them in order as numbered3 #a4 ,if you buy #b tablets at #g cents each and a book for #fe cents1 how much change should you receive from a two-dollar bill8 #b4 ,john sold #d ,saturday ,evening ,posts at #e cents each4 ,he kept #a_/b the money and with the other #a_/b he bought ,sunday papers at #b cents each4 ,how many did he buy8 #c4 ,if ,james had #d times as much money as ,george1 he would have #af4 ,how much money has ,george8 #d4 ,how many pencils can you buy for #ej cents at the rate of #b for #e cents8 #e4 ,the uniforms for a baseball nine cost #b4ej each4 ,the shoes cost #b a pair4 ,what was the total cost of uniforms and shoes for the nine8 #f4 ,in the schools of a certain city there are #bbjj pupils2 #a_/b are in the primary grades1 #a_/d in the grammar grades1 #a_/h in the high school1 and the rest in the night school4 ,how many pupils are there in the night school8 #g4 ,if #c-a_/b tons of coal cost #ba1 what will #e-a_/b tons cost8 #h4 ,a news dealer bought some magazines for #a4 ,he sold them for #a4bj1 gaining #e cents on each magazine4 ,how many magazines were there8 #i4 ,a girl spent #a_/h of her money for car fare1 and three times as much for clothes4 ,half of what she had left was #hj cents4 ,how much money did she have at first8 #aj4 ,two girls receive #b4aj for making buttonholes4 ,one makes #db1 the other #bh4 ,how shall they divide the money8 #aa4 ,mr4 ,brown paid one third of the cost of a building2 ,mr4 ,johnson paid #a_/b the cost4 ,mr4 ,johnson received #ejj more annual rent than ,mr4 ,brown4 ,how much did each receive8 #ab4 ,a freight train left ,albany for ,new ,york at #f o'clock4 ,an express left on the same track at #h o'clock4 ,it went at the rate of #dj miles an hour4 ,at what time of day will it overtake the freight train if the freight train stops after it has gone #ef miles8 ,the criteria he had in mind in selecting the problems were as follows3-- ,the main purpose of the reasoning test is the determination of the ability of ,,vi ,a children to reason in arithmetic4 ,to this end1 the problems1 as selected and arranged1 are meant to embody the following conditions3-- #a4 ,situations equally concrete to all ,,vi ,a children4 #b4 ,graduated difficulties4 a4 ,as to arithmetical thinking4 b4 ,as to familiarity with the situation presented4 #c4 ,the omission of a4 ,large numbers4 b4 ,particular memory requirements4 c4 ,catch problems4 d4 ,all subject matter except whole numbers1 fractions1 and ,united ,states money4 ,the test is purposely so long that only very rarely did any pupil fully complete it in the fifteen minute limit4 ,credits were given of #a1 for each of the first five problems1 #a4d1 #a4b1 and #a4f respectively for problems #f1 #g1 and #h1 and of #b for each of the others4 ,courtis sought to improve the ,stone test of problem-solving1 replacing it by the two tests reproduced below4 ,,arithmetic--,test ,no4 #f4 ,speed ,test--,reasoning ,do not work the following examples4 ,read each example through1 make up your mind what operation you would use if you were going to work it1 then write the name of the operation selected in the blank space after the example4 ,use the following abbreviations3--,add4 for addition1 ,sub4 for subtraction1 ,mul4 for multiplication1 and ,div4 for division4 --------------- ,,operation --------------- #a4 ,a girl brought a collection of #cg colored postal cards to school one day1 and gave away #ai cards to her friends4 ,how many cards did she have left to take home8 --------------- #b4 ,five boys played marbles4 ,when the game was over1 each boy had the same number of marbles4 ,if there were #de marbles altogether1 how many did each boy have8 --------------- #c4 ,a girl1 watching from a window1 saw #bg automobiles pass the school the first hour1 and #cc the second4 ,how many autos passed by the school in the two hours8 --------------- #d4 ,in a certain school there were eight rooms and each room had seats for #ej children4 ,when all the places were taken1 how many children were there in the school8 --------------- #e4 ,a club of boys sent their treasurer to buy baseballs4 ,they gave him #c4ae to spend4 ,how many balls did they expect him to buy1 if the balls cost #de;c4 apiece8 --------------- #f4 ,a teacher weighed all the girls in a certain grade4 ,if one girl weighed #gi pounds and another #aaj pounds1 how many pounds heavier was one girl than the other8 --------------- #g4 ,a girl wanted to buy a #e-pound box of candy to give as a present to a friend4 ,she decided to get the kind worth #ce;c4 a pound4 ,what did she pay for the present8 --------------- #h4 ,one day in vacation a boy went on a fishing trip and caught #ab fish in the morning1 and #g in the afternoon4 ,how many fish did he catch altogether8 --------------- #i4 ,a boy lived #ae blocks east of a school2 his chum lived on the same street1 but #aa blocks west of the school4 ,how many blocks apart were the two boys' houses8 --------------- #aj4 ,a girl was #e times as strong as her small sister4 ,if the little girl could lift a weight of #bj pounds1 how large a weight could the older girl lift8 --------------- #aa4 ,the children of a school gave a sleigh-ride party4 ,there were #bgj children to go on the ride and each sleigh held #cj children4 ,how many sleighs were needed8 --------------- #ab4 ,in ,september there were #dc children in the eighth grade of a certain school2 by ,june there were #ei4 ,how many children entered the grade during the year8 --------------- #ac4 ,a girl who lived #ag blocks away walked to school and back twice a day4 ,what was the total number of blocks the girl walked each day in going to and from school8 --------------- #ad4 ,a boy who made #fg;c4 a day carrying papers1 was hired to run on a long errand for which he received #ej;c4 ,what was the total amount the boy earned that day8 --------------- ,total ,right --------------- 7,two more similar problems follow47 ,test #f and ,test #h are from the ,courtis ,standard ,test4 ,used by permission of ,s4 ,a4 ,courtis4 ,,arithmetic--,test ,no4 #h4 ,reasoning ,in the blank space below1 work as many of the following examples as possible in the time allowed4 ,work them in order as numbered1 entering each answer in the answer column before commencing a new example4 ,do not work on any other paper4 --------- ,,answer --------- #a4 ,the children in a certain school gave a ,christmas party4 ,one of the presents was a box of candy4 ,in filling the boxes1 one grade used #af pounds of candy1 another #ag pounds1 a third #ab pounds1 and a fourth #ac pounds4 ,what did the candy cost at #bf;c4 a pound8 --------- #b4 ,a school in a certain city used #beaf pieces of chalk in #cg school days4 ,three new rooms were opened1 each room holding #ej children1 and the school was then found to use #hd sticks of chalk per day4 ,how many more sticks of chalk were used per day than at first8 --------- #c4 ,several boys went on a bicycle trip of #aejj miles4 ,the first week they rode #cgd miles1 the second week #bfd miles1 the third #dbc miles1 the fourth #dja miles4 ,they finished the trip the next week4 ,how many miles did they ride the last week8 --------- #d4 ,forty-five boys were hired to pick apples from #ae trees in an apple orchard4 ,in #ej minutes each boy had picked #dh choice apples4 ,if all the apples picked were packed away carefully in #h boxes of equal size1 how many apples were put in each box8 --------- #e4 ,in a certain school #baf children gave a sleigh-ride party4 ,they rented #g sleighs at a cost of #cj4jj and paid #bd4jj for the refreshments4 ,the party travelled #ae miles in #b-a_/b hours and had a very pleasant time4 ,what was each child's share of the expense8 --------- #f4 ,a girl found1 by careful counting1 that there were #bdjj letters on one page of her history1 and only #bbie letters on a page of her reader4 ,how many more letters had she read in one book than in the other if she had read #dg pages in each of the books8 --------- #g4 ,each of #ei rooms in the schools of a certain city contributed #be presents to a ,christmas entertainment for poor children4 ,the stores of the city gave #aihf other articles for presents4 ,what was the total number of presents given away at the entertainment8 --------- #h4 ,forty-eight children from a certain school paid #aj;c4 apiece to ride #g miles on the cars to a woods4 ,there in a few hours they gathered #bgfe nuts4 #fje of these were bad1 but the rest were shared equally among the children4 ,how many good nuts did each one get8 --------- ,total --------- ,these proposed measures of ability to apply arithmetic illustrate very nicely the differences of opinion concerning what applied arithmetic and arithmetical reasoning should be4 ,the thinker who emphasizes the fact that in life out of school the situation demanding quantitative treatment is usually real rather than described1 will condemn a test all of whose constituents are described problems4 ,unless we are excessively hopeful concerning the transfer of ideas of method and procedure from one mental function to another we shall protest against the artificiality of ,no4 #c of the ,stone series1 and of the entire ,courtis ,test #h except ,no4 #d4 ,the ,courtis speed-reasoning test 7,no4 #f7 is a striking example of the mixture of ability to understand quantitative relations with the ability to understand words4 ,consider these five1 for example1 in comparison with the revised versions attached4#c #c ,the form of ,test #f quoted here is that given by ,courtis '#aa-'ab1 p4 #bj4 ,this differs a little from the other series of ,test #f1 shown on pages #dc and #dd4 #a4 ,the children of a school gave a sleigh-ride party4 ,there were #i sleighs1 and each sleigh held #cj children4 ,how many children were there in the party8 ,,revision4 ,if one sleigh holds #cj children1 #i sleighs hold 4444 children4 #b4 ,two school-girls played a number-game4 ,the score of the girl that lost was #eg points and she was beaten by #af points4 ,what was the score of the girl that won8 ,,revision4 ,mary and ,nell played a game4 ,mary had a score of #eg4 ,nell beat ,mary by #af4 ,nell had a score of 4444 #c4 ,a girl counted the automobiles that passed a school4 ,the total was #fj in two hours4 ,if the girl saw #bg pass the first hour how many did she see the second8 ,,revision4 ,in two hours a girl saw #fj automobiles4 ,she saw #bg the first hour4 ,she saw 4444 the second hour4 #d4 ,on a playground there were five equal groups of children each playing a different game4 ,if there were #ge children all together1 how many were there in each group8 ,,revision4 #ge pounds of salt just filled five boxes4 ,the boxes were exactly alike4 ,there were 4444 pounds in a box4 #e4 ,a teacher weighed all the children in a certain grade4 ,one girl weighed #gj pounds4 ,her older sister was #di pounds heavier4 ,how many pounds did the sister weigh8 ,,revision4 ,mary weighs #gj lb4 ,jane weighs #di pounds more than ,mary4 ,jane weighs 4444 pounds4 ,the distinction between a problem described as clearly and simply as possible and the same problem put awkwardly or in ill-known words or willfully obscured should be regarded2 and as a rule measurements of ability to apply arithmetic should eschew all needless obscurity or purely linguistic difficulty4 ,for example1 ,a boy bought a two-cent stamp4 ,he gave the man in the store #aj cents4 ,the right change was 4444 cents4 is better as a test than ,if a boy1 purchasing a two-cent stamp1 gave a ten-cent stamp in payment1 what change should he be expected to receive in return8 ,the distinction between the description of a bona fide problem that a human being might be called on to solve out of school and the description of imaginary possibilities or puzzles should also be considered4 ,nos4 #c and #i of ,stone are bad because to frame the problems one must first know the answers1 so that in reality there could never be any point in solving them4 ,it is probably safe to say that nobody in the world ever did or ever will or ever should find the number of apples in a box by the task of ,no4 #d of the ,courtis ,test #h4 ,this attaches no blame to ,dr4 ,stone or to ,mr4 ,courtis4 ,until very recently we were all so used to the artificial problems of the traditional sort that we did not expect anything better2 and so blind to the language demands of described problems that we did not see their very great influence4 ,courtis himself has been active in reform and has pointed out 7'#ac1 p4 #d f47 the defects in his ,tests #f and #h4 ,tests ,nos4 #f and #h1 the so-called reasoning tests1 have proved the least satisfactory of the series4 ,the judgments of various teachers and superintendents as to the inequalities of the units in any one test1 and of the differences between the different editions of the same test1 have proved the need of investigating these questions4 ,tests of adults in many lines of commercial work have yielded in many cases lower scores than those of the average eighth grade children4 ,at the same time the scores of certain individuals of marked ability have been high1 and there appears to be a general relation between ability in these tests and accuracy in the abstract work4 ,the most significant facts1 however1 have been the difficulties experienced by teachers in attempting to remedy the defects in reasoning4 ,it is certain that the tests measure abilities of value but the abilities are probably not what they seem to be4 ,in an attempt to measure the value of different units1 for instance1 as many problems as possible were constructed based upon a single situation4 ,twenty-one varieties were secured by varying the relative form of the question and the relative position of the different phrases4 ,one of these proved nineteen times as hard as another as measured by the number of mistakes made by the children2 yet the cause of the difference was merely the changes in the phrasing4 ,this and other facts of the same kind seem to show that ,tests #f and #h measure mainly the ability to read4 ,the scientific measurement of the abilities and achievements concerned with applied arithmetic or problem-solving is thus a matter for the future4 ,in the case of described problems a beginning has been made in the series which form a part of the ,national ,intelligence ,tests '#bj1 one of which is shown on page #di f4 ,in the case of problems with real situations1 nothing in systematic form is yet available4 ,systematic tests and scales1 besides defining the abilities we are to establish and improve1 are of very great service in measuring the status and improvement of individuals and of classes1 and the effects of various methods of instruction and of study4 ,they are thus helpful to pupils1 teachers1 supervisors1 and scientific investigators2 and are being more and more widely used every year4 ,information concerning the merits of the different tests1 the procedure to follow in giving and scoring them1 the age and grade standards to be used in interpreting results1 and the like1 is available in the manuals of ,educational ,measurement1 such as ,courtis1 ,manual of ,instructions for ,giving and ,scoring the ,courtis ,standard ,tests in the ,three ,r's '#ad2 ,starch1 ,educational ,measurements '#af2 ,chapman and ,rush1 ,scientific ,measurement of ,classroom ,products '#ag2 ,monroe1 ,de,voss1 and ,kelly1 ,educational ,tests and ,measurements '#ag2 ,wilson and ,hoke1 ,how to ,measure '#bj2 and ,mc,call1 ,how to ,measure in ,education '#ba4 ,,test #a ,national ,intelligence ,tests4 ,scale ,a4 ,form #a1 ,edition #a ,find the answers as quickly as you can4 ,write the answers on the dotted lines4 ,use the side of the page to figure on4 ,begin here #a ,five cents make #a nickel4 ,how many nickels make a dime8 ,answer 44444 #b ,john paid #e dollars for a watch and #c dollars for a chain4 ,how many dollars did he pay for the watch and chain8 ,answer 44444 #c ,nell is #ac years old4 ,mary is #i years old4 ,how much younger is ,mary than ,nell8 ,answer 44444 #d ,one quart of ice cream is enough for #e persons4 ,how many quarts of ice cream are needed for #be persons8 ,answer 44444 #e ,john's grandmother is #hf years old4 ,if she lives1 in how many years will she be #ajj years old8 ,answer 44444 #f ,if a man gets #b4ej a day1 what will he be paid for six days' work8 ,answer 44444 #g ,how many inches are there in a foot and a half8 ,answer 44444 #h ,what is the cost of #ab cakes at #f for #e cents8 ,answer 44444 #i ,the uniforms for a baseball team of nine boys cost #b4ej each4 ,the shoes cost #b a pair4 ,what was the total cost of uniforms and shoes for the nine8 ,answer 44444 #aj ,a train that usually arrives at half-past ten was #ag minutes late4 ,when did it arrive8 ,answer 44444 #aa ,at #aj;c a yard1 what is the cost of a piece #aj-a_/b ft4 long8 ,answer 44444 #ab ,a man earns #f a day half the time1 #d4ej a day one fourth of the time1 and nothing on the remaining days for a total period of #dj days4 ,what did he earn in all in the #dj days8 ,answer 44444 #ac ,what per cent of #hjj is #d of #ajjj8 ,answer 44444 #ad ,if #fj men need #aejj lb4 flour per month1 what is the requirement per man per day counting a month as #cj days8 ,answer 44444 #ae ,a car goes at the rate of a mile a minute4 ,a truck goes #bj miles an hour4 ,how many times as far will the car go as the truck in #aj seconds8 ,answer 44444 #af ,the area of the base 7inside measure7 of a cylindrical tank is #ij square feet4 ,how tall must it be to hold #ajj cubic yards8 ,answer 44444 ,from ,national ,intelligence ,tests by ,national ,research ,council4 ,copyright1 #aibj1 by ,the ,world ,book ,company1 ,yonkers-on-,hudson1 ,new ,york4 ,used by permission of the publishers4 ,,chapter ,,iii ,,the ,,constitution ,,of ,,arithmetical ,,abilities ,,the ,,elementary ,,functions ,,of ,,arithmetical ,,learning ,it would be a useful work for some one to try to analyze arithmetical learning into the unitary abilities which compose it1 showing just what1 in detail1 the mind has to do in order to be prepared to pass a thorough test on the whole of arithmetic4 ,these unitary abilities would make a very long list4 ,examination of a well-planned textbook will show that such an ability as multiplication is treated as a composite of the following3 knowledge of the multiplications up to #i x #i2 ability to multiply two 7or more7-place numbers by #b1 #c1 and #d when 'carrying' is not required and no zeros occur in the multiplicand2 ability to multiply by #b1 #c1 444 #i1 with carrying2 the ability to handle zeros in the multiplicand2 the ability to multiply with two-place numbers not ending in zero2 the ability to handle zero in the multiplier as last number2 the ability to multiply with three 7or more7-place numbers not including a zero2 the ability to multiply with three- and four-place numbers with zero in second or third1 or second and third1 as well as in last place2 the ability to save time by annexing zeros2 and so on and on through a long list of further abilities required to multiply with ,united ,states money1 decimal fractions1 common fractions1 mixed numbers1 and denominate numbers4 ,the units or 'steps' thus recognized by careful teaching would make a long list1 but it is probable that a still more careful study of arithmetical ability as a hierarchy of mental habits or connections would greatly increase the list4 ,consider1 for example1 ordinary column addition4 ,the majority of teachers probably treat this as a simple application of the knowledge of the additions to #i #i1 plus understanding of 'carrying4' ,on the contrary there are at least seven processes or minor functions involved in two-place column addition1 each of which is psychologically distinct and requires distinct educational treatment4 ,these are3-- ,a4 ,learning to keep one's place in the column as one adds4 ,b4 ,learning to keep in mind the result of each addition until the next number is added to it4 ,c4 ,learning to add a seen to a thought-of number4 ,d4 ,learning to neglect an empty space in the columns4 ,e4 ,learning to neglect #js in the columns4 ,f4 ,learning the application of the combinations to higher decades may for the less gifted pupils involve as much time and labor as learning all the original addition tables4 ,and even for the most gifted child the formation of the connection '#h and #g #ae' probably never quite insures the presence of the connections '#ch and #g #de' and '#ah #g #be4' ,g4 ,learning to write the figure signifying units rather than the total sum of a column4 ,in particular1 learning to write #j in the cases where the sum of the column is #aj1 #bj1 etc4 ,learning to 'carry' also involves in itself at least two distinct processes1 by whatever way it is taught4 ,we find evidence of such specialization of functions in the results with such tests as ,woody's4 ,for example1 #b #e #a 4444 surely involves abilities in part different from #b #d #c - because only #gg percent of children in grade #c do the former correctly1 whereas #ie percent of children in that grade do the latter correctly4 ,in grade #b the difference is even more marked4 ,in the case of subtraction #d #d - involves abilities different from those involved in #i #c1 - being much less often solved correctly in grades #b and #d4 #f #j - is much harder than either of the above4 #dc #a #ba #b #cc #ac is much harder than #ce4 -- -- ,it may be said that these differences in difficulty are due to different amounts of practice4 ,this is probably not true1 but if it were1 it would not change the argument2 if the two abilities were identical1 the practice of one would improve the other equally4 ,i shall not undertake here this task of listing and describing the elementary functions which constitute arithmetical learning1 partly because what they are is not fully known1 partly because in many cases a final ability may be constituted in several different ways whose descriptions become necessarily tedious1 and partly because an adequate statement of what is known would far outrun the space limits of this chapter4 ,instead1 ,i shall illustrate the results by some samples4 ,,knowledge ,,of ,,the ,,meaning ,,of ,a ,,fraction ,as a first sample1 consider knowledge of the meaning of a fraction4 ,is the ability in question simply to understand that a fraction is a statement of the number of parts1 each of a certain size1 the upper number or numerator telling how many parts are taken and the lower number or denominator telling what fraction of unity each part is8 ,and is the educational treatment required simply to describe and illustrate such a statement and have the pupils apply it to the recognition of fractions and the interpretation of each of them8 ,and is the learning process 7#a7 the formation of the notions of part1 size of part1 number of part1 7#b7 relating the last two to the numbers in a fraction1 and1 as a necessary consequence1 7#c7 applying these notions adequately whenever one encounters a fraction in operation8 ,precisely this was the notion a few generations ago4 ,the nature of fractions was taught as one principle1 in one step1 and the habits of dealing with fractions were supposed to be deduced from the general law of a fraction's nature4 ,as a result the subject of fractions had to be long delayed1 was studied at great cost of time and effort1 and1 even so1 remained a mystery to all save gifted pupils4 ,these gifted pupils probably of their own accord built up the ability piecemeal out of constituent insights and habits4 ,at all events1 scientific teaching now does build up the total ability as a fusion or organization of lesser abilities4 ,what these are will be seen best by examining the means taken to get them4 7#a7 ,first comes the association of #a_/b of a pie1 #a_/b of a cake1 #a_/b of an apple1 and such like with their concrete meanings so that a pupil can properly name a clearly designated half of an obvious unit like an orange1 pear1 or piece of chalk4 ,the same degree of understanding of #a_/d1 #a_/h1 #a_/c1 #a_/f1 and #a_/e is secured4 ,the pupil is taught that #a pie #b #a_/bs1 #c #a_/cs1 #d #a_/ds1 #e #a_/es1 #f #a_/fs1 and #h #a_/hs2 similarly for #a cake1 #a apple1 and the like4 ,so far he understands #a_/x of y in the sense of certain simple parts of obviously unitary ys4 7#b7 ,next comes the association with #a_/b of an inch1 #a_/b of a foot1 #a_/b of a glassful and other cases where y is not so obviously a unitary object whose pieces still show their derivation from it4 ,similarly for #a_/d1 #a_/c1 etc4 7#c7 ,next comes the association with #a_/b of a collection of eight pieces of candy1 #a_/c of a dozen eggs1 #a_/e of a squad of ten soldiers1 etc41 until #a_/b1 #a_/c1 #a_/d1 #a_/e1 #a_/f1 and #a_/h are understood as names of certain parts of a collection of objects4 7#d7 ,next comes the similar association when the nature of the collection is left undefined1 the pupil responding to #a_/b of #f is 4441 #a_/d of #h is 4441 #b is #a_/e of 4441 #a_/c of #f is 4441 #a_/c of #i is 4441 #b is #a_/c of 4441 and the like4 ,each of these abilities is justified in teaching by its intrinsic merits1 irrespective of its later service in helping to constitute the general understanding of the meaning of a fraction4 ,the habits thus formed in grades #c or #d are of constant service then and thereafter in and out of school4 7#e7 ,with these comes the use of #a_/e of #aj1 #ae1 #bj1 etc41 #a_/f of #ab1 #ah1 #db1 etc41 as a useful variety of drill on the division tables1 valuable in itself1 and a means of making the notion of a unit fraction more general by adding #a_/g and #a_/i to the scheme4 7#f7 ,next comes the connection of #c_/d1 #b_/e1 #c_/e1 #d_/e1 #b_/c1 #a_/f1 #e_/f1 #c_/h1 #e_/h1 #g_/h1 #c_/aj1 #g_/aj1 and #i_/aj1 each with its meaning as a certain part of some conveniently divisible unit1 and1 7#g7 and 7#h71 connections between these fractions and their meanings as parts of certain magnitudes 7#g7 and collections 7#h7 of convenient size1 and 7#i7 connections between these fractions and their meanings when the nature of the magnitude or collection is unstated1 as in #d_/e of #ae 4441 #e_/h of #cb 4444 7#aj7 ,that the relation is general is shown by using it with numbers requiring written division and multiplication1 such as #g_/h of #agcf 4441 and with ,united ,states money4 ,elements 7#f7 to 7#aj7 again are useful even if the pupil never goes farther in arithmetic4 ,one of the commonest uses of fractions is in calculating the cost of fractions of yards of cloth1 and fractions of pounds of meat1 cheese1 etc4 ,the next step 7#aa7 is to understand to some extent the principle that the value of any of these fractions is unaltered by multiplying or dividing the numerator and denominator by the same number4 ,the drills in expressing fractions in lower and higher terms which accomplish this are paralleled by 7#ab7 and 7#ac7 simple exercises in adding and subtracting fractions to show that fractions are quantities that can be operated on like any quantities1 and by 7#ad7 simple work with mixed numbers 7addition and subtraction and reductions71 and 7#ae7 improper fractions4 ,all that is done with improper fractions is 7a7 to have the pupil use a few of them as he would any fractions and 7b7 to note their equivalent mixed numbers4 ,in 7#ab71 7#ac71 and 7#ad7 only fractions of the same denominators are added or subtracted1 and in 7#ab7 7#ac71 7#ad71 and 7#ae7 only fractions with #b1 #c1 #d1 #e1 #f1 #h1 or #aj in the denominator are used4 ,as hitherto1 the work of 7#aa7 to 7#ae7 is useful in and of itself4 7#af7 ,definitions are given of the following type3-- ,numbers like #b1 #c1 #d1 #g1 #aa1 #bj1 #cf1 #adj1 #iba are called whole numbers4 ,numbers like #g_/h1 #a_/e1 #b_/c1 #c_/d1 #aa_/h1 #g_/f1 #a_/c1 #d_/c1 #a_/h1 #a_/f are called fractions4 ,numbers like #e-a_/d1 #g-c_/h1 #i-a_/b1 #af-d_/e1 #cae-g_/h1 #a-a_/c1 #a-b_/c are called mixed numbers4 7#ag7 ,the terms numerator and denominator are connected with the upper and lower numbers composing a fraction4 ,building this somewhat elaborate series of minor abilities seems to be a very roundabout way of getting knowledge of the meaning of a fraction1 and is1 if we take no account of what is got along with this knowledge4 ,taking account of the intrinsically useful habits that are built up1 one might retort that the pupil gets his knowledge of the meaning of a fraction at zero cost4 ,,knowledge ,,of ,,the ,,subtraction ,,and ,,division ,,tables ,consider next the knowledge of the subtraction and division ',tables4' ,the usual treatment presupposes that learning them consists of forming independently the bonds3-- #c - #a #b #d _/ #b #b #c - #b #a #f _/ #b #c #d - #a #c #f _/ #c #b 4 4 4 4 4 4 4 4 4 4 4 4 #ah - #i #i #ha _/ #i #i ,in fact1 however1 these #abf bonds are not formed independently4 ,except perhaps in the case of the dullest twentieth of pupils1 they are somewhat facilitated by the already learned additions and multiplications4 ,and by proper arrangement of the learning they may be enormously facilitated thereby4 ,indeed1 we may replace the independent memorizing of these facts by a set of instructive exercises wherein the pupil derives the subtractions from the corresponding additions by simple acts of reasoning or selective thinking4 ,as soon as the additions giving sums of #i or less are learned1 let the pupil attack an exercise like the following3-- ,write the missing numbers3-- ,a ,b ,c ,d #c and 444 are #e4 #e and 444 are #h4 #d and 444 are #e4 #d and 444 are #h4 #c and 444 are #i4 #c and 444 are #f4 #e and 444 are #f4 #a and 444 are #g4 #d and 444 are #g4 #d and 444 are #i4 #f and 444 are #i4 #f and 444 are #g4 #e and 444 are #g4 #b and 444 #f4 #a and 444 are #h4 #h and 444 are #i4 #f and 444 are #h4 #e and 444 #i4 #c and 444 are #g4 #c 444 are #d4 #d and 444 are #f4 #b and 444 #g4 #a 444 are #c4 #g 444 are #h4 #b and 444 are #e4 #c and 444 #h4 #a 444 are #e4 #d 444 are #i4 #b and 444 #h4 #a and 444 #d4 #d 444 are #h4 #b 444 are #c4 #c and 444 #f4 #b and 444 #d4 #g 444 are #i4 #a 444 are #i4 #f and 444 #i4 #c and 444 #h4 #b 444 #d4 #c 444 #f4 #d and 444 #f4 #f and 444 #g4 #c 444 #h4 #e 444 #i4 #d and 444 #g4 #b and 444 #e4 #d 444 #e4 #a 444 #c4 ,the task for reasoning is only to try1 one after another1 numbers that seem promising and to select the right one when found4 ,with a little stimulus and direction children can thus derive the subtractions up to those with #i as the larger number4 ,let them then be taught to do the same with the printed forms3-- ,subtract #i #g #h #e #h #f #c #e #f #b #b #d etc4 - - - - - - and #i - #g 4441 #i - #e 4441 #g - #e 4441 etc4 ,in the case of the divisions1 suppose that the pupil has learned his first table and gained surety in such exercises as3-- #d #es 4444 #f x #e 4444 #i nickels 4444 cents4 #h #es 4444 #d x #e 4444 #f 4444 #c #es 4444 #b x #e 4444 #e 4444 #g #es 4444 #i x #e 4444 #g 4444 ,if one ball costs #e cents1 two balls cost 4444 cents1 three balls cost 4444 cents1 etc4 ,he may then be set at once to work at the answers to exercises like the following3-- ,write the answers and the missing numbers3-- ,a ,b ,c ,d 4444 #es #ae #dj 4444 #es 4444 x #e #be #bj cents 4444 nickels4 4444 #es #bj #bj 4444 #es 4444 x #e #ej #cj cents 4444 nickels4 4444 #es #dj #ae 4444 #es 4444 x #e #ce #ae cents 4444 nickels4 4444 #es #be #de 4444 #es 4444 x #e #aj #dj cents 4444 nickels4 4444 #es #cj #ej 4444 #es 4444 x #e #dj 4444 #es #ce #be 4444 #es 4444 x #e #de ,e ,for #e cents you can buy #a small loaf of bread4 ,for #aj cents you can buy #b small loaves of bread4 ,for #be cents you can buy 4444 small loaves of bread4 ,for #de cents you can buy 4444 small loaves of bread4 ,for #ce cents you can buy 4444 small loaves of bread4 ,f #e cents pays #a car fare4 #ae cents pays 4444 car fares4 #aj cents pays 4444 car fares4 #bj cents pays 4444 car fares4 ,g ,how many #e cent balls can you buy with #cj cents8 4444 ,how many #e cent balls can you buy with #ce cents8 4444 ,how many #e cent balls can you buy with #be cents8 4444 ,how many #e cent balls can you buy with #ae cents8 4444 ,in the case of the meaning of a fraction1 the ability1 and so the learning1 is much more elaborate than common practice has assumed2 in the case of the subtraction and division tables the learning is much less so4 ,in neither case is the learning either mere memorizing of facts or the mere understanding of a principle in abstracto followed by its application to concrete cases4 ,it is 7and this we shall find true of almost all efficient learning in arithmetic7 the formation of connections and their use in such an order that each helps the others to the maximum degree1 and so that each will do the maximum amount for arithmetical abilities other than the one specially concerned1 and for the general competence of the learner4 ,,learning ,,the ,,processes ,,of ,,computation ,as another instructive topic in the constitution of arithmetical abilities1 we may take the case of the reasoning involved in understanding the manipulations of figures in two 7or more7-place addition and subtraction1 multiplication and division involving a two 7or more7-place number1 and the manipulations of decimals in all four operations4 ,the psychology of these is of special interest and importance4 ,for there are two opposite explanations possible here1 leading to two opposite theories of teaching4 ,the common explanation is that these methods of manipulation1 if understood at all1 are understood as deductions from the properties of our system of decimal notation4 ,the other is that they are understood partly as inductions from the experience that they always give the right answer4 ,the first explanation leads to the common preliminary deductive explanations of the textbooks4 ,the other leads to explanations by verification2 e4g41 of addition by counting1 of subtraction by addition1 of multiplication by addition1 of division by multiplication4 ,samples of these two sorts of explanation are given below4 ,,short ,,multiplication ,,without ,,carrying3 ,,deductive ,,explanation ,,multiplication is the process of taking one number as many times as there are units in another number4 ,the ,,product is the result of the multiplication4 ,the ,,multiplicand is the number to be taken4 ,the ,,multiplier is the number denoting how many times the multiplicand is to be taken4 ,the multiplier and multiplicand are the ,,factors4 ,multiply #fbc by #c ,,operation ,multiplicand #fbc ,multiplier #c ---- ,product #ahfi ,,explanation4--,for convenience we write the multiplier under the multiplicand1 and begin with units to multiply4 #c times #c units are #i units4 ,we write the nine units in units' place in the product4 #c times #b tens are #f tens4 ,we write the #f tens in tens' place in the product4 #c times #f hundreds are #ah hundreds1 or #a thousand and #h hundreds4 ,the #a thousand we write in thousands' place and the #h hundreds in hundreds' place in the product4 ,therefore1 the product is #a thousand #h hundreds1 #f tens and #i units1 or #ahfi4 ,,short ,,multiplication ,,without ,,carrying3 ,,inductive ,,explanation #a4 ,the children of the third grade are to have a picnic4 #cb are going4 ,how many sandwiches will they need if each of the #cb children has four sandwiches8 ,here is a quick way to find out3-- #cb ,think #d x #b1 write #h under the #b in the ones column4 #d ,think #d x #c1 write #ab under the #c in the tens column4 -- #b4 ,how many bananas will they need if each of the #cb children has two bananas8 #cb x #b or #b x #cb will give the answer4 #c4 ,how many little cakes will they need if each child has three cakes8 #cb x #c or #c x #cb will give the answer4 #cb #c x #b 4444 ,where do you write the #f8 #c #c x #c 4444 ,where do you write the #i8 -- #d4 ,prove that #abh1 #fd1 and #if are right by adding four #cbs1 two #cbs1 and three #cbs4 #cb #cb #cb #cb #cb #cb #cb #cb #cb -- -- -- ,multiplication ,you multiply when you find the answers to questions like ,how many are #i x #c8 ,how many are #c x #cb8 ,how many are #h x #e8 ,how many are #d x #db8 #a4 ,read these lines4 ,say the right numbers where the dots are3 ,if you add #c to #cb1 you have 4444 #ce is the sum4 ,if you subtract #c from #cb1 the result is 4444 #bi is the difference or remainder4 ,if you multiply #c by #cb or #cb by #c1 you have 4444 #if is the product4 ,find the products4 ,check your answers to the first line by adding4 #b4 #c4 #d4 #e4 #f4 #g4 #h4 #i4 #da #cc #db #dd #ec #dc #cd #bd #c #b #d #b #c #b #b #b -- -- -- -- -- -- -- -- #aj4 #aa4 #ab4 #ac4 #ad4 #ae4 #af4 #dc #eb #cb #bc #da #ea #ad #c #c #c #c #b #d #b -- -- -- -- -- -- -- #ag4 #bac ,write the #i in the ones column4 #c ,write the #c in the tens column4 --- ,write the #f in the hundreds column4 ,check your answer by adding4 ,add #bac #bac #bac --- #ah4 #ai4 #bj4 #ba4 #bb4 #bc4 #bd4 #bad #cab #dcb #bca #acb #cad #bdc #b #c #b #c #c #b #b --- --- --- --- --- --- --- ,,short ,,division3 ,,deductive ,,explanation ,divide #ahbe by #d ,divisor #d #ahbe ,dividend -------- #def-a_/d ,quotient ,,explanation4--,for convenience we write the divisor at the left of the dividend1 and the quotient below it1 and begin at the left to divide4 #d is not contained in #a thousand any thousand times1 therefore the quotient contains no unit of any order higher than hundreds4 ,consequently we find how many times #d is contained in the hundreds of the dividend4 #a thousand and #h hundreds are #ah hundreds4 #d is contained in #ah hundreds #d hundred times and #b hundreds remaining4 ,we write the #d hundreds in the quotient4 ,the #b hundreds we consider as united with the #b tens1 making #bb tens4 #d is contained in #bb tens #e tens times1 and #b tens remaining4 ,we write the #e tens in the quotient1 and the remaining #b tens we consider as united with the #e units1 making #be units4 #d is contained in #be units #f units times and #a unit remaining4 ,we write the #f units in the quotient and indicate the division of the remainder1 #a unit1 by the divisor #d4 ,therefore the quotient of #ahbe divided by #d is #def-a_/d1 or #def and #a remainder4 ,,short ,,division3 ,,inductive ,,explanation ,dividing ,large ,numbers #a4 ,tom1 ,dick1 ,will1 and ,fred put in #b cents each to buy an eight-cent bag of marbles4 ,there are #abh marbles in it4 ,how many should each boy have1 if they divide the marbles equally among the four boys8 ----- #d #abh ,think #ab three #ds4 ,write the #c over the #b in the tens column4 ,think #h two #ds4 ,write the #b over the #h in the ones column4 #cb is right1 because #d x #cb #abh4 #b4 ,mary1 ,nell1 and ,alice are going to buy a book as a present for their ,sunday-school teacher4 ,the present costs #fi cents4 ,how much should each girl pay1 if they divide the cost equally among the three girls8 ---- #cfi ,think #f 4444 #cs4 ,write the #b over the #f in the tens column4 ,think #i 4444 #cs4 ,write the #c over the #i in the ones column4 #bc is right1 for #c x #bc #fi4 #c4 ,divide the cost of a #if-;cent present equally among three girls4 ,how much should each girl pay8 ------ #cif #d4 ,divide the cost of an #hd-;cent present equally among #d girls4 ,how much should each girl pay8 #e4 ,learn this3 7,read _/ as divided by47 #ab #d #af4 #af is the sum4 #ab - #d #h4 #h is the difference or remainder4 #ab x #d #dh4 #dh is the product4 #ab _/ #d #c4 #c is the quotient4 #f4 ,find the quotients4 ,check your answers by multiplying4 ---- ---- ---- ----- ----- ----- #cii #bhf #eaee #fbdf #dafh #cbai ,uneven division is taught by the same general plan1 extended4 ,,long ,,division3 ,,deductive ,,explanation ,to ,divide by ,long ,division #a4 ,let it be required to divide #cdeca by #ae4 ,operation ,divided ,divisor #ae7cdeca7bcjb-a_/ae ,quotient #cj -- #de #de -- #ca #cj -- #a ,remainder ,for convenience we write the divisor at the left and the quotient at the right of the dividend1 and begin to divide as in ,short ,division4 #ae is contained in #c ten-thousands #j ten-thousands times2 therefore1 there will be #j ten-thousands in the quotient4 ,take #cd thousands2 #ae is contained in #cd thousands #b thousands times2 we write the #b thousands in the quotient4 #ae x #b thousands #cj thousands1 which1 subtracted from #cd thousands1 leaves #d thousands #dj hundreds4 ,adding the #e hundreds1 we have #de hundreds4 #ae in #de hundreds #c hundreds times2 we write the #c hundreds in the quotient4 #ae x #c hundreds #de hundreds1 which subtracted from #de hundreds1 leaves nothing4 ,adding the #c tens1 we have #c tens4 #ae in #c tens #j tens times2 we write #j tens in the quotient4 ,adding to the three tens1 which equal #cj units1 the #a unit1 we have #ca units4 #ae in #ca units #b units times2 we write the #b units in the quotient4 #ae x #b units #cj units1 which1 subtracted from #ca units1 leaves #a unit as a remainder4 ,indicating the division of the #a unit1 we annex the fractional expression1 #a_/ae unit1 to the integral part of the quotient4 ,therefore1 #cdeca divided by #ae is equal to #bcjb-a_/ae4 ,b4 ,greenleaf1 ,practical ,arithmetic1 '#gc1 p4 #di4 ,,long ,,division3 ,,inductive ,,explanation ,dividing by ,large ,numbers #a4 ,just before ,christmas ,frank's father sent #cfj oranges to be divided among the children in ,frank's class4 ,there are #bi children4 ,how many oranges should each child receive8 ,how many oranges will be left over8 ,here is the best way to find out3 #ab and #ab ,think how many #bis there are in #cf4 #a is right4 remainder ,write #a over the #f of #cf4 ,multiply #bi by #a4 #bi 7#cfj ,write the #bi under the #cf4 ,subtract #bi from #cf4 #bi ,write the #j of #cfj after the #g4 --- ,think how many #bis there are in #gj4 #b is right4 #gj ,write #b over the #j of #cfj4 ,multiply #bi by #b4 #eh ,write the #eh under #gj4 ,subtract #eh from #gj4 -- ,there is #ab remainder4 #ab ,each child gets #ab oranges1 and there are #ab left over4 ,this is right1 for #ab multiplied by #bi #cdh1 and #cdh #ab #cfj4 99 99 99 99 99 #h4 ,in ,no4 #h1 keep on dividing by #ca until you have used the #e1 the #h1 and the #g1 and have four #ca7ii1ehg figures in the quotient4 #i4 #aj4 #aa4 #ab4 #ac4 #bb7bec #bb7bhie #ba7hhia #bb7bij #cb7af1cfh ,check your results for #i1 #aj1 #aa1 #ab1 and #ac4 #a4 ,the boys and girls of the ,welfare ,club plan to earn money to buy a victrola4 ,there are #bc boys and girls4 ,they can get a good second-hand victrola for #e4ge4 ,how much must each earn if they divide the cost equally8 ,here is the best way to find out3 4#be ,think how many #bcs there are in #eg4 #b is right4 ----- ,write #b over the #g of #eg4 ,multiply #bc by #b4 #bce4ge ,write #df under #eg and subtract4 ,write the #e of #ege #df after the #aa4 ---- ,think how many #bcs there are in #aae4 #e is right4 #aae ,write #e over the #e of #ege4 ,multiply #bc by #e4 #aae ,write the #aae under the #aae that is there and subtract4 ---- ,there is no remainder4 ,put and the decimal point where they belong4 ,each child must earn #be cents4 ,this is right1 for 4#be multiplied by #bc #e4ge4 #b4 ,divide #ga4gf equally among #bc persons4 ,how much is each person's share8 #c4 ,check your result for ,no4 #b by multiplying the quotient by the divisor4 ,find the quotients4 ,check each quotient by multiplying it by the divisor4 #d4 #e4 #f4 #g4 #h4 #bc7ii4ac #be7ah4ej #ba7abi4ae #ac7bi4be #cb7gc4ib #a bushel #cb qt4 #i4 ,how many bushels are there in #bhh qt48 #aj4 ,in #aib qt48 #aa4 ,in #daf qt48 ,crucial experiments are lacking1 but there are several lines of well-attested evidence4 ,first of all1 there can be no doubt that the great majority of pupils learn these manipulations at the start from the placing of units under units1 tens under tens1 etc41 in adding1 to the placing of the decimal point in division with decimals1 by imitation and blind following of specific instructions1 and that a very large proportion of the pupils do not to the end1 that is to the fifth school-year1 understand them as necessary deductions from decimal notation4 ,it also seems probable that this proportion would not be much reduced no matter how ingeniously and carefully the deductions were explained by textbooks and teachers4 ,evidence of this fact will appear abundantly to any one who will observe schoolroom life4 ,it also appears in the fact that after the properties of the decimal notation have been thus used again and again2 e4g41 for deducing 'carrying' in addition1 'borrowing' in subtraction1 'carrying' in multiplication1 the value of the digits in the partial product1 the value of each remainder in short division1 the value of the quotient figures in division1 the addition1 subtraction1 multiplication1 and division of ,united ,states money1 and the placing of the decimal point in multiplication1 no competent teacher dares to rely upon the pupil1 even though he now has four or more years' experience with decimal notation1 to deduce the placing of the decimal point in division with decimals4 ,it may be an illusion1 but one seems to sense in the better textbooks a recognition of the futility of the attempt to secure deductive derivations of those manipulations4 ,i refer to the brevity of the explanations and their insertion in such a form that they will influence the pupils' thinking as little as possible4 ,at any rate the fact is sure that most pupils do not learn the manipulations by deductive reasoning1 or understand them as necessary consequences of abstract principles4 ,it is a common opinion that the only alternative is knowing them by rote4 ,this1 of course1 is one common alternative1 but the other explanation suggests that understanding the manipulations by inductive reasoning from their results is another and an important alternative4 ,the manipulations of 'long' multiplication1 for instance1 learned by imitation or mechanical drill1 are found to give for #be x ,a a result about twice as large as for #ac x ,a1 for #ch or #ci x ,a a result about three times as large2 for #aae x ,a a result about ten times as large as for #aa x ,a4 ,with even the very dull pupils the procedure is verified at least to the extent that it gives a result which the scientific expert in the case--the teacher--calls right4 ,with even the very bright pupils1 who can appreciate the relation of the procedure to decimal notation1 this relation may be used not as the sole deduction of the procedure beforehand1 but as one partial means of verifying it afterward4 ,or there may be the condition of half-appreciation of the relation in which the pupil uses knowledge of the decimal notation to convince himself that the procedure does1 but not that it must give the right answer1 the answer being 'right' because the teacher1 the answer-list1 and collateral evidence assure him of it4 ,i have taken the manipulation of the partial products as an illustration because it is one of the least favored cases for the explanation ,i am presenting4 ,if we take the first case where a manipulation may be deduced from decimal notation1 known merely by rote1 or verified inductively1 namely1 the addition of two-place numbers1 it seems sure that the mental processes just described are almost the universal rule4 ,surely in our schools at present children add the #c of #bc to the #c of #ec and the #b of #bc to the #e of #ec at the start1 in nine cases out of ten because they see the teacher do so and are told to do so4 ,they are protected from adding #c #c #b #e not by any deduction of any sort but because they do not know how to add #h and #e1 because they have been taught the habit of adding figures that stand one above the other1 or with a between them2 and because they are shown or told what they are to do4 ,they are protected from adding #c #e and #b #c1 again1 by no deductive reasoning but for the second and third reasons just given4 ,in nine cases out of ten they do not even think of the possibility of adding in any other way than the '#c #c1 #b #e' way1 much less do they select that way on account of the facts that #ec #ej #c and #bc #bj #c1 that #ej #bj #gj1 that #c #c #f1 and that 7a b7 7c d7 7a c7 7b d76 ,just as surely all but the very dullest twentieth or so of children come in the end to something more than rote knowledge1--to understand1 to know that the procedure in question is right4 ,whether they know why #gf is right depends upon what is meant by why4 ,if it means that #gf is the result which competent people agree upon1 they do4 ,if it means that #gf is the result which would come from accurate counting they perhaps know why as well as they would have1 had they been given full explanations of the relation of the procedure in two-place addition to decimal notation4 ,if why means because #ec #ej #c1 #bc #bj #c1 #ej #bj #gj1 and 7a b7 7c d7 7a c7 7b d71 they do not4 ,nor1 ,i am tempted to add1 would most of them by any sort of teaching whatever4 ,i conclude1 therefore1 that school children may and do reason about and understand the manipulations of numbers in this inductive1 verifying way without being able to1 or at least without1 under present conditions1 finding it profitable to derive them deductively4 ,i believe1 in fact1 that pure arithmetic as it is learned and known is largely an inductive science4 ,at one extreme is a minority to whom it is a series of deductions from principles2 at the other extreme is a minority to whom it is a series of blind habits2 between the two is the great majority1 representing every gradation but centering about the type of the inductive thinker4 ,,chapter ,,iv ,,the ,,constitution ,,of ,,arithmetical ,,abilities 7,,continued73 ,,the ,,selection ,,of ,,the ,,bonds ,,to ,,be ,,formed ,when the analysis of the mental functions involved in arithmetical learning is made thorough it turns into the question1 ',what are the elementary bonds or connections that constitute these functions8' and when the problem of teaching arithmetic is regarded1 as it should be in the light of present psychology1 as