,,! ,,macmillan ,,company ,,new ,,york 4 ,,bo/on 4 ,,*icago 4 ,,dallas ,,atlanta 4 ,,san ,,francisco ,,macmillan ,,co41 ,,limit$ ,,london 4 ,,bombay 4 ,,calcutta ,,melb\rne ,,! ,,macmillan ,,company ,,( ,,canada1 ,,limit$ ,,toronto ,,! ,,psy*ology ,,( ,,ari?metic ,,by ,,edw>d ,l4 ,,?orndike ,,tea*]s ,,college1 ,,columbia ,,univ]sity ,new ,york ,,! ,,macmillan ,,company #aibi ,all rights res]v$ ,,copyright1 #aibb1 ,,by ,,! ,,macmillan ,,company4 ,set up & electrotyp$4 ,publi%$ ,janu>y1 #aibb4 ,repr9t$ ,octob]1 #aibd2 ,may1 #aibf2 ,augu/1 #aibg2 ,octob]1 #aibi4 4 ,,pr9t$ ,,9 ,,! ,,unit$ ,,/ates ,,( ,,am]ica 4 ,,preface ,)9 rec5t ye>s !re h be5 ?ree l9es ( advance 9 psy*ology :i* are ( notable significance = tea*+4 ,! fir/ is ! new po9t ( view conc]n+ ! g5]al process ( le>n+4 ,we n[ und]/& t le>n+ is ess5tially ! =mation ( connections or bonds betwe5 situations & responses1 t ! satisfy+ness ( ! result is ! *ief =ce t =ms !m1 & t habit rules 9 ! realm ( ?\ght z truly & z fully z 9 ! realm ( action4 ,! second is ! great increase 9 k ( ! am\nt1 rate1 & conditions ( improvem5t 9 ^? organiz$ gr\ps or hi]>*ies ( habits :i* we call abilities1 su* z ability to add or ability to read4 ,practice & improvem5t are no long] vague g5]alities1 b conc]n *anges :i* are def9able & measurable by /&>d te/s & scales4 ,! ?ird is ! bett] und]/&+ ( ! s-call$ high] processes ( analysis1 ab/raction1 ! =mation ( g5]al notions1 & reason+4 ,! old] view ( a m5tal *emi/ry :]eby s5sations 7 comp\nd$ into p]cepts1 p]cepts 7 duplicat$ by images1 p]cepts & images 7 amalgamat$ into ab/ractions & concepts1 & ^! 7 manipulat$ by reason+1 has giv5 way to ! und]/&+ ( ! laws ( response to elem5ts or aspects ( situations & to _m situations or elem5ts !re( 9 comb9ation4 ,james' view ( reason+ z selection ( ess5tials & ?9k+ ?+s toge!r 9 a revis$ & cl>ifi$ =m has important applications 9 ! tea*+ ( all ! s*ool subjects4 ,?is book pres5ts ! applications ( ?is new] dynamic psy*ology to ! tea*+ ( ari?metic4 ,its cont5ts are sub/antially :at h be5 includ$ 9 a c\rse ( lectures on ! psy*ology ( ! elem5t>y s*ool subjects giv5 by ! au?or = "s ye>s to /ud5ts ( elem5t>y education at ,tea*]s ,college4 ,_m ( ^! =m] /ud5ts1 n[ 9 sup]visory *>ge ( elem5t>y s*ools1 h urg$ t ^! lectures be made available to tea*]s 9 g5]al4 ,s !y are n[ publi%$ 9 spite ( ! au?or's desire to cl>ify & re9=ce c]ta9 matt]s by fur!r rese>*es4 ,a ^w ( explanation is necess>y conc]n+ ! ex]cises & problems cit$ to illu/rate v>i\s matt]s1 especially errone\s p$agogy4 ,^! are all g5u9e1 hav+ _! s\rce 9 actual textbooks1 c\rses ( /udy1 /ate exam9ations1 & ! l4 ,to avoid any possibility ( invidi\s comp>isons !y are n quotations1 b equival5t problems su* z repres5t accurately ! _s & int5t ( ! orig9als4 ,i take pleasure 9 ackn[l$g+ ! c\rtesy ( ,mr4 ,s4 ,a4 ,c\rtis1 ,g9n & ,company1 ,d4 ,c4 ,hea? & ,company1 ,! ,macmillan ,company1 ,! ,ox=d ,univ]sity ,press1 ,r&1 ,mc,nally & ,company1 ,dr4 ,c4 ,w4 ,/one1 ,! ,tea*]s ,college ,bureau ( ,publications1 & ,! ,_w ,book ,company1 9 p]mitt+ v>i\s quotations4 ,,edw>d ,l4 ,,?orndike4 ,,tea*]s ,,college ,,columbia ,,univ]sity ,april #a1 #aibj ,,cont5ts ,,*apt] ,,page ,,introduction3 ,,! ,,psy*ology ,,( ,,! ,,elem5t>y ,,s*ool ,,subjects xi ,i4 ,,! ,,nature ,,( ,,ari?metical ,,abilities #a ,k ( ! ,mean+s ( ,numb]s ,ari?metical ,language ,problem ,solv+ ,ari?metical ,reason+ ,summ>y ,! ,sociology ( ,ari?metic ,,ii4 ,,! ,,measurem5t ,,( ,,ari?metical ,,abilities #bg ,a ,sample ,measurem5t ( an ,ari?metical ,ability ,ability to ,add ,integ]s ,measurem5ts ( ,ability 9 ,computation ,measurem5ts ( ,ability 9 ,appli$ ,ari?metic3 ! ,solution ( ,problems ,,iii4 ,,! ,,con/itution ,,( ,,ari?metical ,,abilities #ea ,! ,elem5t>y ,functions ( ,ari?metical ,le>n+ ,k ( ! ,mean+ ( a ,fraction ,le>n+ ! ,processes ( ,computation ,,iv4 ,,! ,,con/itution ,,( ,,ari?metical ,,abilities 7cont9u$7 #gj ,! ,selection ( ! ,bonds to ,be ,=m$ ,! ,importance ( ,habit ,=mation ,desirable ,bonds ,n[ ,(t5 ,neglect$ ,wa/eful & ,h>mful ,bonds ,guid+ ,pr9ciples ,v4 ,,! ,,psy*ology ,,( ,,drill ,,9 ,,ari?metic3 ,,! ,,/r5g? ,,( ,,bonds #ajb ,! ,ne$ ( ,/rong] ,elem5t>y ,bonds ,e>ly ,ma/]y ,! ,/r5g? ( ,bonds = ,tempor>y ,s]vice ,! ,/r5g? ( ,bonds ) ,te*nical ,facts & ,t]ms ,! ,/r5g? ( ,bonds ,conc]n+ ! ,reasons = ,ari?metical ,processes ,propa$eutic ,bonds ,,vi4 ,,! ,,psy*ology ,,( ,,drill ,,9 ,,ari?metic3 ,,! ,,am\nt ,,( ,,practice ,,& ,,! ,,organization ,,( ,,abilities #abb ,! ,am\nt ( ,practice ,"u-le>n+ & ,ov]-le>n+ ,! ,organization ( ,abilities ,,vii4 ,,! ,,sequ5ce ,,( ,,topics3 ,,! ,,ord] ,,( ,,=mation ,,( ,,bonds #ada ,conv5tional v]sus ,effective ,ord]s ,decreas+ ,int]f]5ce & ,increas+ ,facilitation ,int]e/ ,g5]al ,pr9ciples ,,viii4 ,,! ,,di/ribution ,,( ,,practice #aef ,! ,problem ,sample ,di/ributions ,possible ,improvem5ts ,,ix4 ,,! ,,psy*ology ,,( ,,?9k+3 ,,ab/ract ,,ideas ,,& ,,g5]al ,,notions ,,9 ,,ari?metic #afi ,responses to ,elem5ts & ,classes ,facilitat+ ! ,analysis ( ,elem5ts ,sy/ematic & ,opportuni/ic ,/imuli to ,analysis ,adaptations to ,elem5t>y-s*ool ,pupils ,x4 ,,! ,,psy*ology ,,( ,,?9k+3 ,,reason+ ,,9 ,,ari?metic #ahe ,! ,ess5tials ( ,ari?metical ,reason+ ,reason+ z ! ,cooep]ation ( ,organiz$ ,habits ,,xi4 ,,orig9al ,,t5d5cies ,,& ,,acquisitions ,,be=e ,,s*ool #aie ,! ,utilization ( ,in/9ctive ,int]e/s ,! ,ord] ( ,developm5t ( ,orig9al ,t5d5cies ,inv5tories ( ,ari?metical ,k & ,skill ,! ,p]ception ( ,numb] & ,quantity ,! ,e>ly ,aw>5ess ( ,numb] ,,xii4 ,,int]e/ ,,9 ,,ari?metic #bji ,c5suses ( ,pupils' ,int]e/s ,reliev+ ,eye ,/ra9 ,significance = ,relat$ ,activities ,intr9sic ,int]e/ 9 ,ari?metical ,le>n+ ,,xiii4 ,,! ,,conditions ,,( ,,le>n+ #bbg ,ext]nal ,conditions ,! ,hygi5e ( ! ,eyes 9 ,ari?metic ,! ,use ( ,concrete ,objects 9 ,ari?metic ,oral1 ,m5tal1 & ,writt5 ,ari?metic ,,xiv4 ,,! ,,conditions ,,( ,,le>n+3 ,,! ,,problem ,,attitude #bff ,illu/rative ,cases ,g5]al ,pr9ciples ,difficulty & ,success z ,/imuli ,false ,inf]5ces ,,xv4 ,,individual ,,diff]5ces #bhe ,nature & ,am\nt ,diff]5ces )9 ,"o ,class ,! ,causes ( ,individual ,diff]5ces ,! ,int]relations ( ,individual ,diff]5ces ,,bibliography ,,( ,,ref]5ces #cjb ,,index #caa ,,g5]al ,,introduction ,,! ,,psy*ology ,,( ,,! ,,elem5t>y ,,s*ool ,,subjects ,! psy*ology ( ! elem5t>y s*ool subjects is conc]n$ ) ! connections :]eby a *ild is able to respond to ! sight ( pr9t$ words by ?\ghts ( _! mean+s1 to ! ?\ght ( six & eight by ?9k+ f\rte51 to c]ta9 sorts ( /ories1 poems1 songs1 & pictures by appreciation !re(1 to c]ta9 situations by acts ( skill1 to c]ta9 o!rs by acts ( c\rtesy & ju/ice1 & s on & on ?r\gh ! s]ies ( situations & responses :i* are provid$ by ! sy/ematic tra9+ ( ! s*ool subjects & ! less sy/ematic tra9+ ( s*ool life dur+ _! /udy4 ,! aims ( elem5t>y education1 :5 fully def9$1 w be f\nd to be ! production ( *anges 9 human nature repres5t$ by an almo/ c\ntless li/ ( connections or bonds :]eby ! pupil ?9ks or feels or acts 9 c]ta9 ways 9 response to ! situations ! s*ool has organiz$ & is influ5c$ to ?9k & feel & act simil>ly to simil> situations :5 life outside ( s*ool confronts him ) !m4 ,we are n at pres5t able to def9e ! "w ( ! elem5t>y s*ool 9 detail z ! =mation ( su* & su* bonds betwe5 c]ta9 deta*$ situations & c]ta9 specifi$ responses4 ,z else:]e 9 human le>n+1 we are at pres5t =c$ to ?9k some:at vaguely 9 t]ms ( m5tal functions1 l ability to read ! v]nacul>1 ability to spell common words1 ability to add1 subtract1 multiply1 & divide ) integ]s1 k ( ! hi/ory ( ! ,unit$ ,/ates1 hone/y 9 exam9ations1 & appreciation ( good music1 def9$ by "s g5]al results obta9$ r ?an by ! elem5t>y bonds :i* con/itute !m4 ,! psy*ology ( ! s*ool subjects beg9s :]e our common s5se k ( ^! functions leaves (f & tries to def9e ! k1 int]e/1 p[]1 skill1 or ideal 9 "q m adequately1 to measure improvem5t 9 x1 to analyze x into its con/itu5t bonds1 to decide :at bonds ne$ to be =m$ & 9 :at ord] z means to ! mo/ economical