a problem in the development of a hierarchy of intellectual habits1 it becomes in large measure a problem of the choice of the bonds to be formed and of the discovery of the best order in which to form them and the best means of forming each in that order4 ,,the ,,importance ,,of ,,habit-,,formation ,the importance of habit-formation or connection-making has been grossly underestimated by the majority of teachers and writers of textbooks4 ,for1 in the first place1 mastery by deductive reasoning of such matters as 'carrying' in addition1 'borrowing' in subtraction1 the value of the digits in the partial products in multiplication1 the manipulation of the figures in division1 the placing of the decimal point after multiplication or division with decimals1 or the manipulation of the figures in the multiplication and division of fractions1 is impossible or extremely unlikely in the case of children of the ages and experience in question4 ,they do not as a rule deduce the method of manipulation from their knowledge of decimal notation4 ,rather they learn about decimal notation by carrying1 borrowing1 writing the last figure of each partial product under the multiplier which gives that product1 etc4 ,they learn the method of manipulating numbers by seeing them employed1 and by more or less blindly acquiring them as associative habits4 ,in the second place1 we1 who have already formed and long used the right habits and are thereby protected against the casual misleadings of unfortunate mental connections1 can hardly realize the force of mere association4 ,when a child writes sixteen as #fa1 or finds #dbh as the sum of #ae #ai #af #ah -- or gives #fdb as an answer to #bg x #cf1 or says that #d divided by #a_/d #a1 we are tempted to consider him mentally perverse1 forgetting or perhaps never having understood that he goes wrong for exactly the same general reason that we go right2 namely1 the general law of habit-formation4 ,if we study the cases of #fa for #af1 we shall find them occurring in the work of pupils who after having been drilled in writing #bf1 #cf1 #df1 #fb1 #fc1 and so on1 in which the order of the six in writing is the same as it is in speech1 return to writing the 'teen numbers4 ,if our language said onety-one for eleven and onety-six for sixteen1 we should probably never find such errors except as 'lapses' or as the results of misperception or lack of memory4 ,they would then be more frequent before the #bjs1 #cjs1 etc41 were learned4 ,if pupils are given much drill on written single column addition involving the higher decades 7each time writing the two-figure sum71 they are forming a habit of writing #bh after the sum of #h1 #f1 #i1 and #e is reached2 and it should not surprise us if the pupil still occasionally writes the two-figure sum for the first column though a second column is to be added also4 ,on the contrary1 unless some counter force influences him1 he is absolutely sure to make this mistake4 ,the last mistake quoted 7#d _/ #a_/d #a7 is interesting because here we have possibly one of the cases where deduction from psychology alone can give constructive aid to teaching4 ,multiplication and division by fractions have been notorious for their difficulty4 ,the former is now alleviated by using of instead of x until the new habit is fixed4 ,the latter is still approached with elaborate caution and with various means of showing why one must 'invert and multiply' or 'multiply by the reciprocal4' ,but in the author's opinion it seems clear that the difficulty in multiplying and dividing by a fraction was not that children felt any logical objections to canceling or inverting4 ,i fancy that the majority of them would cheerfully invert any fraction three times over or cancel numbers at random in a column if they were shown how to do so4 ,but if you are a youngster inexperienced in numerical abstractions and if you have had divide connected with 'make smaller' three thousand times and never once connected with 'make bigger1' you are sure to be somewhat impelled to make the number smaller the three thousand and first time you are asked to divide it4 ,some of my readers will probably confess that even now they feel a slight irritation or doubt in saying or writing that #af_/a _/ #a_/h #abh4 ,the habits that have been confirmed by every multiplication and division by integers are1 in this particular of 'the ratio of result to number operated upon1' directly opposed to the formation of the habits required with fractions4 ,and that is1 ,i believe1 the main cause of the difficulty4 ,its treatment then becomes easy1 as will be shown later4 ,these illustrations could be added to almost indefinitely1 especially in the case of the responses made to the so-called 'catch' problems4 ,the fact is that the learner rarely can1 and almost never does1 survey and analyze an arithmetical situation and justify what he is going to do by articulate deductions from principles4 ,he usually feels the situation more or less vaguely and responds to it as he has responded to it or some situation like it in the past4 ,arithmetic is to him not a logical doctrine which he applies to various special instances1 but a set of rather specialized habits of behavior toward certain sorts of quantities and relations4 ,and in so far as he does come to know the doctrine it is chiefly by doing the will of the master4 ,this is true even with the clearest expositions1 the wisest use of objective aids1 and full encouragement of originality on the pupil's part4 ,lest the last few paragraphs be misunderstood1 ,i hasten to add that the psychologists of to-day do not wish to make the learning of arithmetic a mere matter of acquiring thousands of disconnected habits1 nor to decrease by one jot the pupil's genuine comprehension of its general truths4 ,they wish him to reason not less than he has in the past1 but more4 ,they find1 however1 that you do not secure reasoning in a pupil by demanding it1 and that his learning of a general truth without the proper development of organized habits back of it is likely to be1 not a rational learning of that general truth1 but only a mechanical memorizing of a verbal statement of it4 ,they have come to know that reasoning is not a magic force working in independence of ordinary habits of thought1 but an organization and cooeperation of those very habits on a higher level4 ,the older pedagogy of arithmetic stated a general law or truth or principle1 ordered the pupil to learn it1 and gave him tasks to do which he could not do profitably unless he understood the principle4 ,it left him to build up himself the particular habits needed to give him understanding and mastery of the principle4 ,the newer pedagogy is careful to help him build up these connections or bonds ahead of and along with the general truth or principle1 so that he can understand it better4 ,the older pedagogy commanded the pupil to reason and let him suffer the penalty of small profit from the work if he did not4 ,the newer provides instructive experiences with numbers which will stimulate the pupil to reason so far as he has the capacity1 but will still be profitable to him in concrete knowledge and skill1 even if he lacks the ability to develop the experiences into a general understanding of the principles of numbers4 ,the newer pedagogy secures more reasoning in reality by not pretending to secure so much4 ,the newer pedagogy of arithmetic1 then1 scrutinizes every element of knowledge1 every connection made in the mind of the learner1 so as to choose those which provide the most instructive experiences1 those which will grow together into an orderly1 rational system of thinking about numbers and quantitative facts4 ,it is not enough for a problem to be a test of understanding of a principle2 it must also be helpful in and of itself4 ,it is not enough for an example to be a case of some rule2 it must help review and consolidate habits already acquired or lead up to and facilitate habits to be acquired4 ,every detail of the pupil's work must do the maximum service in arithmetical learning4 ,,desirable ,,bonds ,,now ,,often ,,neglected ,as hitherto1 ,i shall not try to list completely the elementary bonds that the course of study in arithmetic should provide for4 ,the best means of preparing the student of this topic for sound criticism and helpful invention is to let him examine representative cases of bonds now often neglected which should be formed and representative cases of useless1 or even harmful1 bonds now often formed at considerable waste of time and effort4 7#a7 ,numbers as measures of continuous quantities4--,the numbers one1 two1 three1 #a1 #b1 #c1 etc41 should be connected soon after the beginning of arithmetic each with the appropriate amount of some continuous quantity like length or volume or weight1 as well as with the appropriate sized collection of apples1 counters1 blocks1 and the like4 ,lines should be labeled #a foot1 #b feet1 #c feet1 etc42 one inch1 two inches1 three inches1 etc42 weights should be lifted and called one pound1 two pounds1 etc42 things should be measured in glassfuls1 handfuls1 pints1 and quarts4 ,otherwise the pupil is likely to limit the meaning of1 say1 four to four sensibly discrete things and to have difficulty in multiplication and division4 ,measuring1 or counting by insensibly marked off repetitions of a unit1 binds each number name to its meaning as ---- times whatever #a is1 more surely than mere counting of the units in a collection can1 and should reenforce the latter4 7#b7 ,additions in the higher decades4--,in the case of all save the very gifted children1 the additions with higher decades--that is1 the bonds1 #af #g #bc1 #bf #g #cc1 #cf #g #dc1 #ad #h #bb1 #bd #h #cb1 and the like--need to be specifically practiced until the tendency becomes generalized4 ',counting' by #bs beginning with #a1 and with #b1 counting by #cs beginning with #a1 with #b1 and with #c1 counting by #ds beginning with #a1 with #b1 with #c1 and with #d1 and so on1 make easy beginnings in the formation of the decade connections4 ,practice with isolated bonds should soon be added to get freer use of the bonds4 ,the work of column addition should be checked for accuracy so that a pupil will continually get beneficial practice rather than 'practice in error4' 7#c7 ,the uneven divisions4--,the quotients with remainders for the divisions of every number to #ai by #b1 every number to #bi by #c1 every number to #ci by #d1 and so on should be taught as well as the even divisions4 ,a table like the following will be found a convenient means of making these connections3-- #aj 4444 #bs #aj 4444 #cs and 4444 rem4 #aj 4444 #ds and 4444 rem4 #aj 4444 #es #aa 4444 #bs and 4444 rem4 #aa 4444 #cs and 4444 rem4 4 4 4 #hi 4444 #is and 4444 rem4 ,these bonds must be formed before short division can be efficient1 are useful as a partial help toward selection of the proper quotient figures in long division1 and are the chief instruments for one of the important problem series in applied arithmetic1--,how many xs can ,i buy for y cents at z cents per x and how much will ,i have left8 ,that these bonds are at present sadly neglected is shown by ,kirby '#ac1 who found that pupils in the last half of grade #c and the first half of grade #d could do only about four such examples per minute 7in a ten-minute test71 and even at that rate made far from perfect records1 though they had been taught the regular division tables4 ,sixty minutes of practice resulted in a gain of nearly #ge percent in number done per minute1 with an increase in accuracy as well4 7#d7 ,the equation form4--,the equation form with an unknown quantity to be determined1 or a missing number to be found1 should be connected with its meaning and with the problem attitude long before a pupil begins algebra1 and in the minds of pupils who never will study algebra4 ,children who have just barely learned to add and subtract learn easily to do such work as the following3-- ,write the missing numbers3-- #d #h 4444 #e 4444 #ad 4444 #c #aa 4444 #e #b #af #g 4444 #ab 4444 #e ,the equation form is the simplest uniform way yet devised to state a quantitative issue4 ,it is capable of indefinite extension if certain easily understood conventions about parentheses and fraction signs are learned4 ,it should be employed widely in accounting and the treatment of commercial problems1 and would be except for outworn conventions4 ,it is a leading contribution of algebra to business and industrial life4 ,arithmetic can make it nearly as well4 ,it saves more time in the case of drills on reducing fractions to higher and lower terms alone than is required to learn its meaning and use4 ,to rewrite a quantitative problem as an equation and then make the easy selection of the necessary technique to solve the equation is one of the most universally useful intellectual devices known to man4 ,the words 'equals1' 'equal1' 'is1' 'are1' 'makes1' 'make1' 'gives1' 'give1' and their rarer equivalents should therefore early give way on many occasions to the '' which so far surpasses them in ultimate convenience and simplicity4 7#e7 ,addition and subtraction facts in the case of fractions4--,in the case of adding and subtracting fractions1 certain specific bonds--between the situation of halves and thirds to be added and the responses of thinking of the numbers as equal to so many sixths1 between the situation thirds and fourths to be added and thinking of them as so many twelfths1 between fourths and eighths to be added and thinking of them as eighths1 and the like--should be formed separately4 ,the general rule of thinking of fractions as their equivalents with some convenient denominator should come as an organization and extension of such special habits1 not as an edict from the textbook or teacher4 7#f7 ,fractional equivalents4--,efficiency requires that in the end the much used reductions should be firmly connected with the situations where they are needed4 ,they may as well1 therefore1 be so connected from the beginning1 with the gain of making the general process far easier for the dull pupils to master4 ,we shall see later that1 for all save the very gifted pupils1 the economical way to get an understanding of arithmetical principles is not1 usually1 to learn a rule and then apply it1 but to perform instructive operations and1 in the course of performing them1 to get insight into the principles4 7#g7 ,protective habits in multiplying and dividing with fractions4--,in multiplying and dividing with fractions special bonds should be formed to counteract the now harmful influence of the 'multiply get a larger number' and 'divide get a smaller number' bonds which all work with integers has been reenforcing4 ,for example1 at the beginning of the systematic work with multiplication by a fraction1 let the following be printed clearly at the top of every relevant page of the textbook and displayed on the blackboard3-- ,when you multiply a number by anything more than #a the result is larger than the number4 ,when you multiply a number by #a the result is the same as the number4 ,when you multiply a number by anything less than #a the result is smaller than the number4 ,let the pupils establish the new habit by many such exercises as3-- #ah x #d 4444 #i x #b 4444 #d x #d 4444 #f x #b 4444 #b x #d 4444 #c x #b 4444 #a x #d 4444 #a x #b 4444 #a_/b x #d 4444 #a_/c x #b 4444 #a_/d x #d 4444 #a_/f x #b 4444 #a_/h x #d 4444 #a_/i x #b 4444 ,in the case of division by a fraction the old harmful habit should be counteracted and refined by similar rules and exercises as follows3-- ,when you divide a number by anything more than #a the result is smaller than the number4 ,when you divide a number by #a the result is the same as the number4 ,when you divide a number by anything less than #a the result is larger than the number4 ,state the missing numbers3-- #h 4444 #ds #ab 4444 #fs #i 4444 #is #h 4444 #bs #ab 4444 #ds #i 4444 #cs #h 4444 #as #ab 4444 #cs #i 4444 #as #h 4444 #a_/bs #ab 4444 #bs #i 4444 #a_/cs #h 4444 #a_/ds #ab 4444 #as #i 4444 #a_/is #h 4444 #a_/hs #ab 4444 #a_/bs #ab 4444 #a_/cs #ab 4444 #a_/ds #af _/ #af #i _/ #i #aj _/ #aj #ab _/ #f #af _/ #h #i _/ #c #aj _/ #e #ab _/ #d #af _/ #d #i _/ #a #aj _/ #a #ab _/ #c #af _/ #b #i _/ #a_/c #aj _/ #a_/e #ab _/ #b #af _/ #a #i _/ #a_/i #aj _/ #a_/aj #ab _/ #a #af _/ #a_/b #ab _/ #a_/b #af _/ #a_/d #ab _/ #a_/c #af _/ #a_/h #ab _/ #a_/d #ab _/ #a_/f 7#h7 ' of' means 'hundredths times'4--,in the case of percentage a series of bonds like the following should be formed3-- #e percent of 4#je times #bj 4#bj #f 4#jf #be 4#be x #ab 4#ab x #c 4#jc x ,four five-minute drills on such connections between 'x percent of' and 'its decimal equivalent times' are worth an hour's study of verbal definitions of the meaning of percent as per hundred or the like4 ,the only use of the study of such definitions is to facilitate the later formation of the bonds1 and1 with all save the brighter pupils1 the bonds are more needed for an understanding of the definitions than the definitions are needed for the formation of the bonds4 7#i7 ,habits of verifying results4--,bonds should early be formed between certain manipulations of numbers and certain means of checking1 or verifying the correctness of1 the manipulation in question4 ,the additions to #i #i and the subtractions to #ah - #i should be verified by objective addition and subtraction and counting until the pupil has sure command2 the multiplications to #i x #i should be verified by objective multiplication and counting of the result 7in piles of tens and a pile of ones7 eight or ten times1#d and by addition eight or ten times2#d the divisions to #ha _/ #i should be verified by multiplication and occasionally objectively until the pupil has sure command2 column addition should be checked by adding the columns separately and adding the sums so obtained1 and by making two shorter tasks of the given task and adding the two sums2 'short' multiplication should be verified eight or ten times by addition2 'long' multiplication should be checked by reversing multiplier and multiplicand and in other ways2 'short' and 'long' division should be verified by multiplication4 #d ,eight or ten times in all1 not eight or ten times for each fact of the tables4 ,these habits of testing an obtained result are of threefold value4 ,they enable the pupil to find his own errors1 and to maintain a standard of accuracy by himself4 ,they give him a sense of the relations of the processes and the reasons why the right ways of adding1 subtracting1 multiplying1 and dividing are right1 such as only the very bright pupils can get from verbal explanations4 ,they put his acquisition of a certain power1 say multiplication1 to a real and intelligible use1 in checking the results of his practice of a new power1 and so instill a respect for arithmetical power and skill in general4 ,the time spent in such verification produces these results at little cost2 for the practice in adding to verify multiplications1 in multiplying to verify divisions1 and the like is nearly as good for general drill and review of the addition and multiplication themselves as practice devised for that special purpose4 ,early work in adding1 subtracting1 and reducing fractions should be verified by objective aids in the shape of lines and areas divided in suitable fractional parts4 ,early work with decimal fractions should be verified by the use of the equivalent common fractions for 4#be1 4#ge1 4#abe1 4#cge1 and the like4 ,multiplication and division with fractions1 both common and decimal1 should in the early stages be verified by objective aids4 ,the placing of the decimal point in multiplication and division with decimal fractions should be verified by such exercises as3-- #bj ,it cannot be #bjj2 for #bjj x #a4bc is much more than #bd4f4 ,it cannot be #b2 for #b x #a4bc is much less than #bd4f4 #a4bc 7#bd4fj #bdf ---- ,the establishment of habits of verifying results and their use is very greatly needed4 ,the percentage of wrong answers in arithmetical work in schools is now so high that the pupils are often being practiced in error4 ,in many cases they can feel no genuine and effective confidence in the processes1 since their own use of the processes brings wrong answers as often as right4 ,in solving problems they often cannot decide whether they have done the right thing or the wrong1 since even if they have done the right thing1 they may have done it inaccurately4 ,a wrong answer to a problem is therefore too often ambiguous and uninstructive to them4#e #e ,the facts concerning the present inaccuracy of school work in arithmetic will be found on pages #ajb to #aje4 ,these illustrations of the last few pages are samples of the procedures recommended by a consideration of all the bonds that one might form and of the contribution that each would make toward the abilities that the study of arithmetic should develop and improve4 ,it is by doing more or less at haphazard what psychology teaches us to do deliberately and systematically in this respect that many of the past advances in the teaching of arithmetic have been made4 ,,wasteful ,,and ,,harmful ,,bonds ,a scrutiny of the bonds now formed in the teaching of arithmetic with questions concerning the exact service of each1 results in a list of bonds of small value or even no value1 so far as a psychologist can determine4 ,i present here samples of such psychologically unjustifiable bonds with some of the reasons for their deficiencies4 7#a7 ,arbitrary units4--,in drills intended to improve the ability to see and use the meanings of numbers as names for ratios or relative magnitudes1 it is unwise to employ entirely arbitrary units4 ,the procedure in ,,ii 7on page #hd7 is better than that in ,i4 ,inches1 half-inches1 feet1 and centimeters are better as units of length than arbitrary ,as4 ,square inches1 square centimeters1 and square feet are better for areas4 ,ounces and pounds should be lifted rather than arbitrary weights4 ,pints1 quarts1 glassfuls1 cupfuls1 handfuls1 and cubic inches are better for volume4 ,all the real merit in the drills on relative magnitude advocated by ,speer1 ,mc,lellan and ,dewey1 and others can be secured without spending time in relating magnitudes for the sake of relative magnitude alone4 ,the use of units of measure in drills which will never be used in bona fide measuring is like the use of fractions like sevenths1 elevenths1 and thirteenths4 ,a very little of it is perhaps desirable to test the appreciation of certain general principles1 but for regular training it should give place to the use of units of practical significance4 ,illustration3 ,,fig4 #c4 ,a ---------- ,b ------------------------------ ,c -------------------- ,d ---------------------------------------- ,i4 ,if ,a is #a which line is #b8 ,which line is #d8 ,which line is #c8 ,a and ,c together equal what line8 ,a and ,b together equal what line8 ,how much longer is ,b than ,a8 ,how much longer is ,b than ,c8 ,how much longer is ,d than ,a8 ,illustration3 ,,fig4 #d4 ,a ---------- ,b ------------------------------ ,c -------------------- ,d ---------------------------------------- ,,ii4 ,a is #a inch long4 ,which line is #b inches long8 ,which line is #d inches long8 ,which line is #c inches long8 ,a and ,c together make 444 inches8 ,a and ,b together make 444 inches8 ,b is 444 444 longer than ,a8 ,b is 444 444 longer than ,c8 ,d is 444 444 longer than ,a8 7#b7 ,multiples of #aa4--,the multiplications of #b to #ab by #aa and #ab as single connections should be left for the pupil to acquire by himself as he needs them4 ,these connections interfere with the process of learning two-place multiplication4 ,the manipulations of numbers there required can be learned much more easily if #aa and #ab are used as multipliers in just the same way that #gh or #if would be4 ,later the #ab x #b1 #ab x #c1 etc41 may be taught4 ,there is less reason for knowing the multiples of #aa than for knowing the multiples of #ae1 #af1 or #be4 7#c7 ,abstract and concrete numbers4--,the elaborate emphasis of the supposed fact that we cannot multiply #gbf by #h dollars and the still more elaborate explanations of why nevertheless we find the cost of #gbf articles at #h each by multiplying #gbf by #h and calling the answer dollars are wasteful4 ,the same holds of the corresponding pedantry about division4 ,these imaginary difficulties should not be raised at all4 ,the pupil should not think of multiplying or dividing men or dollars1 but simply of the necessary equation and of the sort of thing that the missing number represents4 #h x #gbf 4444 ,answer is dollars1 or #h1 #gbf1 multiply4 ,answer is dollars1 is all that he needs to think1 and is in the best form for his thought4 ,concerning the distinction between abstract and concrete numbers1 both logic and common sense as well as psychology support the contention of ,mc,dougle '#ad1 p4 #bjf;f41 who writes3-- ,the most elementary counting1 even that stage when the counts were not carried in the mind1 but merely in notches on a stick or by ,de,morgan's stones in a pot1 requires some thought2 and the most advanced counting implies memory of things4 ,the terms1 therefore1 abstract and concrete number1 have long since ceased to be used by thinking people4 ,recently the writer visited an arithmetic class in a ,state ,normal ,school and saw a group of practically adult students confused about this very question concerning abstract and concrete numbers1 according to their previous training in the conventionalities of the textbook4 ,their teacher diverted the work of the hour and she and the class spent almost the whole period in reestablishing the requirements 'that the product must always be the same kind of unit as the multiplicand1' and 'addends must all be alike to be added4' ,this is not an exceptional case4 ,throughout the whole range of teaching arithmetic in the public schools pupils are obfuscated by the philosophical encumbrances which have been imposed upon the simplest processes of numerical work4 ,the time is surely ripe1 now that we are readjusting our ideas of the subject of arithmetic1 to revise some of these wasteful and disheartening practices4 ,algebra historically grew out of arithmetic1 yet it has not been laden with this distinction4 ,no pupil in algebra lets x equal the horses2 he lets x equal the number of horses1 and proceeds to drop the idea of horses out of his consideration4 ,he multiplies1 divides1 and extracts the root of the number1 sometimes handling fractions in the process1 and finally interprets the result according to the conditions of his problem4 ,of course1 in the early number work there have been the sense-objects from which number has been perceived1 but the mind retreats naturally from objectivity to the pure conception of number1 and then to the number symbol4 ,the following is taken from the appendix to ,horn's thesis1 where a seventh grade girl gets the population of the ,united ,states in #ahbj3-- #g1hfb1aff whites #bcc1fcd free negroes #a1ech1jbb slaves --------- #i1fcc1hbb ,in this problem three different kinds of addends are combined1 if we accept the usual distinction4 ,some may say that this is a mistake1--that the pupil transformed the 'whites1' 'free negroes1' and 'slaves' into a common unit1 such as 'people' of 'population' and then added these common units4 ,but this 'explanation' is entirely gratuitous1 as one will find if he questions the pupil about the process4 ,it will be found that the child simply added the figures as numbers only and then interpreted the result1 according to the statement of the problem1 without so much mental gymnastics4 ,the writer has questioned hundreds of students in ,normal ,school work on this point1 and he believes that the ordinary mind-movement is correctly set forth here1 no matter how well one may maintain as an academic proposition that this is not logical4 ,many classes in the ,eastern ,kentucky ,state ,normal have been given this problem to solve1 and they invariably get the same result3-- ',in a garden on the ,summit are as many cabbage-heads as the total number of ladies and gentlemen in this class4 ,how many cabbage-heads in the garden8' ,and the blackboard solution looks like this each time3-- #bi ladies #ae gentlemen -- #dd cabbage-heads ,so1 also1 one may say3 ,i have #f times as many sheep as you have cows4 ,if you have #e cows1 how many sheep have ,i8 ,here we would multiply the number of cows1 which is #e1 by #f and call the result #cj1 which must be linked with the idea of sheep because the conditions imposed by the problem demand it4 ,the mind naturally in this work separates the pure number from its situation1 as in algebra1 handles it according to the laws governing arithmetical combinations1 and labels the result as the statement of the problem demands4 ,this is expressed in the following1 which is tacitly accepted in algebra1 and should be accepted equally in arithmetic3 ',in all computations and operations in arithmetic1 all numbers are essentially abstract and should be so treated4 ,they are concrete only in the thought process that attends the operation and interprets the result4' 7#d7 ,least common multiple4--,the whole set of bonds involved in learning 'least common multiple' should be left out4 ,in adding and subtracting fractions the pupil should not find the least common multiple of their denominators but should find any common multiple that he can find quickly and correctly4 ,no intelligent person would ever waste time in searching for the least common multiple of sixths1 thirds1 and halves except for the unfortunate traditions of an oversystematized arithmetic1 but would think of their equivalents in sixths or twelfths or twenty-fourths or any other convenient common multiple4 ,the process of finding the least common multiple is of such exceedingly rare application in science or business or life generally that the textbooks have to resort to purely fantastic problems to give drill in its use4 7#e7 ,greatest common divisor4--,the whole set of bonds involved in learning 'greatest common divisor' should also be left out4 ,in reducing fractions to lowest terms the pupil should divide by anything that he sees that he can divide by1 favoring large divisors1 and continue doing so until he gets the fraction in terms suitable for the purpose in hand4 ,the reader probably never has had occasion to compute a greatest common divisor since he left school4 ,if he has computed any1 the chances are that he would have saved time by solving the problem in some other way6 ,the following problems are taken at random from those given by one of the best of the textbooks that make the attempt to apply the facts of ,greatest ,common ,divisor and ,least ,common ,multiple to problems4#f ,most of these problems are fantastic4 ,the others are trivial1 or are better solved by trial and adaptation4 #a4 ,a certain school consists of #acb pupils in the high school1 #aed in the grammar1 and #aih in the primary grades4 ,if each group is divided into sections of the same number containing as many pupils as possible1 how many pupils will there be in each section8 #b4 ,a farmer has #bdj bu4 of wheat and #ibj bu4 of oats1 which he desires to put into the least number of boxes of the same capacity1 without mixing the two kinds of grain4 ,find how many bushels each box must hold4 #c4 ,four bells toll at intervals of #c1 #g1 #ab1 and #ad seconds respectively1 and begin to toll at the same instant4 ,when will they next toll together8 #d4 ,a1 ,b1 ,c1 and ,d start together1 and travel the same way around an island which is #fjj mi4 in circuit4 ,a goes #bj mi4 per day1 ,b #cj1 ,c #be1 and ,d #dj4 ,how long must their journeying continue1 in order that they may all come together again8 #e4 ,the periods of three planets which move uniformly in circular orbits round the sun1 are respectively #bjj1 #bej1 and #cjj da4 ,supposing their positions relatively to each other and the sun to be given at any moment1 determine how many da4 must elapse before they again have exactly the same relative positions4 #f ,mc,lellan and ,ames1 ,public ,school ,arithmetic #aijj4 7#f7 ,rare and unimportant words4--,the bonds between rare or unimportant words and their meanings should not be formed for the mere sake of verbal variety in the problems of the textbook4 ,a pupil should not be expected to solve a problem that he cannot read4 ,he should not be expected in grades #b and #c1 or even in grade #d1 to read words that he has rarely or never seen before4 ,he should not be given elaborate drill in reading during the time devoted to the treatment of quantitative facts and relations4 ,all this is so obvious that it may seem needless to relate4 ,it is not4 ,with many textbooks it is now necessary to give definite drill in reading the words in the printed problems intended for grades #b1 #c1 and #d1 or to replace them by oral statements1 or to leave the pupils in confusion concerning what the problems are that they are to solve4 ,many good teachers make a regular reading-lesson out of every page of problems before having them solved4 ,there should be no such necessity4 ,to define rare and unimportant concretely1 ,i will say that for pupils up to the middle of grade #c1 such words as the following are rare and unimportant 7though each of them occurs in the very first fifty pages of some well-known beginner's book in arithmetic74 absentees account ,adele admitted ,agnes agreed ,albany ,allen allowed alternate ,andrew ,arkansas arrived assembly automobile baking powder balance barley beggar ,bertie ,bessie bin ,boston bouquet bronze buckwheat ,byron camphor ,carl ,carrie ,cecil ,charlotte charity ,chicago cinnamon ,clara clothespins collect comma committee concert confectioner cranberries crane currants dairyman ,daniel ,david dealer debt delivered ,denver department deposited dictation discharged discover discovery dish-water drug due ,edgar ,eddie ,edwin election electric ,ella ,emily enrolled entertainment envelope ,esther ,ethel exceeds explanation expression generally gentlemen ,gilbert ,grace grading ,graham grammar ,harold hatchet ,heralds hesitation ,horace ,mann impossible income indicated inmost inserts installments instantly insurance ,iowa ,jack ,jennie ,johnny ,joseph journey ,julia ,katherine lettuce-plant library ,lottie ,lula margin ,martha ,matthew ,maud meadow mentally mercury mineral ,missouri molasses ,morton movements muslin ,nellie nieces ,oakland observing obtained offered office onions opposite original package packet palm ,patrick ,paul payments peep ,peter perch phaeton photograph piano pigeons ,pilgrims preserving proprietor purchased ,rachel ,ralph rapidity rather readily receipts register remanded respectively ,robert ,roger ,ruth rye ,samuel ,san ,francisco seldom sheared shingles skyrockets sloop solve speckled sponges sprout stack ,stephen strap successfully suggested sunny supply ,susan ,susie's syllable talcum term test thermometer ,thomas torpedoes trader transaction treasury tricycle tube two-seated united usually vacant various vase velocipede votes walnuts ,walter ,washington watched whistle woodland worsted 7#g7 ,misleading facts and procedures4--,bonds should not be formed between articles of commerce and grossly inaccurate prices therefor1 between events and grossly improbable consequences1 or causes or accompaniments thereof1 nor between things1 qualities1 and events which have no important connections one with another in the real world4 ,in general1 things should not be put together in the pupil's mind that do not belong together4 ,if the reader doubts the need of this warning let him examine problems #a to #e1 all from reputable books that are in common use1 or have been within a few years1 and consider how addition1 subtraction1 and the habits belonging with each are confused by exercise #f4 #a4 ,if a duck flying #c_/e as fast as a hawk flies #ij miles in an hour1 how fast does the hawk fly8 #b4 ,at #e_/h of a cent apiece how many eggs can ,i buy for #fj8 #c4 ,at 4#fh a pair how many pairs of overshoes can you buy for #haf8 #d4 ,at 4#ac a dozen how many dozen bananas can you buy for #c4ab8 #e4 ,how many pecks of beans can be put into a box that will hold just #ba bushels8 #f4 ,write answers3 #ecg ,beginning at the bottom say #aa1 #ah1 and #b 7writing it in #cfe its place7 are #bj4 #e1 #aa1 #ad1 and #f 7writing it7 are #bj1 8 #e1 #aj4 ,the number1 omitted1 is #fb4 #cf ---- #ajjj a4 #eha b4 #fbe c4 #geb d4 #cad e4 8 #ig 8 #dad #dbi #hde #cfd #ij #acj 8 #bbc 8 #dag 8 #gf #ie ---- ---- ---- ---- ---- #ageh #bjej #bdfj #ajjj #bcfg 7#h7 ,trivialities and absurdities4--,bonds should not be formed between insignificant or foolish questions and the labor of answering them1 nor between the general arithmetical work of the school and such insignificant or foolish questions4 ,the following are samples from recent textbooks of excellent standing3-- ,on one side of ,george's slate there are #cb words1 and on the other side #bf words4 ,if he erases #f words from one side1 and #h from the other1 how many words remain on his slate8 ,a certain school has #ad rooms1 and an average of #dj children in a room4 ,if every one in the school should make #ejj straight marks on each side of his slate1 how many would be made in all8 #h times the number of stripes in our flag is the number of years from #ahjj until ,roosevelt was elected ,president4 ,in what year was he elected ,president8 ,from the ,declaration of ,independence to the ,world's ,fair in ,chicago was #i times as many years as there are stripes in the flag4 ,how many years was it8 7#i7 ,useless methods4--,bonds should not be formed between a described situation and a method of treating the situation which would not be a useful one to follow in the case of the real situation4 ,for example1 ,if ,i set #if trees in rows1 sixteen trees in a row1 how many rows will ,i have8 forms the habit of treating by division a problem that in reality would be solved by counting the rows4 ,so also ,i wish to give #be cents to each of a group of boys and find that it will require #b4ge4 ,how many boys are in the group8 forms the habit of answering a question by division whose answer must already have been present to give the data of the problem4 7#aj7 ,problems whose answers would1 in real life1 be already known4--,the custom of giving problems in textbooks which could not occur in reality because the answer has to be known to frame the problem is a natural result of the lazy author's tendency to work out a problem to fit a certain process and a certain answer4 ,such bogus problems are very1 very common4 ,in a random sampling of a dozen pages of ,general ,review problems in one of the most widely used of recent textbooks1 ,i find that about #f percent of the problems are of this sort4 ,among the problems extemporized by teachers these bogus problems are probably still more frequent4 ,such are3-- ,a clerk in an office addressed letters according to a given list4 ,after she had addressed #bejj1 #d_/i of the names on the list had not been used2 how many names were in the entire list8 ,the ,canadian power canal at ,sault ,ste4 ,marie furnished #bj1jjj horse power4 ,the canal on the ,michigan side furnished #b-a_/b times as much4 ,how many horse power does the latter furnish8 ,it may be asserted that the ideal of giving as described problems only problems that might occur and demand the same sort of process for solution with a real situation1 is too exacting4 ,if a problem is comprehensible and serves to illustrate a principle or give useful drill1 that is enough1 teachers may say4 ,for really scientific teaching it is not enough4 ,moreover1 if problems are given merely as tests of knowledge of a principle or as means to make some fact or principle clear or emphatic1 and are not expected to be of direct service in the quantitative work of life1 it is better to let the fact be known4 ,for example1 ,i am thinking of a number4 ,half of this number is twice six4 ,what is the number8 is better than ,a man left his wife a certain sum of money4 ,half of what he left her was twice as much as he left to his son1 who receives #fjjj4 ,how much did he leave his wife8 ,the former is better because it makes no false pretenses4 7#aa7 ,needless linguistic difficulties4--,it should be unnecessary to add that bonds should not be formed between the pupil's general attitude toward arithmetic and needless1 useless difficulty in language or needless1 useless1 wrong reasoning4 ,our teaching is1 however1 still tainted by both of these unfortunate connections1 which dispose the pupil to think of arithmetic as a mystery and folly4 ,consider1 for example1 the profitless linguistic difficulty of problems #a-f1 whose quantitative difficulties are simply those of3-- #a4 #e #h #c #g #b4 #fd _/ #h1 and knowledge that #a peck #h quarts #c4 #ab _/ #d #d4 #f _/ #b #e4 #c x #b #f4 #d x #d #a4 ,what amount should you obtain by putting together #e cents1 #h cents1 #c cents1 and #g cents8 ,did you find this result by adding or multiplying8 #b4 ,how many times must you empty a peck measure to fill a basket holding #fd quarts of beans8 #c4 ,if a girl commits to memory #d pages of history in one day1 in how many days will she commit to memory #ab pages8 #d4 ,if ,fred had #f chickens how many times could he give away #b chickens to his companions8 #e4 ,if a croquet-player drove a ball through #b arches at each stroke1 through how many arches will he drive it by #c strokes8 #f4 ,if mamma cut the pie into #d pieces and gave each person a piece1 how many persons did she have for dinner if she used #d whole pies for dessert8 ,arithmetically this work belongs in the first or second years of learning4 ,but children of grades #b and #c1 save a few1 would be utterly at a loss to understand the language4 ,we are not yet free from the follies illustrated in the lessons of pages #if to #ii1 which mystified our parents4 ,,lesson ,i ,illustration3 ,,fig4 #e4 #a4 ,in this picture1 how many girls are in the swing8 #b4 ,how many girls are pulling the swing8 #c4 ,if you count both girls together1 how many are they8 ,one girl and one other girl are how many8 #d4 ,how many kittens do you see on the stump8 #e4 ,how many on the ground8 #f4 ,how many kittens are in the picture8 ,one kitten and one other kitten are how many8 #g4 ,if you should ask me how many girls are in the swing1 or how many kittens are on the stump1 ,i could answer aloud1 ,one2 or ,i could write ,one2 or thus1 #a4 #h4 ,if ,i write ,one1 this is called the word ,one4 #i4 ,this1 #a1 is named a figure ,one1 because it means the same as the word ,one1 and stands for ,one4 #aj4 ,write #a4 ,what is this named8 ,why8 #aa4 ,a figure #a may stand for one girl1 one kitten1 or one anything4 #ab4 ,when children first attend school1 what do they begin to learn8 ,ans4 ,letters and words4 #ac4 ,could you read or write before you had learned either letters or words8 #ad4 ,if we have all the letters together1 they are named the ,alphabet4 #ae4 ,if we write or speak words1 they are named ,language4 #af4 ,you are commencing to study ,arithmetic2 and you can read and write in ,arithmetic only as you learn the ,alphabet and ,language of ,arithmetic4 ,but little time will be required for this purpose4 ,,lesson ,,ii ,illustration3 ,,fig4 #f4 #a4 ,if we speak or write words1 what do we name them1 when taken together8 #b4 ,what are you commencing to study8 ,ans4 ,arithmetic4 #c4 ,what ,language must you now learn8 #d4 ,what do we name this1 #a8 ,why8 #e4 ,this figure1 #a1 is part of the ,language of ,arithmetic4 #f4 ,if ,i should write something to stand for ,two--two girls1 two kittens1 or two things of any kind--what do you think we would name it8 #g4 ,a figure ,two is written thus3 #b4 ,make a figure two4 #h4 ,why do we name this a figure two8 #i4 ,this figure two 7#b7 is part of the ,language of ,arithmetic4 #aj4 ,in this picture one boy is sitting1 playing a flageolet4 ,what is the other boy doing8 ,if the boy standing should sit down by the other1 how many boys would be sitting together8 ,one boy and one other boy are how many boys8 #aa4 ,you see a flageolet and a violin4 ,they are musical instruments4 ,one musical instrument and one other musical instrument are how many8 #ab4 ,i will write thus3 #a #a #b4 ,we say that #a boy and #a other boy1 counted together1 are #b boys2 or are equal to #b boys4 ,we will now write something to show that the first #a and the other #a are to be counted together4 #ac4 ,we name a line drawn thus1 -1 a horizontal line4 ,draw such a line4 ,name it4 #ad4 ,a line drawn thus1 1 we name a vertical line4 ,draw such a line4 ,name it4 #ae4 ,now ,i will put two such lines together2 thus1 4 ,what kind of a line do we name the first 7-78 ,and what do we name the last8 778 ,are these lines long or short8 ,where do they cross each other8 #af4 ,each of you write thus3 -1 1 4 #ag4 ,this1 1 is named ,plus4 ,plus means more2 and also means more4 #ah4 ,i will write4 ,one and ,one ,more ,equal ,two4 #ai4 ,now ,i will write part of this in the ,language of ,arithmetic4 ,i write the first ,one thus1 #a2 then the other ,one thus1 #a4 ,afterward ,i write1 for the word ,more1 thus1 1 placing the between #a and #a1 so that the whole stands thus3 #a #a4 ,as ,i write1 ,i say1 ,one and ,one more4 #bj4 ,each of you write #a #a4 ,read what you have written4 #ba4 ,this 1 when written between the #as1 shows that they are to be put together1 or counted together1 so as to make #b4 #bb4 ,because shows what is to be done1 it is called a ,sign4 ,if we take its name1 ,plus1 and the word ,sign1 and put both words together1 we have ,sign ,plus1 or ,plus ,sign4 ,in speaking of this we may call it ,sign ,plus1 or ,plus ,sign1 or ,plus4 #bc4 #a1 #b1 1 are part of the ,language of ,arithmetic4 ,write the following in the ,language of ,arithmetic3 #bd4 ,one and one more4 #be4 ,one and two more4 #bf4 ,two and one more4 7#ab7 ,ambiguities and falsities4--,consider the ambiguities and false reasoning of these problems4 #a4 ,if you can earn #d cents a day1 how much can you earn in #f weeks8 7,are ,sundays counted8 ,should a child who earns #d cents some day expect to repeat the feat daily87 #b4 ,how many lines must you make to draw ten triangles and five squares8 7,i can do this with #h lines1 though the answer the book requires is #ej47 #c4 ,a runner ran twice around an #a_/h mile track in two minutes4 ,what distance did he run in #b_/c of a minute8 7,i do not know1 but ,i do know that1 save by chance1 he did not run exactly #b_/c of #a_/h mile47 #d4 ,john earned #d4ce in a week1 and ,henry earned #a4ic4 ,they put their money together and bought a gun4 ,what did it cost8 7,maybe #e1 maybe #aj4 ,did they pay for the whole of it8 ,did they use all their earnings1 or less1 or more87 #e4 ,richard has #ab nickels in his purse4 ,how much more than #ej cents would you give him for them8 7,would a wise child give #fj cents to a boy who wanted to swap #ab nickels therefor1 or would he suspect a trick and hold on to his own coins87 #f4 ,if a horse trots #aj miles in one hour how far will he travel in #i hours8 #g4 ,if a girl can pick #c quarts of berries in #a hour how many quarts can she pick in #c hours8 7,these last two1 with a teacher insisting on the #ij and #i1 might well deprive a matter-of-fact boy of respect for arithmetic for weeks thereafter47 ,the economics and physics of the next four problems speak for themselves4 #h4 ,i lost #ae by selling a horse for #he4 ,what was the value of the horse8 #i4 ,if floating ice has #g times as much of it under the surface of the water as above it1 what part is above water8 ,if an iceberg is #ej ft4 above water1 what is the entire height of the iceberg8 ,how high above water would an iceberg #cjj ft4 high have to be8 #aj4 ,a man's salary is #ajjj a year and his expenses #fbe4 ,how many years will elapse before he is worth #aj1jjj if he is worth #bejj at the present time8 #aa4 ,sound travels #aabj ft4 a second4 ,how long after a cannon is fired in ,new ,york will the report be heard in ,philadelphia1 a distance of #ij miles8 ,,guiding ,,principles ,the reader may be wearied of these special details concerning bonds now neglected that should be formed and useless or harmful bonds formed for no valid reason4 ,any one of them by itself is perhaps a minor matter1 but when we have cured all our faults in this respect and found all the possibilities for wiser selection of bonds1 we shall have enormously improved the teaching of arithmetic4 ,the ideal is such choice of bonds 7and1 as will be shown later1 such arrangement of them7 as will most improve the functions in question at the least cost of time and effort4 ,the guiding principles may be kept in mind in the form of seven simple but golden rules3-- #a4 ,consider the situation the pupil faces4 #b4 ,consider the response you wish to connect with it4 #c4 ,form the bond2 do not expect it to come by a miracle4 #d4 ,other things being equal1 form no bond that will have to be broken4 #e4 ,other things being equal1 do not form two or three bonds when one will serve4 #f4 ,other things being equal1 form bonds in the way that they are required later to act4 #g4 ,favor1 therefore1 the situations which life itself will offer1 and the responses which life itself will demand4 ,,chapter ,v ,,the ,,psychology ,,of ,,drill ,,in ,,arithmetic3 ,,the ,,strength ,,of ,,bonds ,an inventory of the bonds to be formed in learning arithmetic should be accompanied by a statement of how strong each bond is to be made and kept year by year4 ,since1 however1 the inventory itself has been presented here only in samples1 the detailed statement of desired strength for each bond cannot be made4 ,only certain general facts will be noted here4 ,,the ,,need ,,of ,,stronger ,,elementary ,,bonds ,the constituent bonds involved in the fundamental operations with numbers need to be much stronger than they now are4 ,inaccuracy in these operations means weakness of the constituent bonds4 ,inaccuracy exists1 and to a degree that deprives the subject of much of its possible disciplinary value1 makes the pupil's achievements of slight value for use in business or industry1 and prevents the pupil from verifying his work with new processes by some previously acquired process4 ,the inaccuracy that exists may be seen in the measurements made by the many investigators who have used arithmetical tasks as tests of fatigue1 practice1 individual differences and the like1 and in the special studies of arithmetical achievements for their own sake made by ,courtis and others4 ,burgerstein '#ia1 using such examples as #bhgjdeafichbgfedfcig #cehfidbgceiafchbgbfc ---------------------- and similar long numbers to be multiplied by #b or by #c or by #d or by #e or by #f1 found #hea errors in #bh1bfg answer-figures1 or #c per hundred answer-figures1 or #c_/e of an error per example4 ,the children were #i-a_/b to #ae years old4 ,laser '#id1 using the same sort of addition and multiplication1 found somewhat over #c errors per hundred answer-figures in the case of boys and girls averaging #aa-a_/b years1 during the period of their most accurate work4 ,holmes '#ie1 using addition of the sort just described1 found #cdf errors in #bc1gac answer-figures or about #a-a_/b per hundred4 ,the children were from all grades from the third to the eighth4 ,in ,laser's work1 #ba1 #ai1 #ac1 and #aj answer-figures were obtained per minute4 ,friedrich '#ig with similar examples1 giving the very long time of #bj minutes for obtaining about #bjj answer-figures1 found from #a to #b per hundred wrong4 ,king '#jg had children in grade #e do sums1 each consisting of #e two-place numbers4 ,in the most accurate work-period1 they made #a error per #bj columns4 ,in multiplying a four-place by a four-place number they had less than one total answer right out of three4 ,in ,new ,york ,city ,courtis found '#aa-'ab with his ,test #g that in #ab minutes the average achievement of fourth-grade children is #h4h units attempted with #d4b right4 ,in grade #e the facts are #aj4i attempts with #e4h right2 in grade #f1 #ab4e attempts with #g4j right2 in grade #g1 #ae attempts with #h4e right2 in grade #h1 #ae4g attempts with #aj4a right4 ,these results are near enough to those obtained from the country at large to serve as a text here4 ,the following were set as official standards1 in an excellent school system1 ,courtis ,series ,b being used3-- ,,speed ,,percent ,,of ,,grade4 ,,attempts4 ,,correct ,,answers4 ,addition #h #ab #hj #g #aa #hj #f #aj #gj #e #i #gj #d #h #gj ,subtraction #h #ab #ij #g #aa #ij #f #aj #ij #e #i #hj #d #g #hj ,multiplication #h #aa #hj #g #aj #hj #f #i #hj #e #g #gj #d #f #fj ,division #h #aa #ij #g #aj #ij #f #h #hj #e #f #gj #d #d #fj ,kirby '#ac1 pp4 #af ff4 and #ee ff4 found that1 in adding columns like those printed below1 children in grade #d got on the average less than #hj percent of correct answers4 ,their average speed was about #b columns per minute4 ,in doing division of the sort printed below children of grades #c ,b and #d ,a got less than #ie percent of correct answers1 the average speed being #d divisions per minute4 ,in both cases the slower computers were no more accurate than the faster ones4 ,practice improved the speed very rapidly1 but the accuracy remained substantially unchanged4 ,brown '#aa and '#ab found a similar low status of ability and notable improvement from a moderate amount of special practice4 #c #e #f #b #c #h #i #g #d #i #g #i #f #e #e #f #d #e #h #b #c #d #g #h #g #c #g #i #c #g #h #h #d #h #b #f #h #b #i #h #b #b #d #g #f #i #h #e #f #b #f #i #e #g #h #e #b #c #b #d #i #f #d #b #g #b #i #d #d #e #c #c #g #i #i #i #b #h #i #g #f #h #i #f #d #g #g #i #b #d #h #d #f #i #i #b #f #i #h #i -- -- -- -- -- -- -- -- -- -- #bj 4444 #es #ef 4444 #is and 4444 r4 #cj 4444 #gs and 4444 r4 #hi 4444 #is and 4444 r4 #bj 4444 #hs and 4444 r4 #ef 4444 #fs and 4444 r4 #ca 4444 #ds and 4444 r4 #hf 4444 #is and 4444 r4 ,it is clear that numerical work as inaccurate as this has little or no commercial or industrial value4 ,if clerks got only six answers out of ten right as in the ,courtis tests1 one would need to have at least four clerks make each computation and would even then have to check many of their discrepancies by the work of still other clerks1 if he wanted his accounts to show less than one error per hundred accounting units of the ,courtis size4 ,it is also clear that the habits of 444 absolute accuracy1 and satisfaction in truth as a result which arithmetic is supposed to further must be largely mythical in pupils who get right answers only from three to nine times out of ten6 ,,early ,,mastery ,the bonds in question clearly must be made far stronger than they now are4 ,they should in fact be strong enough to abolish errors in computation1 except for those due to temporary lapses4 ,it is much better for a child to know half of the multiplication tables1 and to know that he does not know the rest1 than to half-know them all2 and this holds good of all the elementary bonds required for computation4 ,any bond should be made to work perfectly1 though slowly1 very soon after its formation is begun4 ,speed can easily be added by proper practice4 ,the chief reasons why this is not done now seem to be the following3 7#a7 ,certain important bonds 7like the additions with higher decades7 are not given enough attention when they are first used4 7#b7 ,the special training necessary when a bond is used in a different connection 7as when the multiplications to #i x #i are used in examples like #gbi #h --- where the pupil has also to choose the right number to multiply1 keep in mind what is carried1 use it properly1 and write the right figure in the right place1 and carry a figure1 or remember that he carries none7 is neglected4 7#c7 ,the pupil is not taught to check his work4 7#d7 ,he is not made responsible for substantially accurate results4 ,furthermore1 the requirement of 7#d7 without the training of 7#a71 7#b71 and 7#c7 will involve either a fruitless failure on the part of many pupils1 or an utterly unjust requirement of time4 ,the common error of supposing that the task of computation with integers consists merely in learning the additions to #i #i1 the subtractions to #ah - #i1 the multiplications to #h x #i1 and the divisions to #ha _/ #i1 and in applying this knowledge in connection with the principles of decimal notation1 has had a large share in permitting the gross inaccuracy of arithmetical work4 ,the bonds involved in 'knowing the tables' do not make up one fourth of the bonds involved in real adding1 subtracting1 multiplying1 and dividing 7with integers alone74 ,it should be noted that if the training mentioned in 7#a7 and 7#b7 is well cared for1 the checking of results as recommended in 7#c7 becomes enormously more valuable than it is under present conditions1 though even now it is one of our soundest practices4 ,if a child knows the additions to higher decades so that he can add a seen one-place number to a thought-of two-place number in three seconds or less with a correct answer #aii times out of #bjj1 there is only an infinitesimal chance that a ten-figure column twice added 7once up1 once down7 a few minutes apart with identical answers will be wrong4 ,suppose that1 in long multiplication1 a pupil can multiply to #i x #i while keeping his place and keeping track of what he is 'carrying' and of where to write the figure he writes1 and can add what he carries without losing track of what he is to add it to1 where he is to write the unit figure1 what he is to multiply next and by what1 and what he will then have to carry1 in each case to a surety of #ii percent of correct responses4 ,then two identical answers got by multiplying one three-place number by another a few minutes apart1 and with reversal of the numbers1 will not be wrong more than twice in his entire school career4 ,checks approach proofs when the constituent bonds are strong4 ,if1 on the contrary1 the fundamental bonds are so weak that they do not work accurately1 checking becomes much less trustworthy and also very much more laborious4 ,in fact1 it is possible to show that below a certain point of strength of the fundamental bonds1 the time required for checking is so great that part of it might better be spent in improving the fundamental bonds4 ,for example1 suppose that a pupil has to find the sum of five numbers like #b4di1 #e4be1 #f4ej1 #g4hi1 and #c4ge4 ,counting each act of holding in mind the number to be carried and each writing of a column's result as equivalent in difficulty to one addition1 such a sum equals nineteen single additions4 ,on this basis and with certain additional estimates#g we can compute the practical consequences for a pupil's use of addition in life according to the mastery of it that he has gained in school4 #g ,these concern allowances for two errors occurring in the same example and for the same wrong answer being obtained in both original work and check work4 ,i have so computed the amount of checking a pupil will have to do to reach two agreeing numbers 7out of two1 or three1 or four1 or five1 or whatever the number before he gets two that are alike71 according to his mastery of the elementary processes4 ,the facts appear in ,table #a4 ,it is obvious that a pupil whose mastery of the elements is that denoted by getting them right #if times out of #ajj will require so much time for checking that1 even if he were never to use this ability for anything save a few thousand sums in addition1 he would do well to improve this ability before he tried to do the sums4 ,an ability of #aii out of #bjj1 or #iie out of #ajjj1 seems likely to save much more time than would be taken to acquire it1 and a reasonable defense could be made for requiring #iif or #iig out of #ajjj4 ,a precision of from #iie to #iig out of #ajjj being required1 and ordinary sagacity being used in the teaching1 speed will substantially take care of itself4 ,counting on the fingers or in words will not give that precision4 ,slow recourse to memory of serial addition tables will not give that precision4 ,nothing save sure memory of the facts operating under the conditions of actual examples will give it4 ,and such memories will operate with sufficient speed4 ,,table #a ,,the ,,effect ,,of ,,mastery ,,of ,,the ,,elementary ,,facts ,,of ,,addition ,,upon ,,the ,,labor ,,required ,,to ,,secure ,,two ,,agreeing ,,answers ,,when ,,adding ,,five ,,three-,,figure ,,numbers ,,mastery ,,of ,,approximate ,,approximate ,,approximate ,,approximate ,,the ,,number ,,of ,,number ,,of ,,number ,,of ,,number ,,of ,,elementary ,,wrong ,,answers,,agreeing ,,agreeing ,,checkings ,,additions ,,in ,,sums ,,of #e ,,answers1 ,,answers1 ,,required 7,,over ,,times ,,right ,,three-,,place ,,after ,,one ,,after ,a ,,and ,,above ,,the ,,in #ajjj ,,numbers ,,per ,,checking1 ,,checking ,,of ,,first ,,general #ajjj ,,per #ajjj ,,the ,,first ,,checking ,,of ,,discrepancies,,the #ajjj ,,sums7 ,,to ,,secure ,,two ,,agreeing ,,results ---------------------------------------- -------------------------- #ifj #gjj #ij #baf #dejj #ihj #chj #chd #fgf #abjj #iij #aij #fef #ijf #dgj #iie #ie #hai #ige #baj #iif #gf #hed #ihd #afe #iig #ed #hie #iib #aae #iih #ch #ibe #iif #hj #iii #ai #ifb #iii #dj ---------------------------------------- -------------------------- ,there is one intelligent objection to the special practice necessary to establish arithmetical connections so fully as to give the accuracy which both utilitarian and disciplinary aims require4 ,it may be said that the pupils in grades #c1 #d1 and #e cannot appreciate the need and that consequently the work will be dull1 barren1 and alien1 without close personal appropriation by the pupil's nature4 ,it is true that no vehement life-purpose is directly involved by the problem of perfecting one's power to add #g to #bh in grade #b1 or by the problem of multiplying #bec by #h accurately in grade #c or by precise subtraction in long division in grade #d4 ,it is also true1 however1 that the most humanly interesting of problems--one that the pupil attacks most whole-heartedly--will not be solved correctly unless the pupil has the necessary associative mechanisms in order2 and the surer he is of them1 the freer he is to think out the problem as such4 ,further1 computation is not dull if the pupil can compute4 ,he does not himself object to its barrenness of vital meaning1 so long as the barrenness of failure is prevented4 ,we must not forget that pupils like to learn4 ,in teaching excessively dull individuals1 who has not often observed the great interest which they display in anything that they are enabled to master8 ,there is pathos in their joy in learning to recognize parts of speech1 perform algebraic simplifications1 or translate ,latin sentences1 and in other accomplishments equally meaningless to all their interests save the universal human interest in success and recognition4 ,still further1 it is not very hard to show to pupils the imperative need of accuracy in scoring games1 in the shop1 in the store1 and in the office4 ,finally1 the argument that accurate work of this sort is alien to the pupil in these grades is still stronger against inaccurate work of the same sort4 ,if we are to teach computation with two- and three- and four-place numbers at all1 it should be taught as a reliable instrument1 not as a combination of vague memories and faith4 ,the author is ready to cut computation with numbers above #aj out of the curriculum of grades #a-f as soon as more valuable educational instruments are offered in its place1 but he is convinced that nothing in child-nature makes a large variety of inaccurate computing more interesting or educative or germane to felt needs1 than a smaller variety of accurate computing6 ,,the ,,strength ,,of ,,bonds ,,for ,,temporary ,,service ,the second general fact is that certain bonds are of service for only a limited time and so need to be formed only to a limited and slight degree of strength4 ,the data of problems set to illustrate a principle or improve some habit of computation are1 of course1 the clearest cases4 ,the pupil needs to remember that ,john bought #c loaves of bread and that they were #e-;cent loaves and that he gave #be cents to the baker only long enough to use the data to decide what change ,john should receive4 ,the connections between the total described situation and the answer obtained1 supposing some considerable computation to intervene1 is a bond that we let expire almost as soon as it is born4 ,it is sometimes assumed that the bond between a certain group of features which make a problem a ',buy a things at b per thing1 find total cost' problem or a ',buy a things at b per thing1 what change from c' problem or a ',what gain on buying for a and selling for b' problem or a ',how many things at a each can ,i buy for b cents' problem--it is assumed that the bond between these essential defining features and the operation or operations required for solution is as temporary as the bonds with the name of the buyer or the price of the thing4 ,it is assumed that all problems are and should be solved by some pure act of reasoning without help or hindrance from bonds with the particular verbal structure and vocabulary of the problems4 ,whether or not they should be1 they are not4 ,every time that a pupil solves a 'bought-sold' problem by subtraction he strengthens the tendency to respond to any problem whatsoever that contains the words 'bought for' and 'sold for' by subtraction2 and he will by no means surely stop and survey every such problem in all its elements to make sure that no other feature makes inapplicable the tendency to subtract which the 'bought sold' evokes4 ,to prevent pupils from responding to the form of statement rather than the essential facts1 we should then not teach them to forget the form of statement1 but rather give them all the common forms of statement to which the response in question is an appropriate response1 and only such4 ,if a certain form of statement does in life always signify a certain arithmetical procedure1 the bond between it and that procedure may properly be made very strong4 ,another case of the formation of bonds to only a slight degree of strength concerns the use of so-called 'crutches' such as writing 1 -1 and x in copying problems like those below3-- ,add ,subtract ,multiply #bc #gi #cb #fa #bd #c -- -- -- or altering the figures when 'borrowing' in subtraction1 and the like4 ,since it is undesirable that the pupil should regard the 'crutch' response as essential to the total procedure1 or become so used to having it that he will be disturbed by its absence later1 it is supposed that the bond between the situation and the crutch should not be fully formed4 ,there is a better way out of the difficulty1 in case crutches are used at all4 ,this is to associate the crutch with a special 'set1' and its non-use with the general set which is to be the permanent one4 ,for example1 children may be taught from the start never to write the crutch sign or crutch figure unless the work is accompanied by ,write 444 to help you to4444 ,write - to help you to ,find the differences3-- remember that you must #ci #fg #gh #ef #de subtract in this row4 #bc #dd #cf #bf #bd -- -- -- -- -- ,remember that you must ,find the differences3-- subtract in this row4 #he #bg #if #ch #gh #fc #ad #ea #de #cb -- -- -- -- -- ,the bond evoking the use of the crutch may then be formed thoroughly enough so that there is no hesitation1 insecurity1 or error1 without interfering to any harmful extent with the more general bond from the situation to work without the crutch4 ,,the ,,strength ,,of ,,bonds ,,with ,,technical ,,facts ,,and ,,terms ,another instructive case concerns the bonds between certain words and their meanings1 and between certain situations of commerce1 industry1 or agriculture and useful facts about these situations4 ,illustrations of the former are the bonds between cube root1 hectare1 brokerage1 commission1 indorsement1 vertex1 adjacent1 nonagon1 sector1 draft1 bill of exchange1 and their meanings4 ,illustrations of the latter are the bonds from ,money being lent 'with interest' at no specified rate1 what rate is charged8 to ,the legal rate of the state1 from ,x per ,m as a rate for lumber to ,means ,x per thousand board feet1 a board foot being #a ft4 by #a ft4 by #a in4 ,it is argued by many that such bonds are valuable for a short time2 namely1 while arithmetical procedures in connection with which they serve are learned1 but that their value is only to serve as a means for learning these procedures and that thereafter they may be forgotten4 ,they are formed only as accessory means to certain more purely arithmetical knowledge or discipline2 after this is acquired they may be forgotten4 ,everybody does in fact forget them1 relearning them later if life requires4 ,so runs the argument4 ,in some cases learning such words and facts only to use them in solving a certain sort of problems and then forget them may be profitable4 ,the practice is1 however1 exceedingly risky4 ,it is true that everybody does in fact forget many such meanings and facts1 but this commonly means either that they should not have been learned at all at the time that they were learned1 or that they should have been learned more permanently1 or that details should have been learned with the expectation that they themselves would be forgotten but that a general fact or attitude would remain4 ,for example1 duodecagon should not be learned at all in the elementary school2 indorsement should either not be learned at all there1 or be learned for permanence of a year or more2 the details of the metric system should be so taught as to leave for several years at least knowledge of the facts that there is a system so named that is important1 whose tables go by tens1 hundreds1 or thousands1 and a tendency 7not necessarily strong7 to connect meter1 kilogram1 and liter with measurement by the metric system and with approximate estimates of their several magnitudes4 ,if an arithmetical procedure seems to require accessory bonds which are to be forgotten1 once the procedure is mastered1 we should be suspicious of the value of the procedure itself4 ,if pupils forget what compound interest is1 we may be sure that they will usually also have forgotten how to compute it4 ,surely there is waste if they have learned what it is only to learn how to compute it only to forget how to compute it6 ,,the ,,strength ,,of ,,bonds ,,concerning ,,the ,,reasons ,,for ,,arithmetical ,,processes ,the next case of the formation of bonds to slight strength is the problematic one of forming the bonds involved in understanding the reasons for certain processes only to forget them after the process has become a habit4 ,should a pupil1 that is1 learn why he inverts and multiplies1 only to forget it as soon as he can be trusted to divide by a fraction8 ,should he learn why he puts the units figure of each partial product in multiplication under the figure that he multiplies by1 only to forget the reason as soon as he has command of the process8 ,should he learn why he gets the number of square inches in a rectangle by multiplying the length by the width1 both being expressed in linear inches1 and forget why as soon as he is competent to make computations of the areas of rectangles8 ,on general psychological grounds we should be suspicious of forming bonds only to let them die of starvation later1 and tend to expect that elaborate explanations learned only to be forgotten either should not be learned at all1 or should be learned at such a time and in such a way that they would not be forgotten4 ,especially we should expect that the general principles of arithmetic1 the whys and wherefores of its fundamental ways of manipulating numbers1 ought to be the last bonds of all to be forgotten4 ,details of how you arranged numbers to multiply might vanish1 but the general reasons for the placing would be expected to persist and enable one to invent the detailed manipulations that had been forgotten4 ,this suspicion is1 ,i think1 justified by facts4 ,the doctrine that the customary deductive explanations of why we invert and multiply1 or place the partial products as we do before adding1 may be allowed to be forgotten once the actual habits are in working order1 has a suspicious source4 ,it arose to meet the criticism that so much time and effort were required to keep these deductive explanations in memory4 ,the fact was that the pupil learned to compute correctly irrespective of the deductive explanations4 ,they were only an added burden4 ,his inductive learning that the procedure gave the right answer really taught him4 ,so he wisely shuffled off the extra burden of facts about the consequences of the nature of a fraction or the place values of our decimal notation4 ,the bonds weakened because they were not used4 ,they were not used because they were not useful in the shape and at the time that they were formed1 or because the pupil was unable to understand the explanations so as to form them at all4 ,the criticism was valid and should have been met in part by replacing the deductive explanations by inductive verifications1 and in part by using the deductive reasoning as a check after the process itself is mastered4 ,the very same discussions of place-value which are futile as proof that you must do a certain thing before you have done it1 often become instructive as an explanation of why the thing that you have learned to do and are familiar with and have verified by other tests works as well as it does4 ,the general deductive theory of arithmetic should not be learned only to be forgotten4 ,much of it should1 by most pupils1 not be learned at all4 ,what is learned should be learned much later than now1 as a synthesis and rationale of habits1 not as their creator4 ,what is learned of such deductive theory should rank among the most rather than least permanent of a pupil's stock of arithmetical knowledge and power4 ,there are bonds which are formed only to be lost1 and bonds formed only to be lost in their first form1 being used in a new organization as material for bonds of a higher order2 but the bonds involved in deductive explanations of why certain processes are right are not such3 they are not to be formed just to be forgotten1 nor as mere propaedeutics to routine manipulations4 ,,propaedeutic ,,bonds ,the formation of bonds to a limited strength because they are to be lost in their first form1 being worked over in different ways in other bonds to which they are propaedeutic or contributing is the most important case of low strength1 or rather low permanence1 in bonds4 ,the bond between four #es in a column to be added and the response of thinking '#aj1 #ae1 #bj' is worth forming1 but it is displaced later by the multiplication bond or direct connection of 'four #es to be added' with '#bj4' ,counting by #bs from #b1 #cs from #c1 #ds from #d1 #es from #e1 etc41 forms serial bonds which as series might well be left to disappear4 ,their separate steps are kept as permanent bonds for use in column addition1 but their serial nature is changed from #b 7and #b7 #d1 7and #b7 #f1 7and #b7 #h1 etc41 to two #bs #d1 three #bs #f1 four #bs #h1 etc42 after playing their part in producing the bonds whereby any multiple of #b by #b to #i1 can be got1 the original serial bonds are1 as series1 needed no longer4 ,the verbal response of saying 'and' in adding1 after helping to establish the bonds whereby the general set of the mind toward adding cooeperates with the numbers seen or thought of to produce their sum1 should disappear2 or remain so slurred in inner speech as to offer no bar to speed4 ,the rule for such bonds is1 of course1 to form them strongly enough so that they work quickly and accurately for the time being and facilitate the bonds that are to replace them1 but not to overlearn them4 ,there is a difference between learning something to be held for a short time1 and the same amount of energy spent in learning for long retention4 ,the former sort of learning is1 of course1 appropriate with many of these propaedeutic bonds4 ,the bonds mentioned as illustrations are not purely propaedeutic1 nor formed only to be transmuted into something else4 ,even the saying of 'and' in addition has some genuine1 intrinsic value in distinguishing the process of addition1 and may perhaps be usefully reviewed for a brief space during the first steps in adding common fractions4 ,some such propaedeutic bonds may be worth while apart from their value in preparing for other bonds4 ,consider1 for example1 exercises like those shown below which are propaedeutic to long division1 giving the pupil some basis in experience for his selection of the quotient figures4 ,these multiplications are intrinsically worth doing1 especially the #abs and #bes4 ,whatever the pupil remembers of them will be to his advantage4 #a4 ,count by #aas to #acb1 beginning #aa1 #bb1 #cc4 #b4 ,count by #abs to #add1 beginning #ab1 #bd1 #cf4 #c4 ,count by #bes to #cjj1 beginning #be1 #ej1 #ge4 #d4 ,state the missing numbers3-- ,a4 ,b4 ,c4 ,d4 #c #aas #e #aas #h ft4 4444 in4 #b dozen #d #abs #c #abs #aj ft4 4444 in4 #d dozen #e #abs #f #abs #g ft4 4444 in4 #aj dozen #f #aas #ab #aas #d ft4 4444 in4 #e dozen #i #aas #b #abs #f ft4 4444 in4 #g dozen #g #abs #i #abs #i ft4 4444 in4 #ab dozen #h #abs #g #aas #aa ft4 4444 in4 #i dozen #aa #aas #ab #abs #e ft4 4444 in4 #f dozen #e4 ,count by #bes to #b4ej1 saying1 #be cents1 #ej cents1 #ge cents1 one dollar1 and so on4 #f4 ,count by #aes to #a4ej4 #g4 ,find the products4 ,do not use pencil4 ,think what they are4 ,a4 ,b4 ,c4 ,d4 ,e4 #b x #be #c x #ae #b x #ab #d x #aa #f x #be #c x #be #aj x #ae #b x #ae #d x #ae #f x #ae #e x #be #d x #ae #b x #be #d x #ab #f x #ab #aj x #be #b x #ae #b x #aa #d x #be #f x #aa #d x #be #g x #ae #c x #be #e x #aa #g x #ab #f x #be #i x #ae #c x #ae #e x #ab #g x #ae #h x #be #e x #ae #c x #aa #e x #ae #g x #be #g x #be #h x #ae #c x #ab #e x #be #g x #aa #i x #be #f x #ae #h x #ab #i x #ab #h x #be ,state the missing numbers3-- ,a4 #cf 4444 #abs ,b4 #dd 4444 #aas ,c4 #ej 4444 #bes #fj 4444 #abs #hh 4444 #aas #abe 4444 #bes #bd 4444 #abs #gg 4444 #aas #ge 4444 #bes #dh 4444 #abs #ee 4444 #aas #bjj 4444 #bes #add 4444 #abs #ii 4444 #aas #bej 4444 #bes #ajh 4444 #abs #aaj 4444 #aas #age 4444 #bes #gb 4444 #abs #cc 4444 #aas #bbe 4444 #bes #if 4444 #abs #ff 4444 #aas #aej 4444 #bes #hd 4444 #abs #bb 4444 #aas #ajj 4444 #bes ,find the quotients and remainders4 ,if you need to use paper and pencil to find them1 you may4 ,but find as many as you can without pencil and paper4 ,do ,row ,a first4 ,then do ,row ,b4 ,then ,row ,c1 etc4 ,row ,a4 #aade #abde #bede #aede #bade #bbde ,row ,b4 #beee #aaee #abee #aeee #bbee #cjee ,row ,c4 #abfj #befj #aefj #aafj #cjfj #bafj ,row ,d4 #abge #aage #aege #bege #cjge #cege ,row ,e4 #aaajj #abajj #beajj #aeajj #cjajj #bbajj ,row ,f4 #aaif #abif #beif #aeif #cjif #bbif ,row ,g4 #beaje #aaaje #aeaje #abaje #bbaje #ceaje ,row ,h4 #abfd #aefd #befd #aafd #bbfd #bafd ,row ,i4 #aahj #abhj #aehj #behj #cehj #bahj ,row ,j4 #bebjj #cjbjj #gebjj #fcbjj #febjj #ffbjj ,do this section again4 ,do all the first column first4 ,then do the second column1 then the third1 and so on4 ,consider1 from the same point of view1 exercises like 7#c x #d7 #b1 7#g x #f7 #e1 7#i x #d7 #f1 given as a preparation for written multiplication4 ,the work of #dh #fh #dg #c #g #i -- -- -- and the like is facilitated if the pupil has easy control of the process of getting a product1 and keeping it in mind while he adds a one-place number to it4 ,the practice with 7#c x #d7 #b and the like is also good practice intrinsically4 ,so some teachers provide systematic preparatory drills of this type just before or along with the beginning of short multiplication4 ,in some cases the bonds are purely propaedeutic or are formed only for later reconstruction4 ,they then differ little from 'crutches4' ,the typical crutch forms a habit which has actually to be broken1 whereas the purely propaedeutic bond forms a habit which is left to rust out from disuse4 ,for example1 as an introduction to long division1 a pupil may be given exercises using one-figure divisors in the long form1 as3-- #ggc and #e remainder #g7edaf #di -- #ea #di -- #bf #ba -- #e ,the important recommendation concerning these purely propaedeutic bonds1 and bonds formed only for later reconstruction1 is to be very critical of them1 and not indulge in them when1 by the exercise of enough ingenuity1 some bond worthy of a permanent place in the individual's equipment can be devised which will do the work as well4 ,arithmetical teaching has done very well in this respect1 tending to err by leaving out really valuable preparatory drills rather than by inserting uneconomical ones4 ,it is in the teaching of reading that we find the formation of propaedeutic bonds of dubious value 7with letters1 phonograms1 diacritical marks1 and the like7 often carried to demonstrably wasteful extremes4 ,,chapter ,,vi ,,the ,,psychology ,,of ,,drill ,,in ,,arithmetic3 ,,the ,,amount ,,of ,,practice ,,and ,,the ,,organization ,,of ,,abilities ,,the ,,amount ,,of ,,practice ,it will be instructive if the reader will perform the following experiment as an introduction to the discussion of this chapter1 before reading any of the discussion4 ,suppose that a pupil does all the work1 oral and written1 computation and problem-solving1 presented for grades #a to #f inclusive 7that is1 in the first two books of a three-book series7 in the average textbook now used in the elementary school4 ,how many times will he have exercised each of the various bonds involved in the four operations with integers shown below8 ,that is1 how many times will he have thought1 #a and #a are #b1 #a and #b are #c1 etc48 ,every case of the action of each bond is to be counted4 ,,the ,,fundamental ,,bonds #a #a #b - #a #a x #a #b _/ #a #a #b #b - #b #b x #a #b _/ #b #a #c #c x #a #a #d #d x #a #a #e #c - #a #e x #a #c _/ #a #a #f #c - #b #f x #a #c _/ #b #a #g #c - #c #g x #a #c _/ #c #a #h #h x #a #a #i #i x #a #d - #a #d _/ #a #d - #b #d _/ #b #aa 7or #ba or #ca1 etc47 #a #d - #c #a x #b #d _/ #c #aa #b #d - #d #b x #b #d _/ #d #aa #c #c x #b #aa #d #d x #b #aa #e #e - #a #e x #b #e _/ #a #aa #f #e - #b #f x #b #e _/ #b #aa #g #e - #c #g x #b #e _/ #c #aa #h #e - #d #h x #b #e _/ #d #aa #i #e - #e #i x #b #e _/ #e #f - #a #a x #c #f _/ #a #b #a #f - #b #b x #c #f _/ #b #b #b #f - #c #c x #c #f _/ #c #b #c #f - #d #d x #c #f _/ #d #b #d #f - #e #e x #c #f _/ #e #b #e #f - #f #f x #c #f _/ #f #b #f #g x #c #b #g #h x #c #b #h #g - #a #i x #c #g _/ #a #b #i #g - #b #g _/ #b #g - #c #g _/ #c #g - #d #a x #d #g _/ #d #ab 7or #bb or #cb1 etc47 #a #g - #e #b x #d #g _/ #e #ab #b #g - #f and so on #g _/ #f #g - #g to #i x #i #g _/ #g and so on to and so on and so on to #i #i to #ah - #i #hb _/ #i #ai 7or #bi or #ci1 etc47 #i #hc _/ #i1 etc4 ,if estimating for the entire series is too long a task1 it will be sufficient to use eight or ten from each1 say3-- #c #b #ac1 #bc1 etc4 #b #g #b #ag1 #bg1 etc4 #b #c #c #c #c #d #d #d #d #e #e #e #e #f #f #f #f #g #g #g #g #h #h #h #h #i #i #i #i #c - #c #g - #g #i x #g #fc _/ #i #d #h #g x #i #fd #e #i #h x #f #fe #f #aj #f x #h #ff #g #aa #fg #h #ab #fh #i #ac #fi #aj #ad #gj #aa #ae #ga #ab #af ,,table #b ,,estimates ,,of ,,the ,,amount ,,of ,,practice ,,provided ,,in ,,books ,i ,,and ,,ii ,,of ,,the ,,average ,,three-,,book ,,text ,,in ,,arithmetic2 ,,by #ej ,,experienced ,,teachers ,,lowest ,,median ,,highest ,,range ,,required ,,to ,,arithmetical ,,fact ,,estimate,,estimate,,estimate ,,include ,,half ,,of ,,the ,,estimates ---------------------------------------- -------------------------- #c or #ac or #bc1 etc4 #b #be #aejj #a1jjj1jjj #hjj-ejjj #c #bd #adej #hj1jjj #dge-ejjj #d #bc #aaej #ej1jjj #gej-ejjj #e #bb #adjj #dd1jjj #gjj-ejjj #f #ba #acej #da1jjj #gjj-dejj #g #ba #aejj #cg1jjj #fjj-djjj #h #bj #adjj #cc1jjj #eej-dajj #i #bj #aaej #bh1jjj #fej-dejj #g or #ag or #bg1 etc4 #b #bj #abej #b1jjj1jjj #fjj-ejjj #c #ai #aajj #a1jjj1jjj #fej-dijj #d #ah #ajjj #hj1jjj #fej-dijj #e #ag #acjj #hj1jjj #fej-ddjj #f #af #aajj #bi1jjj #fej-dejj #g #ae #aajj #be1jjj #ejj-dejj #h #ac #aajj #ba1jjj #fej-chjj #i #aj #abge #ag1jjj #ejj-djjj #c - #c #be #ajjj #ajj1jjj #ejj-djjj #d - #c #bj #ajej #ejj1jjj #ebe-cjjj #e - #c #bj #aajj #b1ejj1jjj #fej-dbjj #f - #c #aj #ajej #ba1jjj #fej-cbej #g - #c #bb #aajj #ae1jjj #eej-cjej #h - #c #ba #ajge #ae1jjj #fej-cjjj #i - #c #ba #ajjj #ae1jjj #gjj-bfjj #aj - #c #bj #ajjj #bj1jjj #fjj-bejj #aa - #c #bj #ajjj #ae1jjj #dfe-beej #ab - #c #ah #ajjj #ae1jjj #fej-bajj #g - #g #aj #ajjj #ah1jjj #dbe-cjjj #h - #g #ae #ajjj #ah1jjj #dac-cajj #i - #g #ae #iej #ah1jjj #eej-cjjj #aj - #g #ae #iej #ah1jjj #fjj-ciej #aa - #g #aj #ijj #ah1jjj #eej-cjjj #ab - #g #aj #ibe #ah1jjj #ebe-cajj #ac - #g #aj #ijj #ah1jjj #ejj-bfjj #ad - #g #aj #ijj #ah1jjj #ejj-cajj #ae - #g #aj #ibe #ah1jjj #ejj-cjjj #af - #g #aj #hge #ah1jjj #ejj-bejj #i x #g #aj #gjj #bj1jjj #ejj-bjjj #g x #i #aj #gjj #bj1jjj #ejj-agej #h x #f #aj #gej #bj1jjj #ejj-bejj #f x #h #i #gjj #bj1jjj #ejj-bejj #fc _/ #i #i #ejj #d1ejj #cjj-bejj #fd _/ #i #i #bjj #d1jjj #ajj- #gjj #fe _/ #i #h #bjj #d1jjj #ajj- #fjj #ff _/ #i #g #bjj #d1jjj #ajj- #eej #fg _/ #i #g #bjj #d1jjj #ge- #dej #fh _/ #i #f #bjj #d1jjj #hg- #ege #fi _/ #i #f #bjj #d1jjj #hg- #dej #gj _/ #i #e #bjj #d1jjj #ge- #ege #ga _/ #i #e #bjj #d1jjj #ge- #gjj ,,xx #dj #eej #a1jjj1jjj #cjj-bjjj ,,xo #bj #ejj #aa1ejj #aej-bjjj ,,xxx #ae #dej #ab1jjj #ajj-ajjj ,,xxo #be #djj #ae1jjj #aej-ajjj ,,xoo #ae #djj #e1jjj #ajj-ajjj ,,xox #aj #djj #aj1jjj #ajj- #ige ,having made his estimates the reader should compare them first with similar estimates made by experienced teachers 7shown on page #abd f471 and then with the results of actual counts for representative textbooks in arithmetic 7shown on pages #abf to #acb74 ,it will be observed in ,table #b that even experienced teachers vary enormously in their estimates of the amount of practice given by an average textbook in arithmetic1 and that most of them are in serious error by overestimating the amount of practice4 ,in general it is the fact that we use textbooks in arithmetic with very vague and erroneous ideas of what is in them1 and think they give much more practice than they do4 ,the authors of the textbooks as a rule also probably had only very vague and erroneous ideas of what was in them4 ,if they had known1 they would almost certainly have revised their books4 ,surely no author would intentionally provide nearly four times as much practice on #b #b as on #h #h1 or eight times as much practice on #b x #b as on #i x #h1 or eleven times as much practice on #b - #b as on #ag - #h1 or over forty times as much practice on #b _/ #b as on #ge _/ #h and #ge _/ #i1 both together4 ,surely no author would have provided intentionally only twenty to thirty occurrences each of #af - #g1 #af - #h1 #af - #i1 #ag - #h1 #ag - #i1 and #ah - #i for the entire course through grade #f2 or have left the practice on #fj _/ #g1 #fj _/ #h1 #fj _/ #i1 #fa _/ #g1 #fa _/ #h1 #fa _/ #i1 and the like to occur only about once a year6 ,,table #c ,,amount ,,of ,,practice3 ,,addition ,,bonds ,,in ,a ,,recent ,,textbook 7,a7 ,,of ,,excellent ,,repute4 ,,books ,i ,,and ,,ii1 ,,all ,,save ,,four ,,sections ,,of ,,supplementary ,,material1 ,,to ,,be ,,used ,,at ,,the ,,teacher',s ,,discretion ,the ,table reads3 #b #b was used #bbf times1 #ab #b was used #gd times1 #bb #b1 #cb #b1 #db #b1 and so on were used #ej times4 #b #c #d #e #f #g #h #i ,,total ---------------------------------------- --------------------- #b #bbf #aed #afb #aej #ig #hg #ff #de #ab #gd #ec #gf #df #ea #cg #cf #cc #bb1 etc4 #ej #fj #fh #fc #db #ej #ch #bf #c #baf #ada #abg #hi #hb #ed #eh #dj #ac #dc #dc #fj #gj #eb #cj #bb #ah #bc1 etc4 #ae #cj #ea #ej #db #cb #bi #cj #g #he #ij #ajc #ajc #hd #ha #fa #dg #ag #ce #be #db #cb #ce #ba #bi #af #bg1 etc4 #cj #bc #cb #bi #bd #bc #be #bh #h #ahe #aab #adf #ii #ge #ga #gc #fa #ah #bh #ce #eb #df #bh #bi #bd #ad #bh1 etc4 #ec #cf #cd #ch #bc #cf #bg #bg #i #ajd #ha #aab #if #fc #gd #eh #eg #ai #ac #aa #ca #ch #be #ad #bb #aa #bi1 etc4 #ai #ag #bg #bj #cb #cb #ai #ah #b1 #ab1 #bb1 etc4 #cej #bgg #cjf #bfj #aij #agd #adj #ajd #ahja #c1 #ac1 #bc1 etc4 #bgd #bad #bcj #bji #agf #aaf #aji #hh #adjf #g1 #ag1 #bg1 etc4 #adh #ach #ahg #afd #ada #abe #aae #ia #aaji #h1 #ah1 #bh1 etc4 #bff #ahc #bcb #ahe #abf #acf #abd #ajb #aced #i1 #ai1 #bi1 etc4 #acf #aji #agj #aed #abj #abj #ii #hf #iid ,totals #aafd #iba #aabe #igb #gec #fga #fhg #dga ,,table #d ,,amount ,,of ,,practice3 ,,subtraction ,,bonds ,,in ,a ,,recent ,,textbook 7,a7 ,,of ,,excellent ,,repute4 ,,books ,i ,,and ,,ii1 ,,all ,,save ,,four ,,sections ,,of ,,supplementary ,,material1 ,,to ,,be ,,used ,,at ,,the ,,teacher',s ,,discretion ,,subtrahends ,,minuends ---------------------------------------- ------------- #a #b #c #d #e #f #g #h #i ---------------------------------------- --------------- #a #cgb #b #bad #caa #c #acf #adi #ahi #d #adf #adb #ajc #bje #e #aga #ia #ib #afd #acf #f #hj #ei #fi #ga #ha #aib #g #ajf #eg #ee #fg #ei #aef #hj #h #gc #ej #ej #ge #ej #fb #dh #aeb #i #ga #ge #ed #gd #dh #ee #ee #abd #acc #aj #bfa #hd #fc #ajj #aic #hc #eg #abd #ia #aa #dh #ca #ej #cf #da #cb #df #ce #ab #dh #gg #eg #ea #ce #hj #cj #ac #ce #bb #dj #bi #ce #bh #ad #be #cg #cf #di #cb #ae #cc #ai #dh #bj #af #af #cf #bf #ag #bg #bj #ah #ai ,total excluding #a-a1 #b-b1 etc4 #abeh #gee #efe #gac #ega #eeh #cbg #efi #cja ,,table #e ,,frequencies ,,of ,,subtractions ,,not ,,included ,,in ,,table #d ,these are cases where the pupil would1 by reason of his stage of advancement1 probably operate #ce-cj1 #df-df1 etc41 as one bond4 ,,subtrahends ---------------------------------------- #a #b #c #d #e #f #g #h #i ,,minuends #aa #ab #ac #ad #ae #af #ag #ah #ai #aj #ba #bb #bc #bd #be #bf #bg #bh #bi #bj etc4etc4etc4etc4etc4etc4etc4etc4etc4etc4 ---------------------------------------- -------------------- #aj1 #bj1 #cj1 #dj1 etc4 #aa #bi #af #eb #cb #ea #g #cj #bb #fj #aa1 #ba1 #ca1 #da1 etc4 #db #ad #bb #cb #ab #bf #ai #eb #ag #aj #ab1 #bb1 #cb1 #db1 etc4 #dg #ig #e #ac #i #ba #aa #bd #ai #ag #ac1 #bc1 #cc1 #dc1 etc4 #g #dj #g #ad #ae #ac #ai #ai #bb #c #ad1 #bd1 #cd1 #dd1 etc4 #h #bh #ad #eh #ac #af #ad #bf #ai #g #ae1 #be1 #ce1 #de1 etc4 #ba #bh #bi #ed #ea #ae #ba #ab #bd #h #af1 #bf1 #cf1 #df1 etc4 #e #ah #ab #bg #ce #fi #ac #ag #ai #b #ag1 #bg1 #cg1 #dg1 etc4 #e #i #ab #dj #cb #ed #bd #ab #ab #a #ah1 #bh1 #ch1 #dh1 etc4 #b #af #aj #bc #bb #cf #ah #dg #af #j #ai1 #bi1 #ci1 etc4 #e #g #g #aj #ac #bh #ad #bc #af #j ,totals #aec #bhf #acd #cbc #bcd #cbi #afj #bfb #ahf #ajh ,,table #f ,,amount ,,of ,,practice3 ,,multiplication ,,bonds ,,in ,,another ,,recent ,,textbook 7,b7 ,,of ,,excellent ,,repute4 ,,books ,i ,,and ,,ii ,,multiplicands ,,multipliers ---------------------------------------- ----------------- #j #a #b #c #d #e #f #g #h #i ,totals ---------------------------------------- ------------------- #a #bii #ecd #dgb #bga #caj #bic #bfa #agh #aie #ii #biab #b #cej #fdd #ffh #dhj #deh #cgg #ccb #bch #bci #aee #cida #c #bhj #dhg #eji #chh #cah #cjb #bdg #aii #bbg #aeb #caji #d #ahf #cge #cih #bdb #bjc #bfe #aig #afc #aei #ic #bbha #e #bfh #cei #cic #bcd #bfc #bdc #bag #aib #aig #aad #bdhj #f #ahj #bhd #bfe #aii #aif #aia #afh #afi #afe #ajf #aibc #g #ace #bhc #bgg #agf #ahg #aeh #aee #aba #ade #aah #agee #h #acg #bgb #bib #age #aib #afd #aeh #aeg #abf #abf #agii #i #ga #agc #adj #abb #ig #ajb #aja #ajj #hb #aaj #ajih ,totals #aijfcdaacdadbbhgbbbdbjieahcfaeagaeceajg c ,,table #g ,,amount ,,of ,,practice3 ,,divisions ,,without ,,remainder ,,in ,,textbook ,b1 ,,parts ,i ,,and ,,ii ,,divisors ,,dividends ---------------------------------------- ------ #b #c #d #e #f #g #h #i ,totals ---------------------------------------- --------------------- ,integral #cig #bbd #bej #acj #ic #dd #ih #bc #abei multiples of #b to #i #bef #abd #aeb #gi #bh #dc #fa #be #gfh in sequence2 i4e41 #d _/ #b #cah #abc #acj #fe #ej #ai #ci #ai #gfc occurred #cig times1 #beh #ih #hf #aje #be #bd #cd #bj #fej #f _/ #b occurred #bef times1 #aih #di #gf #bg #bb #cj #cc #af #dea #f _/ #c1 #bbd times1 #i _/ #c1 #abd times4 #gg #ed #cf #ca #bh #bg #af #i #bgh #ahj #ia #ej #ch #ag #ac #bb #af #dbg #fi #df #cg #bd #ab #ag #af #ae #bcf ,totals #agec #hji #hag #dii #bge #bag #cai #adb ,,table #h ,,division ,,bonds1 ,,with ,,and ,,without ,,remainders4 ,,book ,b ,all work through grade #f1 except estimates of quotient figures in long division4 ,dividend #b #c #d #e ,divisor #a #b #a #b #c #a #b #c #d #a #b #c #d #e ,number of ,occurrences #da #chf #bg #ahi #bdj #bf #cig #ff #ahe #bc #acf #dc #ec #ace ,dividend #f #g ,divisor #a #b #c #d #e #f #a #b #c #d #e #f #g ,number of ,occurrences #ba #bef #bbd #fh #dc #hc #bc #gb #ee #ch #df #cb #ed ,dividend #h #i ,divisor #a #b #c #d #e #f #g #h #a #b #c #d #e #f #g #h #i ,number of ,occurrences #ag #cah #cj #bej #bb #bh #ci #ia #ai #ej #abd #di #be #ae #ah #cj #ch ,dividend #aj #aa ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,number of ,occurrences #beh #ch #df #abj #ai #i #bd #bd #cb #ba #af #c #g #aa #ad #c ,dividend #ab #ac ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,number of ,occurrences #aih #abc #aeb #bi #ic #i #af #g #de #af #ae #aa #g #d #e #c ,dividend #ad #ae ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,number of ,occurrences #gg #bj #ac #e #h #dd #h #f #fi #ih #af #gi #h #h #d #f ,dividend #af #ag ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,number of ,occurrences #ahj #ai #acj #ad #f #i #ih #c #fa #i #ae #ad #f #f #ab #c ,dividend #ah #ai ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,number of ,occurrences #fi #di #ac #f #bh #g #g #bc #ba #f #aj #e #c #d #aj #d ,dividend #bj #ba ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,number of ,occurrences #bd #hf #fe #aa #c #bc #e #ed #ab #h #e #dc #aj #e ,dividend #bb #bc ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,number of ,occurrences #ag #af #ae #h #ac #f #ae #g #h #aa #h #f #c #b ,dividend #bd #be ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,number of ,occurrences #ia #gf #ah #ej #e #fa #a #aa #ac #aje #e #f #e #c ,dividend #bf #bg ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,number of ,occurrences #e #f #c #c #d #f #c #df #h #aj #d #b #f #be ,dividend #bh #bi ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,number of ,occurrences #d #cf #h #c #ai #c #g #f #h #j #e #aa #b #c ,dividend #cj #ca #cb ,divisor #d #e #f #g #h #i #d #e #f #g #h #i #d #e #f #g #h #i ,number of ,occurrences #ba #bg #be #f #g #ac #d #c #a #a #d #b #ej #aa #c #f #ci #e ,dividend #cc #cd #ce ,divisor #d #e #f #g #h #i #d #e #f #g #h #i #d #e #f #g #h #i ,number of ,occurrences #h #g #g #b #f #a #h #c #e #b #a #a #aj #ca #e #bd #e #c ,dividend #cf #cg #ch ,divisor #d #e #f #g #h #i #d #e #f #g #h #i #d #e #f #g #h #i ,number of ,occurrences #cg #af #bb #b #f #ai #ab #h #g #e #c #i #g #h #g #a #a #e ,dividend #ci #dj #da #db ,divisor #d #e #f #g #h #i #e #f #g #h #i #e #f #g #h #i #e #f #g #h #i ,number of ,occurrences #d #c #g #d #c #a #ch #i #b #cd #b #f #f #c #g #e #g #bh #cj #aj #c ,dividend #dc #dd #de #df ,divisor #e #f #g #h #i #e #f #g #h #i #e #f #g #h #i #e #f #g #h #i ,number of ,occurrences #g #e #aj #c #b #g #f #d #e #j #bd #f #g #aj #bj #c #c #b #b #b ,dividend #dg #dh #di #ej ,divisor #e #f #g #h #i #e #f #g #h #i #e #f #g #h #i #f #g #h #i ,number of ,occurrences #f #b #b #j #c #g #ag #d #cc #b #d #g #bg #i #b #d #f #c #h ,dividend #ea #eb #ec #ed ,divisor #f #g #h #i #f #g #h #i #f #g #h #i #f #g #h #i ,number of ,occurrences #b #c #a #b #e #e #e #c #d #c #b #b #ab #e #a #af ,dividend #ee #ef #eg #eh #ei ,divisor #f #g #h #i #f #g #h #i #f #g #h #i #f #g #h #i #f #g #h #i ,number of ,occurrences #e #c #d #b #j #ac #af #h #j #c #a #c #b #b #c #a #b #c #j #c ,dividend #fj #fa #fb #fc #fd #fe ,divisor #g #h #i #g #h #i #g #h #i #g #h #i #g #h #i #g #h #i ,number of ,occurrences #c #i #a #a #b #e #d #f #a #ag #e #i #e #bb #j #a #aj #a ,dividend #ff #fg #fh #fi #gj #ga ,divisor #g #h #i #g #h #i #g #h #i #g #h #i #h #i #h #i ,number of ,occurrences #b #a #d #j #a #a #a #c #b #j #f #a #f #b #a #j ,dividend #gb #gc #gd #ge #gf #gg #gh #gi ,divisor #h #i #h #i #h #i #h #i #h #i #h #i #h #i #h #i ,number of ,occurrences #af #aj #g #e #c #c #e #c #c #b #c #j #d #a #j #b ,dividend #hj #ha #hb #hc #hd #he #hf #hg #hh #hi ,divisor #i #i #i #i #i #i #i #i #i #i ,number of ,occurrences #d #ae #b #d #a #b #j #c #b #g ,tables #c to #h show that even gifted authors make instruments for instruction in arithmetic which contain much less practice on certain elementary facts than teachers suppose2 and which contain relatively much more practice on the more easily learned facts than on those which are harder to learn4 ,how much practice should be given in arithmetic8 ,how should it be divided among the different bonds to be formed8 ,below a certain amount there is waste because1 as has been shown in ,chapter ,,vi1 the pupil will need more time to detect and correct his errors than would have been required to give him mastery4 ,above a certain amount there is waste because of unproductive overlearning4 ,if #ffh is just enough for #b x #b1 #hb is not enough for #i x #h4 ,if #hb is just enough for #i x #h1 #ffh is too much for #b x #b4 ,it is possible to find the answers to these questions for the pupil of median ability 7or any stated ability7 by suitable experiments4 ,the amount of practice will1 of course1 vary according to the ability of the pupil4 ,it will also vary according to the interest aroused in him and the satisfaction he feels in progress and mastery4 ,it will also vary according to the amount of practice of other related bonds2 #g #g #ad and #fj _/ #g #h and #d remainder will help the formation of #g #h #ae and #fa _/ #g #h and #e remainder4 ,it will also1 of course1 vary with the general difficulty of the bond1 #ag - #h #i being under ordinary conditions of teaching harder to form than #g - #b #e4 ,until suitable experiments are at hand we may estimate for the fundamental bonds as follows1 assuming that by the end of grade #f a strength of #aii correct out of #bjj is to be had1 and that the teaching is by an intelligent person working in accord with psychological principles as to both ability and interest4 ,for one of the easier bonds1 most facilitated by other bonds 7such as #b x #e #aj1 or #aj - #b #h1 or the double bond #g two #cs and #a remainder7 in the case of the median or average pupil1 twelve practices in the week of first learning1 supported by twenty-five practices during the two months following1 and maintained by thirty practices well spread over the later periods should be enough4 ,for the more gifted pupils lesser amounts down to six1 twelve1 and fifteen may suffice4 ,for the less gifted pupils more may be required up to thirty1 fifty1 and a hundred4 ,it is to be doubted1 however1 whether pupils requiring nearly two hundred repetitions of each of these easy bonds should be taught arithmetic beyond a few matters of practical necessity4 ,for bonds of ordinary difficulty1 with average facilitation from other bonds 7such as #aa - #c1 #d x #g1 or #dh _/ #h #f7 in the case of the median or average pupil1 we may estimate twenty practices in the week of first learning1 supported by thirty1 and maintained by fifty practices well spread over the later periods4 ,gifted pupils may