atta9m5t ( ! desir$ improvem5t1 to survey ! orig9al t5d5cies & ! t5d5cies already acquir$ be=e entrance to s*ool :i* help or h9d] progress 9 ! elem5t>y s*ool subjects1 to exam9e ! motives t are or may be us$ to make ! desir$ connections satisfy+1 to exam9e any o!r special conditions ( improvem5t1 & to note any facts conc]n+ individual diff]5ces t are ( special importance to ! conduct ( elem5t>y s*ool "w4 ,put 9 t]ms ( problems1 ! task ( ! psy*ology ( ! elem5t>y s*ool subjects is1 9 ea* case3-- 7#a7 ,:at is ! function8 ,= example1 j :at is ability to read8 ,j :at does ! und]/&+ ( decimal notation mean8 ,j :at are ! moral effects to be s\ght f ! tea*+ ( lit]ature8 7#b7 ,h[ are degrees ( ability or atta9m5t1 & degrees ( progress or improvem5t 9 ! function or a "p ( ! function measur$8 ,= example1 h[ c we det]m9e h[ well a pupil %\ld write1 or h[ h>d words we expect him to spell1 or :at good ta/e we expect him to %[8 ,h[ c we def9e to ourselves :at k ( ! mean+ ( a fraction we %all try to secure 9 grade #d8 7#c7 ,:at c be done t[>d r$uc+ ! function to t]ms ( p>ticul> situation-response connections1 ^: =mation c be m surely & easily controll$8 ,= example1 h[ f> does ability to spell involve ! =mation "o by "o ( bonds betwe5 ! ?\ght ( almo/ e ^w 9 ! language & ! ?\ght ( t ^w's lett]s 9 _! correct ord]2 & h[ f> does1 say1 ! bond lead+ f ! situation ( ! s\nd ( ceive 9 receive & deceive to _! correct spell+ insure ! correct spell+ ( t "p ( p]ceive8 ,does ability to add involve special bonds lead+ f #bg & #d to #ca1 f #bg & #e to #cb1 & #bg & #f to #cc2 or w ! bonds lead+ f #g & #d to #aa1 #g & #e to #ab & #g & #f to #ac 7ea* plus a simple inf]5ce7 s]ve z well8 ,:at are ! situations & responses t repres5t 9 actual behavior ! quality t we call s*ool patriotism8 7#d7 ,9 almo/ e case a c]ta9 desir$ *ange ( k or skill or p[] c be atta9$ by any "o ( sev]al sets ( bonds4 ,:i* ( !m is ! be/8 ,:at are ! advantages ( ea*8 ,= example1 le>n+ to add may include ! bonds #j & #j are #j1 #j & #a are #a1 #j & #b are #b1 #a & #j are #a1 #b & #j are #b1 etc42 or ^! may be all left un=m$1 ! pupil be+ taught ! habits ( ent]+ #j z ! sum ( a column t is compos$ ( z]os & o!rwise neglect+ #j 9 addition4 ,are ! rules ( usage wor? tea*+ z a means t[>d correct spee*1 or is ! "t bett] sp5t 9 detail$ practice 9 correct spee* itself8 7#e7 ,a bond to be =m$ may be =m$ 9 any "o ( _m degrees ( /r5g?4 ,:i* ( ^! is1 at any giv5 /age ( le>n+ ! subject1 ! mo/ desirable1 all ?