gain and keep mastery with twelve1 fifteen1 and twenty practices respectively4 ,pupils dull at arithmetic may need up to twenty1 sixty1 and two hundred4 ,here1 again1 it is to be doubted whether a pupil for whom arithmetical facts1 well taught and made interesting1 are so hard to acquire as this1 should learn many of them4 ,for bonds of greater difficulty1 less facilitated by other bonds 7such as #ag - #i1 #h x #g1 or #ab-a_/b of #a_/h of71 the practice may be from ten to a hundred percent more than the above4 ,,underlearning ,,and ,,overlearning ,if we accept the above provisional estimates as reasonable1 we may consider the harm done by giving less and by giving more than these reasonable amounts4 ,giving less is indefensible4 ,the pupil's time is wasted in excessive checking to find his errors4 ,he is in danger of being practiced in error4 ,his attention is diverted from the learning of new facts and processes by the necessity of thinking out these supposedly mastered facts4 ,all new bonds are harder to learn than they should be because the bonds which should facilitate them are not strong enough to do so4 ,giving more does harm to some extent by using up time that could be spent better for other purposes1 and 7though not necessarily7 by detracting from the pupil's interest in arithmetic4 ,in certain cases1 however1 such excess practice and overlearning are actually desirable4 ,three cases are of special importance4 ,the first is the case of a bond operating under a changed mental set or adjustment4 ,a pupil may know #g x #h adequately as a thing by itself1 but need more practice to operate it in #bhe #g --- where he has to remember that #c is to be added to the #ef when he obtains it1 and that only the #i is to be written down1 the #e to be held in mind for later use4 ,the practice required to operate the bond efficiently in this new set is desirable1 even though it is excess from a narrower point of view1 and causes the straightforward 'seven eights are fifty-six' to be overlearned4 ,so also a pupil's work with #bd1 #cd1 #dd1 etc41 #i may react to give what would be excess practice from the point of view of #d #i alone2 his work in estimating approximate quotient figures in long division may give excess practice on the division tables4 ,there are many such cases4 ,even adding the #e and #g in #e_/ab #g_/ab is not quite the same task as adding #e and #g undisturbed by the fact that they are twelfths4 ,we know far too little about the amount of practice needed to adapt arithmetical bonds to efficient operation in these more complicated conditions to estimate even approximately the allowances to be made4 ,but some allowance1 and often a rather large allowance1 must be made4 ,the second is the case where the computation in general should be made very easy and sure for the pupil except for some one new element that is being learned4 ,for example1 in teaching the meaning and uses of ',averages' and of uneven division1 we may deliberately use #b1 #c1 and #d as divisors rather than #g and #i1 so as to let all the pupil's energy be spent in learning the new facts1 and so that the fraction in the quotient may be something easily understood1 real1 and significant4 ,in teaching the addition of mixed numbers1 we may use1 in the early steps1 #aa-a_/b #ac-a_/b #bd ------ rather than #gi-a_/b #ih-a_/b #fg ------ so as to save attention for the new process itself4 ,in cancellation1 we may give excess practice to divisions by #b1 #c1 #d1 and #e in order to make the transfer to the new habits of considering two numbers together from the point of view of their divisibility by some number4 ,in introducing trade discount1 we may give excess practice on '#e of' and '#aj of' deliberately1 so that the meaning of discount may not be obscured by difficulties in the computation itself4 ,excess practice on1 and overlearning of1 certain bonds is thus very often justifiable4 ,the third case concerns bonds whose importance for practical uses in life or as notable facilitators of other bonds is so great that they may profitably be brought to a greater strength than #aii correct out of #bjj at a speed of #b sec4 or less1 or be brought to that degree of strength very early4 ,examples of bonds of such special practical use are the subtractions from #aj1 #a_/b #a_/b1 #a_/b #a_/d1 #a_/b of #fj1 #a_/d of #fj1 and the fractional parts of #ab and of #a4jj4 ,examples of notable facilitating bonds are ten #ajs #ajj1 ten #ajjs #ajjj1 additions like #b #b1 #c #c1 and #d #d1 and all the multiplication tables to #i x #i4 ,in consideration of these three modifying cases or principles1 a volume could well be written concerning just how much practice to give to each bond1 in each of the types of complex situations where it has to operate4 ,there is evidently need for much experimentation to expose the facts1 and for much sagacity and inventiveness in making sure of effective learning without wasteful overlearning4 ,the facts of primary importance are3-- 7#a7 ,the textbook or other instrument of instruction which is a teacher's general guide may give far too little practice on certain bonds4 7#b7 ,it may divide the practice given in ways that are apparently unjustifiable4 7#c7 ,the teacher needs therefore to know how much practice it does give1 where to supplement it1 and what to omit4 7#d7 ,the omissions1 on grounds of apparent excess practice1 should be made only after careful consideration of the third principle described above4 7#e7 ,the amount of practice should always be considered in the light of its interest and appeal to the pupil's tendency to work with full power and zeal4 ,mere repetition of bonds when the learner does not care whether he is improving is rarely justifiable on any grounds4 7#f7 ,practice that is actually in excess is not a very grave defect if it is enjoyed and improves the pupil's attitude toward arithmetic4 ,not much time is lost2 a hundred practices for each of a thousand bonds after mastery to #aii in #bjj at #b seconds will use up less than #fj hours1 or #ae hours per year in grades #c to #f4 7#g7 ,by the proper division of practice among bonds1 the arrangement of learning so that each bond helps the others1 the adroit shifting of practice of a bond to each new type of situation requiring it to operate under changed conditions1 and the elimination of excess practice where nothing substantial is gained1 notable improvements over the past hit-and-miss customs may be expected4 7#h7 ,unless the material for practice is adequate1 well balanced and sufficiently motivated1 the teacher must keep close account of the learning of pupils4 ,otherwise disastrous underlearning of many bonds is almost sure to occur and retard the pupil's development4 ,,the ,,organization ,,of ,,abilities ,there is danger that the need of brevity and simplicity which has made us speak so often of a bond or an ability1 and of the amount of practice it requires1 may mislead the reader into thinking that these bonds and abilities are to be formed each by itself alone and kept so4 ,they should rarely be formed so and never kept so4 ,this we have indicated from time to time by references to the importance of forming a bond in the way in which it is to be used1 to the action of bonds in changed situations1 to facilitation of one bond by others1 to the cooeperation of abilities1 and to their integration into a total arithmetical ability4 ,as a matter of fact1 only a small part of drill work in arithmetic should be the formation of isolated bonds4 ,even the very young pupil learning #e and #c are #h should learn it with '#e and #e #aj1' '#e and #b #g1' at the back of his mind1 so to speak4 ,even so early1 #e #c #h should be part of an organized1 cooeperating system of bonds4 ,later #ej #cj #hj should become allied to it4 ,each bond should be considered1 not simply as a separate tool to be put in a compartment until needed1 but also as an improvement of one total tool or machine1 arithmetical ability4 ,there are differences of course4 ,knowledge of square root can be regarded somewhat as a separate tool to be sharpened1 polished1 and used by itself1 whereas knowledge of the multiplication tables cannot4 ,yet even square root is probably best made more closely a part of the total ability1 being taught as a special case of dividing where divisor is to be the same as quotient1 the process being one of estimating and correcting4 ,in general we do not wish the pupil to be a repository of separated abilities1 each of which may operate only if you ask him the sort of questions which the teacher used to ask him1 or otherwise indicate to him which particular arithmetical tool he is to use4 ,rather he is to be an effective organization of abilities1 cooeperating in useful ways to meet the quantitative problems life offers4 ,he should not as a rule have to think in such fashion as3 ,is this interest or discount8 ,is it simple interest or compound interest8 ,what did ,i do in compound interest8 ,how do ,i multiply by #b percent8 ,the situation that calls up interest should also call up the kind of interest that is appropriate1 and the technique of operating with percents should be so welded together with interest in his mind that the right cooeperation will occur almost without supervision by him4 ,as each new ability is acquired1 then1 we seek to have it take its place as an improvement of a thinking being1 as a cooeperative member of a total organization1 as a soldier fighting together with others1 as an element in an educated personality4 ,such an organization of bonds will not form itself any more than any one bond will create itself4 ,if the elements of arithmetical ability are to act together as a total organized unified force they must be made to act together in the course of learning4 ,what we wish to have work together we must put together and give practice in teamwork4 ,we can do much to secure such cooeperative action when and where and as it is needed by a very simple expedient2 namely1 to give practice with computation and problems such as life provides1 instead of making up drills and problems merely to apply each fact or principle by itself4 ,though a pupil has solved scores of problems reading1 ,a triangle has a base of a feet and an altitude of b feet1 what is its area8 he may still be practically helpless in finding the area of a triangular plot of ground2 still more helpless in using the formula for a triangle which is one of two into which a trapezoid is divided4 ,though a pupil has learned to solve problems in trade discount1 simple interest1 compound interest1 and bank discount one at a time1 stated in a few set forms1 he may be practically helpless before the actual series of problems confronting him in starting in business1 and may take money out of the savings bank when he ought to borrow on a time loan1 or delay payment on his bills when by paying cash he could save money as well as improve his standing with the wholesaler4 ,instead of making up problems to fit the abilities given by school instruction1 we should preferably modify school instruction so that arithmetical abilities will be organized into an effective total ability to meet the problems that life will offer4 ,still more generally1 every bond formed should be formed with due consideration of every other bond that has been or will be formed2 every ability should be practiced in the most effective possible relations with other abilities4 ,,chapter ,,vii ,,the ,,sequence ,,of ,,topics3 ,,the ,,order ,,of ,,formation ,,of ,,bonds ,the bonds to be formed having been chosen1 the next step is to arrange for their most economical order of formation--to arrange to have each help the others as much as possible--to arrange for the maximum of facilitation and the minimum of inhibition4 ,the principle is obvious enough and would probably be admitted in theory by any intelligent teacher1 but in practice we are still wedded to conventional usages which arose long before the psychology of arithmetic was studied4 ,for example1 we inherit the convention of studying addition of integers thoroughly1 and then subtraction1 and then multiplication1 and then division1 and many of us follow it though nobody has ever given a proof that this is the best order for arithmetical learning4 ,we inherit also the opposite convention of studying in a so-called spiral plan1 a little addition1 subtraction1 multiplication1 and division1 and then some more of each1 and then some more1 and many of us follow this custom1 with an unreasoned faith that changing about from one process to another is per se helpful4 ,such conventions are very strong1 illustrating our common tendency to cherish most those customs which we cannot justify6 ,the reductions of denominate numbers ascending and descending were1 until recently1 in most courses of study1 kept until grade #d or grade #e was reached1 although this material is of far greater value for drills on the multiplication and division tables than the customary problems about apples1 eggs1 oranges1 tablets1 and penholders4 ,by some historical accident or for good reasons the general treatment of denominate numbers was put late2 by our naive notions of order and system we felt that any use of denominate numbers before this time was heretical2 we thus became blind to the advantages of quarts and pints for the tables of #bs2 yards and feet for the tables of #cs2 gallons and quarts for the tables of #ds2 nickels and cents for the #es2 weeks and days for the #gs2 pecks and quarts for the #hs2 and square yards and square feet for the #is4 ,problems like #e yards feet or #ae feet yards have not only the advantages of brevity1 clearness1 practical use1 real reference1 and ready variation1 but also the very great advantage that part of the data have to be thought of in a useful way instead of read off from the page4 ,in life1 when a person has twenty cents with which to buy tablets of a certain sort1 he thinks of the price in making his purchase1 asking it of the clerk only in case he does not know it1 and in planning his purchases beforehand he thinks of prices as a rule4 ,in spite of these and other advantages1 not one textbook in ten up to #aijj made early use of these exercises with denominate numbers4 ,so strong is mere use and wont4 ,besides these conventional customs1 there has been1 in those responsible for arithmetical instruction1 an admiration for an arrangement of topics that is easy for a person1 after he knows the subject1 to use in thinking of its constituent parts and their relations4 ,such arrangements are often called 'logical' arrangements of subject matter1 though they are often far from logical in any useful sense4 ,now the easiest order in which to think of a hierarchy of habits after you have formed them all may be an extremely difficult order in which to form them4 ,the criticism of other orders as 'scrappy1' or 'unsystematic1' valid enough if the course of study is thought of as an object of contemplation1 may be foolish if the course of study is regarded as a working instrument for furthering arithmetical learning4 ,we must remember that all our systematizing and labeling is largely without meaning to the pupils4 ,they cannot at any point appreciate the system as a progression from that point toward this and that1 since they have no knowledge of the 'this or that4' ,they do not as a rule think of their work in grade #d as an outcome of their work in grade #c with extensions of a to a#a1 and additions of b#b and b#c to b and b#a1 and refinements of c and d by c#d and d#e4 ,they could give only the vaguest account of what they did in grade #c1 much less of why it should have been done then4 ,they are not much disturbed by a lack of so-called 'system' and 'logical' progression for the same reason that they are not much helped by their presence4 ,what they need and can use is a dynamically effective system or order1 one that they can learn easily and retain long by1 regardless of how it would look in a museum of arithmetical systems4 ,unless their actual arithmetical habits are usefully related it does no good to see the so-called logical relations2 and if their habits are usefully related1 it does not very much matter whether or not they do see these2 finally1 they can be brought to see them best by first acquiring the right habits in a dynamically effective order4 ,,decreasing ,,interference ,,and ,,increasing ,,facilitation ,psychology offers no single1 easy1 royal road to discovering this dynamically best order4 ,it can only survey the bonds1 think what each demands as prerequisite and offers as future help1 recommend certain orders for trial1 and measure the efficiency of each order as a means of attaining the ends desired4 ,the ingenious thought and careful experimentation of many able workers will be required for many years to come4 ,psychology can1 however1 even now1 give solid constructive help in many instances1 either by recommending orders that seem almost certainly better than those in vogue1 or by proposing orders for trial which can be justified or rejected by crucial tests4 ,consider1 for example1 the situation1 'a column of one-place numbers to be added1 whose sum is over #i1' and the response 'writing down the sum4' ,this bond is commonly firmly fixed before addition with two-place numbers is undertaken4 ,as a result the pupil has fixed a habit that he has to break when he learns two-place addition4 ,if oral answers only are given with such single columns until two-place addition is well under way1 the interference is avoided4 ,in many courses of study the order of systematic formation of the multiplication table bonds is3 #a x #a1 #b x #a1 etc41 #a x #b1 #b x #b1 etc41 #a x #c1 #b x #c1 etc41 #a x #i1 #b x #i1 etc4 ,this is probably wrong in two respects4 ,there is abundant reason to believe that the x #es should be learned first1 since they are easier to learn than the #as or the #bs1 and give the idea of multiplying more emphatically and clearly4 ,there is also abundant reason to believe that the #a x #e1 #a x #b1 #a x #c1 etc41 should be put very late--after at least three or four tables are learned1 since the question ,what is #a times #b8 7or #c or #e7 is unnecessary until we come to multiplication of two- and three-place numbers1 seems a foolish question until then1 and obscures the notion of multiplication if put early4 ,also the facts are best learned once for all as the habits #a times k is the same as k1 and k times #a is the same as k4#h #h ,the very early learning of #b x #b1 #b x #c1 #c x #b1 #b x #d1 #d x #b1 #c x #c1 and perhaps a few more multiplications is not considered here4 ,it is advisable4 ,the treatment of #j x #j1 #j x #a1 #a x #j1 etc41 is not considered here4 ,it is probably best to defer the 'x #j' bonds until after all the others are formed and are being used in short multiplication1 and to form them in close connection with their use in short multiplication4 ,the '#j x' bonds may well be deferred until they are needed in 'long' multiplication1 #j x #j coming last of all4 ,in another connection it was recommended that the divisions to #ha _/ #i be learned by selective thinking or reasoning from the multiplications4 ,this determines the order of bonds so far as to place the formation of the division bonds soon after the learning of the multiplications4 ,for other reasons it is well to make the proximity close4 ,one of the arbitrary systematizations of the order of formation of bonds restricts operations at first to the numbers #a to #aj1 then to numbers under #ajj1 then to numbers under #ajjj1 then to numbers under #aj1jjj4 ,apart from the avoidance of unreal and pedantic problems in applied arithmetic to which work with large numbers in low grades does somewhat predispose a teacher1 there is little merit in this restriction of the order of formation of bonds4 ,its demerits are many4 ,for example1 when the pupil is learning to 'carry' in addition he can be given better practice by soon including tasks with sums above #ajj1 and can get a valuable sense of the general use of the process by being given a few examples with three- and four-place numbers to be added4 ,the same holds for subtraction4 ,indeed1 there is something to be said in favor of using six- or seven-place numbers in subtraction1 enforcing the 'borrowing' process by having it done again and again in the same example1 and putting it under control by having the decision between 'borrowing' and 'not borrowing' made again and again in the same example4 ,when the multiplication tables are learned the most important use for them is not in tedious reviews or trivial problems with answers under #ajj1 but in regular 'short' multiplication of two- and three- and even four-place numbers4 ,just as the addition combinations function mainly in the higher-decade modifications of them1 so the multiplication combinations function chiefly in the cases where the bond has to operate while the added tasks of keeping one's place1 adding what has been carried1 writing down the right figure in the right place1 and holding the right number for later addition1 are also taken care of4 ,it seems best to introduce such short multiplication as soon as the x #es1 x #bs1 x #cs1 and x #ds are learned and to put the x #fs1 x #gs1 and the rest to work in such short multiplication as soon as each is learned4 ,still surer is the need for four-1 five-1 and six-place numbers when two-place numbers are used in multiplying4 ,when the process with a two-place multiplier is learned1 multiplications by three-place numbers should soon follow4 ,they are not more difficult then than later4 ,on the contrary1 if the pupil gets used to multiplying only as one does with two-place multipliers1 he will suffer more by the resulting interference than he does from getting six- or seven-place answers whose meaning he cannot exactly realize4 ,they teach the rationale and the manipulations of long multiplication with especial economy because the principles and the procedures are used two or three times over and the contrasts between the values which the partial products have in adding become three instead of one4 ,the entire matter of long multiplication with integers and ,united ,states money should be treated as a teaching unit and the bonds formed in close organization1 even though numbers as large as #ijj1jjj are occasionally involved4 ,the reason is not that it is more logical1 or less scrappy1 but that each of the bonds in question thus gets much help from1 and gives much help to1 the others4 ,in sharp contrast to a topic like 'long multiplication' stands a topic like denominate numbers4 ,it most certainly should not be treated as a large teaching unit1 and all the bonds involved in adding1 subtracting1 multiplying1 and dividing with all the ordinary sorts of measures should certainly not be formed in close sequence4 ,the reductions ascending and descending for many of the measures should be taught as drills on the appropriate multiplication and division tables4 ,the reduction of feet and inches to inches1 yards and feet to yards1 gallons and quarts to quarts1 and the like are admirable exercises in connection with the 7a x b7 c 4444 problems1--the ',bought #c lbs4 of sugar at #g cents and #e cents worth of matches' problems4 ,the reductions of inches to feet and inches and the like are admirable exercises in the d 74444 x b7 c or 'making change' problem1 which in its small-number forms is an excellent preparatory step for short division4 ,they are also of great service in early work with fractions4 ,the feet-mile1 square-foot-square-inch1 and other simple relations give a genuine and intelligible demand for multiplication with large numbers4 ,knowledge of the metric system for linear and square measure would perhaps1 as an introduction to decimal fractions1 more than save the time spent to learn it4 ,it would even perhaps be worth while to invent a measure 7call it the twoqua7 midway between the quart and gallon and teach carrying in addition and borrowing in subtraction by teaching first the addition and subtraction of 'gallon1 twoqua1 quart1 and pint' series6 ,many of the bonds which a system-made tradition huddled together uselessly in a chapter on denominate numbers should thus be formed as helpful preparations for and applications of other bonds all the way from the first to the eighth half-year of instruction in arithmetic4 ,the bonds involved in the ability to respond correctly to the series3-- #e 4444 #bs and 4444 remainder #e 4444 #cs and 4444 remainder #hh 4444 #is and 4444 remainder should be formed before1 not during1 the training in short division4 ,they are admirable at that point as practice on the division tables2 are of practical service in the making-change problems of the small purchase and the like2 and simplify the otherwise intricate task of keeping one's place1 choosing the quotient figure1 multiplying by it1 subtracting and holding in mind the new number to be divided1 which is composed half of the remainder and half of a figure in the written dividend4 ,this change of order is a good illustration of the nearly general rule that ,when the practice or review required to perfect or hold certain bonds can1 by an inexpensive modification1 be turned into a useful preparation for new bonds1 that modification should be made4 ,the bonds involved in the four operations with ,united ,states money should be formed in grades #c and #d along with or very soon after the corresponding bonds with three-place and four-place integers4 ,this statement would have seemed preposterous to the pedagogues of fifty years ago4 ,united ,states money1 they would have said1 is an application of decimals4 ,how can it be learned until the essentials of decimal fractions are known8 ,how will the child understand when multiplying 4#ge by #c that #c times #e cents is #a dime and #e cents1 or that #c times #gj cents is #b dollars and #a dime8 ,why perplex the young pupils with the difficulties of placing the decimal point8 ,why disturb the learning of the four operations with integers by adding at each step a second 'procedure with ,united ,states money'8 ,the case illustrates very well the error of the older oversystematic treatment of the order of topics and the still more important error of confusing the logic of proof with the psychology of learning4 ,to prove that #c x 4#ge #b4be to the satisfaction of certain arithmeticians1 you may need to know the theory of decimal fractions2 but to do such multiplication all a child needs is to do just what he has been doing with integers and then ,put a before the answer to show that it means dollars and cents1 and put a decimal point in the answer to show which figures mean dollars and which figures mean cents4 ,and this is general4 ,the ability to operate with integers plus the two habits of prefixing and separating dollars from cents in the result will enable him to operate with ,united ,states money4 ,consequently good practice came to use ,united ,states money not as a consequence of decimal fractions1 learned by their aid1 but as an introduction to decimal fractions which aids the pupil to learn them4 ,so it has gradually pushed work with ,united ,states money further and further back1 though somewhat timidly4 ,we need not be timid4 ,the pupil will have no difficulty in adding1 subtracting1 multiplying1 and dividing with ,united ,states money--unless we create it by our explanations6 ,if we simply form the two bonds described above and show by proper verification that the procedure always gives the right answer1 the early teaching of the four operations with ,united ,states money will in fact actually show a learning profit6 ,it will save more time in the work with integers than was spent in teaching it6 ,for1 in the first place1 it will help to make work with four-place and five-place numbers more intelligible and vital4 ,a pupil can understand #af4ge or #bh4gi more easily than #afge or #bhgi4 ,the former may be the prices of a suit or sewing machine or bicycle4 ,in the second place1 it permits the use of a large stock of genuine problems about spending1 saving1 sharing1 and the like with advertisements and catalogues and school enterprises4 ,in the third place1 it permits the use of common-sense checks4 ,a boy may find one fourth of #cjjj as #gjej or #ge and not be disturbed1 but he will much more easily realize that one fourth of #cj4jj is not over #gj or less than #a4 ,even the decimal point of which we used to be so afraid may actually help the eye to keep its place in adding4 ,,interest ,so far1 the illustrations of improvements in the order of bonds so as to get less interference and more facilitation than the customary orders secure have sought chiefly to improve the mechanical organization of the bonds4 ,any gain in interest which the changes described effected would be largely due to the greater achievement itself4 ,dewey and others have emphasized a very different principle of improving the order of formation of bonds--the principle of determination of the bonds to be formed by some vital1 engaging problem which arouses interest enough to lighten the labor and which goes beyond or even against cut-and-dried plans for sequences in order to get effective problems4 ,for example1 the work of the first month in grade #b;,b might sacrifice facilitations of the mechanical sort in order to put arithmetic to use in deciding what dimensions a rabbit's cage should have to give him #ab square feet of floor space1 how much bread he should have per meal to get #f ounces a day1 how long a ten-cent loaf would last1 how many loaves should be bought per week1 how much it costs to feed the rabbit1 how much he has gained in weight since he was brought to the school1 and so on4 ,such sacrifices of the optimal order if interest were equal1 in order to get greater interest or a healthier interest1 are justifiable4 ,vital problems as nuclei around which to organize arithmetical learning are of prime importance4 ,it is even safe probably to insist that some genuine problem-situation requiring a new process1 such as addition with carrying1 multiplication by two-place numbers1 or division with decimals1 be provided in every case as a part of the introduction to that process4 ,the sacrifice should not be too great1 however2 the search for vital problems that fit an economical order of subject matter is as much needed as the amendment of that order to fit known interests2 and the assurance that a problem helps the pupil to learn arithmetic is as important as the assurance that arithmetic is used to help the pupil solve his personal problems4 ,much ingenuity and experimentation will be required to find the order that is satisfactory in both quality and quantity of interest or motive and helpfulness of the bonds one to another4 ,the difficulty of organizing arithmetic around attractive problems is much increased by the fact of class instruction4 ,for any one pupil vital1 personal problems or projects could be found to provide for many arithmetical abilities2 and any necessary knowledge and technique which these projects did not develop could be somehow fitted in along with them4 ,but thirty children1 half boys and half girls1 varying by five years in age1 coming from different homes1 with different native capacities1 will not1 in ,september1 #aibj1 unanimously feel a vital need to solve any one problem1 and then conveniently feel another on1 say1 ,october #ae6 ,in the mechanical laws of learning children are much alike1 and the gain we may hope to make from reducing inhibitions and increasing facilitations is1 for ordinary class-teaching1 probably greater than that to be made from the discovery of attractive central problems4 ,we should1 however1 get as much as possible of both4 ,,general ,,principles ,the reader may by now feel rather helpless before the problem of the arrangement of arithmetical subject matter4 ,sometimes you complete a topic1 sometimes you take it piecemeal months or years apart1 often you make queer twists and shifts to get a strategic advantage over the enemy1 he may think1 but are there no guiding principles1 no general rules8 ,there is only one that is absolutely general1 to take the order that works best for arithmetical learning4 ,there are particular rules1 but there are so many and they are so limited by an 'other things being equal' clause1 that probably a general eagerness to think out the pros and cons for any given proposal is better than a stiff attempt to adhere to these rules4 ,i will state and illustrate some of them1 and let the reader judge4 ,other things being equal1 one new sort of bonds should not be started until the previous set is fairly established1 and two different sets should not be started at once4 ,thus1 multiplication of two- and three-place numbers by #b1 #c1 #d1 and #e will first use numbers such that no carrying is required1 and no zero difficulties are encountered1 then introduce carrying1 then introduce multiplicands like #bjf and #cbj4 ,if other things were equal1 the carrying would be split into two steps--first drills with 7#d x #f7 #b1 7#c x #g7 #c1 7#e x #d7 #a1 and the like1 and second the actual use of these habits in the multiplication4 ,the objection to this separation of the double habit is that the first part of it in isolation is too artificial--that it may be better to suffer the extra difficulty of forming the two together than to teach so rarely used habits as the 7a x b7 c series4 ,experimental tests are needed to decide this point4 ,other things being equal1 bonds should be formed in such order that none will have to be broken later4 ,for example1 there is a strong argument for teaching long division first1 or very early1 with remainders1 letting the case of zero remainder come in as one of many4 ,if the pupils have been familiarized with the remainder notion by the drills recommended as preparation for short division1#i the use of remainders in long division will offer little difficulty4 ,the exclusive use of examples without remainders may form the habit of not being exact in computation1 of trusting to 'coming out even' as a sole check1 and even of writing down a number to fit the final number to be divided instead of obtaining it by honest multiplication4 #i ,see page #gf4 ,for similar reasons additions with #b and #c as well as #a to be 'carried' have much to recommend them in the very first stages of column addition with carrying4 ,there is here the added advantage that a pupil will be more likely to remember to carry if he has to think what to carry4 ,the present common practice of using small numbers for ease in the addition itself teaches many children to think of carrying as adding one4 ,other things being equal1 arrange to have variety4 ,thus it is probably1 though not surely1 wise to interrupt the monotony of learning the multiplication and division tables1 by teaching the fundamentals of 'short' multiplication and perhaps of division after the #es1 #bs1 #cs1 and #ds are learned4 ,this makes a break of several weeks4 ,the facts for the #fs1 #gs1 #hs1 and #is can then be put to varied use as fast as learned4 ,it is almost certainly wise to interrupt the first half-year's work with addition and subtraction1 by teaching #b x #b1 #b x #c1 #c x #b1 #b x #d1 #d x #b1 #b x #e1 later by #b x #aj1 #c x #aj1 #d x #aj1 #e x #aj1 later by #a_/b #a_/b1 #a-a_/b #a_/b1 #a_/b of #b1 #a_/b of #d1 #a_/b of #f1 and at some time by certain profitable exercises wherein a pupil tells all he knows about certain numbers which may be made nuclei of important facts 7say1 #e1 #h1 #aj1 #ab1 #ae1 and #bj74 ,other things being equal1 use objective aids to verify an arithmetical process or inference after it is made1 as well as to provoke it4 ,it is well at times to let pupils do everything that they can with relations abstractly conceived1 testing their results by objective counting1 measuring1 adding1 and the like4 ,for example1 an early step in adding should be to show three things1 put them under a book1 show two more1 put these under the book1 and then ask how many there are under the book1 letting the objective counting come later as the test of the correctness of the addition4 ,other things being equal1 reserve all explanations of why a process must be right until the pupils can use the process accurately1 and have verified the fact that it is right4 ,except for the very gifted pupils1 the ordinary preliminary deductive explanations of what must be done are probably useless as means of teaching the pupils what to do4 ,they use up much time and are of so little permanent effect that1 as we have seen1 the very arithmeticians who advocate making them1 admit that after a pupil has mastered the process he may be allowed to forget the reasons for it4 ,i am not sure that the deductive proofs of why we place the decimal point as we do in division by a decimal1 or invert and multiply in dividing by a fraction1 and the like1 are worth teaching at all4 ,if they are to be taught at all1 the time to teach them is 7except for the very gifted7 after the pupil has mastered the process and has confidence in it4 ,he then at least knows what process he is to prove is right1 and that it is right1 and has had some chance of seeing why it is right from his experience with it4 ,one more principle may be mentioned without illustration4 ,arrange the order of bonds with due regard for the aims of the other studies of the curriculum and the practical needs of the pupil outside of school4 ,arithmetic is not a book or a closed system of exercises4 ,it is the quantitative work of the pupils in the elementary school4 ,no narrower view of it is adequate4 ,,chapter ,,viii ,,the ,,distribution ,,of ,,practice ,,the ,,problem ,the same amount of practice may be distributed in various ways4 ,figures #g to #aj1 for example1 show #bjj practices with division by a fraction distributed over three and a half years of #aj months in four different ways4 ,in ,fig4 #g1 practice is somewhat equally distributed over the whole period4 ,in ,fig4 #h the practice is distributed at haphazard4 ,in ,fig4 #i there is a first main learning period1 a review after about ten weeks1 a review at the beginning of the seventh grade1 another review at the beginning of the eighth grade1 and some casual practice rather at random4 ,in ,fig4 #aj there is a main learning period1 with reviews diminishing in length and separated by wider and wider intervals1 with occasional practice thereafter to keep the ability alive and healthy4 ,plans ,i and ,,ii are obviously inferior to ,plans ,,iii and ,,iv2 and ,plan ,,iv gives promise of being more effective than ,plan ,,iii1 since there seems danger that the pupil working by ,plan ,,iii might in the ten weeks lose too much of what he had gained in the initial practice1 and so again in the next ten weeks4 ,it is not wise1 however1 to try now to make close decisions in the case of practice with division by a fraction2 or to determine what the best distribution of practice is for that or any other ability to be improved4 ,the facts of psychology are as yet not adequate for very close decisions1 nor are the types of distribution of practice that are best adapted to different abilities even approximately worked out4 ,illustration3 ,,fig4 #g4--,plan ,i4 #bjj practices distributed somewhat evenly over #c-a_/b years of #aj months4 ,in ,figs4 #g1 #h1 #i1 and #aj1 each tenth of an inch along the base line represents one month4 ,each hundredth of a square inch represents four practices1 a little square #a_/bj of an inch wide and #a_/bj inch high representing one practice4 ,illustration3 ,,fig4 #h4--,plan ,,ii4 #bjj practices distributed haphazard over #c-a_/b years of #aj months4 ,illustration3 ,,fig4 #i4--,plan ,,iii4 ,a learning period1 three reviews1 and incidental practice4 ,illustration3 ,,fig4 #aj4--,plan ,,iv4 ,a learning period with reviews of decreasing length at increasing intervals4 ,,sample ,,distributions ,let us rather examine some actual cases of distribution of practice found in school work and consider1 not the attainment of the best possible distribution1 but simply the avoidance of gross blunders and the attainment of reasonable1 defensible procedures in this regard4 ,figures #aa to #ah show the distribution of examples in multiplication with multipliers of various sorts4 ,x stands for any digit except zero4 ,o stands for #j4 ,,xxo thus means a multiplier like #cej or #bgj or #afj2 ,,xox means multipliers like #djg1 #ije1 or #bjf2 ,,xx means multipliers like #be1 #ag1 #ch4 ,each of these diagrams covers approximately #c-a_/b years of school work1 or from about the middle of grade #c to the end of grade #f4 ,they are made from counts of four textbooks 7,a1 ,b1 ,c1 and ,d71 the count being taken for each successive #h pages4#aj ,each tenth of an inch along the base line equals #h pages of the text in question4 ,each 4#ja sq4 in4 equals one example4 ,the books1 it will be observed1 differ in the amount of practice given1 as well as in the way in which it is distributed4 #aj ,at the end of a volume or part1 the count may be from as few as #e or as many as #ab pages4 ,these distributions are worthy of careful study2 we shall note only a few salient facts about them here4 ,of the distributions of multiplications with multipliers of the ,,xx type1 that of book ,d 7,fig4 #ad7 is perhaps the best4 ,a 7,fig4 #aa7 has too much of the practice too late2 ,b 7,fig4 #ab7 gives too little practice in the first learning2 ,c 7,fig4 #ac7 gives too much in the first learning and in grade #f4 ,among the distributions of multiplication with multipliers of the ,,xox type1 that of book ,d 7,fig4 #ah7 is again probably the best4 ,a1 ,b1 and ,c 7,figs4 #ae1 #af1 and #ag7 have too much practice early and too long intervals between reviews4 ,book ,c 7,fig4 #ag7 by a careless oversight has one case of this very difficult process1 without any explanation1 weeks before the process is taught6 ,illustration3 ,,fig4 #aa4--;,distribution of practise with multipliers of the ,,xx type in the first two books of the three-book text ,a4 ,illustration3 ,,fig4 #ab4--,same as ,fig4 #aa1 but for text ,b4 ,following this period come certain pages of computation to be used by the teacher at her discretion1 containing #bd ,,xx multiplications4 ,illustration3 ,,fig4 #ac4--,same as ,fig4 #aa1 but for text ,c4 ,illustration3 ,,fig4 #ad4--,same as ,fig4 #aa1 but for text ,d4 ,illustration3 ,,fig4 #ae4--;,distribution of practice with multipliers of the ,,xox type in the first two books of the three-book text ,a4 ,illustration3 ,,fig4 #af4--,same as ,fig4 #ae1 but for text ,b4 ,following this period come certain pages of computation to be used by the teacher at her discretion1 containing #ag ,,xox multiplications4 ,illustration3 ,,fig4 #ag4--,same as ,fig4 #af1 but for text ,c4 ,illustration3 ,,fig4 #ah4--,same as ,fig4 #af1 but for text ,d4 ,figures #ai1 #bj1 #ba1 #bb1 and #bc all concern the first two books of the three-book text ,e4 ,figure #ai shows the distribution of practice on #e x #e in the first two books of text ,e4 ,the plan is the same as in ,figs4 #aa to #ah1 except that each tenth of an inch along the base line represents ten pages4 ,figure #bj shows the distribution of practice on #g x #g2 ,fig4 #ba shows it for #f x #g and #g x #f together4 ,in ,figs4 #bj and #ba also1 #j4a inch along the base line equals ten pages4 ,figures #bb and #bc show the distribution of practice on the divisions of #gb1 #gc1 #gd1 #ge1 #gf1 #gg1 #gh1 and #gi by either #h or #i1 and on the divisions of #ha1 #hb 444 #hi by #i4 ,each tenth of an inch along the base line represents ten pages here also4 ,figures #ai to #bc show no consistent plan for distributing practice4 ,with #e x #e 7,fig4 #ai7 the amount of practice increases from the first treatment in grade #c to the end of grade #f1 so that the distribution would be better if the pupil began at the end and went backward6 ,with #g x #g 7,fig4 #bj7 the practice is distributed rather evenly and in small doses4 ,with #f x #g and #g x #f 7,fig4 #ba7 much of it is in very large doses4 ,with the divisions 7,figs4 #bb and #bc7 the practice is distributed more suitably1 though in ,fig4 #bc there is too much of it given at one time in the middle of the period4 ,illustration3 ,,fig4 #ai4--;,distribution of practice with #e x #e in the first two books of the three-book text ,e4 ,illustration3 ,,fig4 #bj4--;,distribution of practice with #g x #g in the first two books of text ,e4 ,illustration3 ,,fig4 #ba4--;,distribution of practice with #f x #g or #g x #f in the first two books of text ,e4 ,illustration3 ,,fig4 #bb4--;,distribution of practice with #gb1 #gc 444 #gi _/ #h or #i in the first two books of text ,e4 ,illustration3 ,,fig4 #bc4--;,distribution of practice with #ha1 #hb 444 #hi _/ #i in the first two books of text ,e4 ,,possible ,,improvements ,even if we knew what the best distribution of practice was for each ability of the many to be inculcated by arithmetical instruction1 we could perhaps not provide it for all of them4 ,for1 in the first place1 the allotments for some of them might interfere with those for others4 ,in the second place1 there are many other considerations of importance in the ordering of topics besides giving the optimal distribution of practice to each ability4 ,such are considerations of interest1 of welding separate abilities into an integrated total ability1 and of the limitations due to the school schedule with its ,saturdays1 ,sundays1 holidays1 and vacations4 ,improvement can1 however1 be made over present practice in many respects4 ,a scientific examination of the teaching of almost any class for a year1 or of many of our standard instruments of instruction1 will reveal opportunities for improving the distribution of practice with no sacrifice of interest1 and with an actual gain in integrated functioning arithmetical power4 ,in particular it will reveal cases where an ability is given practice and then1 never being used again1 left to die of inactivity4 ,it will reveal cases where an ability is given practice and then left so long without practice that the first effect is nearly lost4 ,there will be cases where practice is given and reviews are given1 but all in such isolation from everything else in arithmetic that the ability1 though existent1 does not become a part of the pupil's general working equipment4 ,there will be cases where more practice is given in the late than the earlier periods for no apparent extrinsic advantage2 and cases where the practice is put where it is for no reason that is observable save that the teacher or author in question has decided to have some drill work at that time6 ,each ability has its peculiar needs in this matter1 and no set rules are at present of much value4 ,it will be enough for the present if we are aroused to the problem of distribution1 avoid obvious follies like those just noted1 and exercise what ingenuity we have4 ,,chapter ,,ix ,,the ,,psychology ,,of ,,thinking3 ,,abstract ,,ideas ,,and ,,general ,,notions ,,in ,,arithmetic#aa #aa ,certain paragraphs in this and the following chapter are taken from the author's ,educational ,psychology1 with slight modifications4 ,,responses ,,to ,,elements ,,and ,,classes ,the plate which you see1 the egg before you at the breakfast table1 and this page are concrete things1 but whiteness1 whether of plate1 egg1 or paper1 is1 we say1 an abstract quality4 ,to be able to think of whiteness irrespective of any concrete white object is to be able to have an abstract idea or notion of white2 to be able to respond to whiteness1 irrespective of whether it is a part of china1 eggshell1 paper or whatever object1 is to be able to respond to the abstract element of whiteness4 ,learning arithmetic involves the formation of very many such ideas1 the acquisition of very many such powers of response to elements regardless of the gross total situations in which they appear4 ,to appreciate the fiveness of five boys1 five pencils1 five inches1 five rings of a bell2 to understand the division into eight equal parts of #dj cents1 #cb feet1 #fd minutes1 or #af ones2 to respond correctly to the fraction relation in #b_/c1 #e_/f1 #c_/d1 #g_/ab1 #a_/h1 or any other2 to be sensitive to the common element of #i #c x #c1 #af #d x #d1 #fbe #be x #be1 4#jd 4#b x 4#b1 #a_/d #a_/b x #a_/b1--these are obvious illustrations4 ,all the numbers which the pupil learns to understand and manipulate are in fact abstractions2 all the operations are abstractions2 percent1 discount1 interest1 height1 length1 area1 volume1 are abstractions2 sum1 difference1 product1 quotient1 remainder1 average1 are facts that concern elements or aspects which may appear with countless different concrete surroundings or concomitants4 ,towser is a particular dog2 your house lot on ,elm ,street is a particular rectangle2 ,mr4 and ,mrs4 ,i4,s4 ,peterson and their daughter ,louise are a particular family of three4 ,in contrast to these particulars1 we mean by a dog1 a rectangle1 and a family of three1 any specimens of these classes of facts4 ,the idea of a dog1 of rectangles in general1 of any family of three is a general notion1 a concept or idea of a class or species4 ,the ability to respond to any dog1 or rectangle1 or family of three1 regardless of which particular one it may be1 is the general notion in action4 ,learning arithmetic involves the formation of very many such general notions1 such powers of response to any member of a certain class4 ,thus a hundred different sized lots may all be responded to as rectangles2 #i_/ah1 #ab_/bg1 #ae_/bd1 and #bg_/cf may all be responded to as members of the class1 'both members divisible by #c4' ,the same fact may be responded to in different ways according to the class to which it is assigned4 ,thus #d in #c_/d1 #d_/e1 #de1 #ed1 and #dje is classed respectively as 'a certain sized part of unity1' 'a certain number of parts of the size shown by the #e1' 'a certain number of tens1' 'a certain number of ones1' and 'a certain number of hundreds4' ,each abstract quality may become the basis of a class of facts4 ,so fourness as a quality corresponds to the class 'things four in number or size'2 the fractional quality or relation corresponds to the class 'fractions4' ,the bonds formed with classes of facts and with elements or features by which one whole class of facts is distinguished from another1 are in fact1 a chief concern of arithmetical learning4#ab #ab ,it should be noted that just as concretes give rise to abstractions1 so these in turn give rise to still more abstract abstractions4 ,thus fourness1 fiveness1 twentyness1 and the like give rise to 'integral-number-ness4' ,similarly just as individuals are grouped into general classes1 so classes are grouped into still more general classes4 ,half1 quarter1 sixth1 and tenth are general notions1 but 'one 444th' is more general2 and 'fraction' is still more general4 ,,facilitating ,,the ,,analysis ,,of ,,elements ,abstractions and generalizations then depend upon analysis and upon bonds formed with more or less subtle elements rather than with gross total concrete situations4 ,the process involved is most easily understood by considering the means employed to facilitate it4 ,the first of these is having the learner respond to the total situations containing the element in question with the attitude of piecemeal examination1 and with attentiveness to one element after another1 especially to so near an approximation to the element in question as he can already select for attentive examination4 ,this attentiveness to one element after another serves to emphasize whatever appropriate minor bonds from the element in question the learner already possesses4 ,thus1 in teaching children to respond to the 'fiveness' of various collections1 we show five boys or five girls or five pencils1 and say1 ,see how many boys are standing up4 ,is ,jack the only boy that is standing here8 ,are there more than two boys standing8 ,name the boys while ,i point at them and count them4 7,jack7 is one1 and 7,fred7 is one more1 and 7,henry7 is one more4 ,jack and ,fred make 7two7 boys4 ,jack and ,fred and ,henry make 7three7 boys4 7,and so on with the attentive counting47 ,the mental set or attitude is directed toward favoring the partial and predominant activity of 'how-many-ness' as far as may be2 and the useful bonds that the 'fiveness1' the 'one and one and one and one and one-ness1' already have1 are emphasized as far as may be4 ,the second of the means used to facilitate analysis is having the learner respond to many situations each containing the element in question 7call it ,a71 but with varying concomitants 7call these ,v4 ,c47 his response being so directed as1 so far as may be1 to separate each total response into an element bound to the ,a and an element bound to the ,v4 ,c4 ,thus the child is led to associate the responses--',five boys1' ',five girls1' ',five pencils1' ',five inches1' ',five feet1' ',five books1' ',he walked five steps1' ',i hit my desk five times1' and the like--each with its appropriate situation4 ,the ',five' element of the response is thus bound over and over again to the 'fiveness' element of the situation1 the mental set being ',how many81' but is bound only once to any one of the concomitants4 ,these concomitants are also such as have preferred minor bonds of their own 7the sight of a row of boys per se tends strongly to call up the ',boys' element of the response74 ,the other elements of the responses 7boys1 girls1 pencils1 etc47 have each only a slight connection with the 'fiveness' element of the situations4 ,these slight connections also in large part#ac counteract each other1 leaving the field clear for whatever uninhibited bond the 'fiveness' has4 #ac ,they may1 of course1 also result in a fusion or an alternation of responses1 but only rarely4 ,the third means used to facilitate analysis is having the learner respond to situations which1 pair by pair1 present the element in a certain context and present that same context with the opposite of the element in question1 or with something at least very unlike the element4 ,thus1 a child who is being taught to respond to 'one fifth' is not only led to respond to 'one fifth of a cake1' 'one fifth of a pie1' 'one fifth of an apple1' 'one fifth of ten inches1' 'one fifth of an army of twenty soldiers1' and the like2 he is also led to respond to each of these in contrast with 'five cakes1' 'five pies1' 'five apples1' 'five times ten inches1' 'five armies of twenty soldiers4' ,similarly the 'place values' of tenths1 hundredths1 and the rest are taught by contrast with the tens1 hundreds1 and thousands4 ,these means utilize the laws of connection-forming to disengage a response element from gross total responses and attach it to some situation element4 ,the forces of use1 disuse1 satisfaction1 and discomfort are so maneuvered that an element which never exists by itself in nature can influence man almost as if it did so exist1 bonds being formed with it that act almost or quite irrespective of the gross total situation in which it inheres4 ,what happens can be most conveniently put in a general statement by using symbols4 ,denote by a b1 a g1 a l1 a q1 a v1 and a ,b certain situations alike in the element a and different in all else4 ,suppose that1 by original nature or training1 a child responds to these situations respectively by r#a r#b1 r#a r#g1 r#a r#ab1 r#a r#ag1 r#a r#bb1 r#a r#bg4 ,suppose that man's neurones are capable of such action that r#a1 r#b1 r#g1 r#ab1 r#bb1 and r#bg1 can each be made singly4 ,case ,i4 ,varying ,concomitants ,suppose that a b1 a g1 a l1 etc41 occur once each4 ,we have a b responded to by r#a r#b1 a g r#a r#g1 a l r#a r#ab1 a q r#a r#ag1 a v r#a r#bb1 and a ,b r#a r#bg1 as shown in ,scheme ,i4 ,scheme ,i a b g l q v ,b r#a #f #a #a #a #a #a #a r#b #a #a r#g #a #a r#ab #a #a r#ag #a #a r#bb #a #a r#bg #a #a a is thus responded to by r#a 7that is1 connected with r#a7 each time1 or six in all1 but only once each with b1 g1 l1 q1 v1 and ,b4 b1 g1 l1 q1 v1 and ,b are connected once each with r#a and once respectively with r#b1 r#g1 r#ab1 etc4 ,the bond from a to r#a1 has had six times as much exercise as the bond from a to r#b1 or from a to r#g1 etc4 ,in any new gross situation1 a #j1 a will be more predominant in determining response than it would otherwise have been2 and r#a will be more likely to be made than r#b1 r#g1 r#ab1 etc41 the other previous associates in the response to a situation containing a4 ,that is1 the bond from the element a to the response r#a has been notably strengthened4 ,case ,,ii4 ,contrasting ,concomitants ,now suppose that b and g are very dissimilar elements 7e4g41 white and black71 that l and q are very dissimilar 7e4g41 long and short71 and that v and ,b are also very dissimilar4 ,to be very dissimilar means to be responded to very differently1 so that r#g1 the response to g1 will be very unlike r#b1 the response to b4 ,so r#g may be thought of as rnot #b or r-#b4 ,in the same way r#ab may be thought of as rnot #ab or r-#ab1 and r#bg may be called rnot #bb or r-#bb4 ,then1 if the situations a b1 a g1 a l1 a q1 a v1 and a ,b are responded to1 each once1 we have3-- a b responded to by r#a r#b1 a g r#a rnot #b1 a l r#a r#ab1 a q r#a rnot #ab1 a v r#a r#bb1 and a ,b r#a rnot #bb1 as shown in ,scheme ,,ii4 ,scheme ,,ii a b g l q v ,b 7opp4 of b7 7opp4 of l7 7opp4 of v7 r#a #f #a #a #a #a #a #a rnot #a r#b #a #a rnot #b #a #a r#ab #a #a rnot #ab #a #a r#bb #a #a rnot #bb #a #a r#a is connected to a by #f repetitions4 r#b and rnot #b are each connected to a by #a repetition1 but since they interfere1 canceling each other so to speak1 the net result is for a to have zero tendency to call up r#b or rnot #b4 r#ab and rnot #ab are each connected to a by #a repetition1 but they interfere with or cancel each other with the net result that a has zero tendency to call up r#ab or rnot #ab4 ,so with r#bb and rnot #bb4 ,here then the net result of the six connections of a b1 a g1 a l1 a q1 a v1 and a ,b is to connect a with r1 and with nothing else4 ,case ,,iii4 ,contrasting ,concomitants and ,contrasting ,element ,suppose now that the facts are as in ,case ,,ii1 but with the addition of six experiences where a certain element which is the opposite of1 or very dissimilar to1 a is connected with the response rnot #a1 or r-#a which is opposite to1 or very dissimilar to r#a4 ,call this opposite of a1 - a4 ,that is1 we have not only a b responded to by r#a r#b1 a g r#a rnot #b1 a l r#a r#ab1 a q r#a rnot #ab1 a v r#a r#bb1 and a ,b r#a rnot #bb1 but also - a b responded to by rnot #a r#b1 - a g rnot #a rnot #b1 - a l rnot #a r#ab1 - a q rnot #a rnot #ab1 - a v rnot #a r#bb1 and - a ,b rnot #a rnot #bb1 as shown in ,scheme ,,iii4 ,scheme ,,iii a opp4 b g l q v ,b of a 7opp4 of b7 7opp4 of l7 7opp4 of v7 r#a #f #a #a #a #a #a #a rnot #a #f #a #a #a #a #a #a r#b #a #a #b rnot #b #a #a #b r#ab #a #a #b rnot #ab #a #a #b r#bb #a #a #b rnot #bb #a #a #b ,in this series of twelve experiences a connects with r#a six times and the opposite of a connects with rnot #a six times4 a connects equally often with three pairs of mutual destructives r#b and rnot #b1 r#ab and rnot #ab1 r#bb and rnot #bb1 and so has zero tendency to call them up4 - a has also zero tendency to call up any of these responses except its opposite1 rnot #a4 b1 g1 l1 q1 v1 and ,b are made to connect equally often with r#a and rnot #a4 ,so1 of these elements1 a is the only one left with a tendency to call up r#a4 ,thus1 by the mere action of frequency of connection1 r#a is connected with a2 the bonds from a to anything except r#a are being counteracted1 and the slight bonds from anything except a to r#a are being counteracted4 ,the element a becomes predominant in situations containing it2 and its bond toward r#a becomes relatively enormously strengthened and freed from competition4 ,these three processes occur in a similar1 but more complicated1 form if the situations a b1 a g1 etc41 are replaced by a b c d e f1 a g h i j k1 etc41 and the responses r#a r#b1 r#a r#g1 r#a r#ab1 etc41 are replaced by r#a r#b r#c r#d r#e r#f1 r#a r#g r#h r#i r#aj r#aa1 etc4--provided the r#a1 r#b1 r#c1 r#d1 etc41 can be made singly4 ,in so far as any one of the responses is necessarily co-active with any one of the others 7so that1 for example1 r#ac always brings r#bf with it and vice versa71 the exact relations of the numbers recorded in schemes like schemes ,i1 ,,ii1 and ,,iii on pages #agb to #agd will change2 but1 unless r#a has such an inevitable co-actor1 the general results of schemes ,i1 ,,ii1 and ,,iii will hold good4 ,if r#a does have such an inseparable co-actor1 say r#b1 then1 of course1 a can never acquire bonds with r#a alone1 but everywhere that r#a or r#b appears in the preceding schemes the other element must appear also4 r#a r#b would then have to be used as a unit in analysis4 ,the 'a b1' 'a g1' 'a l1' 444 'a ,b' situations may occur unequal numbers of times1 altering the exact numerical relations of the connections formed and presented in schemes ,i1 ,,ii1 and ,,iii2 but the process in general remains the same4 ,so much for the effect of use and disuse in attaching appropriate response elements to certain subtle elements of situations4 ,there are three main series of effects of satisfaction and discomfort4 ,they serve1 first1 to emphasize1 from the start1 the desired bonds leading to the responses r#a r#b1 r#a r#g1 etc41 to the total situations1 and to weed out the undesirable ones4 ,they also act to emphasize1 in such comparisons and contrasts as have been described1 every action of the bond from a to r#a2 and to eliminate every tendency of a to connect with aught save r#a1 and of aught save a to connect with r#a4 ,their third service is to strengthen the bonds produced of appropriate responses to a wherever it occurs1 whether or not any formal comparisons and contrasts take place4 ,the process of learning to respond to the difference of pitch in tones from whatever instrument1 to the 'square-root-ness' of whatever number1 to triangularity in whatever size or combination of lines1 to equality of whatever pairs1 or to honesty in whatever person or instance1 is thus a consequence of associative learning1 requiring no other forces than those of use1 disuse1 satisfaction1 and discomfort4 ,what happens in such cases is that the response1 by being connected with many situations alike in the presence of the element in question and different in other respects1 is bound firmly to that element and loosely to each of its concomitants4 ,conversely any element is bound firmly to any one response that is made to all situations containing it and very1 very loosely to each of those responses that are made to only a few of the situations containing it4 ,the element of triangularity1 for example1 is bound firmly to the response of saying or thinking 'triangle' but only very loosely to the response of saying or thinking white1 red1 blue1 large1 small1 iron1 steel1 wood1 paper1 and the like4 ,a situation thus acquires bonds not only with some response to it as a gross total1 but also with responses to any of its elements that have appeared in any other gross totals4 ,appropriate response to an element regardless of its concomitants is a necessary consequence of the laws of exercise and effect if an animal learns to make that response to the gross total situations that contain the element and not to make it to those that do not4 ,such prepotent determination of the response by one or another element of the situation is no transcendental mystery1 but1 given the circumstances1 a general rule of all learning4 ,such are at bottom only extreme cases of the same learning as a cat exhibits that depresses a platform in a certain box whether it faces north or south1 whether the temperature is #ej or #hj degrees1 whether one or two persons are in sight1 whether she is exceedingly or moderately hungry1 whether fish or milk is outside the box4 ,all learning is analytic1 representing the activity of elements within a total situation4 ,in man1 by virtue of certain instincts and the course of his training1 very subtle elements of situations can so operate4 99 99 99 99 99 ,learning by analysis does not often proceed in the carefully organized way represented by the most ingenious marshaling of comparing and contrasting activities4 ,the associations with gross totals1 whereby in the end an element is elevated to independent power to determine response1 may come in a haphazard order over a long interval of time4 ,thus a gifted three-year-old boy will have the response element of 'saying or thinking two1' bound to the 'two-ness' element of very many situations in connection with the 'how-many' mental set2 and he will have made this analysis without any formal1 systematic training4 ,an imperfect and inadequate analysis already made is indeed usually the starting point for whatever systematic abstraction the schools direct4 ,thus the kindergarten exercises in analyzing out number1 color1 size1 and shape commonly assume that 'one-ness' versus 'more-than-one-ness1' black and white1 big and little1 round and not round are1 at least vaguely1 active as elements responded to in some independence of their contexts4 ,moreover1 the tests of actual trial and success in further undirected exercises usually cooeperate to confirm and extend and refine what the systematic drills have given4 ,thus the ordinary child in school is left1 by the drills on decimal notation1 with only imperfect power of response to the 'place-values4' ,he continues to learn to respond properly to them by finding that #d x #dj #afj1 #d x #djj #afjj1 #hjj - #hj #gbj1 #hjj - #h #gib1 #hjj-hjj #j1 #db x #dh #bjaf1 #bd x #dh #aaeb1 and the like1 are satisfying2 while #d x #dj #af1 #bc x #dh #hcb1 #hjj - #h #j1 and the like1 are not4 ,the process of analysis is the same in such casual1 unsystematized formation of connections with elements as in the deliberately managed1 piecemeal inspection1 comparison1 and contrast described above4 ,,systematic ,,and ,,opportunistic ,,stimuli ,,to ,,analysis ,the arrangement of a pupil's experiences so as to direct his attention to an element1 vary its concomitants instructively1 stimulate comparison1 and throw the element into relief by contrast may be by fixed1 formal1 systematic exercises4 ,or it may be by much less formal exercises1 spread over a longer time1 and done more or less incidentally in other connections4 ,we may call these two extremes the 'systematic' and 'opportunistic1' since the chief feature of the former is that it systematically provides experiences designed to build up the power of correct response to the element1 whereas the chief feature of the latter is that it uses especially such opportunities as occur by reason of the pupil's activities and interests4 ,each method has its advantages and disadvantages4 ,the systematic method chooses experiences that are specially designed to stimulate the analysis2 it provides these at a certain fixed time so that they may work together2 it can then and there test the pupils to ascertain whether they really have the power to respond to the element or aspect or feature in question4 ,its disadvantages are1 first1 that many of the pupils will feel no need for and attach no interest or motive to these formal exercises2 second1 that some of the pupils may memorize the answers as a verbal task instead of acquiring insight into the facts2 third1 that the ability to respond to the element may remain restricted to the special cases devised for the systematic training1 and not be available for the genuine uses of arithmetic4 ,the opportunistic method is strong just where the systematic is weak4 ,since it seizes upon opportunities created by the pupil's abilities and interests1 it has the attitude of interest more often4 ,since it builds up the experiences less formally and over a wider space of time1 the pupils are less likely to learn verbal answers4 ,since its material comes more from the genuine uses of life1 the power acquired is more likely to be applicable to life4 ,its disadvantage is that it is harder to manage4 ,more thought and experimentation are required to find the best experiences2 greater care is required to keep track of the development of an abstraction which is taught not in two days1 but over two months2 and one may forget to test the pupils at the end4 ,in so far as the textbook and teacher are able to overcome these disadvantages by ingenuity and care1 the opportunistic method is better4 ,,adaptations ,,to ,,elementary ,,school ,,pupils ,we may expect much improvement in the formation of abstract and general ideas in arithmetic from the application of three principles in addition to those already described4 ,they are3 7#a7 ,provide enough actual experiences before asking the pupil to understand and use an abstract or general idea4 7#b7 ,develop such ideas gradually1 not attempting to give complete and perfect ideas all at once4 7#c7 ,develop such ideas so far as possible from experiences which will be valuable to the pupil in and of themselves1 quite apart from their merit as aids in developing the abstraction or general notion4 ,consider these three principles in order4 ,children1 especially the less gifted intellectually1 need more experiences as a basis for and as applications of an arithmetical abstraction or concept than are usually given them4 ,for example1 in paving the way for the principle1 ,any number times #j equals #j1 it is not safe to say1 ,john worked #h days for #j minutes per day4 ,how many minutes did he work8 and ,how much is #j times #d cents8 ,it will be much better to spend ten or fifteen minutes as follows3#ad ,what does zero mean8 7,not any4 ,no47 ,how many feet are there in eight yards8 ,in #e yards8 ,in #c yards8 ,in #b yards8 ,in #a yard8 ,in #j yard8 ,how many inches are there in #d ft48 ,in #b ft48 ,in #j ft48 #g pk4 4444 qt4 #e pk4 4444 qt4 #j pk4 4444 qt4 ,a boy receives #fj cents an hour when he works4 ,how much does he receive when he works #c hr48 #h hr48 #f hr48 #j hr48 ,a boy received #fj cents a day for #j days4 ,how much did he receive8 ,how much is #j times #fjj8 ,how much is #j times #ejjj8 ,how much is #j times a million dollars8 #j times any number equals4444 #bcb 7,at the blackboard47 #j time #bcb equals what8 #cj ,i write #j under the #j4ae #c times #bcb equals what8 ---- #fifj ,continue at the blackboard with #gcd #cba #cab #da #bj #dj #cj #fj etc4 --- --- --- -- #ad ,the more gifted children may be put to work using the principle after the first minute or two4 #ae #bcb #cj ,if desired this form may be used1 with the appropriate --- difference in the form of the questions and statements4 #jjj #fif ---- #fifj ,pupils in the elementary school1 except the most gifted1 should not be expected to gain mastery over such concepts as common fraction1 decimal fraction1 factor1 and root quickly4 ,they can learn a definition quickly and learn to use it in very easy cases1 where even a vague and imperfect understanding of it will guide response correctly4 ,but complete and exact understanding commonly requires them to take1 not one intellectual step1 but many2 and mastery in use commonly comes only as a slow growth4 ,for example1 suppose that pupils are taught that 4#a1 4#b1 4#c1 etc41 mean #a_/aj1 #b_/aj1 #c_/aj1 etc41 that 4#ja1 4#jb1 4#jc1 etc41 mean #a_/ajj1 #b_/ajj1 #c_/ajj1 etc41 that 4#jja1 4#jjb1 4#jjc1 etc41 mean #a_/ajjj1 #b_/ajjj1 #c_/ajjj1 etc41 and that 4#a1 4#jb1 4#jja1 etc41 are decimal fractions4 ,they may then respond correctly when asked to write a decimal fraction1 or to state which of these1--#a_/d1 4#d1 #c_/h1 4#jg1 4#jjb1 #e_/f1--;are common fractions and which are decimal fractions4 ,they may be able1 though by no means all of them will be1 to write decimal fractions which equal #a_/b and #a_/e1 and the common fractions which equal 4#a and 4#ji4 ,most of them will not1 however1 be able to respond correctly to ,write a decimal mixed number2 or to state which of these1--#a_/ajj1 4#d-a_/b1 4#jjg_/cej1 4#be1--;are common fractions1 and which are decimals2 or to write the decimal fractions which equal #c_/d and #a_/c4 ,if now the teacher had given all at once the additional experiences needed to provide the ability to handle these more intricate and subtle features of decimal-fraction-ness1 the result would have been confusion for most pupils4 ,the general meaning of 4#cb1 4#ad1 4#ii1 and the like requires some understanding of 4#cj1 4#aj1 4#ij1 and 4#jb1 4#jd1 4#jh2 but it is not desirable to disturb the child with 4#cj while he is trying to master #b4c1 #d4c1 #f4c1 and the like4 ,decimals in general require connection with place value and the contrasts of 4#da with #da1 #daj1 #d4a1 and the like1 but if the relation to place values in general is taught in the same lesson with the relation to _/#ajs1 _/#ajjs1 _/#ajjjs1 the mind will suffer from violent indigestion4 ,a wise pedagogy in fact will break up the process of learning the meaning and use of decimal fractions into many teaching units1 for example1 as follows3-- 7#a7 ,such familiarity with fractions with large denominators as is desirable for pupils to have1 as by an exercise in reducing to lowest terms1 #h_/aj1 #cf_/fd1 #bj_/be1 #ah_/bd1 #bd_/cb1 #ba_/cj1 #be_/ajj1 #dj_/ajj1 and the like4 ,this is good as a review of cancellation1 and as an extension of the idea of a fraction4 7#b7 ,objective work1 showing #a_/aj sq4 ft41 #a_/ej sq4 ft41 #a_/ajj sq4 ft41 and #a_/ajjj sq4 ft41 and having these identified and the forms #a_/aj sq4 ft41 #a_/ajj sq4 ft41 and #a_/ajjj sq4 ft4 learned4 ,finding how many feet #a_/aj mile and #a_/ajj mile4 7#c7 ,familiarity with _/#ajjs and _/#ajjjs by reductions of #gej_/ajjj1 #ej_/ajj1 etc41 to lowest terms and by writing the missing numerators in #ejj_/ajjj _/#ajj _/#aj and the like1 and by finding #a_/aj1 #a_/ajj1 and #a_/ajjj of #cjjj1 #fjjj1 #ijjj1 etc4 7#d7 ,writing #a_/aj as 4#a and #a_/ajj as 4#ja1 #aa_/ajj1 #ab_/ajj1 #ac_/ajj1 etc41 as 4#aa1 4#ab1 4#ac4 ,united ,states money is used as the introduction4 ,application is made to miles4 7#e7 ,mixed numbers with a first decimal place4 ,the cyclometer or speedometer4 ,adding numbers like #i4a1 #ad4g1 #aa4d1 etc4 7#f7 ,place value in general from thousands to hundredths4 7#g7 ,review of 7#a7 to 7#f74 7#h7 ,tenths and hundredths of a mile1 subtraction when both numbers extend to hundredths1 using a railroad table of distances4 7#i7 ,thousandths4 ,the names 'decimal fractions or decimals1' and 'decimal mixed numbers or decimals4' ,drill in reading any number to thousandths4 ,the work will continue with gradual extension and refinement of the understanding of decimals by learning how to operate with them in various ways4 ,such may seem a slow progress1 but in fact it is not1 and many of these exercises whereby the pupil acquires his mastery of decimals are useful as organizations and applications of other arithmetical facts4 ,that1 it will be remembered1 was the third principle3--,develop abstract and general ideas by experiences which will be intrinsically valuable4 ,the reason is that1 even with the best of teaching1 some pupils will not1 within any reasonable limits of time expended1 acquire ideas that are fully complete1 rigorous when they should be1 flexible when they should be1 and absolutely exact4 ,many children 7and adults1 for that matter7 could not within any reasonable limits of time be so taught the nature of a fraction that they could decide unerringly in original exercises like3-- ,is #b4ge_/be a common fraction8 ,is 4#be a decimal fraction8 ,is one xth of y a fraction8 ,can the same words mean both a common fraction and a decimal fraction8 ,express #a as a common fraction4 ,express #a as a decimal fraction4 ,these same children can1 however1 be taught to operate correctly with fractions in the ordinary uses thereof4 ,and that is the chief value of arithmetic to them4 ,they should not be deprived of it because they cannot master its subtler principles4 ,so we seek to provide experiences that will teach all pupils something of value1 while stimulating in those who have the ability the growth of abstract ideas and general principles4 ,finally1 we should bear in mind that working with qualities and relations that are only partly understood or even misunderstood does under certain conditions give control over them4 ,the general process of analytic learning in life is to respond as well as one can2 to get a clearer idea thereby2 to respond better the next time2 and so on4 ,for instance1 one gets some sort of notion of what #a_/e means2 he then answers such questions as #a_/e of #aj 8 #a_/e of #e 8 #a_/e of #bj 82 by being told when he is right and when he is wrong1 he gets from these experiences a better idea of #a_/e2 again he does his best with #a_/e _/#aj1 #a_/e _/#ae1 etc41 and as before refines and enlarges his concept of #a_/e4 ,he adds #a_/e to #b_/e1 etc41 #a_/e to #c_/aj1 etc41 #a_/e to #a_/b1 etc41 and thereby gains still further1 and so on4 ,what begins as a blind habit of manipulation started by imitation may thus grow into the power of correct response to the essential element4 ,the pupil who has at the start no notion at all of 'multiplying' may learn what multiplying is by his experience that '#d #f multiplying gives #bd'2 '#c #i multiplying gives #bg1' etc4 ,if the pupil keeps on doing something with numbers and differentiates right results1 he will often reach in the end the abstractions which he is supposed to need in the beginning4 ,it may even be the case with some of the abstractions required in arithmetic that elaborate provision for comprehension beforehand is not so efficient as the same amount of energy devoted partly to provision for analysis itself beforehand and partly to practice in response to the element in question without full comprehension4 ,it certainly is not the best psychology and not the best educational theory to think that the pupil first masters a principle and then merely applies it--first does some thinking and then computes by mere routine4 ,on the contrary1 the applications should help to establish1 extend1 and refine the principle--the work a pupil does with numbers should be a main means of increasing his understanding of the principles of arithmetic as a science4 ,,chapter ,x ,,the ,,psychology ,,of ,,thinking3 ,,reasoning ,,in ,,arithmetic ,,the ,,essentials ,,of ,,arithmetical ,,reasoning ,we distinguish aimless reverie1 as when a child dreams of a vacation trip1 from purposive thinking1 as when he tries to work out the answer to ,how many weeks of vacation can a family have for #abj if the cost is #bb a week for board1 #b4be a week for laundry1 and #a4ge a week for incidental expenses1 and if the railroad fares for the round trip are #ab8 ,we distinguish the process of response to familiar situations1 such as five integral numbers to be added1 from the process of response to novel situations1 such as 7for a child who has not been trained with similar problems73--,a man has four pieces of wire4 ,the lengths are #abj yd41 #acb meters1 #afj feet1 and #a_/h mile4 ,how much more does he need to have #ajjj yd4 in all8 ,we distinguish 'thinking things together1' as when a diagram or problem or proof is understood1 from thinking of one thing after another as when a number of words are spelled or a poem in an unknown tongue is learned4 ,in proportion as thinking is purposive1 with selection from the ideas that come up1 and in proportion as it deals with novel problems for which no ready-made habitual response is available1 and in proportion as many bonds act together in an organized way to produce response1 we call it reasoning4 ,when the conclusion is reached as the effect of many particular experiences1 the reasoning is called inductive4 ,when some principle already established leads to another principle or to a conclusion about some particular fact1 the reasoning is called deductive4 ,in both cases the process involves the analysis of facts into their elements1 the selection of the elements that are deemed significant for the question at hand1 the attachment of a certain amount of importance or weight to each of them1 and their use in the right relations4 ,thought may fail because it has not suitable facts1 or does not select from them the right ones1 or does not attach the right amount of weight to each1 or does not put them together properly4 ,in the world at large1 many of our failures in thinking are due to not having suitable facts4 ,some of my readers1 for example1 cannot solve the problem--,what are the chances that in drawing a card from an ordinary pack of playing-cards four times in succession1 the same card will be drawn each time8 ,and it will be probably because they do not know certain facts about the theory of probabilities4 ,the good thinkers among such would look the matter up in a suitable book4 ,similarly1 if a person did not happen to know that there were fifty-two cards in all and that no two were alike1 he could not reason out the answer1 no matter what his mastery of the theory of probabilities4 ,if a competent thinker1 he would first ask about the size and nature of the pack4 ,in the actual practice of reasoning1 that is1 we have to survey our facts to see if we lack any that are necessary4 ,if we do1 the first task of reasoning is to acquire those facts4 ,this is specially true of the reasoning about arithmetical facts in life4 ,will #c-a_/b yards of this be enough for a dress8 ,reason directs you to learn how wide it is1 what style of dress you intend to make of it1 how much material that style normally calls for1 whether you are a careful or a wasteful cutter1 and how big the person is for whom the dress is to be made4 ,how much cheaper as a diet is bread alone1 than bread with butter added to the extent of #aj of the weight of the bread8 ,reason directs you to learn the cost of bread1 the cost of butter1 the nutritive value of bread1 and the nutritive value of butter4 ,in the arithmetic of the school this feature of reasoning appears in cases where some fact about common measures must be brought to bear1 or some table of prices or discounts must be consulted1 or some business custom must be remembered or looked up4 ,thus ,how many badges1 each #i inches long1 can be made from #b-a_/b yd4 ribbon8 cannot be solved without getting into mind #a yd4 #cf inches4 ,at ,jones' prices1 which costs more1 #c-c_/d lb4 butter or #f-a_/b lb4 lard8 ,how much more8 is a problem which directs the thinker to ascertain ,jones' prices4 ,it may be noted that such problems are1 other things being equal1 somewhat better training in thinking than problems where all the data are given in the problem itself 7e4g41 ,which costs more1 #c-c_/d lb4 butter at #dh;c per lb4 or #f-a_/b lb4 lard at #bg;c per lb48 ,how much more874 ,at least it is unwise to have so many problems of the latter sort that the pupil may come to think of a problem in applied arithmetic as a problem where everything is given and he has only to manipulate the data4 ,life does not present its problems so4 ,the process of selecting the right elements and attaching proper weight to them may be illustrated by the following problem3--,which of these offers would you take1 supposing that you wish a ,d4,c4,k4 upright piano1 have #ej saved1 can save a little over #bj per month1 and can borrow from your father at #f interest8 ,a ,a ,reliable ,piano4 ,the ,famous ,d4,c4,k4 ,upright4 ,you pay #ej cash down and #ba a month for only a year and a half4 ,no interest to pay4 ,we ask you to pay only for the piano and allow you plenty of time4 ,b ,we offer the well-known ,d4,c4,k4 ,piano for #cij4 #ej cash and #bj a month thereafter4 ,regular interest at #f4 ,the interest soon is reduced to less than #a a month4 ,c ,the ,d4,c4,k4 ,piano4 ,special ,offer1 #cge1 cash4 ,compare our prices with those of any reliable firm4 ,if you consider chiefly the only1 ,no interest to pay1 only1 and plenty of time in offer ,a1 attaching much weight to them and little to the thought1 ,how much will #ej plus 7#ah x #ba7 be81 you will probably decide wrongly4 ,the situations of life are often complicated by many elements of little or even of no relevance to the correct solution4 ,the offerer of ,a may belong to your church2 your dearest friend may urge you to accept offer ,b2 you may dislike to talk with the dealer who makes offer ,c2 you may have a prejudice against owing money to a relative2 that prejudice may be wise or foolish2 you may have a suspicion that the ,b piano is shopworn2 that suspicion may be well-founded or groundless2 the salesman for ,c says1 ,you don't want your friends to say that you bought on the installment plan4 ,only low-class persons do that1 etc4 ,the statement of arithmetical problems in school usually assists the pupil to the extent of ruling out all save definitely quantitative elements1 and of ruling out all quantitative elements except those which should be considered4 ,the first of the two simplifications is very beneficial1 on the whole1 since otherwise there might be different correct solutions to a problem according to the nature and circumstances of the persons involved4 ,the second simplification is often desirable1 since it will often produce greater improvement in the pupils1 per hour of time spent1 than would be produced by the problems requiring more selection4 ,it should not1 however1 be a universal custom2 for in that case the pupils are tempted to think that in every problem they must use all the quantities given1 as one must use all the pieces in a puzzle picture4 ,it is obvious that the elements selected must not only be right but also be in the right relations to one another4 ,for example1 in the problems below1 the #f must be thought of in relation to a dozen and as being half of a dozen1 and also as being #f times #a4 #a must be mentally tied to each4 ,the #f as half of a dozen must be related to the #a4jj1 #a4fj1 etc4 ,the #f as #f times #a must be related to the 4#ji1 4#ad1 etc4 ,buying in ,quantity ,these are a grocer's prices for certain things by the dozen and for a single one4 ,he sells a half dozen at half the price of a dozen4 ,find out how much you save by buying #f all at one time instead of buying them one at a time4 ,doz4 ,each #a4 ,evaporated ,milk #a4jj 4#ji #b4 ,puffed ,rice #a4fj 4#ad #c4 ,puffed ,wheat #a4aj 4#aj #d4 ,canned ,soup #a4ij 4#ag #e4 ,sardines #a4hj 4#af #f4 ,beans 7,no4 #b cans7 #a4ej 4#ac #g4 ,pork and ,beans #a4gj 4#ae #h4 ,peas 7,no4 #b cans7 #a4dj 4#ab #i4 ,tomatoes 7extra cans7 #c4bj 4#bh #aj4 ,ripe olives 7qt4 cans7 #g4bj 4#fe ,it is obvious also that in such arithmetical work as we have been describing1 the pupil1 to be successful1 must 'think things together4' ,many bonds must cooeperate to determine his final response4 ,as a preface to reasoning about a problem we often have the discovery of the problem and the classification of just what it is1 and as a postscript we have the critical inspection of the answer obtained to make sure that it is verified by experiment or is consistent with known facts4 ,during the process of searching for1 selecting1 and weighting facts1 there may be similar inspection and validation1 item by item4 ,,reasoning ,,as ,,the ,c,o,oe,p,e,r,a,t,i,o,n ,,of ,,organized ,,habits ,the pedagogy of the past made two notable errors in practice based on two errors about the psychology of reasoning4 ,it considered reasoning as a somewhat magical power or essence which acted to counteract and overrule the ordinary laws of habit in man2 and it separated too sharply the 'understanding of principles' by reasoning from the 'mechanical' work of computation1 reading problems1 remembering facts and the like1 done by 'mere' habit and memory4 ,reasoning or selective1 inferential thinking is not at all opposed to1 or independent of1 the laws of habit1 but really is their necessary result under the conditions imposed by man's nature and training4 ,a closer examination of selective thinking will show that no principles beyond the laws of readiness1 exercise1 and effect are needed to explain it2 that it is only an extreme case of what goes on in associative learning as described under the 'piecemeal' activity of situations2 and that attributing certain features of learning to mysterious faculties of abstraction or reasoning gives no real help toward understanding or controlling them4 ,it is true that man's behavior in meeting novel problems goes beyond1 or even against1 the habits represented by bonds leading from gross total situations and customarily abstracted elements thereof4 ,one of the two reasons therefor1 however1 is simply that the finer1 subtle1 preferential bonds with subtler and less often abstracted elements go beyond1 and at times against1 the grosser and more usual bonds4 ,one set is as much due to exercise and effect as the other4 ,the other reason is that in meeting novel problems the mental set or attitude is likely to be one which rejects one after another response as their unfitness to satisfy a certain desideratum appears4 ,what remains as the apparent course of thought includes only a few of the many bonds which did operate1 but which1 for the most part1 were unsatisfying to the ruling attitude or adjustment4 ,successful responses to novel data1 associations by similarity and purposive behavior are in only apparent opposition to the fundamental laws of associative learning4 ,really they are beautiful examples of it4 ,man's successful responses to novel data--as when he argues that the diagonal on a right triangle of #gif4bgh mm4 base and #acg4bid mm4 altitude will be #hjh4jbb mm41 or that ,mary ,jones1 born this morning1 will sometime die--are due to habits1 notably the habits of response to certain elements or features1 under the laws of piecemeal activity and assimilation4 ,nothing is less like the mysterious operations of a faculty of reasoning transcending the laws of connection-forming1 than the behavior of men in response to novel situations4 ,let children who have hitherto confronted only such arithmetical tasks1 in addition and subtraction with one- and two-place numbers and multiplication with one-place numbers1 as those exemplified in the first line below1 be told to do the examples shown in the second line4 ,,add ,,add ,,add ,,subt4 ,,subt4 ,,multiply ,,multiply ,,multiply #h #cg #ce #h #cg #h #i #f #e #bd #fh #e #bd #e #g #c -- -- #bc -- -- -- -- -- #ai -- ,,multiply ,,multiply ,,multiply #cb #dc #cd #bc #bb #bf -- -- -- ,they will add the numbers1 or subtract the lower from the upper number1 or multiply #c x #b and #b x #c1 etc41 getting #ff1 #hf1 and #fbd1 or respond to the element of ',multiply' attached to the two-place numbers by ,i can't or ,i don't know what to do1 or the like2 or1 if one is a child of great ability1 he may consider the ',multiply' element and the bigness of the numbers1 be reminded by these two aspects of the situation of the fact that '#i #i multiply' -- gave only #ha1 and that '#aj #aj multiply' -- gave only #ajj1 or the like2 and so may report an intelligent and justified ,i can't1 or reject the plan of #c x #b and #b x #c1 with #ff1 #hf1 and #fbd for answers1 as unsatisfactory4 ,what the children will do will1 in every case1 be a product of the elements in the situation that are potent with them1 the responses which these evoke1 and the further associates which these responses in turn evoke4 ,if the child were one of sufficient genius1 he might infer the procedure to be followed as a result of his knowledge of the principles of decimal notation and the meaning of ',multiply1' responding correctly to the 'place-value' element of each digit and adding his #f tens and #i tens1 #bj twos and #c thirties2 but if he did thus invent the shorthand addition of a collection of twenty-three collections1 each of #cb units1 he would still do it by the operation of bonds1 subtle but real4 ,association by similarity is1 as ,james showed long ago1 simply the tendency of an element to provoke the responses which have been bound to it4 abcde leads to vwxyz because a has been bound to vwxyz by original nature1 exercise1 or effect4 ,purposive behavior is the most important case of the influence of the attitude or set or adjustment of an organism in determining 7#a7 which bonds shall act1 and 7#b7 which results shall satisfy4 ,james early described the former fact1 showing that the mechanism of habit can give the directedness or purposefulness in thought's products1 provided that mechanism includes something paralleling the problem1 the aim1 or need1 in question4 ,the second fact1 that the set or attitude of the man helps to determine which bonds shall satisfy1 and which shall annoy1 has commonly been somewhat obscured by vague assertions that the selection and retention is of what is in point1 or is the right one1 or is appropriate1 or the like4 ,it is thus asserted1 or at least hinted1 that the will1 the voluntary attention1 the consciousness of the problem1 and other such entities are endowed with magic power to decide what is the right or useful bond and to kill off the others4 ,the facts are that in purposive thinking and action1 as everywhere else1 bonds are selected and retained by the satisfyingness1 and are killed off by the discomfort1 which they produce2 and that the potency of the man's set or attitude to make this satisfy and that annoy--to put certain conduction-units in readiness to act and others in unreadiness--is in every way as important as its potency to set certain conduction-units in actual operation4 ,reasoning is not a radically different sort of force operating against habit but the organization and cooeperation of many habits1 thinking facts together4 ,reasoning is not the negation of ordinary bonds1 but the action of many of them1 especially of bonds with subtle elements of the situation4 ,some outside power does not enter to select and criticize2 the pupil's own total repertory of bonds relevant to the problem is what selects and rejects4 ,an unsuitable idea is not killed off by some actus purus of intellect1 but by the ideas which it itself calls up1 in connection with the total set of mind of the pupil1 and which show it to be inadequate4 ,almost nothing in arithmetic need be taught as a matter of mere unreasoning habit or memory1 nor need anything1 first taught as a principle1 ever become a matter of mere habit or memory4 #e x #d #bj should not be learned as an isolated fact1 nor remembered as we remember that ,jones' telephone number is #fdh ,j #b4 ,almost everything in arithmetic should be taught as a habit that has connections with habits already acquired and will work in an organization with other habits to come4 ,the use of this organized hierarchy of habits to solve novel problems is reasoning4 ,,chapter ,,xi ,,original ,,tendencies ,,and ,,acquisitions ,,before ,,school ,,the ,,utilization ,,of ,,instinctive ,,interests ,the activities essential to acquiring ability in arithmetic can rely on little in man's instinctive equipment beyond the purely intellectual tendencies of curiosity and the satisfyingness of thought for thought's sake1 and the general enjoyment of success rather than failure in an enterprise to which one sets oneself4 ,it is only by a certain amount of artifice that we can enlist other vehement inborn interests of childhood in the service of arithmetical knowledge and skill4 ,when this can be done at no cost the gain is great4 ,for example1 marching in files of two1 in files of three1 in files of four1 etc41 raising the arms once1 two times1 three times1 showing a foot1 a yard1 an inch with the hands1 and the like are admirable because learning the meanings of numbers thus acquires some of the zest of the passion for physical action4 ,even in late grades chances to make pictures showing the relations of fractional parts1 to cut strips1 to fold paper1 and the like will be useful4 ,various social instincts can be utilized in matches after the pattern of the spelling match1 contests between rows1 certain number games1 and the like4 ,the scoring of both the play and the work of the classroom is a useful field for control by the teacher of arithmetic4 ,hunt '#ab has noted the more important games which have some considerable amount of arithmetical training as a by-product and which are more or less suitable for class use4 ,flynn '#ab has described games1 most of them for home use1 which give very definite arithmetical drill1 though in many cases the drills are rather behind the needs of children old enough to understand and like the game itself4 ,it is possible to utilize the interests in mystery1 tricks1 and puzzles so as to arouse a certain form of respect for arithmetic and also to get computational work done4 ,i quote one simple case from ,miss ,selkin's admirable collection '#ab1 p4 #fi f43-- ,i4 ,,addition ,we must admit that there is nothing particularly interesting in a long column of numbers to be added4 ,let the teacher1 however1 suggest that he can write the answer at sight1 and the task will assume a totally different aspect4 ,a very simple number trick of this kind can be performed by making use of the principle of complementary addition4 ,the arithmetical complement of a number with respect to a larger number is the difference between these two numbers4 ,most interesting results can be obtained by using complements with respect to #i4 ,the children may be called upon to suggest several numbers of two1 three1 or more digits4 ,below these write an equal number of addends and immediately announce the answer4 ,the children1 impressed by this apparently rapid addition1 will set to work enthusiastically to test the results of this lightning calculation4 ,example3-- #ceg #iii #fhb ,a x #c #gic ---- #biig #fdb #cag ,b #bjf ,explanation3--,the addends in group ,a are written down at random or suggested by the class4 ,those in group ,b are their complements4 ,to write the first number in group ,b we look at the first number in group ,a and1 starting at the left write #f1 the complement of #c with respect to #i2 #d1 the complement of #e2 #b1 the complement of #g4 ,the second and third addends in group ,b are derived in the same way4 ,since we have three addends in each group1 the problem reduces itself to multiplying #iii by #c1 or to taking #cjjj - #c4 ,any number of addends may be used and each addend may consist of any number of digits4 ,respect for arithmetic as a source of tricks and magic is very much less important than respect for its everyday services2 and computation to test such tricks is likely to be undertaken zealously only by the abler pupils4 ,consequently this source of interest should probably be used only sparingly1 and perhaps the teacher should give such exhibitions only as a reward for efficiency in the regular work4 ,for example1 if the work for a week is well done in four days the fifth day might be given up to some semi-arithmetical entertainment1 such as the demonstration of an adding machine1 the story of primitive methods of counting1 team races in computation1 an exhibition of lightning calculation and intellectual sleight-of-hand by the teacher1 or the voluntary study of arithmetical puzzles4 ,the interest in achievement1 in success1 mentioned above is stronger in children than is often realized and makes advisable the systematic use of the practice experiment as a method of teaching much of arithmetic4 ,children who thus compete with their own past records1 keeping an exact score from week to week1 make notable progress and enjoy hard work in making it4 ,,the ,,order ,,of ,,development ,,of ,,original ,,tendencies ,negatively the difficulty of the work that pupils should be expected to do is conditioned by the gradual maturing of their capacities4 ,other things being equal1 the common custom of reserving hard things for late in the elementary school course is1 of course1 sound4 ,it seems probable that little is gained by using any of the child's time for arithmetic before grade #b1 though there are many arithmetical facts that he can learn in grade #a4 ,postponement of systematic work in arithmetic to grade #c or even grade #d is allowable if better things are offered4 ,with proper textbooks and oral and written exercises1 however1 a child in grades #b and #c can spend time profitably on arithmetical work4 ,when all children can be held in school through the eighth grade it does not much matter whether arithmetic is begun early or late4 ,if1 however1 many children are to leave in grades #e and #f as now1 we may think it wise to provide somehow that certain minima of arithmetical ability be given them4 ,there are1 so far as is known1 no special times and seasons at which the human animal by inner growth is specially ripe for one or another section or aspect of arithmetic1 except in so far as the general inner growth of intellectual powers makes the more abstruse and complex tasks suitable to later and later years4 ,indeed1 very few of even the most enthusiastic devotees of the recapitulation theory or culture-epoch theory have attempted to apply either to the learning of arithmetic1 and ,branford is the only mathematician1 so far as ,i know1 who has advocated such application1 even tempered by elaborate shiftings and reversals of the racial order4 ,he says3-- ,thus1 for each age of the individual life--infancy1 childhood1 school1 college--may be selected from the racial history the most appropriate form in which mathematical experience can be assimilated4 ,thus the capacity of the infant and early childhood is comparable with the capacity of animal consciousness and primitive man4 ,the mathematics suitable to later childhood and boyhood 7and1 of course1 girlhood7 is comparable with ,archaean mathematics passing on through ,greek and ,hindu to mediaeval ,european mathematics2 while the student is become sufficiently mature to begin the assimilation of modern and highly abstract ,european thought4 ,the filling in of details must necessarily be left to the individual teacher1 and also1 within some such broadly marked limits1 the precise order of the marshalling of the material for each age4 ,for1 though1 on the whole1 mathematical development has gone forward1 yet there have been lapses from advances already made4 ,witness the practical world-loss of much valuable ,hindu thought1 and1 for long centuries1 the neglect of ,greek thought3 witness the world-loss of the invention by the ,babylonians of the ,zero1 until re-invented by the ,hindus1 passed on by them to the ,arabs1 and by these to ,europe4 ,moreover1 many blunders and false starts and false principles have marked the whole course of development4 ,in a phrase1 rivers have their backwaters4 ,but it is precisely the teacher's function to avoid such racial mistakes1 to take short cuts ultimately discovered1 and to guide the young along the road ultimately found most accessible with such halts and retracings--returns up side-cuts--as the mental peculiarities of the pupils demand4 ,all this1 the practical realization of the spirit of the principle1 is to be wisely left to the mathematical teacher1 familiar with the history of mathematical science and with the particular limitations of his pupils and himself4 '#jh1 p4 #bde4 ,the latitude of modification suggested by ,branford reduces the guidance to be derived from racial history to almost nil4 ,also it is apparent that the racial history in the case of arithmetical achievement is entirely a matter of acquisition and social transmission4 ,man's original nature is destitute of all arithmetical ideas4 ,the human germs do not know even that one and one make two6 ,,inventories ,,of ,,arithmetical ,,knowledge ,,and ,,skill ,a scientific plan for teaching arithmetic would begin with an exact inventory of the knowledge and skill which the pupils already possessed4 ,our ordinary notions of what a child knows at entrance to grade #a1 or grade #b1 or grade #c1 and of what a first-grade child or second-grade child can do1 are not adequate4 ,if they were1 we should not find reputable textbooks arranging to teach elaborately facts already sufficiently well known to over three quarters of the pupils when they enter school4 ,nor should we find other textbooks presupposing in their first fifty pages a knowledge of words which not half of the children can read even at the end of the #b ,b grade4 ,we do find just such evidence that ordinary ideas about the abilities of children at the beginning of systematic school training in arithmetic may be in gross error4 ,for example1 a reputable and in many ways admirable recent book has fourteen pages of exercises to teach the meaning of two and the fact that one and one make two6 ,as an example of the reverse error1 consider putting all these words in the first twenty-five pages of a beginner's book3--absentees1 attendance1 blanks1 continue1 copy1 during1 examples1 grouped1 memorize1 perfect1 similar1 splints1 therefore1 total6 ,little1 almost nothing1 has been done toward providing an exact inventory compared with what needs to be done4 ,we may note here 7#a7 the facts relevant to arithmetic found by ,stanley ,hall1 ,hartmann1 and others in their general investigations of the knowledge possessed by children at entrance to school1 7#b7 the facts concerning the power of children to perceive differences in length1 area1 size of collection1 and organization within a collection such as is shown in ,fig4 #bd1 and certain facts and theories about early awareness of number4 ,in the ,berlin inquiry of #ahfi1 knowledge of the meaning of two1 three1 and four appeared in #gd1 #gd1 and #gc percent of the children upon entrance to school4 ,some of those recorded as ignorant probably really knew1 but failed to understand that they were expected to reply or were shy4 ,only #he percent were recorded as knowing their fathers' names4 ,seven eighths as many children knew the meanings of two1 three1 and four as knew their fathers' names4 ,in a similar but more careful experiment with ,boston children in ,september1 #ahhj1 ,stanley ,hall found that #ib percent knew three1 #hc percent knew four1 and #ga-a_/b percent knew five4 ,three was known about as well as the color red2 four was known about as well as the color blue or yellow or green4 ,hartmann '#ij found that two thirds of the children entering school in ,annaberg could count from one to ten4 ,this is about as many as knew money1 or the familiar objects of the town1 or could repeat words spoken to them4 ,illustration3 ,,fig4 #bd4--,objective presentation4 ,in the ,stanford form of the ,binet tests counting four pennies is given as an ability of the typical four-year-old4 ,counting #ac pennies correctly in at least one out of two trials1 and knowing three of the four coins1--penny1 nickel1 dime1 and quarter1--are given as abilities of the typical six-year-old4 ,,the ,,perception ,,of ,,number ,,and ,,quantity ,we know that educated adults can tell how many lines or dots1 etc41 they see in a single glance 7with an exposure too short for the eye to move7 up to four or more1 according to the clearness of the objects and their grouping4 ,for example1 ,nanu '#jd reports that when a number of bright circles on a dark background are shown to educated adults for only 4#jcc second1 ten can be counted when arranged to form a parallelogram1 but only five when arranged in a row4 ,with certain groupings1 of course1 their 'perception' involves much inference1 even conscious addition and multiplication4 ,similarly they can tell1 up to twenty and beyond1 the number of taps1 notes1 or other sounds in a series too rapid for single counting if the sounds are grouped in a convenient rhythm4 ,these abilities are1 however1 the product of a long and elaborate learning1 including the learning of arithmetic itself4 ,elementary psychology and common experience teach us that the mere observation of groups or quantities1 no matter how clear their number quality appears to the person who already knows the meanings of numbers1 does not of itself create the knowledge of the meanings of numbers in one who does not4 ,the experiments of ,messenger '#jc and ,burnett '#jf showed that there is no direct intuitive apprehension even of two as distinct from one4 ,we have to learn to feel the two touches or see the two dots or lines as two4 ,we do not know by exact measurements the growth in children of this ability to count or infer the number of elements in a collection seen or series heard4 ,still less do we know what the growth would be without the influence of school training in counting1 grouping1 adding1 and multiplying4 ,many textbooks and teachers seem to overestimate it greatly4 ,not all educated adults can1 apart from measurement1 decide with surety which of these lines is the longer1 or which of these areas is the larger1 or whether this is a ninth or a tenth or an eleventh of a circle4 ,illustration ,children upon entering school have not been tested carefully in respect to judgments of length and area1 but we know from such studies as ,gilbert's '#id that the difference required in their case is probably over twice that required for children of #ac or #ad4 ,in judging weights1 for example1 a difference of #f is perceived as easily by children #ac to #ae years of age as a difference of #ae by six-year-olds4 ,a teacher who has adult powers of estimating length or area or weight and who also knows already which of the two is longer or larger or heavier1 may use two lines to illustrate a difference which they really hide from the child4 ,it is unlikely1 for example1 that the first of these lines would be recognized as shorter than the second by every child in a fourth-grade class1 and it is extremely unlikely that it would be recognized as being #g_/h of the length of the latter1 rather than #c_/d of it or #e_/f of it or #i_/aj of it or #aa_/ab of it4 ,if the two were shown to a second grade1 with the question1 ,the first line is #g4 ,how long is the other line8 there would be very many answers of #g or #i2 and these might be entirely correct arithmetically1 the pupils' errors being all due to their inability to compare the lengths accurately4 ,a ,b ,c ------- -------- ,d ,e 4'_/4 4'_/4 _/_/ _/_/ ----------- ----------- _/_/ _/_/ 4_/4' 4_/4' ,the quantities used should be such that their mere discrimination offers no difficulty even to a child of blunted sense powers4 ,if #g_/h and #a are to be compared1 ,a and ,b are not allowable4 ,c1 ,d1 and ,e are much better4 ,teachers probably often underestimate or neglect the sensory difficulties of the tasks they assign and of the material they use to illustrate absolute and relative magnitudes4 ,the result may be more pernicious when the pupils answer correctly than when they fail4 ,for their correct answering may be due to their divination of what the teacher wants2 and they may call a thing an inch larger to suit her which does not really seem larger to them at all4 ,this1 of course1 is utterly destructive of their respect for arithmetic as an exact and matter-of-fact instrument4 ,for example1 if a teacher drew a series of lines #bj1 #ba1 #bb1 #bc1 #bd1 and #be inches long on the blackboard in this form-- and asked1 ,this is #bj inches long1 how long is this8 she might1 after some errors and correction thereof1 finally secure successful response to all the lines by all the children4 ,but their appreciation of the numbers #bj1 #ba1 #bb1 #bc1 #bd1 and #be would be actually damaged by the exercise4 ,,the ,,early ,,awareness ,,of ,,number ,there has been some disagreement concerning the origin of awareness of number in the individual1 in particular concerning the relative importance of the perception of how-many-ness and that of how-much-ness1 of the perception of a defined aggregate and the perception of a defined ratio4 7,see ,mc,lellan and ,dewey '#ie1 ,phillips '#ig and '#ih1 and ,decroly and ,degand '#ab47 ,the chief facts of significance for practice seem to be these3 7#a7 ,children with rare exceptions hear the names one1 two1 three1 four1 half1 twice1 two times1 more1 less1 as many as1 again1 first1 second1 and third1 long before they have analyzed out the qualities and relations to which these words refer so as to feel them at all clearly4 7#b7 ,their knowledge of the qualities and relations is developed in the main in close association with the use of these words to the child and by the child4 7#c7 ,the ordinary experiences of the first five years so develop in the child awareness of the 'how many somethings' in various groups1 of the relative magnitudes of two groups or quantities of any sort1 and of groups and magnitudes as related to others in a series4 ,for instance1 if fairly gifted1 a child comes1 by the age of five1 to see that a row of four cakes is an aggregate of four1 seeing each cake as a part of the four and the four as the sum of its parts1 to know that two of them are as many as the other two1 that half of them would be two1 and to think1 when it is useful for him to do so1 of four as a step beyond three on the way to five1 or to think of hot as a step from warm on the way to very hot4 ,the degree of development of these abilities depends upon the activity of the law of analysis in the individual and the character of his experiences4 7#d7 ,he gets certain bad habits of response from the ambiguity of common usage of #b1 #c1 #d1 etc41 for second1 third1 fourth4 ,thus he sees or hears his parents or older children or others count pennies or rolls or eggs by saying one1 two1 three1 four1 and so on4 ,he himself is perhaps misled into so counting4 ,thus the names properly belonging to a series of aggregations varying in amount come to be to him the names of the positions of the parts in a counted whole4 ,this happens especially with numbers above #c or #d1 where the correct experience of the number as a name for the group has rarely been present4 ,this attaching to the cardinal numbers above three or four the meanings of the ordinal numbers seems to affect many children on entrance to school4 ,the numbering of pages in books1 houses1 streets1 etc41 and bad teaching of counting often prolong this error4 7#e7 ,he also gets the habit1 not necessarily bad1 but often indirectly so1 of using many names such as eight1 nine1 ten1 eleven1 fifteen1 a hundred1 a million1 without any meaning4 7#f7 ,the experiences of half1 twice1 three times as many1 three times as long1 etc41 are rarer2 even if they were not1 they would still be less easily productive of the analysis of the proper abstract element than are the experiences of two1 three1 four1 etc41 in connection with aggregates of things each of which is usually called one1 such as boys1 girls1 balls1 apples4 ,experiences of the names1 two1 three1 and four1 in connection with two twos1 two threes1 two fours1 are very rare4 ,hence1 the names1 two1 three1 etc41 mean to these children in the main1 one something and one something1 one something usually called one1 and one something usually called one1 and another something usually called one1 and more rarely and imperfectly two times anything1 three times anything1 etc4 ,with respect to ,mr4 ,phillips' emphasis of the importance of the series-idea in children's minds1 the matters of importance are3 first1 that the knowledge of a series of number names in order is of very little consequence to the teaching of arithmetic and of still less to the origin of awareness of number4 ,second1 the habit of applying this series of words in counting in such a way that #h is associated with the eighth thing1 #i with the ninth thing1 etc41 is of consequence because it does so much mischief4 ,third1 the really valuable idea of the number series1 the idea of a series of groups or of magnitudes varying by steps1 is acquired later1 as a result1 not a cause1 of awareness of numbers4 ,with respect to the ,mc,lellan-,dewey doctrine1 the ratio aspect of numbers should be emphasized in schools1 not because it is the main origin of the child's awareness of number1 but because it is not1 and because the ordinary practical issues of child life do not adequately stimulate its action4 ,it also seems both more economical and more scientific to introduce it through multiplication1 division1 and fractions rather than to insist that #d and #e shall from the start mean #d or #e times anything that is called #a1 for instance1 that #h inches shall be called #d two-inches1 or #aj cents1 #e two-cents4 ,if ,i interpret ,professor ,dewey's writings correctly1 he would agree that the use of inch1 foot1 yard1 pint1 quart1 ounce1 pound1 glassful1 cupful1 handful1 spoonful1 cent1 nickel1 dime1 and dollar gives a sufficient range of units for the first two school years4 ,teaching the meanings of #a_/b of #d1 #a_/b of #f1 #a_/b of #h1 #a_/b of #aj1 #a_/b of #bj1 #a_/c of #f1 #a_/c of #i1 #a_/c of #cj1 #a_/d of #h1 two #bs1 five #bs1 and the like1 in early grades1 each in connection with many different units of measure1 provides a sufficient assurance that numbers will connect with relationships as well as with collections4 ,,chapter ,,xii ,,interest ,,in ,,arithmetic ,,censuses ,,of ,,pupils' ,,interests ,arithmetic1 although it makes little or no appeal to collecting1 muscular manipulation1 sensory curiosity1 or the potent original interests in things and their mechanisms and people and their passions1 is fairly well liked by children4 ,the censuses of pupils' likes and dislikes that have been made are not models of scientific investigation1 and the resulting percentages should not be used uncritically4 ,they are1 however1 probably not on the average over-favorable to arithmetic in any unfair way4 ,some of their results are summarized below4 ,in general they show arithmetic to be surpassed in interest clearly by only the manual arts 7shopwork and manual training for boys1 cooking and sewing for girls71 drawing1 certain forms of gymnastics1 and history4 ,it is about on a level with reading and science4 ,it clearly surpasses grammar1 language1 spelling1 geography1 and religion4 ,lobsien '#jc1 who asked one hundred children in each of the first five grades 7,stufen7 of the elementary schools of ,kiel1 ,which part of the school work 7literally1 'which instruction period'7 do you like best8 found arithmetic led only by drawing and gymnastics in the case of the boys1 and only by handwork in the case of the girls4 ,this is an exaggerated picture of the facts1 since no count is made of those who especially dislike arithmetic4 ,arithmetic is as unpopular with some as it is popular with others4 ,when full allowance is made for this1 arithmetic still has popularity above the average4 ,stern '#je asked1 ,which subject do you like most8 and ,which subject do you like least8 ,the balance was greatly in favor of gymnastics for boys 7#bh--a71 handwork for girls 7#cb--a-a_/b71 and drawing for both 7#af-a_/b--f74 ,writing 7#f-a_/b--d71 arithmetic 7#ad-a_/b--ac71 history 7#i--f-a_/b71 reading 7#h-a_/b--h71 and singing 7#f--g-a_/b7 come next4 ,religion1 nature study1 physiology1 geography1 geometry1 chemistry1 language1 and grammar are low4 ,mc,knight '#jg found with boys and girls in grades #g and #h of certain ,american cities that arithmetic was liked better than any of the school subjects except gymnastics and manual training4 ,the vote as compared with history was3-- ,arithmetic #cbg liked greatly1 #if disliked greatly4 ,history #afd liked greatly1 #aac disliked greatly4 ,in a later study ,lobsien '#ji had #fbdh pupils from #i to #ae years old representing all grades of the elementary school report1 so far as they could1 the subject most disliked1 the subject most liked1 the subject next most liked1 and the subject next in order4 ,no child was forced to report all of these four judgments1 or even any of them4 ,lobsien counts the likes and the dislikes for each subject4 ,gymnastics1 handwork1 and cooking are by far the most popular4 ,history and drawing are next1 followed by arithmetic and reading4 ,below these are geography1 writing1 singing1 nature study1 biblical history1 catechism1 and three minor subjects4 ,lewis '#ac secured records from ,english children in elementary schools of the order of preference of all the studies listed below4 ,he reports the results in the following table of percents3 ,,top ,,third ,,of ,,middle ,,third ,,of ,,lowest ,,third ,,of ,,studies ,,for ,,studies ,,for ,,studies ,,for ,,interest ,,interest ,,interest ---------------------------------------- ------------------------ ,drawing #gh #bj #b ,manual ,subjects #ff #bf #h ,history #fd #bd #ab ,reading #ec #ch #i ,singing #cb #dh #bj ,drill #bj #ee #be ,arithmetic #af #ec #ca ,science #bc #cg #dj ,nature ,study #af #cf #dh ,dictation #d #eg #ci ,composition #ah #bh #ed ,scripture #d #ch #eh ,recitation #i #bc #fh ,geography #d #bd #gb ,grammar -- #f #id ,brandell '#ac obtained data from #bacg ,swedish children in ,stockholm 7#cbg71 ,norrkoeping 7#hgj71 and ,gothenburg 7#idj74 ,in general he found1 as others have1 that handwork1 shopwork for boys and household work for girls1 and drawing were reported as much better liked than arithmetic4 ,so also was history1 and 7in this he differs from most students of this matter7 so were reading and nature study4 ,gymnastics he finds less liked than arithmetic4 ,religion1 geography1 language1 spelling1 and writing are1 as in other studies1 much less popular than arithmetic4 ,other studies are by ,lilius '#aa in ,finland1 ,walsemann '#jg1 ,wiederkehr '#jg1 ,pommer '#ad1 ,seekel '#ad1 and ,stern '#ac and '#ad1 in ,germany4 ,they confirm the general results stated4 ,the reasons for the good showing that arithmetic makes are probably the strength of its appeal to the interest in definite achievement1 success1 doing what one attempts to do2 and of its appeal1 in grades #e to #h1 to the practical interest of getting on in the world1 acquiring abilities that the world pays for4 ,of these1 the former is in my opinion much the more potent interest4 ,arithmetic satisfies it especially well1 because1 more than any other of the 'intellectual' studies of the elementary school1 it permits the pupil to see his own progress and determine his own success or failure4 ,the most important applications of the psychology of satisfiers and annoyers to arithmetic will therefore be in the direction of utilizing still more effectively this interest in achievement4 ,next in importance come the plans to attach to arithmetical learning the satisfyingness of bodily action1 play1 sociability1 cheerfulness1 and the like1 and of significance as a means of securing other desired ends than arithmetical abilities themselves4 ,next come plans to relieve arithmetical learning from certain discomforts such as the eyestrain of some computations and excessive copying of figures4 ,these will be discussed here in the inverse order4 ,,relieving ,,eyestrain ,at present arithmetical work is1 hour for hour1 probably more of a tax upon the eyes than reading4 ,the task of copying numbers from a book to a sheet of paper is one of the very hardest tasks that the eyes of a pupil in the elementary schools have to perform4 ,a certain amount of such work is desirable to teach a child to write numbers1 to copy exactly1 and to organize material in shape for computation4 ,but beyond that1 there is no more reason for a pupil to copy every number with which he is to compute than for him to copy every word he is to read4 ,the meaningless drudgery of copying figures should be mitigated by arranging much work in the form of exercises like those shown on pages #baf1 #bag1 and #bah1 and by having many of the textbook examples in addition1 subtraction1 and multiplication done with a slip of paper laid below the numbers1 the answers being written on it4 ,there is not only a resulting gain in interest1 but also a very great saving of time for the pupil 7very often copying an example more than quadruples the time required to get its answer71 and a much greater efficiency in supervision4 ,arithmetical errors are not confused with errors of copying1#af and the teacher's task of following a pupil's work on the page is reduced to a minimum1 each pupil having put the same part of the day's work in just the same place4 ,the use of well-printed and well-spaced pages of exercises relieves the eyestrain of working with badly made gray figures1 unevenly and too closely or too widely spaced4 ,i reproduce in ,fig4 #be specimens taken at random from one hundred random samples of arithmetical work by pupils in grade #h4 ,contrast the task of the eyes in working with these and their task in working with pages #baf to #bah4 ,the customary method of always copying the numbers to be used in computation from blackboard or book to a sheet of paper is an utterly unjustifiable cruelty and waste4 #af ,courtis finds in the case of addition that of all the individuals making mistakes at any given time in a class1 at least one third1 and usually two thirds1 will be making mistakes in carrying or copying4 ,illustration3 ,,fig4 #be;a4--,specimens taken at random from the computation work of eighth-grade pupils4 ,this computation occurred in a genuine test4 ,in the original gray of the pencil marks the work is still harder to make out4 ,illustration3 ,,fig4 #be;b4--,specimens taken at random from the computation work of eighth-grade pupils4 ,this computation occurred in a genuine test4 ,in the original gray of the pencil marks the work is still harder to make out4 ,write the products3-- ,a4 #c #ds ,b4 #e #gs ,c4 #i #bs #e #bs #h #cs #d #ds #g #bs #d #bs #b #gs #a #f #d #es #f #ds #a #c #d #gs #e #es #c #gs #e #is #c #fs #d #as #g #es #c #bs #f #hs #g #as #c #is #i #hs #f #cs #e #as #d #cs #d #is #h #fs #b #ds #c #es #h #ds #b #bs #i #fs #h #es #h #gs #b #es #g #is #e #hs #e #ds #f #bs #g #fs #h #bs #g #ds #g #cs #h #is #i #cs ,d4 #d #bjs ,e4 #i #fjs ,f4 #dj x #b #hj #d #bjjs #i #fjjs #bj x #b #f #cjs #e #cjs #cj x #b #f #cjjs #e #cjjs #dj x #b #g x #ej #h x #bj #bj x #c #g x #ejj #h x #bjj #cj x #c #c x #dj #b x #gj #cjj x #c #ijj #c x #djj #b x #gjj #cjj x #b ,write the missing numbers3 7r stands for remainder47 #be 4444 #cs and 4444 r4 #be 4444 #ds 4444 r4 #be 4444 #es 4444 r4 #be 4444 #fs 4444 r4 #be 4444 #gs 4444 r4 #be 4444 #hs 4444 r4 #be 4444 #is 4444 r4 #bf 4444 #cs and 4444 r4 #bf 4444 #ds 4444 r4 #bf 4444 #es 4444 r4 #bf 4444 #fs 4444 r4 #bf 4444 #gs 4444 r4 #bf 4444 #hs 4444 r4 #bf 4444 #is 4444 r4 #cj 4444 #ds and 4444 r4 #cj 4444 #es 4444 r4 #cj 4444 #fs 4444 r4 #cj 4444 #gs 4444 r4 #cj 4444 #hs 4444 r4 #cj 4444 #is 4444 r4 #ca 4444 #ds and 4444 r4 #ca 4444 #es 4444 r4 #ca 4444 #fs 4444 r4 #ca 4444 #gs 4444 r4 #ca 4444 #hs 4444 r4 #ca 4444 #is 4444 r4 ,write the whole numbers or mixed numbers which these fractions equal3-- #e #d #i #d #g - - - - - #d #c #e #b #c #g #e #aa #c #h - - -- - - #d #c #h #b #h #h #f #i #i #af - - - - -- #d #c #h #d #h #aa #g #ac #h #f -- - -- - - #d #e #h #e #f ,write the missing figures3-- #f #b #h #a #b - - - - -- - - -- - - #h #d #d #b #aj #e #e #aj #c #f ,write the missing numerators3-- #a - -- - -- - -- - -- #b #ab #h #aj #d #af #f #ad #a - -- - -- - -- -- -- #c #ab #i #ah #f #ae #bd #ba #a - -- -- - -- -- -- -- #d #ab #af #h #bd #bj #bh #cb #a - -- -- -- -- -- -- -- #e #aj #bj #ae #be #dj #ce #cj #b - -- -- -- - -- -- - #c #ab #ah #ba #f #ae #bd #i #c - - -- -- -- -- -- -- #d #h #af #ab #bj #bd #cb #bh ,find the products4 ,cancel when you can3-- #e #aa #b -- x #d -- x #c - x #e #af #ab #c #g #h #a -- x #h - x #ae - x #h #ab #e #f ,,significance ,,for ,,related ,,activities ,the use of bodily action1 social games1 and the like was discussed in the section on original tendencies4 ,significance as a means of securing other desired ends than arithmetical learning itself is therefore our next topic4 ,such significance can be given to arithmetical work by using that work as a means to present and future success in problems of sports1 housekeeping1 shopwork1 dressmaking1 self-management1 other school studies than arithmetic1 and general school life and affairs4 ,significance as a means to future ends alone can also be more clearly and extensively attached to it than it now is4 ,whatever is done to supply greater strength of motive in studying arithmetic must be carefully devised so as not to get a strong but wrong motive1 so as not to get abundant interest but in something other than arithmetic1 and so as not to kill the goose that after all lays the golden eggs--the interest in intellectual activity and achievement itself4 ,it is easy to secure an interest in laying out a baseball diamond1 measuring ingredients for a cake1 making a balloon of a certain capacity1 or deciding the added cost of an extra trimming of ribbon for one's dress4 ,the problem is to attach that interest to arithmetical learning4 ,nor should a teacher be satisfied with attaching the interest as a mere tail that steers the kite1 so long as it stays on1 or as a sugar-coating that deceives the pupil into swallowing the pill1 or as an anodyne whose dose must be increased and increased if it is to retain its power4 ,until the interest permeates the arithmetical activity itself our task is only partly done1 and perhaps is made harder for the next time4 ,one important means of really interfusing the arithmetical learning itself with these derived interests is to lead the pupil to seek the help of arithmetic himself--to lead him1 in ,dewey's phrase1 to 'feel the need'--to take the 'problem' attitude--and thus appreciate the technique which he actively hunts for to satisfy the need4 ,in so far as arithmetical learning is organized to satisfy the practical demands of the pupil's life at the time1 he should1 so to speak1 come part way to get its help4 ,even if we do not make the most skillful use possible of these interests derived from the quantitative problems of sports1 housekeeping1 shopwork1 dressmaking1 self-management1 other school studies1 and school life and affairs1 the gain will still be considerable4 ,to have them in mind will certainly preserve us from giving to children of grades #c and #d problems so devoid of relation to their interests as those shown below1 all found 7in #aiaj7 in thirty successive pages of a book of excellent repute3-- ,a chair has #d legs4 ,how many legs have #h chairs8 #e chairs8 ,a fly has #f legs4 ,how many legs have #c flies8 #i flies8 #g flies8 7,eight more of the same sort47 ,in #ahij ,new ,york had #a1eac1eja inhabitants1 ,milwaukee had #bjf1cjh1 ,boston had #ddg1gbj1 ,san ,francisco #big1iij4 ,how many had these cities together8 7,five more of the same sort47 ,milton was born in #afjh and died in #afgd4 ,how many years did he live8 7,several others of the same sort47 ,the population of a certain city was #ce1fbi in #ahhj and #ajf1fgj in #ahij4 ,find the increase4 7,several others of this sort47 ,a number of others about the words in various inaugural addresses and the ,psalms in the ,bible4 ,it also seems probable that with enough care other systematic plans of textbooks can be much improved in this respect4 ,from every point of view1 for example1 the early work in arithmetic should be adapted to some extent to the healthy childish interests in home affairs1 the behavior of other children1 and the activities of material things1 animals1 and plants4 ,,table #i ,,frequency ,,of ,,appearance ,,of ,,certain ,,words ,,about ,,family ,,life1 ,,play1 ,,and ,,action ,,in ,,eight ,,elementary ,,textbooks ,,in ,,arithmetic1 pp4 #a-ej4 ,a ,b ,c ,d ,e ,f ,g ,h ---------------------------------------- ---------------- baby #b #d brother #b #f #a #a #a family #b #b #d father #a #c #e #b #a help home #b #d #d #b #b #g #a mother #d #b #i #e #e #a #g sister #a #b #b #i #a #a fork knife plate #d #b #b #a spoon doll #aj #a #aj #f #aj #i game #a #c #e #e jump #d marbles #aj #d #aj #aj #a play #a #c run #a #c sing tag toy #a car #b #d #b #c #a cut #aj #f #b #h dig #b flower #a #d #a #a #b grow #a plant #b seed #c #a string #a #aj #a #a wheel #e #aj ,the words used by textbooks give some indication of how far this aim is being realized1 or rather of how far short we are of realizing it4 ,consider1 for example1 the words home1 mother1 father1 brother1 sister1 help1 plate1 knife1 fork1 spoon1 play1 game1 toy1 tag1 marbles1 doll1 run1 jump1 sing1 plant1 seed1 grow1 flower1 car1 wheel1 string1 cut1 dig4 ,the frequency of appearance in the first fifty pages of eight beginners' arithmetics was as shown in ,table #i4 ,the eight columns refer to the eight books 7the first fifty pages of each74 ,the numbers refer to the number of times the word in question appeared1 the number #aj meaning #aj or more times in the fifty pages4 ,plurals1 past tenses1 and the like were counted4 ,help1 fork1 knife1 spoon1 jump1 sing1 and tag did not appear at all6 ,toy and grow appeared each once in the #djj pages6 ,play1 run1 dig1 plant1 and seed appeared once in a hundred or more pages4 ,baby did not appear as often as buggy4 ,family appeared no oftener than fence or ,friday4 ,father appears about a third as often as farmer4 ,book ,a shows only #aj of these thirty words in the fifty pages2 book ,b only #d2 book ,c only #ab2 and books ,d1 ,e1 ,f1 ,g1 and ,h only #ac1 #h1 #ad1 #ac1 #aj1 respectively4 ,the total number of appearances 7counting the #ajs as only #aj in each case7 is #dj for ,a1 #i for ,b1 #fj for ,c1 #db for ,d1 #be for ,e1 #fb for ,f1 #cj for ,g1 and #cg for ,h4 ,the five words--apple1 egg1 ,mary1 milk1 and orange--are used oftener than all these thirty together4 ,if it appeared that this apparent neglect of childish affairs and interests was deliberate to provide for a more systematic treatment of pure arithmetic1 a better gradation of problems1 and a better preparation for later genuine use than could be attained if the author of the textbook were tied to the child's apron strings1 the neglect could be defended4 ,it is not at all certain that children in grade #b get much more enjoyment or ability from adding the costs of purchases for ,christmas or ,fourth of ,july1 or multiplying the number of cakes each child is to have at a party by the number of children who are to be there1 than from adding gravestones or multiplying the number of hairs of bald-headed men4 ,when1 however1 there is nothing gained by substituting remote facts for those of familiar concern to children1 the safe policy is surely to favor the latter4 ,in general1 the neglect of childish data does not seem to be due to provision for some other end1 but to the same inertia of tradition which has carried over the problems of laying walls and digging wells into city schools whose children never saw a stone wall or dug well4 99 99 99 99 99 ,i shall not go into details concerning the arrangement of courses of study1 textbooks1 and lesson-plans to make desirable connections between arithmetical learning and sports1 housework1 shopwork1 and the rest4 ,it may be worth while1 however1 to explain the term self-management1 since this source of genuine problems of real concern to the pupils has been overlooked by most writers4 ,by self-management is meant the pupil's use of his time1 his abilities1 his knowledge1 and the like4 ,by the time he reaches grade #e1 and to some extent before then1 a boy should keep some account of himself1 of how long it takes him to do specified tasks1 of how much he gets done in a specified time at a certain sort of work and with how many errors1 of how much improvement he makes month by month1 of which things he can do best1 and the like4 ,such objective1 matter-of-fact1 quantitative study of one's behavior is not a stimulus to morbid introspection or egotism2 it is one of the best preventives of these4 ,to treat oneself impersonally is one of the essential elements of mental balance and health4 ,it need not1 and should not1 encourage priggishness4 ,on the contrary1 this matter-of-fact study of what one is and does may well replace a certain amount of the exhortations and admonitions concerning what one ought to do and be4 ,all this is still truer for a girl4 ,the demands which such an accounting of one's own activities make of arithmetic have the special value of connecting directly with the advanced work in computation4 ,they involve the use of large numbers1 decimals1 averaging1 percentages1 approximations1 and other facts and processes which the pupil has to learn for later life1 but to which his childish activities as wage-earner1 buyer and seller1 or shopworker from #aj to #ad do not lead4 ,children have little money1 but they have time in thousands of units6 ,they do not get discounts or bonuses from commercial houses1 but they can discount their quantity of examples done for the errors made1 and credit themselves with bonuses of all sorts for extra achievements4 ,,intrinsic ,,interest ,,in ,,arithmetical ,,learning ,there remains the most important increase of interest in arithmetical learning--an increase in the interest directly bound to achievement and success in arithmetic itself4 ,arithmetic1 says ,david ,eugene ,smith1 is a game and all boys and girls are players4 ,it should not be a mere game for them and they should not merely play1 but their unpractical interest in doing it because they can do it and can see how well they do do it is one of the school's most precious assets4 ,any healthy means to give this interest more and better stimulus should therefore be eagerly sought and cherished4 ,two such means have been suggested in other connections4 ,the first is the extension of training in checking and verifying work so that the pupil may work to a standard of approximately #ajj success1 and may know how nearly he is attaining it4 ,the second is the use of standardized practice material and tests1 whereby the pupil may measure himself against his own past1 and have a clear1 vivid1 and trustworthy idea of just how much better or faster he can do the same tasks than he could do a month or a year ago1 and of just how much harder things he can do now than then4 ,another means of stimulating the essential interest in quantitative thinking itself is the arrangement of the work so that real arithmetical thinking is encouraged more than mere imitation and assiduity4 ,this means the avoidance of long series of applied problems all of one type to be solved in the same way1 the avoidance of miscellaneous series and review series which are almost verbatim repetitions of past problems1 and in general the avoidance of excessive repetition of any one problem-situation4 ,stimulation to real arithmetical thinking is weak when a whole day's problem work requires no choice of methods1 or when a review simply repeats without any step of organization or progress1 or when a pupil meets a situation 7say the 'buy x things at y per thing1 how much pay' situation7 for the five-hundredth time4 ,another matter worthy of attention in this connection is the unwise tendency to omit or present in diluted form some of the topics that appeal most to real intellectual interests1 just because they are hard4 ,the best illustration1 perhaps1 is the problem of ratio or ,how many times as large 7long1 heavy1 expensive1 etc47 as x is y8 ,mastery of the 'times as' relation is hard to acquire1 but it is well worth acquiring1 not only because of its strong intellectual appeal1 but also because of its prime importance in the applications of arithmetic to science4 ,in the older arithmetics it was confused by pedantries and verbal difficulties and penalized by unreal problems about fractions of men doing parts of a job in strange and devious times4 ,freed from these1 it should be reinstated1 beginning as early as grade #e with such simple exercises as those shown below and progressing to the problems of food values1 nutritive ratios1 gears1 speeds1 and the like in grade #h4 ,john is #d years old4 ,fred is #f years old4 ,mary is #h years old4 ,nell is #aj years old4 ,alice is #ab years old4 ,bert is #ae years old4 ,who is twice as old as ,john8 ,who is half as old as ,alice8 ,who is three times as old as ,john8 ,who is one and one half times as old as ,nell8 ,who is two thirds as old as ,fred8 etc41 etc41 etc4 ,alice is 4444 times as old as ,john4 ,john is 4444 as old as ,mary4 ,fred is 4444 times as old as ,john4 ,alice is 4444 times as old as ,fred4 ,fred is 4444 as old as ,mary4 etc41 etc41 etc4 ,finally it should be remembered that all improvements in making arithmetic worth learning and helping the pupil to learn it will in the long run add to its interest4 ,pupils like to learn1 to achieve1 to gain mastery4 ,success is interesting4 ,if the measures recommended in the previous chapters are carried out1 there will be little need to entice pupils to take arithmetic or to sugar-coat it with illegitimate attractions4 ,,chapter ,,xiii ,,the ,,conditions ,,of ,,learning ,we shall consider in this chapter the influence of time of day1 size of class1 and amount of time devoted to arithmetic in the school program1 the hygiene of the eyes in arithmetical work1 the use of concrete objects1 and the use of sounds1 sights1 and thoughts as situations and of speech and writing and thought as responses4#ag #ag ,facts concerning the conditions of learning in general will be found in the author's ,educational ,psychology1 ,vol4 #b1 ,chapter #h1 or in the ,educational ,psychology1 ,briefer ,course1 ,chapter #ae4 ,,external ,,conditions ,computation of one or another sort has been used by several investigators as a test of efficiency at different times in the day4 ,when freed from the effects of practice on the one hand and lack of interest due to repetition on the other1 the results uniformly show an increase in speed late in the school session with a falling off in accuracy that about balances it4#ah ,there is no wisdom in putting arithmetic early in the session because of its difficulty4 ,lively and sociable exercises in mental arithmetic with oral answers in fact seem to be admirably fitted for use late in the session4 ,except for the general principles 7#a7 of starting the day with work that will set a good standard of cheerful1 efficient production and 7#b7 of getting the least interesting features of the day's work done fairly early in the day1 psychology permits practical exigencies to rule the program1 so far as present knowledge extends4 ,adequate measurements of the effect of time of day on improvement have not been made1 but there is no reason to believe that any one time between #i ,a4,m4 and #d ,p4,m4 is appreciably more favorable to arithmetical learning than to learning geography1 history1 spelling1 and the like4 #ah ,see ,thorndike '#jj1 ,king '#jg1 and ,heck '#ac4 ,the influence of size of class upon progress in school studies is very difficult to measure because 7#a7 within the same city system the average of the six 7or more7 sizes of class that a pupil has experienced will tend to approximate closely to the corresponding average for any other child2 because further 7#b7 there may be a tendency of supervisory officers to assign more pupils to the better teachers2 and because 7#c7 separate systems which differ in respect to size of class probably differ in other respects also so that their differences in achievement may be referable to totally different differences4 ,elliott '#ad has made a beginning by noting size of class during the year of test in connection with his own measures of the achievements of seventeen hundred pupils1 supplemented by records from over four hundred other classes4 ,as might be expected from the facts just stated1 he finds no appreciable difference between classes of different sizes within the same school system1 the effect of the few months in a small class being swamped by the antecedents or concomitants thereof4 ,the effect of the amount of time devoted to arithmetic in the school program has been studied extensively by ,rice '#jb and '#jc and ,stone '#jh4 ,dr4 ,rice '#jb measured the arithmetical ability of some #fjjj children in #ah different schools in #g different cities4 ,the results of these measurements are summarized in ,table #aj4 ,this table gives two averages for each grade as well as for each school as a whole4 ,thus1 the school at the top shows averages of #hj4j and #hc4a1 and the one at the bottom1 #be4c and #ca4e4 ,the first represents the percentage of answers which were absolutely correct2 the second shows what per cent of the problems were correct in principle1 i4e4 the average that would have been received if no mechanical errors had been made4 ,the facts of ,dr4 ,rice's table show that there is a positive relation between the general standing of a school system in the tests and the amount of time devoted to arithmetic by its program4 ,the relation is not close1 however1 being that expressed by a correlation coefficient of 4#cf-a_/b4 ,within any one school system there is no relation between the standing of a particular school and the amount of time devoted to arithmetic in that school's program4 ,it must be kept in mind that the amount of time given in the school program may be counterbalanced by emphasizing work at home and during study periods1 or1 on the other hand1 may be a symptom of correspondingly small or great emphasis on arithmetic in work set for the study periods at home4 ,a still more elaborate investigation of this same topic was made by ,stone '#jh4 ,i quote somewhat fully from it1 since it is an instructive sample of the sort of studies that will doubtless soon be made in the case of every elementary school subject4 ,he found that school systems differed notably in the achievements made by their sixth-grade pupils in his tests of computation 7the so-called 'fundamentals'7 and of the solution of verbally described problems 7the so-called 'reasoning'74 ,the facts were as shown in ,table #aa4 ,,table #aj ,,averages ,,for ,,individual ,,schools ,,in ,,arithmetic ,,key ,a3 ,,city ,b3 ,,school ,c3 ,result ,d3 ,principle ,e3 ,percent of ,mechanical ,errors ,f3 ,minutes ,daily #f,,th ,,year #g,,th ,,year #h,,th ,,year ,,school ,,average ---------------------------------------- -- ,a ,b ,c ,d ,c ,d ,c ,d ,c ,d ,e ,f ---------------------------------------- -------- ,,iii #a #gi4chj4cha4ahb4cia4gic4ihj4jhc4a #c4g #ec ,i #a #hj4dha4efd4bfg4bhj4ihb4hgf4fhj4c #d4f #fj ,i #b #hj4ihc4ddc4eej4igb4ggi4afi4cge4a #g4g #be ,i #c #gb4bgd4jfc4eff4bgd4egf4ffg4hgb4b #f4a #de ,i #d #fi4igb4bed4feg4hff4efi4afd4cgj4c #h4e #de ,,ii #a #ga4bge4ccc4fce4gcf4hdj4jfj4bfd4h #g4a #fj ,,iii #b #dc4gde4jec4ief4gea4aec4aed4eeh4i #g4d #fj ,,iv #a #eh4ifj4dca4bcd4ada4fdc4eee4aeh4d #e4f #fj ,,iv #b #ei4hfc4a -- -- #bb4ebb4eec4ieh4h #h4c -- ,,iv #c #ed4ieh4ace4bch4fdc4ede4jea4eeg4faj4e #fj ,,iv #d #db4cde4aaf4aai4bdh4gdh4gdb4hdh4baa4b -- ,v #a #dd4adh4gbi4bcb4eea4aeh4cde4iea4caj4e #dj ,,vi #a #fh4cga4ccc4ecf4fbf4icj4gci4jdb4i #i4j #cc ,,vi #b #df4adi4eai4ebd4bcj4bdj4fcf4edc4faf4b #cj ,,vi #c #cd4ecf4dcj4ece4abc4cbd4acf4jdb4eae4b #dh ,,vii #a #ce4bcg4gbi4acb4ebe4abg4bdj4ede4iaa4g #db ,,vii #b #ce4bch4gae4jaf4dai4fba4bcf4edj4faj4a #ge ,,vii #c #bg4fcc4g #h4iaj4aaa4caa4cbe4cca4eai4f #de ,high achievement by a system in computation went with high achievement in solving the problems1 the correlation being about 4#ej2 and the system that scored high in addition or subtraction or multiplication or division usually showed closely similar excellence in the other three1 the correlations being about 4#ij4 ,,table #aa ,,scores ,,made ,,by ,,the ,,sixth-,,grade ,,pupils ,,of ,,each ,,of ,,twenty-,,six ,,school ,,systems ,,system ,,score ,,in ,,tests ,,with ,,score ,,in ,,tests ,,in ,,problems ,,computing ---------------------------------------- ------- #bc #cef #ahda #bd #dbi #ceac #ag #ddd #cjdb #d #dfd #cefc #be #dfd #bafg #bb #dfh #bcaa #af #dfi #cgjg #bj #dia #bafh #ah #eji #cgeh #ae #ecb #bggi #c #ecc #bhde #h #ech #bgdg #f #eej #cagc #a #eeb #bice #aj #fja #bgdi #b #fae #bieh #ba #fbg #biea #ac #fcf #cjdi #ad #ffa #cefa #i #fia #cdjd #g #gcd #cghb #ab #gcf #cdaj #aa #gei #cbfa #bf #gia #cfhb #ai #hdh #djii #e #iad #cefi ,of the conditions under which arithmetical learning took place1 the one most elaborately studied was the amount of time devoted to arithmetic4 ,on the basis of replies by principals of schools to certain questions1 he gave each of the twenty-six school systems a measure for the probable time spent on arithmetic up through grade #f4 ,leaving home study out of account1 there seems to be little or no correlation between the amount of time a system devotes to arithmetic and its score in problem-solving1 and not much more between time expenditure and score in computation4 ,with home study included there is little relation to the achievement of the system in solving problems1 but there is a clear effect on achievement in computation4 ,the facts as given by ,stone are3-- ,,table #ab ,,correlation ,,of ,,time ,,expenditures ,,with ,,abilities ,without ,home ,study ,reasoning and ,time ,expenditure -4#ja ,fundamentals and ,time ,expenditure 4#ji ,including ,home ,study ,reasoning and ,time ,expenditure 4#ac ,fundamentals and ,time ,expenditure 4#di ,these correlations1 it should be borne in mind1 are for school systems1 not for individual pupils4 ,it might be that1 though the system which devoted the most time to arithmetic did not show corresponding superiority in the product over the system devoting only half as much time1 the pupils within the system did achieve in exact proportion to the time they gave to study4 ,neither correlation would permit inference concerning the effect of different amounts of time spent by the same pupil4 ,stone considered also the printed announcements of the courses of study in arithmetic in these twenty-six systems4 ,nineteen judges rated these announced courses of study for excellence according to the instructions quoted below3-- ,,concerning ,,the ,,rating ,,of ,,courses ,,of ,,study ,judges please read before scoring ,i4 ,some ,factors ,determining ,relative ,excellence4 7,n4 ,b4 ,the following enumeration is meant to be suggestive rather than complete or exclusive4 ,and each scorer is urged to rely primarily on his own judgment47 #a4 ,helpfulness to the teacher in teaching the subject matter outlined4 #b4 ,social value or concreteness of sources of problems4 #c4 ,the arrangement of subject matter4 #d4 ,the provision made for adequate drill4 #e4 ,a reasonable minimum requirement with suggestions for valuable additional work4 #f4 ,the relative values of any predominating so-called methods--such as ,speer1 ,grube1 etc4 #g4 ,the place of oral or so-called mental arithmetic4 #h4 ,the merit of textbook references4 ,,ii4 ,cautions and ,directions4 7,judges please follow as implicitly as possible47 #a4 ,include references to textbooks as parts of the ,course of ,study4 ,this necessitates judging the parts of the texts referred to4 #b4 ,as far as possible become equally familiar with all courses before scoring any4 #c4 ,when you are ready to begin to score1 7#a7 arrange in serial order according to excellence1 7#b7 starting with the middle one score it #ej1 then score above and below #ej according as courses are better or poorer1 indicating relative differences in excellence by relative differences in scores1 i4e4 in so far as you find that the courses differ by about equal steps1 score those better than the middle one #ea1 #eb1 etc41 and those poorer #di1 #dh1 etc41 but if you find that the courses differ by unequal steps show these inequalities by omitting numbers4 #d4 ,write ratings on the slip of paper attached to each course4 ,the systems whose courses of study were thus rated highest did not manifest any greater achievement in ,stone's tests than the rest4 ,the thirteen with the most approved announcements of courses of study were in fact a little inferior in achievement to the other thirteen1 and the correlation coefficients were slightly negative4 ,stone also compared eighteen systems where there was supervision of the work by superintendents or supervisors as well as by principals with four systems where the principals and teachers had no such help4 ,the scores in his tests were very much lower in the four latter cities4 ,,the ,,hygiene ,,of ,,the ,,eyes ,,in ,,arithmetic ,we have already noted that the task of reading and copying numbers is one of the hardest that the eyes have to perform in the elementary school1 and that it should be alleviated by arranging much of the work so that only answers need be written by the pupil4 ,the figures to be read and copied should obviously be in type of suitable size and style1 so arranged and spaced on the page or blackboard as to cause a minimum of effort and strain4 ,illustration3 ,,fig4 #bf4--,type too large4 ,illustration3 ,,fig4 #bg4--ab-point1 #aa-point1 and #aj-point type4 ,size4--,type may be too large as well as too small1 though the latter is the commoner error4 ,if it is too large1 as in ,fig4 #bf1 which is a duplicate of type actually used in a form of practice pad1 the eye has to make too many fixations to take in a given content4 ,all things considered1 #ab-point type in grades #c and #d1 #aa-point in grades #e and #f1 and #aj-point in grades #g and #h seem the most desirable sizes4 ,these are shown in ,fig4 #bg4 ,too small type occurs oftenest in fractions and in the dimension-numbers or scale numbers of drawings4 ,figures #bh1 #bi1 and #cj are samples from actual school practice4 ,samples of the desirable size are shown in ,figs4 #ca and #cb4 ,the technique of modern typesetting makes it very difficult and expensive to make fractions of the horizontal type 7#a #c #e - - - #d1 #h1 #f7 large enough without making the whole-number figures with which they are mingled too large or giving an uncouth appearance to the total4 ,consequently fractions somewhat smaller than are desirable may have to be used occasionally in textbooks4#ai ,there is no valid excuse1 however1 for the excessively small fractions which often are made in blackboard work4 #ai ,a special type could be constructed that would use a large type body1 say #ad point1 with integers in #aj or #ab point and fractions much larger than now4 ,illustration3 ,,fig4 #bh4--,type of measurements too small4 ,this is a picture of ,mary's garden4 ,how many feet is it around the garden8 ,illustration3 ,,fig4 #bi4--,type too small4 ,illustration3 ,,fig4 #cj4--,numbers too small and badly designed4 ,illustration3 ,,fig4 #ca4--;,figure #bh with suitable numbers4 ,illustration3 ,,fig4 #cb4--;,figure #cj with suitable numbers4 ,style4--,the ordinary type forms often have #c and #h so made as to require strain to distinguish them4 #e is sometimes easily confused with #c and even with #h4 #a1 #d1 and #g may be less easily distinguishable than is desirable4 ,figure #cc shows a specially good type in which each figure is represented by its essential#bj features without any distracting shading or knobs or turns4 ,figure #cd shows some of the types in common use4 ,there are no demonstrably great differences amongst these4 ,in fractions there is a notable gain from using the slant form 7#b_/c1 #c_/d7 for exercises in addition and subtraction1 and for almost all mixed numbers4 ,this appears clearly to the eye in the comparison of ,fig4 #ce below1 where the same fractions all in #aj-point type are displayed in horizontal and in slant form4 ,the figures in the slant form are in general larger and the space between them and the fraction-line is wider4 ,also the slant form makes it easier for the eye to examine the denominators to see whether reductions are necessary4 ,except for a few cases to show that the operations can be done just as truly with the horizontal forms1 the book and the blackboard should display mixed numbers and fractions to be added or subtracted in the slant form4 ,the slant line should be at an angle of approximately #de degrees4 ,pupils should be taught to use this form in their own work of this sort4 #bj ,it will be still better if the #d is replaced by an open-top #d4 ,when script figures are presented they should be of simple design1 showing clearly the essential features of the figure1 the line being everywhere of equal or nearly equal width 7that is1 without shading1 and without ornamentation or eccentricity of any sort74 ,the opening of the #c should be wide to prevent confusion with #h2 the top of the #c should be curved to aid its differentiation from #e2 the down stroke of the #i should be almost or quite straight2 the #a1 #d1 #g1 and #i should be clearly distinguishable4 ,there are many ways of distinguishing them clearly1 the best probably being to use the straight line for #a1 the open #d with clear angularity1 a wide top to the #g1 and a clearly closed curve for the top of the #i4 ,illustration3 ,,fig4 #cc4--;,block type2 a very desirable type except that it is somewhat too heavy4 ,illustration3 ,,fig4 #cd4--;,common styles of printed numbers4 ,illustration3 ,,fig4 #ce4--;,diagonal and horizontal fractions compared4 ,illustration3 ,,fig4 #cf4--;,good vertical spacing4 ,illustration3 ,,fig4 #cg4--;,bad vertical spacing4 ,illustration3 ,,figs4 #ch 7above7 and #ci 7below74--,good and bad left-right spacing4 ,the pupil's writing of figures should be clear4 ,he will thereby be saved eyestrain and errors in his school work as well as given a valuable ability for life4 ,handwriting of figures is used enormously in spite of the development of typewriters2 illegible figures are commonly more harmful than illegible letters or words1 since the context far less often tells what the figure is intended to be2 the habit of making clear figures is not so hard to acquire1 since they are written unjoined and require only the automatic action of ten minor acts of skill4 ,the schools have missed a great opportunity in this respect4 ,whereas the hand writing of words is often better than it needs to be for life's purposes1 the writing of figures is usually much worse4 ,the figures presented in books on penmanship are also commonly bad1 showing neglect or misunderstanding of the matter on the part of leaders in penmanship4 ,spacing4--,spacing up and down the column is rarely too wide1 but very often too narrow4 ,the specimens shown in ,figs4 #cf and #cg show good practice contrasted with the common fault4 ,spacing from right to left is generally fairly satisfactory in books1 though there is a bad tendency to adopt some one routine throughout and so to miss chances to use reductions and increases of spacing so as to help the eye and the mind in special cases4 ,specimens of good and bad spacing are shown in ,figs4 #ch and #ci4 ,in the work of the pupils1 the spacing from right to left is often too narrow4 ,this crowding of letters1 together with unevenness of spacing1 adds notably to the task of eye and mind4 ,the composition or make-up of the page4--,other things being equal1 that arrangement of the page is best which helps a child most to keep his place on a page and to find it after having looked away to work on the paper on which he computes1 or for other good reasons4 ,a good page and a bad page in this respect are shown in ,figs4 #dj and #da4 ,illustration3 ,,fig4 #dj4--;,a page well made up to suit the action of the eye4 ,illustration3 ,,fig4 #da4--,the same matter as in ,fig4 #dj1 much less well made up4 ,objective presentations4--,pictures1 diagrams1 maps1 and other presentations should not tax the eye unduly1 7a7 by requiring too fine distinctions1 or 7b7 by inconvenient arrangement of the data1 preventing easy counting1 measuring1 comparison1 or whatever the task is1 or 7c7 by putting too many facts in one picture so that the eye and mind1 when trying to make out any one1 are confused by the others4 ,illustrations of bad practices in these respects are shown in ,figs4 #db to #eb4 ,a few specimens of work well arranged for the eye are shown in ,figs4 #ec to #ef4 ,good rules to remember are3-- ,other things being equal1 make distinctions by the clearest method1 fit material to the tendency of the eye to see an 'eyeful' at a time 7roughly #a-a_/b inch by #a_/b inch in a book2 #a-a_/b ft4 by #a_/b ft4 on the blackboard71 and let one picture teach only one fact or relation1 or such facts and relations as do not interfere in perception4 ,the general conditions of seating1 illumination1 paper1 and the like are even more important when the eyes are used with numbers than when they are used with words4 ,illustration3 ,,fig4 #db4--,try to count the rungs on the ladder1 or the shocks in the wagon4 ,illustration3 ,,fig4 #dc4--;,how many oars do you see8 ,how many birds8 ,how many fish8 ,illustration3 ,,fig4 #dd4--;,count the birds in each of the three flocks of birds4 ,illustration3 ,,fig4 #de4--,note the lack of clear division of the hundreds4 ,consider the difficulty of counting one of these columns of dots4 ,illustration3 ,,fig4 #df4--,what do you suppose these pictures are intended to show8 ,illustration3 ,,fig4 #dg4--,would a beginner know that after ,,thirteen he was to switch around and begin at the other end8 ,could you read the ,,six of ,,twenty-,,six if you did not already know what it ought to be8 ,what meaning would all the brackets have for a little child in grade #b8 ,does this picture illustrate or obfuscate8 ,illustration3 ,,fig4 #dh4--;,how long did it take you to find out what these pictures mean8 ,illustration3 ,,fig4 #di4--;,count the figures in the first row1 using your eyes alone2 have some one make lines of #aj1 #aa1 #ab1 #ac1 and more repetitions of this figure spaced closely as here4 ,count #bj or #cj such lines1 using the eye unaided by fingers1 pencil1 etc4 ,illustration3 ,,fig4 #ej4--;,can you answer the question without measuring8 ,could a child of seven or eight8 ,illustration3 ,,fig4 #ea4--,what are these drawings intended to show8 ,why do they show the facts only obscurely and dubiously8 ,illustration3 ,,fig4 #eb4--,what are these drawings intended to show8 ,what simple change would make them show the facts much more clearly8 ,illustration3 ,,fig4 #ec4--;,arranged in convenient eye-fulls4 ,illustration3 ,,fig4 #ed4--;,clear1 simple1 and easy of comparison4 ,illustration3 ,,fig4 #ee4--;,clear1 simple1 and well spaced4 ,illustration3 ,,fig4 #ef4--,well arranged1 though a little wider spacing between the squares would make it even better4 ,,the ,,use ,,of ,,concrete ,,objects ,,in ,,arithmetic ,we mean by concrete objects actual things1 events1 and relations presented to sense1 in contrast to words and numbers and symbols which mean or stand for these objects or for more abstract qualities and relations4 ,blocks1 tooth-picks1 coins1 foot rules1 squared paper1 quart measures1 bank books1 and checks are such concrete things4 ,a foot rule put successively along the three thirds of a yard rule1 a bell rung five times1 and a pound weight balancing sixteen ounce weights are such concrete events4 ,a pint beside a quart1 an inch beside a foot1 an apple shown cut in halves display such concrete relations to a pupil who is attentive to the issue4 ,concrete presentations are obviously useful in arithmetic to teach meanings under the general law that a word or number or sign or symbol acquires meaning by being connected with actual things1 events1 qualities1 and relations4 ,we have also noted their usefulness as means to verifying the results of thinking and computing1 as when a pupil1 having solved1 ,how many badges each #e inches long can be made from #c-a_/c yd4 of ribbon8 by using #aj x #ab_/e1 draws a line #c-a_/c yd4 long and divides it into #e-;inch lengths4 ,concrete experiences are useful whenever the meaning of a number1 like #i or #g_/h or 4#jjd1 or of an operation1 like multiplying or dividing or cubing1 or of some term1 like rectangle or hypothenuse or discount1 or some procedure1 like voting or insuring property against fire or borrowing money from a bank1 is absent or incomplete or faulty4 ,concrete work thus is by no means confined to the primary grades but may be appropriate at all stages when new facts1 relations1 and procedures are to be taught4 ,how much concrete material shall be presented will depend upon the fact or relation or procedure which is to be made intelligible1 and the ability and knowledge of the pupil4 ,thus 'one half' will in general require less concrete illustration than 'five sixths'2 and five sixths will require less in the case of a bright child who already knows #b_/c1 #c_/d1 #c_/h1 #e_/h1 #g_/h1 #b_/e1 #c_/e1 and #d_/e than in the case of a dull child or one who only knows #b_/c and #c_/d4 ,as a general rule the same topic will require less concrete material the later it appears in the school course4 ,if the meanings of the numbers are taught in grade #b instead of grade #a1 there will be less need of blocks1 counters1 splints1 beans1 and the like4 ,if #a-a_/b #a_/b #b is taught early in grade #c1 there will be more gain from the use of #a-a_/b inches and #a_/b inch on the foot rule than if the same relations were taught in connection with the general addition of like fractions late in grade #d4 ,sometimes the understanding can be had either by connecting the idea with the reality directly1 or by connecting the two indirectly via some other idea4 ,the amount of concrete material to be used will depend on its relative advantage per unit of time spent4 ,thus it might be more economical to connect #e_/ab1 #g_/ab1 and #aa_/ab with real meanings indirectly by calling up the resemblance to the #b_/c1 #c_/d1 #c_/h1 #e_/h1 #g_/h1 #b_/e1 #c_/e1 #d_/e1 and #e_/f already studied1 than by showing #e_/ab of an apple1 #g_/ab of a yard1 #aa_/ab of a foot1 and the like4 ,in general the economical course is to test the understanding of the matter from time to time1 using more concrete material if it is needed1 but being careful to encourage pupils to proceed to the abstract ideas and general principles as fast as they can4 ,it is wearisome and debauching to pupils' intellects for them to be put through elaborate concrete experiences to get a meaning which they could have got themselves by pure thought4 ,we should also remember that the new idea1 say of the meaning of decimal fractions1 will be improved and clarified by using it 7see page #ahc f471 so that the attainment of a perfect conception of decimal fractions before doing anything with them is unnecessary and probably very wasteful4 ,a few illustrations may make these principles more instructive4 7a7 ,very large numbers1 such as #ajjj1 #aj1jjj1 #ajj1jjj1 and #a1jjj1jjj1 need more concrete aids than are commonly given4 ,guessing contests about the value in dollars of the school building and other buildings1 the area of the schoolroom floor and other surfaces in square inches1 the number of minutes in a week1 and year1 and the like1 together with proper computations and measurements1 are very useful to reenforce the concrete presentations and supply genuine problems in multiplication and subtraction with large numbers4 7b7 ,numbers very much smaller than one1 such as #a_/cb1 #a_/fd1 4#jd1 and 4#jjb1 also need some concrete aids4 ,a diagram like that of ,fig4 #eg is useful4 7c7 ,majority and plurality should be understood by every citizen4 ,they can be understood without concrete aid1 but an actual vote is well worth while for the gain in vividness and surety4 ,illustration3 ,,fig4 #eg4--;,concrete aid to understanding fractions with large denominators4 ,a #a_/ajjj sq4 ft42 ,b #a_/ajj sq4 ft42 ,c #a_/ej sq4 ft42 ,d #a_/aj sq4 ft4 7d7 ,insurance against loss by fire can be taught by explanation and analogy alone1 but it will be economical to have some actual insuring and payment of premiums and a genuine loss which is reimbursed4 7e7 ,four play banks in the corners of the room1 receiving deposits1 cashing checks1 and later discounting notes will give good educational value for the time spent4 7f7 ,trade discount1 on the contrary1 hardly requires more concrete illustration than is found in the very problems to which it is applied4 7g7 ,the process of finding the number of square units in a rectangle by multiplying with the appropriate numbers representing length and width is probably rather hindered than helped by the ordinary objective presentation as an introduction4 ,the usual form of objective introduction is as follows3-- ,illustration3 ,,fig4 #eh4 ,how long is this rectangle8 ,how large is each square8 ,how many square inches are there in the top row8 ,how many rows are there8 ,how many square inches are there in the whole rectangle8 ,since there are three rows each containing #d square inches1 we have #c x #d square inches #ab square inches4 ,draw a rectangle #g inches long and #b inches wide4 ,if you divide it into inch squares how many rows will there be8 ,how many inch squares will there be in each row8 ,how many square inches are there in the rectangle8 ,illustration3 ,,fig4 #ei4 ,it is better actually to hide the individual square units as in ,fig4 #ei4 ,there are four reasons3 7#a7 ,the concrete rows and columns rather distract attention from the essential thing to be learned4 ,this is not that x rows one square wide1 y squares in a row will make xy squares in all1 but that by using proper units and the proper operation the area of any rectangle can be found from its length and width4 7#b7 ,children have little difficulty in learning to multiply rather than add1 subtract1 or divide when computing area4 7#c7 ,the habit so formed holds good for areas like #a-b_/c by #d-a_/b1 with fractional dimensions1 in which any effort to count up the areas of rows is very troublesome and confusing4 7#d7 ,the notion that a square inch is an area #a' by #a' rather than #a_/b' by #b' or #a_/c in4 by #c in4 or #a-a_/b in4 by #b_/c in4 is likely to be formed too emphatically if much time is spent upon the sort of concrete presentation shown above4 ,it is then better to use concrete counting of rows of small areas as a means of verification after the procedure is learned1 than as a means of deriving it4 ,there has been1 especially in ,germany1 much argument concerning what sort of number-pictures 7that is1 arrangement of dots1 lines1 or the like1 as shown in ,fig4 #fj7 is best for use in connection with the number names in the early years of the teaching of arithmetic4 ,lay '#ih and '#jg1 ,walsemann '#jg1 ,freeman '#aj1 ,howell '#ad1 and others have measured the accuracy of children in estimating the number of dots in arrangements of one or more of these different types4#ba ,many writers interpret a difference in favor of estimating1 say1 the square arrangements of ,born or ,lay as meaning that such is the best arrangement to use in teaching4 ,the inference is1 however1 unjustified4 ,that certain number-pictures are easier to estimate numerically does not necessarily mean that they are more instructive in learning4 ,one set may be easier to estimate just because they are more familiar1 having been oftener experienced4 ,even if the favored set was so after equal experience with all sets1 accuracy of estimation would be a sign of superiority for use in instruction only if all other things were equal 7or in favor of the arrangement in question74 ,obviously the way to decide which of these is best to use in teaching is by using them in teaching and measuring all relevant results1 not by merely recording which of them are most accurately estimated in certain time exposures4 #ba ,for an account in ,english of their main findings see ,howell '#ad1 pp4 #adi-bea4 ,it may be noted that the ,born1 ,lay1 and ,freeman pictures have claims for special consideration on grounds of probable instructiveness4 ,since they are also superior in the tests in respect to accuracy of estimate1 choice should probably be made from these three by any teacher who wishes to connect one set of number-pictures systematically with the number names1 as by drills with the blackboard or with cards4 ,illustration3 ,,fig4 #fj4--,various proposed arrangements of dots for use in teaching the meanings of the numbers #a to #aj4 ,such drills are probably useful if undertaken with zeal1 and if kept as supplementary to more realistic objective work with play money1 children marching1 material to be distributed1 garden-plot lengths to be measured1 and the like1 and if so administered that the pupils soon get the generalized abstract meaning of the numbers freed from dependence on an inner picture of any sort4 ,this freedom is so important that it may make the use of many types of number-pictures advisable rather than the use of the one which in and of itself is best4 ,as ,meumann says3 ,perceptual reckoning can be overdone4 ,it had its chief significance for the surety and clearness of the first foundation of arithmetical instruction4 ,if1 however1 it is continued after the first operations become familiar to the child1 and extended to operations which develop from these elementary ones1 it necessarily works as a retarding force and holds back the natural development of arithmetic4 ,this moves on to work with abstract number and with mechanical association and reproduction4 '#jg1 ,vol4 #b1 p4 #ceg4 ,such drills are commonly overdone by those who make use of them1 being given too often1 and continued after their instructiveness has waned1 and used instead of more significant1 interesting1 and varied work in counting and estimating and measuring real things4 ,consequently1 there is now rather a prejudice against them in our better schools4 ,they should probably be reinstated but to a moderate and judicious use4 ,,oral1 ,,mental1 ,,and ,,written ,,arithmetic ,there has been much dispute over the relative merits of oral and written work in arithmetic--a question which is much confused by the different meanings of 'oral' and 'written4' ,oral has meant 7#a7 work where the situations are presented orally and the pupil's final responses are given orally1 or 7#b7 work where the situations are presented orally and the pupils' final responses are written or partly written and partly oral1 or 7#c7 work where the situations are presented in writing or print and the final responses are oral4 ,written has meant 7#a7 work where the situations are presented in writing or print and the final responses are made in writing1 or 7#b7 work where also many of the intermediate responses are written1 or 7#c7 work where the situations are presented orally but the final responses and a large percentage of the intermediate computational responses are written4 ,there are other meanings than these4 ,it is better to drop these very ambiguous terms and ask clearly what are the merits and demerits1 in the case of any specified arithmetical work1 of auditory and of visual presentation of the situations1 and of saying and of writing each specified step in the response4 ,the disputes over mental versus written arithmetic are also confused by ambiguities in the use of 'mental4' ,mental has been used to mean done without pencil and paper and also done with few overt responses1 either written or spoken1 between the setting of the task and the announcement of the answer4 ,in neither case is the word mental specially appropriate as a description of the total fact4 ,as before1 we should ask clearly1 ,what are the merits and demerits of making certain specified intermediate responses in inner speech or imaged sounds or visual images or imageless thought--that is1 without actual writing or overt speech8 ,it may be said at the outset that oral1 written1 and inner presentations of initial situations1 oral1 written1 and inner announcements of final responses1 and oral1 written1 and inner management of intermediate processes have varying degrees of merit according to the particular arithmetical exercise1 pupil1 and context4 ,devotion to oralness or mentalness as such is simply fanatical4 ,various combinations1 such as the written presentation of the situation with inner management of the intermediate responses and oral announcement of the final response have their special merits for particular cases4 ,these merits the reader can evaluate for himself for any given sort of work for a given class by considering3 7#a7 ,the amount of practice received by the class per hour spent2 7#b7 the ease of correction of the work2 7#c7 the ease of understanding the tasks2 7#d7 the prevention of cheating2 7#e7 the cheerfulness and sociability of the work2 7#f7 the freedom from eyestrain1 and other less important desiderata4 ,it should be noted that the stock schemes ,a1 ,b1 ,c1 and ,d below are only a few of the many that are possible and that schemes ,e1 ,f1 ,g1 and ,h have special merits4 ,,presentation ,,of ,,management ,,of ,,announcement ,,of ,,initial ,,situation ,,intermediate ,,processes ,,final ,,response ,a4 ,printed or written ,written ,written ,b4 ,inner ,oral by one pupil1 inner by the rest ,c4 ,oral 7by teacher7 ,written ,written ,d4 ,inner ,oral by one pupil1 inner by the rest ,e4 ,as in ,a or ,c ,a mixture1 the pupil ,as in ,a or ,b or ,h writing what he needs ,f4 ,the real situation ,as in ,e ,as in ,a or ,b or ,h itself1 in part at least ,g4 ,both read by the pupil ,as in ,e ,as in ,a or ,b or ,h and put orally by the teacher ,h4 ,as in ,a or ,c or ,g ,as in ,e ,written by all pupils1 announced orally by one pupil ,the common practice of either having no use made of pencil and paper or having all computations and even much verbal analysis written out elaborately for examination is unfavorable for learning4 ,the demands which life itself will make of arithmetical knowledge and skill will range from tasks done with every percentage of written work from zero up to the case where every main result obtained by thought is recorded for later use by further thought4 ,in school the best way is that which1 for the pupils in question1 has the best total effect upon quality of product1 speed1 and ease of production1 reenforcement of training already given1 and preparation for training to be given4 ,there is nothing intellectually criminal about using a pencil as well as inner thought2 on the other hand there is no magical value in writing out for the teacher's inspection figures that the pupil does not need in order to attain1 preserve1 verify1 or correct his result4 ,the common practice of having the final responses of all easy tasks given orally has no sure justification4 ,on the contrary1 the great advantage of having all pupils really do the work should be secured in the easy work more than anywhere else4 ,if the time cost of copying the figures is eliminated by the simple plan of having them printed1 and if the supervision cost of examining the papers is eliminated by having the pupils correct each other's work in these easy tasks1 written answers are often superior to oral except for the elements of sociability and 'go' and freedom from eyestrain of the oral exercise4 ,such written work provides the gifted pupils with from two to ten times as much practice as they would get in an oral drill on the same material1 supposing them to give inner answers to every exercise done by the class as a whole2 it makes sure that the dull pupils who would rarely get an inner answer at the rate demanded by the oral exercise1 do as much as they are able to do4 ,two arguments often made for the oral statement of problems by the teacher are that problems so put are better understood1 especially in the grades up through the fifth1 and that the problems are more likely to be genuine and related to the life the pupils know4 ,when these statements are true1 the first is a still better argument for having the pupils read the problems aided by the teacher's oral statement of them4 ,for the difficulty is largely that the pupils cannot read well enough2 and it is better to help them to surmount the difficulty rather than simply evade it4 ,the second is not an argument for oralness versus writtenness1 but for good problems versus bad2 the teacher who makes up such good problems should1 in fact1 take special care to write them down for later use1 which may be by voice or by the blackboard or by printed sheet1 as is best4 ,,chapter ,,xiv ,,the ,,conditions ,,of ,,learning3 ,,the ,,problem ,,attitude ,dewey1 and others following him1 have emphasized the desirability of having pupils do their work as active seekers1 conscious of problems whose solution satisfies some real need of their own natures4 ,other things being equal1 it is unwise1 they argue1 for pupils to be led along blindfold as it were by the teacher and textbook1 not knowing where they are going or why they are going there4 ,they ought rather to have some living purpose1 and be zealous for its attainment4 ,this doctrine is in general sound1 as we shall see1 but it is often misused as a defense of practices which neglect the formation of fundamental habits1 or as a recommendation to practices which are quite unworkable under ordinary classroom conditions4 ,so it seems probable that its nature and limitations are not thoroughly known1 even to its followers1 and that a rather detailed treatment of it should be given here4 ,,illustrative ,,cases ,consider first some cases where time spent in making pupils understand the end to be attained before attacking the task by which it is attained1 or care about attaining the end 7well or ill understood7 is well spent4 ,it is well for a pupil who has learned 7#a7 the meanings of the numbers one to ten1 7#b7 how to count a collection of ten or less1 and 7#c7 how to measure in inches a magnitude of ten1 nine1 eight inches1 etc41 to be confronted with the problem of true adding without counting or measuring1 as in 'hidden' addition and measurement by inference4 ,for example1 the teacher has three pencils counted and put under a book2 has two more counted and put under the book2 and asks1 ,how many pencils are there under the book8 ,answers1 when obtained1 are verified or refuted by actual counting and measuring4 ,the time here is well spent because the children can do the necessary thinking if the tasks are well chosen2 because they are thereby prevented from beginning their study of addition by the bad habit of pseudo-adding by looking at the two groups of objects and counting their number instead of real adding1 that is1 thinking of the two numbers and inferring their sum2 and further1 because facing the problem of adding as a real problem is in the end more economical for learning arithmetic and for intellectual training in general than being enticed into adding by objective or other processes which conceal the difficulty while helping the pupil to master it4 ,the manipulation of short multiplication may be introduced by confronting the pupils with such problems as1 ,how to tell how many ,uneeda biscuit there are in four boxes1 by opening only one box4 ,correct solutions by addition should be accepted4 ,correct solutions by multiplication1 if any gifted children think of this way1 should be accepted1 even if the children cannot justify their procedure4 7,inferring the manipulation from the place-values of numbers is beyond all save the most gifted and probably beyond them47 ,correct solution by multiplication by some child who happens to have learned it elsewhere should be accepted4 ,let the main proof of the trustworthiness of the manipulation be by measurement and by addition4 ,proof by the stock arguments from the place-values of numbers may also be used4 ,if no child hits on the manipulation in question1 the problem of finding the length without adding may be set4 ,if they still fail1 the problem may be made easier by being put as #d times #bb gives the answer4 ,write down what you think #d times #bb will be4 ,other reductions of the difficulty of the problem may be made1 or the teacher may give the answer without very great harm being done4 ,the important requirement is that the pupils should be aware of the problem and treat the manipulation as a solution of it1 not as a form of educational ceremonial which they learn to satisfy the whims of parents and teachers4 ,in the case of any particular class a situation that is more appealing to the pupils' practical interests than the situation used here can probably be devised4 ,the time spent in this way is well spent 7#a7 because all but the very dull pupils can solve the problem in some way1 7#b7 because the significance of the manipulation as an economy over addition is worth bringing out1 and 7#c7 because there is no way of beginning training in short multiplication that is much better4 ,in the same fashion multiplication by two-place numbers may be introduced by confronting pupils with the problem of the number of sheets of paper in #gb pads1 or pieces of chalk in #bd boxes1 or square inches in #ce square feet1 or the number of days in #cb years1 or whatever similar problem can be brought up so as to be felt as a problem4 ,suppose that it is the #ce square feet4 ,solutions by 7#e x #add7 7#cj x #add71 however arranged1 or by 7#aj x #add7 7#aj x #add7 7#aj x #add7 7#e x #add71 or by #cejj 7#ce x #dj7 7#ce x #d71 or by #g x 7#e x #add71 however arranged1 should all be listed for verification or rejection4 ,the pupils need not be required to justify their procedures by a verbal statement4 ,answers like #dcb1gbj1 or #gbj1dcb1 or #aaeb1 or #dbbj1 or #cbbj should be listed for verification or rejection4 ,verification may be by a mixture of short multiplication and objective work1 or by a mixture of short multiplication and addition1 or by addition abbreviated by taking ten #adds as #addj1 or even 7for very stupid pupils7 by the authority of the teacher4 ,or the manipulation in cases like #ec x #i or #hd x #g may be verified by the reverse short multiplication4 ,the deductive proof of the correctness of the manipulation may be given in whole or in part in connection with exercises like #aj x #b #cj x #ad #aj x #c #c x #dd #aj x #d #cj x #dd #aj x #ad #c x #add #aj x #dd #bj x #add #aj x #add #dj x #add #bj x #b #cj x #add #bj x #c #e x #add #cj x #c #ce #cj 4444 #cj x #d #cj x #add added to #e x #add #c x #ad ,certain wrong answers may be shown to be wrong in many ways2 e4g41 #dcb1gbj is too big1 for #ce times a thousand square inches is only #ce1jjj2 #aaeb is too small1 for #ce times a hundred square inches would be #cejj1 or more than #aaeb4 ,the time spent in realizing the problem here is fairly well spent because 7#a7 any successful original manipulation in this case represents an excellent exercise of thought1 because 7#b7 failures show that it is useless to juggle the figures at random1 and because 7#c7 the previous experience with short multiplication makes it possible for the pupils to realize the problem in a very few minutes4 ,it may1 however1 be still better to give the pupils the right method just as soon as the problem is realized1 without having them spend more time in trying to solve it4 ,thus3-- #a square foot has #add square inches4 ,how many square inches are there in #ce square feet 7marked out in chalk on the floor as a piece #aj ft4 x #c ft4 plus a piece #e ft4 x #a ft478 #a yard #cf inches4 ,how many inches long is this wall 7found by measure to be #ac yards78 ,here is a quick way to find the answers3-- #add #ce ---- #gbj #dcb ---- #ejdj sq4 inches in #ce sq4 ft4 #cf #ac --- #ajh #cf --- #dfh inches in #ac yd4 ,consider now the following introduction to dividing by a decimal3-- ,dividing by a ,decimal #a4 ,how many minutes will it take a motorcycle1 to go #ab4fge miles at the rate of 4#ge mi4 per minute8 #af4i 4#geab4fge #g #e --- #e #ag #d #ej ---- #fge #fge --- #b4 ,check by multiplying #af4i by 4#ge4 #c4 ,how do you know that the quotient cannot be as little as #a4fi8 #d4 ,how do you know that the quotient cannot be as large as #afi8 #e4 ,find the quotient for #c4ge _/ #a4e4 #f4 ,check your result by multiplying the quotient by the divisor4 #g4 ,how do you know that the quotient cannot be 4#be or #be8 #h4 ,look at this problem4 4#beg4e ,how do you know that #c4j is wrong for the quotient8 ,how do you know that #cjj is wrong for the quotient8 ,state which quotient is right for each of these3-- 4#jba or 4#ba or #b4a or #ba or #baj #i4 #a4hc4gh 4#jba or 4#ba or #ba or #baj #aj4 #a4hcg4h 4#jc or 4#c or #c or #cj or #cjj #aa4 #a4becg4e 4#jc or 4#c or #c or #cj or #cjj #ab4 #ab4ecg4e 4#je or 4#e or #e or #ej or #ejj #ac4 #a4bef4be 4#je or 4#e or #e or #ej or #ejj #ad4 #ab4ef4be #ae4 ,is this rule true8 ,if it is true1 learn it4 ,in a correct result1 the number of decimal places in the divisor and quotient together equals the number of decimal places in the dividend4 ,these and similar exercises excite the problem attitude in children who have a general interest in getting right answers4 ,such a series carefully arranged is a desirable introduction to a statement of the rule for placing the decimal point in division with decimals4 ,for it attracts attention to the general principle 7divisor x quotient should equal dividend71 which is more important than the rule for convenient location of the decimal point1 and it gives training in placing the point by inspection of the divisor1 quotient1 and dividend1 which suffices for nineteen out of twenty cases that the pupil will ever encounter outside of school4 ,he is likely to remember this method by inspection long after he has forgotten the fixed rule4 ,it is well for the pupil to be introduced to many arithmetical facts by way of problems about their common uses4 ,the clockface1 the railroad distance table in hundredths of a mile1 the cyclometer and speedometer1 the recipe1 and the like offer problems which enlist his interest and energy and also connect the resulting arithmetical learning with the activities where it is needed4 ,there is no time cost1 but a time-saving1 for the learning as a means to the solution of the problems is quicker than the mere learning of the arithmetical facts by themselves alone4 ,a few samples of such procedure are shown below3-- ,,grade #c ,to be ,done at ,home ,look at a watch4 ,has it any hands besides the hour hand and the minute hand8 ,find out all that you can about how a watch tells seconds1 how long a second is1 and how many seconds make a minute4 ,,grade #e ,measuring ,rainfall ,rainfall per ,week 7cu4 in4 per sq4 in4 of area7 ,june #a-g #a4jef #h-ad #a4ajc #ae-ba #a4jdj #bb-bh 4#ifj #bi-;,july #e 4#iae ,july #f-ab 4#ghb #ac-ai 4#gij #bj-bf 4#fgj #bg-;,aug4 #b 4#ejc ,aug4 #c-i 4#eab #aj-af 4#bdj #ag-bc 4#bae #bd-cj 4#haa #a4 ,in which weeks was the rainfall #a or more8 #b4 ,which week of ,august had the largest rainfall for that month8 #c4 ,which was the driest week of the summer8 7,driest means with the least rainfall47 #d4 ,which week was the next to the driest8 #e4 ,in which weeks was the rainfall between 4#hjj and #a4jjj8 #f4 ,look down the table and estimate whether the average rainfall for one week was about 4#e1 or about 4#f1 or about 4#g1 or about 4#h1 or about 4#i4 ,dairy ,records ,record of ,star ,elsie ,pounds of ,milk ,butter-,fat per ,pound of ,milk ,jan4 #agdb 4#jdfa ,feb4 #afij 4#jdhe ,mar4 #aegd 4#jejd ,apr4 #abbf 4#jdij ,may #abjb 4#jdff ,june #abea 4#jdha ,read this record of the milk given by the cow ,star ,elsie4 ,the first column tells the number of pounds of milk given by ,star ,elsie each month4 ,the second column tells what fraction of a pound of butter-fat each pound of milk contained4 #a4 ,read the first line1 saying1 ,in ,january this cow gave #agdb pounds of milk4 ,there were #dfa ten thousandths of a pound of butter-fat per pound of milk4 ,read the other lines in the same way4 #b4 ,how many pounds of butter-fat did the cow produce in ,jan48 #c4 ,in ,feb48 #d4 ,in ,mar48 #e4 ,in ,apr48 #f4 ,in ,may8 #g4 ,in ,june8 ,,grade #e ,,or ,,later ,using ,recipes to ,make ,larger or ,smaller ,quantities ,i4 ,state how much you would use of each material in the following recipes3 7a7 ,to make double the quantity4 7b7 ,to make half the quantity4 7c7 ,to make #a-a_/b times the quantity4 ,you may use pencil and paper when you cannot find the right amount mentally4 #a4 ,,peanut ,,penuche #a tablespoon butter #b cups brown sugar #a_/c cup milk or cream #c_/d cup chopped peanuts #a_/c teaspoon salt #b4 ,,molasses ,,candy #a_/b cup butter #b cups sugar #a cup molasses #a-a_/b cups boiling water #c4 ,,raisin ,,opera ,,caramels #b cups light brown sugar #g_/h cup thin cream #a_/b cup raisins #d4 ,,walnut ,,molasses ,,squares #b tablespoons butter #a cup molasses #a_/c cup sugar #a_/b cup walnut meats #e4 ,,reception ,,rolls #a cup scalded milk #a-a_/b tablespoons sugar #a teaspoon salt #a_/d cup lard #a yeast cake #a_/d cup lukewarm water ,white of #a egg #c-a_/b cups flour #f4 ,,graham ,,raised ,,loaf #b cups milk #f tablespoons molasses #a-a_/b teaspoons salt #a_/c yeast cake #a_/d cup lukewarm water #b cups sifted ,graham flour #a_/b cup ,graham bran #c_/d cup flour 7to knead7 ,,ii4 ,how much would you use of each material in the following recipes3 7a7 ,to make #b_/c as large a quantity8 7b7 ,to make #a-a_/c times as much8 7c7 ,to make #b-a_/b times as much8 #a4 ,,english ,,dumplings #a_/b pound beef suet #a-a_/d cups flour #c teaspoons baking powder #a teaspoon salt #a_/b teaspoon pepper #a teaspoon minced parsley #a_/b cup cold water #b4 ,,white ,,mountain ,,angel ,,cake #a-a_/b cups egg whites #a-a_/b cups sugar #a teaspoon cream of tartar #a cup bread flour #a_/d teaspoon salt #a teaspoon vanilla ,in many cases arithmetical facts and principles can be well taught in connection with some problem or project which is not arithmetical1 but which has special potency to arouse an intellectual activity in the pupil which by some ingenuity can be directed to arithmetical learning4 ,playing store is the most fundamental case4 ,planning for a party1 seeing who wins a game of bean bag1 understanding the calendar for a month1 selecting ,christmas presents1 planning a picnic1 arranging a garden1 the clock1 the watch with second hand1 and drawing very simple maps are situations suggesting problems which may bring a living purpose into arithmetical learning in grade #b4 ,these are all available under ordinary conditions of class instruction4 ,a sample of such problems for a higher grade 7#f7 is shown below4 ,estimating ,areas ,the children in the geography class had a contest in estimating the areas of different surfaces4 ,each child wrote his estimates for each of these maps1 ,a1 ,b1 ,c1 ,d1 and ,e4 7,only ,c and ,d are shown here47 ,in the arithmetic class they learned how to find the exact areas4 ,then they compared their estimates with the exact areas to find who came nearest4 ,illustration ,write your estimates for ,a1 ,b1 ,c1 ,d1 and ,e4 ,then study the next #f pages and learn how to find the exact areas4 7,the next #f pages comprise training in the mensuration of parallelograms and triangles47 ,in some cases the affairs of individual pupils include problems which may be used to guide the individual in question to a zealous study of arithmetic as a means of achieving his purpose--of making a canoe1 surveying an island1 keeping the accounts of a ,girls' ,canning ,club1 or the like4 ,it requires much time and very great skill to direct the work of thirty or more pupils each busy with a special type of his own1 so as to make the work instructive for each1 but in some cases the expense of time and skill is justified4 ,,general ,,principles ,in general what should be meant when one says that it is desirable to have pupils in the problem-attitude when they are studying arithmetic is substantially as follows3-- ,first4--,information that comes as an answer to questions is better attended to1 understood1 and remembered than information that just comes4 ,second4--,similarly1 movements that come as a step toward achieving an end that the pupil has in view are better connected with their appropriate situations1 and such connections are longer retained1 than is the case with movements that just happen4 ,third4--,the more the pupil is set toward getting the question answered or getting the end achieved1 the greater is the satisfyingness attached to the bonds of knowledge or skill which mean progress thereto4 ,fourth4--,it is bad policy to rely exclusively on the purely intellectualistic problems of ,how can ,i do this8 ,how can ,i get the right answer8 ,what is the reason for this8 ,is there a better way to do that8 and the like4 ,it is bad policy to supplement these intellectualistic problems by only the remote problems of ,how can ,i be fitted to earn a higher wage8 ,how can ,i make sure of graduating8 ,how can ,i please my parents8 and the like4 ,the purely intellectualistic problems have too weak an appeal for many pupils2 the remote problems are weak so long as they are remote and1 what is worse1 may be deprived of the strength that they would have in due time if we attempt to use them too soon4 ,it is the extreme of bad policy to neglect those personal and practical problems furnished by life outside the class in arithmetic the solution of which can really be furthered by arithmetic then and there4 ,it is good policy to spend time in establishing certain mental sets--stimulating1 or even creating1 certain needs--setting up problems themselves--when the time so spent brings a sufficient improvement in the quality and quantity of the pupils' interest in arithmetical learning4 ,fifth4--,it would be still worse policy to rely exclusively on problems arising outside arithmetic4 ,to learn arithmetic is itself a series of problems of intrinsic interest and worth to healthy-minded children4 ,the need for ability to multiply with ,united ,states money or to add fractions or to compute percents may be as truly vital and engaging as the need for skill to make a party dress or for money to buy it or for time to play baseball4 ,the intellectualistic needs and problems should be considered along with all others1 and given whatever weight their educational value deserves4 ,,difficulty ,,and ,,success ,,as ,,stimuli ,there are certain misconceptions of the doctrine of the problem-attitude4 ,the most noteworthy is that difficulty--temporary failure--an inadequacy of already existing bonds--is the essential and necessary stimulus to thinking and learning4 ,dewey himself does not1 as ,i understand him1 mean this1 but he has been interpreted as meaning it by some of his followers4#bb #bb ,in his ,how ,we ,think4 ,difficulty--temporary failure1 inadequacy of existing bonds--on the contrary does nothing whatsoever in and of itself2 and what is done by the annoying lack of success which sometimes accompanies difficulty sometimes hinders thinking and learning4 ,mere difficulty1 mere failure1 mere inadequacy of existing bonds1 does nothing4 ,it is hard for me to add three eight-place numbers at a glance2 ,i have failed to find as effective illustrations for pages #bgf to #bgg as ,i wished2 my existing sensori-motor connections are inadequate to playing a golf course in #fe4 ,but these events and conditions have done nothing to stimulate me in respect to the behavior in question4 ,in the first of the three there is no annoying lack and no dynamic influence at all2 in the second there was to some degree an annoying lack--a slight irritation at not getting just what ,i wanted1--and this might have impelled me to further thinking 7though it did not1 and getting one tiptop illustration would as a rule stimulate me to hunt for others more than failing to get such74 ,in the third case the lack of the #fe does not annoy me or have any noteworthy dynamic effect4 ,the lack of #ij instead of #ie-ajj is annoying and is at times a stimulus to further learning1 though not nearly so strong a stimulus as the attainment of the #ij would be6 ,at other times this annoying lack is distinctly inhibitory--a stimulus to ceasing to learn4 ,in the intellectual life the inhibitory effect seems far the commoner of the two4 ,not getting answers seems as a rule to make us stop trying to get them4 ,the annoying lack of success with a theoretical problem most often makes us desert it for problems to whose solution the existing bonds promise to be more adequate4 ,the real issue in all this concerns the relative strength1 in the pupil's intellectual life1 of the negative reaction of behavior in general4 ,an animal whose life processes are interfered with so that an annoying state of affairs is set up1 changes his behavior1 making one after another responses as his instincts and learned tendencies prescribe1 until the annoying state of affairs is terminated1 or the animal dies1 or suffers the annoyance as less than the alternatives which his responses have produced4 ,when the annoying state of affairs is characterized by the failure of things as they are to minister to a craving--as in cases of hunger1 loneliness1 sex-pursuit1 and the like1--we have stimulus to action by an annoying lack or need1 with relief from action by the satisfaction of the need4 ,such is in some measure true of man's intellectual life4 ,in recalling a forgotten name1 in solving certain puzzles1 or in simplifying an algebraic complex1 there is an annoying lack of the name1 solution1 or factor1 a trial of one after another response1 until the annoyance is relieved by success or made less potent by fatigue or distraction4 ,even here the difficulty does not do anything--but only the annoying interference with our intellectual peace by the problem4 ,further1 although for the particular problem1 the annoying lack stimulates1 and the successful attainment stops thinking1 the later and more important general effect on thinking is the reverse4 ,successful attainment stops our thinking on that problem but makes us more predisposed later to thinking in general4 ,overt negative reaction1 however1 plays a relatively small part in man's intellectual life4 ,filling intellectual voids or relieving intellectual strains in this way is much less frequent than being stimulated positively by things seen1 words read1 and past connections acting under modified circumstances4 ,the notion of thinking as coming to a lack1 filling it1 meeting an obstacle1 dodging it1 being held up by a difficulty and overcoming it1 is so one-sided as to verge on phantasy4 ,the overt lacks1 strains1 and difficulties come perhaps once in five hours of smooth straightforward use and adaptation of existing connections1 and they might as truly be called hindrances to thought--barriers which past successes help the thinker to surmount4 ,problems themselves come more often as cherished issues which new facts reveal1 and whose contemplation the thinker enjoys1 than as strains or lacks or 'problems which ,i need to solve4' ,it is just as true that the thinker gets many of his problems as results from1 or bonuses along with1 his information1 as that he gets much of his information as results of his efforts to solve problems4 ,as between difficulty and success1 success is in the long run more productive of thinking4 ,necessity is not the mother of invention4 ,knowledge of previous inventions is the mother2 original ability is the father4 ,the solutions of previous problems are more potent in producing both new problems and their solutions than is the mere awareness of problems and desire to have them solved4 ,in the case of arithmetic1 learning to cancel instead of getting the product of the dividends and the product of the divisors and dividing the former by the latter1 is a clear case of very valuable learning1 with ease emphasized rather than difficulty1 with the adequacy of existing bonds 7when slightly redirected7 as the prime feature of the process rather than their inadequacy1 and with no sense of failure or lack or conflict4 ,it would be absurd to spend time in arousing in the pupil1 before beginning cancellation1 a sense of a difficulty--viz41 that the full multiplying and dividing takes longer than one would like4 ,a pupil in grade #d or #e might well contemplate that difficulty for years to no advantage4 ,he should at once begin to cancel and prove by checking that errorless cancellation always gives the right answer4 ,to emphasize before teaching cancellation the inadequacy of the old full multiplying and dividing would1 moreover1 not only be uneconomical as a means to teaching cancellation2 it would amount to casting needless slurs on valuable past acquisitions1 and it would1 scientifically1 be false4 ,for1 until a pupil has learned to cancel1 the old full multiplying is not inadequate2 it is admirable in every respect4 ,the issue of its inadequacy does not truly appear until the new method is found4 ,it is the best way until the better way is mastered4 ,in the same way it is unwise to spend time in making pupils aware of the annoying lacks to be supplied by the multiplication tables1 the division tables1 the casting out of nines1 or the use of the product of the length and breadth of a rectangle as its area1 the unit being changed to the square erected on the linear unit as base4 ,the annoying lack will be unproductive1 while the learning takes place readily as a modification of existing habits1 and is sufficiently appreciated as soon as it does take place4 ,the multiplication tables come when instead of merely counting by #gs from #j up saying #g1 #ad1 #ba1 etc41 the pupil counts by #gs from #j up saying ,two sevens make #ad1 three sevens make #ba1 four sevens make #bh1 etc4 ,the division tables come as easy selections from the known multiplications2 the casting out of nines comes as an easy device4 ,the computation of the area of a rectangle is best facilitated1 not by awareness of the lack of a process for doing it1 but by awareness of the success of the process as verified objectively4 ,in all these cases1 too1 the pupil would be misled if we aroused first a sense of the inadequacy of counting1 adding1 and objective division1 an awareness of the difficulties which the multiplication and division tables and nines device and area theorem relieve4 ,the displaced processes are admirable and no unnecessary fault should be found with them1 and they are not inadequate until the shorter ways have been learned4 ,,false ,,inferences ,one false inference about the problem-attitude is that the pupil should always understand the aim or issue before beginning to form the bonds which give the method or process that provides the solution4 ,on the contrary1 he will often get the process more easily and value it more highly if he is taught what it is for gradually while he is learning it4 ,the system of decimal notation1 for example1 may better be taken first as a mere fact1 just as we teach a child to talk without trying first to have him understand the value of verbal intercourse1 or to keep clean without trying first to have him understand the bacteriological consequences of filth4 ,a second inference--that the pupil should always be taught to care about an issue and crave a process for managing it before beginning to learn the process--is equally false4 ,on the contrary1 the best way to become interested in certain issues and the ways of handling them is to learn the process--even to learn it by sheer habituation--and then note what it does for us4 ,such is the case with 4#afff-b_/c x divide by #f1 4#ccc-a_/c x divide by #c1 multiply by 4#hge divide the number by #h and subtract the quotient from the number4 ,a third unwise tendency is to degrade the mere giving of information--to belittle the value of facts acquired in any other way than in the course of deliberate effort by the pupil to relieve a problem or conflict or difficulty4 ,as a protest against merely verbal knowledge1 and merely memoriter knowledge1 and neglect of the active1 questioning search for knowledge1 this tendency to belittle mere facts has been healthy1 but as a general doctrine it is itself equally one-sided4 ,mere facts not got by the pupil's thinking are often of enormous value4 ,they may stimulate to active thinking just as truly as that may stimulate to the reception of facts4 ,in arithmetic1 for example1 the names of the numbers1 the use of the fractional form to signify that the upper number is divided by the lower number1 the early use of the decimal point in ,u4 ,s4 money to distinguish dollars from cents1 and the meanings of each1 whole1 part1 together1 in all1 sum1 difference1 product1 quotient1 and the like are self-justifying facts4 ,a fourth false inference is that whatever teaching makes the pupil face a question and think out its answer is thereby justified4 ,this is not necessarily so unless the question is a worthy one and the answer that is thought out an intrinsically valuable one and the process of thinking used one that is appropriate for that pupil for that question4 ,merely to think may be of little value4 ,to rely much on formal discipline is just as pernicious here as elsewhere4 ,the tendency to emphasize the methods of learning arithmetic at the expense of what is learned is likely to lead to abuses different in nature but as bad in effect as that to which the emphasis on disciplinary rather than content value has led in the study of languages and grammar1 or in the old puzzle problems of arithmetic4 ,the last false inference that ,i shall discuss here is the inference that most of the problems by which arithmetical learning is stimulated had better be external to arithmetic itself--problems about ,noah's ,ark or ,easter ,flowers or the ,merry ,go ,round or ,a ,trip down the ,rhine4 ,outside interests should be kept in mind1 as has been abundantly illustrated in this volume1 but it is folly to neglect the power1 even for very young or for very stupid children1 of the problem ,how can ,i get the right answer8 ,children do have intellectual interests4 ,they do like dominoes1 checkers1 anagrams1 and riddles as truly as playing tag1 picking flowers1 and baking cake4 ,with carefully graded work that is within their powers they like to learn to add1 subtract1 multiply1 and divide with integers1 fractions1 and decimals1 and to work out quantitative relations4 ,in some measure1 learning arithmetic is like learning to typewrite4 ,the learner of the latter has little desire to present attractive-looking excuses for being late1 or to save expense for paper4 ,he has no desire to hoard copies of such and such literary gems4 ,he may gain zeal from the fact that a school party is to be given and invitations are to be sent out1 but the problem ,to typewrite better is after all his main problem4 ,learning arithmetic is in some measure a game whose moves are motivated by the general set of the mind toward victory--winning right answers4 ,as a ball-player learns to throw the ball accurately to first-base1 not primarily because of any particular problem concerning getting rid of the ball1 or having the man at first-base possess it1 or putting out an opponent against whom he has a grudge1 but because that skill is required by the game as a whole1 so the pupil1 in some measure1 learns the technique of arithmetic1 not because of particular concrete problems whose solutions it furnishes1 but because that technique is required by the game of arithmetic--a game that has intrinsic worth and many general recommendations4 ,,chapter ,,xv ,,individual ,,differences ,the general facts concerning individual variations in abilities--that the variations are large1 that they are continuous1 and that for children of the same age they usually cluster around one typical or modal ability1 becoming less and less frequent as we pass to very high or very low degrees of the ability--are all well illustrated by arithmetical abilities4 ,,nature ,,and ,,amount ,the surfaces of frequency shown in ,figs4 #fa1 #fb1 and #fc are samples4 ,in these diagrams each space along the baseline represents a certain score or degree of ability1 and the height of the surface above it represents the number of individuals obtaining that score4 ,thus in ,fig4 #fa1 #fc out of #ajjj soldiers had no correct answer1 #cf out of #ajjj had one correct answer1 #di had two1 #ee had three1 #fg had four1 and so on1 in a test with problems 7stated in words74 ,figure #fa shows that these adults varied from no problems solved correctly to eighteen1 around eight as a central tendency4 ,figure #fb shows that children of the same year-age 7they were also from the same neighborhood and in the same school7 varied from under #dj to over #bjj figures correct4 ,figure #fc shows that even among children who have all reached the same school grade and so had rather similar educational opportunities in arithmetic1 the variation is still very great4 ,it requires a range from #ae to over #cj examples right to include even nine tenths of them4 ,illustration3 ,,fig4 #fa4--,the scores of #ajjj soldiers in the ,national ,army born in ,english-speaking countries1 in ,test #b of the ,army ,alpha4 ,the score is the number of correct answers obtained in five minutes4 ,probably #aj to #ae percent of these men were unable to read or able to read only very easy sentences at a very slow rate4 ,data furnished by the ,division of ,psychology in the office of the ,surgeon ,general4 ,it should1 however1 be noted that if each individual had been scored by the average of his work on eight or ten different days instead of by his work in just one test1 the variability would have been somewhat less than appears in ,figs4 #fa1 #fb1 and #fc4 ,illustration3 ,,fig4 #fb4--,the scores of #ajj #aa-year-old pupils in a test of computation4 ,estimated from the data given by ,burt '#ag1 p4 #fh for #aj-1 #aa-1 and #ab-year-olds4 ,the score equals the number of correct figures4 ,it is also the case that if each individual had been scored1 not in problem-solving alone or division alone1 but in an elaborate examination on the whole field of arithmetic1 the variability would have been somewhat less than appears in ,figs4 #fa1 #fb1 and #fc4 ,on the other hand1 if the officers and the soldiers rejected for feeblemindedness had been included in ,fig4 #fa1 if the #aa-year-olds in special classes for the very dull had been included in ,fig4 #fb1 and if all children who had been to school six years had been included in ,fig4 #fc1 no matter what grade they had reached1 the effect would have been to increase the variability4 ,illustration3 ,,fig4 #fc4--,the scores of pupils in grade #f in city schools in the ,woody ,division ,test ,a4 ,the score is the number of correct answers obtained in #bj minutes4 ,from ,woody '#af1 p4 #fa4 ,in spite of the effort by school officers to collect in any one school grade those somewhat equal in ability or in achievement or in a mixture of the two1 the population of the same grades in the same school system shows a very wide range in any arithmetical ability4 ,this is partly because promotion is on a more general basis than arithmetical ability so that some very able arithmeticians are deliberately held back on account of other deficiencies1 and some very incompetent arithmeticians are advanced on account of other excellencies4 ,it is partly because of general inaccuracy in classifying and promoting pupils4 ,in a composite score made up of the sum of the scores in ,woody tests1--,add4 ,a1 ,subt4 ,a1 ,mult4 ,a1 and ,div4 ,a1 and two tests in problem-solving 7ten and six graded problems1 with maximum attainable credits of #cj and #ah71 ,kruse '#ah found facts from which ,i compute those of ,table #ac1 and ,figs4 #fd to #ff1 for pupils all having the training of the same city system1 one which sought to grade its pupils very carefully4 ,illustration3 ,,figs4 #fd1 #fe1 and #ff4--,the scores of pupils in grade #f 7,fig4 #fd71 grade #g 7,fig4 #fe71 and grade #h 7,fig4 #ff7 in a composite of tests in computation and problem-solving4 ,the time was about #abj minutes4 ,the maximum score attainable was #aif4 ,the overlapping of grade upon grade should be noted4 ,of the pupils in grade #f about #ah percent do better than the average pupil in grade #g1 and about #g percent do better than the average pupil in grade #h4 ,of the pupils in grade #h about #cc percent do worse than the average pupil in grade #g and about #ab percent do worse than the average pupil in grade #f4 ,,table #ac ,,relative ,,frequencies ,,of ,,scores ,,in ,,an ,,extensive ,,team ,,of ,,arithmetical ,,tests4#bc ,,in ,,percents ,,score ,,grade #f ,,grade #g ,,grade #h ---------------------------------------- --- #gj to #gi #a4c 4#i 4#d #hj #hi #e4e #b4c 4#d #ij #ii #aj4f #d4c #b4i #ajj #aji #ai4d #e4b #d4d #aaj #aai #ai4h #ah4e #e4h #abj #abi #bc4e #af4b #af4h #acj #aci #ab4f #ag4e #af4h #adj #adi #d4f #ac4i #bb4i #aej #aei #a4g #ac4f #ag4a #afj #afi #a4b #d4h #i4d #agj #agi #b4e #c4c #bc ,compiled from data on p4 #hi of ,kruse '#ah4 ,,differences ,,within ,,one ,,class ,the variation within a single class for which a single teacher has to provide is great4 ,even when teaching is departmental and promotion is by subjects1 and when also the school is a large one and classification within a grade is by ability--there may be a wide range for any given special component ability4 ,under ordinary circumstances the range is so great as to be one of the chief limiting conditions for the teaching of arithmetic4 ,many methods appropriate to the top quarter of the class will be almost useless for the bottom quarter1 and vice versa4 ,illustration3 ,,figs4 #fg and #fh4--,the scores of ten #f ,b classes in a #ab-minute test in computation with integers 7the ,courtis ,test #g74 ,the score is the number of units done4 ,certain long tasks are counted as two units4 ,figures #fg and #fh show the scores of ten classes taken at random from ninety #f ,b classes in one city by ,courtis '#ac1 p4 #fd in amount of computation done in #ab minutes4 ,observe the very wide variation present in the case of every class4 ,the variation within a class would be somewhat reduced if each pupil were measured by his average in eight or ten such tests given on different days4 ,if a rather generous allowance is made for this we still have a variation in speed as great as that shown in ,fig4 #fi1 as the fact to be expected for a class of thirty-two #f ,b pupils4 ,illustration3 ,,fig4 #fi4--;,a conservative estimate of the amount of variation to be expected within a single class of #cb pupils in grade #f1 in the number of units done in ,courtis ,test #g when all chance variations are eliminated4 ,the variations within a class in respect to what processes are understood so as to be done with only occasional errors may be illustrated further as follows3--,a teacher in grade #d at or near the middle of the year in a city doing the customary work in arithmetic will probably find some pupil in her class who cannot do column addition even without carrying1 or the easiest written subtraction 7#h #i #gh7 7#e #c or #cg7 7- - --71 who does not know his multiplication tables or how to derive them1 or understand the meanings of - x and _/1 or have any useful ideas whatever about division4 ,there will probably be some child in the class who can do such work as that shown below1 and with very few errors4 ,add #c_/h #e_/h #g_/h #a_/h #b-a_/b #a_/f #c_/h #f-c_/h #c-c_/d ----- ,subtract #aj4jj #d yd4 #a ft4 #f in4 #c4di #b yd4 #b ft4 #c in4 ----- ---------------------- ,multiply #a-a_/d x #h #af #ade #b-e_/h #bjf ------ --- ,divide #b7ac4ej #be7igej ,the invention of means of teaching thirty so different children at once with the maximum help and minimum hindrance from their different capacities and acquisitions is one of the great opportunities for applied science4 ,courtis1 emphasizing the social demand for a certain moderate arithmetical attainment in the case of nearly all elementary school children of1 say1 grade #f1 has urged that definite special means be taken to bring the deficient children up to certain standards1 without causing undesirable 'overlearning' by the more gifted children4 ,certain experimental work to this end has been carried out by him and others1 but probably much more must be done before an authoritative program for securing certain minimum standards for all or nearly all pupils can be arranged4 ,,the ,,causes ,,of ,,individual ,,differences ,the differences found among children of the same grade in the same city are due in large measure to inborn differences in their original natures4 ,if1 by a miracle1 the children studied by ,courtis1 or by ,woody1 or by ,kruse had all received exactly the same nurture from birth to date1 they would still have varied greatly in arithmetical ability1 perhaps almost as much as they now do vary4 ,the evidence for this is the general evidence that variation in original nature is responsible for much of the eventual variation found in intellectual and moral traits1 plus certain special evidence in the case of arithmetical abilities themselves4 ,thorndike found '#je that in tests with addition and multiplication twins were very much more alike than siblings#bd two or three years apart in age1 though the resemblance in home and school training in arithmetic should be nearly as great for the latter as for the former4 ,also the young twins 7#i-aa7 showed as close a resemblance in addition and multiplication as the older twins 7#ab-ae71 although the similarities of training in arithmetic have had twice as long to operate in the latter case4 #bd ,siblings is used for children of the same parents4 ,if the differences found1 say among children in grade #f in addition1 were due to differences in the quantity and quality of training in addition which they have had1 then by giving each of them #bjj minutes of additional identical training the differences should be reduced4 ,for the #bjj minutes of identical training is a step toward equalizing training4 ,it has been found in many investigations of the matter that when we make training in arithmetic more nearly equal for any group the variation within the group is not reduced4 ,on the contrary1 equalizing training seems rather to increase differences4 ,the superior individual seems to have attained his superiority by his own superiority of nature rather than by superior past training1 for1 during a period of equal training for all1 he increases his lead4 ,for example1 compare the gains of different individuals due to about #cjj minutes of practice in mental multiplication of a three-place number by a three-place number shown in ,table #ad below1 from data obtained by the author '#jh4be #be ,similar results have been obtained in the case of arithmetical and other abilities by ,thorndike '#jh1 '#aj1 '#ae1 '#af1 ,whitley '#aa1 ,starch '#aa1 ,wells '#ab1 ,kirby '#ac1 ,donovan and ,thorndike '#ac1 ,hahn and ,thorndike '#ad1 and on a very large scale by ,race in a study as yet unpublished4 ,,table #ad ,,the ,,effect ,,of ,,equal ,,amounts ,,of ,,practice ,,upon ,,individual ,,difference ,,in ,,the ,,multiplication ,,of ,,three-,,place ,,numbers ,,amount ,,percentage ,,of ,,correct ,,figures ------------------------------- ,initial ,initial ,score ,gain ,score ,gain ---------------------------------------- ------------------------ ,initially highest five individuals #he #fa #gj #ah next five #ef #ea #fh #aj next six #df #bb #gd #h next six #ch #h #eh #ab next six #bi #bd #ef #ad ,,the ,,interrelations ,,of ,,individual ,,differences ,achievement in arithmetic depends upon a number of different abilities4 ,for example1 accuracy in copying numbers depends upon eyesight1 ability to perceive visual details1 and short-term memory for these4 ,long column addition depends chiefly upon great strength of the addition combinations especially in higher decades1 'carrying1' and keeping one's place in the column4 ,the solution of problems framed in words requires understanding of language1 the analysis of the situation described into its elements1 the selection of the right elements for use at each step and their use in the right relations4 ,since the abilities which together constitute arithmetical ability are thus specialized1 the individual who is the best of a thousand of his age or grade in respect to1 say1 adding integers1 may occupy different stations1 perhaps from #ast to #fjjth1 in multiplying with integers1 placing the decimal point in division with decimals1 solving novel problems1 copying figures1 etc41 etc4 ,such specialization is in part due to his having had1 relatively to the others in the thousand1 more or better training in certain of these abilities than in others1 and to various circumstances of life which have caused him to have1 relatively to the others in the thousand1 greater interest in certain of these achievements than in others4 ,the specialization is not wholly due thereto1 however4 ,certain inborn characteristics of an individual predispose him to different degrees of superiority or inferiority to other men in different features of arithmetic4 ,we measure the extent to which ability of one sort goes with or fails to go with ability of some other sort by the coefficient of correlation between the two4 ,if every individual keeps the same rank in the second ability--if the individual who is the best of the thousand in one is the best of the group in the other1 and so on down the list--the correlation is #a4jj4 ,in proportion as the ranks of individuals vary in the two abilities the coefficient drops from #a4jj1 a coefficient of #j meaning that the best individual in ability ,a is no more likely to be in first place in ability ,b than to be in any other rank4 ,the meanings of coefficients of correlation of 4#ij1 4#gj1 4#ej1 and #j are shown by ,tables #ae1 #af1 #ag and #ah4bf #bf ,unless he has a thorough understanding of the underlying theory1 the student should be very cautious in making inferences from coefficients of correlation4 ,,table #ae ,,distribution ,,of ,,arrays ,,in ,,successive ,,tenths ,,of ,,the ,,group ,,when r 4#ij #aj,,th #i,,th #h,,th #g,,th #f,,th #e,,th #d,,th #c;,d #b;,d #a,,st ---------------------------------------- -------------------- #ast tenth 4#a 4#d #a4h #f4f #bb4d #fh4g #b;d tenth 4#a 4#d #a4d #d4g #aa4e #bc4e #cf4j #bb4d #c;d tenth 4#a 4#e #b4a #e4h #ab4h #ba4a #bg4d #bc4e #f4f #dth tenth 4#d #b4a #f4d #ab4h #bj4a #bc4h #ba4b #aa4e #a4h #eth tenth 4#a #a4d #e4h #ab4h #ai4c #bb4f #bj4a #ab4h #d4g 4#d #fth tenth 4#d #d4g #ab4h #bj4a #bb4f #ai4c #ab4h #e4h #a4d 4#a #gth tenth #a4h #aa4e #ba4b #bc4h #bj4a #ab4h #f4d #b4a 4#d #hth tenth #f4f #bc4e #bg4d #ba4a #ab4h #e4h #b4a 4#e 4#a #ith tenth #bb4d #cf4j #bc4e #aa4e #d4g #a4d 4#d 4#a #ajth tenth#fh4g #bb4d #f4f #a4h 4#d 4#a ,,table #af ,,distribution ,,of ,,arrays ,,in ,,successive ,,tenths ,,of ,,the ,,group ,,when r 4#gj #aj,,th #i,,th #h,,th #g,,th #f,,th #e,,th #d,,th #c;,d #b;,d #a,,st ---------------------------------------- -------------------- #ast tenth 4#b 4#g #a4e #b4h #d4h #h4j #ac4j #bb4c #df4g #b;d tenth 4#b #a4b #b4f #d4e #g4j #i4h #ac4d #ag4c #ba4g #bb4c #c;d tenth 4#g #b4f #e4j #g4c #aj4j #ab4e #ad4i #af4g #ag4c #ac4j #dth tenth #a4e #d4e #g4c #i4h #ab4j #ac4g #ad4h #ad4i #ac4d #h4j #eth tenth #b4h #g4j #aj4j #ab4j #ac4d #ad4j #ac4g #ab4e #i4h #d4h #fth tenth #d4h #i4h #ab4e #ac4g #ad4j #ac4d #ab4j #aj4j #g4j #b4h #gth tenth #h4j #ac4d #ad4i #ad4h #ac4g #ab4j #i4h #g4c #d4e #a4e #hth tenth #ac4j #ag4c #af4g #ad4i #ab4e #aj4j #g4c #e4j #b4f 4#g #ith tenth #bb4c #ba4g #ag4c #ac4d #i4h #g4j #d4e #b4f #a4b 4#b #ajth tenth#df4g #bb4c #ac4j #h4j #d4h #b4h #a4e 4#g 4#b ,,table #ag ,,distribution ,,of ,,arrays ,,of ,,successive ,,tenths ,,of ,,the ,,group ,,when r 4#ej #aj,,th #i,,th #h,,th #g,,th #f,,th #e,,th #d,,th #c;,d #b;,d #a,,st ---------------------------------------- -------------------- #ast tenth 4#h #b4j #c4b #d4f #f4b #h4a #aj4e #ac4i #ah4j #ca4h #b;d tenth #b4j #d4a #e4g #g4c #h4h #aj4e #ab4b #ad4a #af4d #ah4i #c;d tenth #c4b #e4g #g4d #h4i #aj4j #aa4b #ab4c #ac4c #ad4a #ac4i #dth tenth #d4f #g4c #h4h #i4i #aj4h #aa4f #ab4j #ab4c #ab4b #aj4e #eth tenth #f4b #h4h #aj4j #aj4h #aa4c #aa4e #aa4f #aa4b #aj4e #h4a #fth tenth #h4a #aj4e #aa4b #aa4f #aa4e #aa4c #aj4h #aj4j #h4h #f4b #gth tenth #aj4e #ab4b #ab4c #ab4j #aa4f #aj4h #i4i #h4h #g4e #d4f #hth tenth #ac4i #ad4a #ac4c #ab4c #aa4b #aj4j #h4h #g4d #e4g #c4b #ith tenth #ah4i #af4d #ad4a #ab4b #aj4e #h4h #g4c #e4g #d4a #b4j #ajth tenth#ca4h #ah4i #ac4i #aj4e #h4a #f4b #d4f #c4b #b4j 4#h ,,table #ah ,,distribution ,,of ,,arrays1 ,,in ,,successive ,,tenths ,,of ,,the ,,group ,,when r 4#j #aj,,th #i,,th #h,,th #g,,th #f,,th #e,,th #d,,th #c;,d #b;,d #a,,st ---------------------------------------- -------------------- #ast tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #b;d tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #c;d tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #dth tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #eth tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #fth tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #gth tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #hth tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #ith tenth #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #ajth tenth#aj #aj #aj #aj #aj #aj #aj #aj #aj #aj ,the significance of any coefficient of correlation depends upon the group of individuals for which it is determined4 ,a correlation of 4#dj between computation and problem-solving in eighth-grade pupils of #ad years would mean a much closer real relation than a correlation of 4#dj in all #ad-year-olds1 and a very1 very much closer relation than a correlation of 4#dj for all children #h to #ae4 ,unless the individuals concerned are very elaborately tested on several days1 the correlations obtained are attenuated toward #j by the accidental errors in the original measurements4 ,this effect was not known until #aijd2 consequently the correlations in the earlier studies of arithmetic are all too low4 ,in general1 the correlation between ability in any one important feature of computation and ability in any other important feature of computation is high4 ,if we make enough tests to measure each individual exactly in3-- 7,a7 ,subtraction with integers and decimals1 7,b7 ,multiplication with integers and decimals1 7,c7 ,division with integers and decimals1 7,d7 ,multiplication and division with common fractions1 and 7,e7 ,computing with percents1 we shall probably find the intercorrelations for a thousand #ad-year-olds to be near 4#ij4 ,addition of integers 7,f7 will1 however1 correlate less closely with any of the above1 being apparently dependent on simpler and more isolated abilities4 ,the correlation between problem-solving 7,g7 and computation will be very much less1 probably not over 4#fj4 ,it should be noted that even when the correlation is as high as 4#ij1 there will be some individuals very high in one ability and very low in the other4 ,such disparities are to some extent1 as ,courtis '#ac1 pp4 #fg-ge and ,cobb '#ag have argued1 due to inborn characteristics of the individual in question which predispose him to very special sorts of strength and weakness4 ,they are often due1 however1 to defects in his learning whereby he has acquired more ability than he needs in one line of work or has failed to acquire some needed ability which was well within his capacity4 ,in general1 all correlations between an individual's divergence from the common type or average of his age for one arithmetical function1 and his divergences from the average for any other arithmetical function1 are positive4 ,the correlation due to original capacity more than counterbalances the effects that robbing ,peter to pay ,paul may have4 ,speed and accuracy are thus positively correlated4 ,the individuals who do the most work in ten minutes will be above the average in a test of accuracy4 ,the common notion that speed is opposed to accuracy is correct when it means that the same person will tend to make more errors if he works at too rapid a rate2 but it is entirely wrong when it means that the kind of person who works more rapidly than the average person is likely to be less accurate than the average person4 ,interest in arithmetic and ability at arithmetic are probably correlated positively in the sense that the pupil who has more interest than other pupils of his age tends in the long run to have more ability than they4 ,they are certainly correlated in the sense that the pupil who 'likes' arithmetic better than geography or history tends to have relatively more ability in arithmetic1 or1 in other words1 that the pupil who is more gifted at arithmetic than at drawing or ,english tends also to like it better than he likes these4 ,these correlations are high4 ,it is correct then to think of mathematical ability as1 in a sense1 a unitary ability of which any one individual may have much or little1 most individuals possessing a moderate amount of it4 ,this is consistent1 however1 with the occasional appearance of individuals possessed of very great talents for this or that particular feature of mathematical ability and equally notable deficiencies in other features4 ,finally it may be noted that ability in arithmetic1 though occasionally found in men otherwise very stupid1 is usually associated with superior intelligence in dealing with ideas and symbols of all sorts1 and is one of the best early indications thereof4 ,,bibliography ,,of ,,references ,,made ,,in ,,the ,,text ,ames1 ,a4 ,f41 and ,mc,lellan1 ,j4 ,f42 '#jj ,public ,school ,arithmetic4 ,ballou1 ,f4 ,w42 '#af ,determining the ,achievements of ,pupils in the ,addition of ,fractions4 ,school ,document ,no4 #c1 #aiaf1 ,boston ,public ,schools4 ,brandell1 ,g42 '#ac ,skolbarns intressen4 ,translated '#ae by ,w4 ,stern as1 ,das ,interesse der ,schulkinder an den ,unterrichtsfaechern4 ,brandford1 ,b42 '#jh ,a ,study of ,mathematical ,education4 ,brown1 ,j4 ,c42 '#aa1 '#ab ,an ,investigation on the ,value of ,drill ,work in the ,fundamental ,operations in ,arithmetic4 ,journal of ,educational ,psychology1 vol4 #b1 pp4 #ha-hh1 vol4 #c1 pp4 #dhe-dib and #efa-egj4 ,brown1 ,j4 ,c4 and ,coffman1 ,l4 ,d42 '#ad ,how to ,teach ,arithmetic4 ,burgerstein1 ,l42 '#ia ,die ,arbeitscurve einer ,schulstunde4 ,zeitschrift fuer ,schulgesundheitspflege1 vol4 #d1 pp4 #edc-efb and #fjg-fbg4 ,burnett1 ,c4 ,j42 '#jf ,the ,estimation of ,number4 ,harvard ,psychological ,studies1 vol4 #b1 pp4 #cdi-djd4 ,burt1 ,c42 '#ag ,the ,distribution and ,relations of ,educational ,abilities4 ,report of ,the ,london ,county ,council1 ,no4 #ahfh4 ,chapman1 ,j4 ,c42 '#ad ,individual ,differences in ,ability and ,improvement and ,their ,correlations4 ,teachers ,college ,contributions to ,education1 ,no4 #fc4 ,chapman1 ,j4 ,c42 '#ag ,the ,scientific ,measurement of ,classroom ,products4 7,with ,g4 ,p4 ,rush47 ,cobb1 ,m4 ,v42 '#ag ,a ,preliminary ,study of the ,inheritance of ,arithmetical ,abilities4 ,jour4 of ,educational ,psychology1 vol4 #h1 pp4 #a-bj4 ,jan41 #aiag4 ,coffman1 ,l4 ,d41 and ,brown1 ,j4 ,c42 '#ad ,how to ,teach ,arithmetic4 ,coffman1 ,l4 ,d41 and ,jessup1 ,w4 ,a42 '#af ,the ,supervision of ,arithmetic4 ,courtis1 ,s4 ,a42 '#ji1 '#aj1 '#aa ,measurement of ,growth and ,efficiency in ,arithmetic4 ,elementary ,school ,teacher1 vol4 #aj1 pp4 #eh-gd and #agg-aii1 vol4 #aa1 pp4 #aga-ahe1 #cfj-cgj1 and #ebh-eci4 ,courtis1 ,s4 ,a42 '#aa1 '#ab ,report on ,educational ,aspects of the ,public ,school ,system of the ,city of ,new ,york4 ,part ,,ii1 ,subdivision #a1 ,section ,d4 ,report on the ,courtis ,tests in ,arithmetic4 ,courtis1 ,s4 ,a42 '#ac ,courtis ,standard ,tests4 ,second ,annual ,accounting4 ,courtis1 ,s4 ,a42 '#ad ,manual of ,instructions for ,giving and ,scoring the ,courtis ,standard ,tests in the ,three ,r's4 ,department of ,comparative ,research1 #hb ,elliot ,st41 ,detroit1 ,mich41 #aiad4 ,decroly1 ,m41 and ,degand1 ,j42 '#ab ,l'evolution des notions de quantites continues et discontinues chez l'enfant4 ,archives de psychologie1 vol4 #ab1 pp4 #ha-aba4 ,degand1 ,j4 ,see ,decroly4 ,de ,voss1 ,j4 ,c4 ,see ,monroe1 ,de ,voss1 and ,kelly4 ,dewey1 ,j42 '#aj ,how ,we ,think4 ,dewey1 ,j41 and ,mc,lellan1 ,j4 ,a42 '#ie ,psychology of ,number and ,its ,applications to ,methods of ,teaching ,arithmetic4 ,donovan1 ,m4 ,e41 and ,thorndike1 ,e4 ,l42 '#ac ,improvement in a ,practice ,experiment under ,school ,conditions4 ,american ,journal of ,psychology1 vol4 #bd1 pp4 #dbf-dbh4 ,elliott1 ,c4 ,h42 '#ad ,variation in the ,achievements of ,pupils4 ,teachers ,college1 ,columbia ,university1 ,contributions to ,education1 ,no4 #gb4 ,flynn1 ,f4 ,j42 '#ab ,mathematical ,games--,adaptations from ,games ,old and ,new4 ,teachers ,college ,record1 vol4 #ac1 pp4 #cii-dab4 ,freeman1 ,f4 ,n42 '#aj ,untersuchungen ueber den ,aufmerksamkeitsumfang und die ,zahlauffassung4 ,paedagogische-,psychologische ,arbeiten1 ,i1 #hh-afh4 ,friedrich1 ,j42 '#ig ,untersuchungen ueber die ,einfluesse der ,arbeitsdauer und die ,arbeitspausen auf die geistige ,leistungsfaehigkeit der ,schulkinder4 ,zeitschrift fuer ,psychologie1 vol4 #ac1 pp4 #a-ec4 ,gilbert1 ,j4 ,a42 '#id ,researches on the ,mental and ,physical ,development of ,school ,children4 ,studies from the ,yale ,psychological ,laboratory1 vol4 #b1 pp4 #dj-ajj4 ,greenleaf1 ,b42 '#gc ,practical ,arithmetic4 ,hahn1 ,h4 ,h41 and ,thorndike1 ,e4 ,l42 '#ad ,some ,results of ,practice in ,addition under ,school ,conditions4 ,journal of ,educational ,psychology1 vol4 #e1 ,no4 #b1 pp4 #fe-hd4 ,hall1 ,g4 ,s42 '#hc ,the ,contents of ,children's ,minds on ,entering ,school4 ,princeton ,review1 vol4 ,,ii1 pp4 #bdi-bgb1 ,may1 #ahhc4 ,reprinted in ,aspects of ,child ,life and ,education1 #aijg4 ,hartmann1 ,b42 '#ij ,die ,analyze des ,kindlichen ,gedanken-,kreises als die ,naturgemaessedes ,ersten ,schulunterrichts1 #ahij4 ,heck1 ,w4 ,h42 '#ac ,a ,study of ,mental ,fatigue4 ,heck1 ,w4 ,h42 '#ac ,a ,second ,study in ,mental ,fatigue in the ,daily ,school ,program4 ,psychological ,clinic1 vol4 #g1 pp4 #bi-cd4 ,hoffmann1 ,p42 '#aa ,das ,interesse der ,schueler an den ,unterrichtsfaechern4 ,zeitschrift fuer paedagogische ,psychologie1 ,,xii1 #deh-dgj4 ,hoke1 ,k4 ,j41 and ,wilson1 ,g4 ,m42 '#bj ,how to ,measure4 ,holmes1 ,m4 ,e42 '#ie ,the ,fatigue of a ,school ,hour4 ,pedagogical ,seminary1 vol4 #c1 pp4 #bac-bcd4 ,howell1 ,h4 ,b42 '#ad ,a ,foundation ,study in the ,pedagogy of ,arithmetic4 ,hunt1 ,c4 ,w42 '#ab ,play and ,recreation in ,arithmetic4 ,teachers ,college ,record1 vol4 #ac1 pp4 #chh-cih4 ,jessup1 ,w4 ,a41 and ,coffman1 ,l4 ,d42 '#af ,the ,supervision of ,arithmetic4 ,kelly1 ,f4 ,j4 ,see ,monroe1 ,de ,voss and ,kelly4 ,king1 ,a4 ,c42 '#jg ,the ,daily ,program in ,elementary ,schools4 ,,mss4 ,kirby1 ,t4 ,j42 '#ac ,practice in the ,case of ,school ,children4 ,teachers ,college ,contributions to ,education1 ,no4 #eh4 ,klapper1 ,p42 '#af ,the ,teaching of ,arithmetic4 ,kruse1 ,p4 ,j42 '#ah ,the ,overlapping of ,attainments in ,certain ,sixth1 ,seventh1 and ,eighth ,grades4 ,teachers ,college1 ,columbia ,university1 ,contributions to ,education1 ,no4 #ib4 ,laser1 ,h42 '#id ,ueber geistige ,ermuedung beim ,schulunterricht4 ,zeitschrift fuer ,schulgesundheitspflege1 vol4 #g1 pp4 #b-bb4 ,lay1 ,w4 ,a42 '#ih ,fuehrer durch den ersten ,rechenunterricht4 ,lay1 ,w4 ,a42 '#jg ,fuehrer durch den ,rechenunterricht der ,unterstufe4 ,lewis1 ,e4 ,o42 '#ac ,popular and ,unpopular ,school-,subjects4 ,the ,journal of ,experimental ,pedagogy1 vol4 #b1 pp4 #hi-ih4 ,lobsien1 ,m42 '#jc ,kinderideale4 ,zeitschrift fuer paedagogische ,psychologie1 ,v1 #cbc-cdd and #deg-did4 ,lobsien1 ,m42 '#ji ,beliebtheit und ,unbeliebtheit der ,unterrichtsfaecher4 ,paedagogisches ,magazin1 ,heft #cfa4 ,mc,call1 ,w4 ,a42 '#ba ,how to ,measure in ,education4 ,mc,dougle1 ,e4 ,c42 '#ad ,a ,contribution to the ,pedagogy of ,arithmetic4 ,pedagogical ,seminary1 vol4 #ba1 pp4 #afa-bah4 ,mc,knight1 ,j4,a42 '#jg ,differentiation of the ,curriculum in the ,upper ,grammar ,grades4 ,,mss4 in the library of ,teachers ,college1 ,columbia ,university4 ,mc,lellan1 ,j4,a41 and ,dewey1 ,j42 '#ie ,psychology of ,number and ,its ,applications to ,methods of ,teaching4 ,mc,lellan1 ,j4,a41 and ,ames1 ,a4,f42 '#jj ,public ,school ,arithmetic4 ,messenger1 ,j4,f42 '#jc ,the ,perception of ,number4 ,psychological ,review1 ,monograph ,supplement ,no4 #bb4 ,meumann1 ,e42 '#jg ,vorlesungen zur ,einfuehrung in die experimentelle ,paedagogik4 ,mitchell1 ,h4,e42 '#bj ,unpublished studies of the uses of arithmetic in factories1 shops1 farms1 and the like4 ,monroe1 ,w4,s41 ,de ,voss1 ,j4,c41 and ,kelly1 ,f4,j42 '#ag ,educational ,tests and ,measurements4 ,nanu1 ,h4,a42 '#jd ,zur ,psychologie der ,zahl ,auffassung4 ,national ,intelligence ,tests2 '#bj ,scale ,a1 ,form #a1 ,edition #a4 ,phillips1 ,d4,e42 '#ig ,number and ,its ,application ,psychologically ,considered4 ,pedagogical ,seminary1 vol4 #e1 pp4 #bba-bha4 ,pommer1 ,o42 '#ad ,die ,erforschung der ,beliebtheit der ,unterrichtsfaecher4 ,ihre psychologischen ,grundlagen und ihre paedagog4 ,bedeutung4 ,,vii4 ,jahresber4 des k4k4 ,ssaatsgymn4 im ,,xviii ,bez4 v4 ,wien4 ,rice1 ,j4,m42 '#jb ,test in ,arithmetic4 ,forum1 vol4 #cd1 pp4 #bha-big4 ,rice1 ,j4,m42 '#jc ,causes of ,success and ,failure in ,arithmetic4 ,forum1 vol4 #cd1 pp4 #dcg-deb4 ,rush1 ,g4,p42 '#ag ,the ,scientific ,measurement of ,classroom ,products4 7,with ,j4 ,c4 ,chapman47 ,seekel1 ,e42 '#ad ,ueber die ,beziehung zwischen der ,beliebtheit und der ,schwierigkeit der ,schulfaecher4 ,ergebnisse einer ,erhebung4 ,zeitschrift fuer ,angewandte ,psychologie #i4 ,s4 #bfh-bgg4 ,selkin1 ,f4 ,b42 '#ab ,number ,games ,bordering on ,arithmetic and ,algebra4 ,teachers ,college ,record1 vol4 #ac1 pp4 #deb-dic4 ,smith1 ,d4 ,e42 '#ja ,the ,teaching of ,elementary ,mathematics4 ,smith1 ,d4 ,e42 '#aa ,the ,teaching of ,arithmetic4 ,speer1 ,w4 ,w42 '#ig ,arithmetic3 ,elementary for ,pupils4 ,starch1 ,d42 '#aa ,transfer of ,training in ,arithmetical ,operations4 ,journal of ,educational ,psychology1 vol4 #b1 pp4 #cjf-caj4 ,starch1 ,d42 '#af ,educational ,measurements4 ,stern1 ,w42 '#je ,ueber ,beliebtheit und ,unbeliebtheit der ,schulfaecher4 ,zeitschrift fuer paedagogische ,psychologie1 ,,vii1 #bfg-bif4 ,stern1 ,c41 and ,stern1 ,w42 '#ac ,beliebtheit und ,schwierigkeit der ,schulfaecher4 7,freie ,schulgemeinde ,wickersdorf47 ,auf ,grund der von ,herrn ,luserke beschafften ,materialien bearbeitet4 ,in3 ,die ,ausstellung zur vergleichenden ,jungendkunde der ,geschlechter in ,breslau4 ,arbeit #g des ,bundes fuer ,schulreform4 ,s4 #bd-bf4 ,stern1 ,w42 '#ad ,zur vergleichenden ,jugendkunde der ,geschlechter4 ,vortrag4 ,,iii4 ,deutsch4 ,kongr4 f4 ,jugendkunde usw4 ,arbeiten #h des ,bundes fuer ,schulreform4 ,s4 #ag-ch4 ,stone1 ,c4,w42 '#jh ,arithmetical ,abilities and ,some ,factors ,determining ,them4 ,teachers ,college ,contributions to ,education1 ,no4 #ai4 ,suzzallo1 ,h42 '#aa ,the ,teaching of ,primary ,arithmetic4 ,thorndike1 ,e4,l42 '#jj ,mental ,fatigue4 ,psychological ,review1 vol4 #g1 pp4 #dff-dhb and #edg-egi4 ,thorndike1 ,e4,l42 '#jh ,the ,effect of ,practice in the ,case of a ,purely ,intellectual ,function4 ,american ,journal of ,psychology1 vol4 #aj1 pp4 #cgd-chd4 ,thorndike1 ,e4,l42 '#aj ,practice in the case of ,addition4 ,american ,journal of ,psychology1 vol4 #ba1 pp4 #dhc-dhf4 ,thorndike1 ,e4,l41 and ,donovan1 ,m4,e42 '#ac ,improvement in a ,practice ,experiment under ,school ,conditions4 ,american ,journal of ,psychology1 vol4 #bd1 pp4 #dbf-dbh4 ,thorndike1 ,e4,l41 and ,donovan1 ,m4,e41 and ,hahn1 ,h4,h42 '#ad ,some ,results of ,practice in ,addition under ,school ,conditions4 ,journal of ,educational ,psychology1 vol4 #e1 ,no4 #b1 pp4 #fe-hd4 ,thorndike1 ,e4,l42 '#ae ,the ,relation between ,initial ,ability and ,improvement in a ,substitution ,test4 ,school and ,society1 vol4 #ab1 p4 #dbi4 ,thorndike1 ,e4,l42 '#af ,notes on ,practice1 ,improvability and the ,curve of ,work4 ,american ,journal of ,psychology1 vol #bg1 pp4 #eej-efe4 ,walsh1 ,j4,h42 '#jf ,grammar ,school ,arithmetic4 ,wells1 ,f4,l42 '#ab ,the ,relation of ,practice to ,individual ,differences4 ,american ,journal of ,psychology1 vol4 #bc1 pp4 #ge-hh4 ,white1 ,e4 ,e42 '#hc ,a ,new ,elementary ,arithmetic4 ,whitley1 ,m4 ,t42 '#aa ,an ,empirical ,study of ,certain ,tests for ,individual ,differences4 ,archives of ,psychology1 ,no4 #ai4 ,wiederkehr1 ,g42 '#jg ,statistiche ,untersuchungen ueber die ,art und den ,grad des ,interesses bei ,kindern der ,volksschule4 ,neue ,bahnen1 vol4 #ai1 pp4 #bda-bea1 #bhi-bii4 ,wilson1 ,g4 ,m42 '#ai ,a ,survey of the ,social and ,business ,usage of ,arithmetic4 ,teachers ,college ,contributions to ,education1 ,no4 #ajj4 ,wilson1 ,g4 ,m41 and ,hoke1 ,k4 ,j42 '#bj ,how to ,measure4 ,woody1 ,c42 '#aj ,measurements of ,some ,achievements in ,arithmetic4 ,teachers ,college ,contributions to ,education1 ,no4 #hj4 ,,index ,abilities1 arithmetical1 nature of1 #a ff42 measurement of1 #bg ff42 constitution of1 #ea ff42 organization of1 #acg ff4 ,abstract numbers1 #he ff4 ,abstraction1 #afi ff4 ,accuracy1 in relation to speed1 #ca2 in fundamental operations1 #ajb ff4 ,addition1 measurement of1 #bg ff41 #cd2 constitution of1 #eb f42 habit in relation to1 #ga f42 in 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#di ,,holmes1 ,m4 ,e41 #ajc ,,howell1 ,h4 ,b41 #bei ,,hunt1 ,c4 ,w41 #aif ,hygiene of arithmetic1 #bab ff41 #bcd ff4 ,individual differences1 #bhe ff4 ,inductive reasoning1 #fj ff41 #afi ff4 ,insurance1 #bef ,interest as a principle determining the order of topics1 #aej ff4 ,interests1 instinctive #aie ff42 censuses of1 #bji ff42 neglect of childish1 #bbf ff42 in self-management1 #bbc f42 intrinsic1 #bbd ff4 ,interference1 #adc ff4 ,inventories of arithmetical knowledge and skill1 #aii ff4 ,,jessup1 ,w4 ,a41 xvi ,,kelly1 ,f4 ,j41 #di ,,king1 ,a4 ,c41 #ajc1 #bbg ,,kirby1 ,t4 ,j41 #gf f41 #ajd1 #bie ,,klapper1 ,p41 xvi ,,kruse1 ,p4 ,j41 #bhi1 #bic ,ladder tests1 #bh ff41 #cf ff4 ,language in arithmetic1 #h f41 #ai1 #hi ff41 #id ff4 ,,laser1 ,h41 #ajc ,,lay1 ,w4 ,a41 #bei1 #bfa ,learning1 nature of arithmetical1 #a ff4 ,least common multiple1 #hh f4 ,,lewis1 ,e4 ,o41 #baj f4 ,,lobsien1 ,m41 #bji f4 ,,mccall1 ,w4 ,a41 #di ,,mcdougle1 ,e4 ,c41 #he ff4 ,,mcknight1 ,j4 ,a41 #baj ,,mclellan1 ,j4 ,a41 #c1 #hc1 #hi1 #bje1 #bjg ,manipulation of numbers1 #fj ff4 ,meaning1 of numbers1 #b ff41 #aga2 of a fraction1 #ed ff42 of decimals1 #aha f4 ,measurement of arithmetical abilities1 #bg ff4 ,mental arithmetic1 #bfb ff4 ,,messenger1 ,j4 ,f41 #bjb ,metric system1 #adg ,,meumann1 ,e41 #bfa ,,mitchell1 ,h4 ,e41 #bd ,,monroe1 ,w4 ,s41 #di ,multiplication1 measurement of1 #ce1 #cf2 constitution of1 #ea2 deductive explanations of1 #fa2 inductive explanations of1 #fa f42 with fractions1 #gh ff42 by eleven1 #he2 amount of practice in1 #abb ff42 order of learning the elementary facts of1 #add f42 distribution of practice in1 #aeh ff42 use of the problem attitude in teaching1 #bfg ff4 ,,nanu1 ,h4 ,a41 #bjb ,national ,intelligence ,tests1 #di f4 ,negative reaction in intellectual life1 #bgh f4 ,number pictures1 #bei ff4 ,numbers1 meaning of1 #b2 as measures of continuous quantities1 #ge2 abstract and concrete1 #he ff42 denominate1 #ada f41 #adg f42 use of large1 #ade f42 perception of1 #bje ff42 early awareness of1 #bje ff42 confusion of cardinal and ordinal1 #bjf4 ,see also ,decimal numbers and ,fractions4 ,objective aids1 used for verification1 #aed2 in general1 #bdc ff4 ,oral arithmetic1 #bfb ff4 ,order of topics1 #ada ff4 ,ordinal numbers1 confused with cardinal1 #bjf ,original tendencies and arithmetic1 #aie ff4 ,overlearning1 #acd ff4 ,percents1 #hj f4 ,perception of number1 #bjb ff4 ,,phillips1 ,d4 ,e41 #c1 #d1 #bje1 #bjg ,pictures1 hygiene of1 #bdf ff42 number1 #bei ff4 ,,pommer1 ,o41 #bab ,practice1 amount of1 #abb ff42 distribution of1 #aef ff4 ,precision in fundamental operations1 #ajb ff4 ,problem attitude1 #bff ff4 ,problems1 #i ff42 catch1 #ba ff42 measurement of ability with1 #db ff42 whose answer must be known in order to frame them1 #ic f42 verbal form of1 #aaa f42 interest in1 #bbj ff42 as introductions to arithmetical learning1 #bff ff4 ,propaedeutic bonds1 #aag ff4 ,purposive thinking1 #aic ff4 ,quantity1 number and1 #he ff42 perception of1 #bjb ff4 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#di1 #bie ,,stern1 ,w41 #baj1 #bab ,,stone1 ,c4 ,w41 #bg ff41 #db ff41 #bbh ff4 ,subtraction1 measurement of1 #cd f42 constitution of1 #eg f42 amount of practice in1 #abb ff4 ,supervision1 #bcc f4 ,,suzzallo1 ,h41 xvi ,temporary bonds1 #aaa ff4 ,terms1 #aac f4 ,tests of arithmetical abilities1 #bg ff4 ,,thorndike1 ,e4 ,l41 #cd1 #ch ff41 #bbg1 #bid ,time1 devoted to arithmetic1 #bbh ff42 of day1 in relation to arithmetical learning1 #bbg f4 ,type1 hygiene of1 #bce ff4 ,underlearning1 #acd ff4 ,united ,states money1 #adh ff4 ,units of measure1 arbitrary1 #e1 #hc f4 ,variation1 among individuals1 #bhe ff4 ,variety1 in teaching1 #aec ,verification1 #ha f42 aided by greater strength of the fundamental bonds1 #ajg ff4 ,,walsh1 ,j4 ,h41 #aa ,,wells1 ,f4 ,l41 #bie ,,white1 ,e4 ,e41 #e ,,whitley1 ,m4 ,t41 #bie ,,wiederkehr1 ,g41 #bab ,,wilson1 ,g4 ,m41 #bd1 #di ,,woody1 ,c41 #bi ff41 #eb1 #bhg1 #bic ,words4 ,see ,language and ,terms4 ,written arithmetic1 #bfb ff4 ,zero in multiplication1 #agi f4 ,,transcriber',s ,,notes3 #a4 ,passages in italics are surrounded by underscores4 #b4 ,passages in bold are indicated by bold4 #c4 ,mixed fractions are represented using forward slash and hyphen in this text version4 ,for example1 #e-a_/b represents five and a half4 #d4 ,images and footnotes have been moved from the middle of a paragraph to the closest paragraph break4 #e4 ,obvious errors in spelling and punctuation have been silently closed4