+s consid]$8 ,= example1 %all ! dates ( all ! e>ly settlem5ts ( ,nor? ,am]ica be le>n$ s t ! exact ye> w be rememb]$ = t5 ye>s1 or s t ! exact date w be rememb]$ = t5 m9utes & ! date ) an error plus or m9us ( t5 ye>s w be rememb]$ = a ye> or two8 ,%all ! tables ( in*es1 feet1 & y>ds1 & p9ts1 qu>ts1 & gallons be le>n$ at _! fir/ appe>ance s z to be rememb]$ = a ye>1 or %all !y be le>n$ only well en\gh to be usable 9 ! "w ( t week1 :i* 9 turn fixes !m to la/ = a mon? or s8 ,%\ld a pupil 9 ! fir/ ye> ( /udy ( ,fr5* h su* p]fect connections betwe5 ! s\nds ( ,fr5* words & _! mean+s t he c und]/& simple s5t5ces conta9+ !m spok5 at an ord9>y rate ( speak+8 ,or is sl[ spee* p]missible1 & ev5 imp]ative1 on ! "p ( ! tea*]1 ) gradual increase ( rate8 7#f7 ,9 almo/ e case1 any set ( bonds may produce ! desir$ *ange :5 pres5t$ 9 any "o ( sev]al ord]s4 ,:i* is ! be/ ord]8 ,:at are ! advantages ( ea*8 ,c]ta9 sy/ems = tea*+ h&writ+ p]fect ! elem5t>y movem5ts "o at a "t & !n tea* _! comb9ation 9 words & s5t5ces4 ,o!rs beg9 & cont9ue ) ! complex movem5t-s]ies t actual words require4 ,:at d ! latt] lose & ga98 ,! bonds con/itut+ k ( ! metric sy/em are n[ =m$ late 9 ! pupil's c\rse4 ,w\ld x be bett] if !y 7 =m$ e>ly z a means ( facilitat+ k ( decimal fractions8 7#g7 ,:at are ! orig9al t5d5cies & pre-s*ool acquisitions ^u :i* ! connection-=m+ ( ! elem5t>y s*ool may be bas$ or :i* x has to c\nt]act8 ,= example1 if a pupil kn[s ! mean+ ( a he>d ^w1 he may read x und]/&+ly f gett+ its s\nd1 z by phonic recon/ruction4 ,:at words does ! av]age beg9n] s "k8 ,:at are ! individual diff]5ces 9 ?is respect8 ,:at d ! in/9cts ( greg>i\sness1 att5tion-gett+1 approval1 & helpfulness recomm5d conc]n+ gr\p-"w v]sus individual-"w1 & conc]n+ ! size ( a gr\p t is mo/ desirable8 ,! orig9al t5d5cy ( ! eyes is c]ta9ly n to move along a l9e f left to "r ( a page1 !n back 9 "o sweep & along ! next l9e4 ,:at is _! orig9al t5d5cy :5 confront$ ) ! pr9t$ page1 & :at mu/ we d ) x 9 tea*+ read+8 7#h7 ,:at armam5t ( satisfi]s & annoy]s1 ( positive & negative int]e/s & motives1 /&s ready = use 9 ! =mation ( ! intr9sically un9t]e/+ connections betwe5 black m>ks & mean+s1 num]ical ex]cises & _! answ]s1 words & _! spell+1 & ! l8 ,s*ool practice has tri$1 m or less at r&om1 inc5tives & det]r5ts f quasi-physical pa9 to ! mo/ s5tim5tal fondl+1 f %e] cajol]y to philosophical argum5t1 f appeals to assum$ savage & primitive traits to appeals to ! int]e/ 9 automobiles1 fly+-ma*9es1 & wireless telegraphy4 ,c n psy*ology give "s rules = guidance1 or at lea/ limit exp]im5tation to its m hopeful fields8 7#i7 ,! g5]al conditions ( effici5t le>n+ are describ$ 9 manuals ( educational psy*ology4 ,h[ d ^! apply 9 ! case ( ea* task ( ! elem5t>y s*ool8 ,= example1 ! arrangem5t ( s*ool drills 9 addition & 9 %ort division 9 ! =m ( practice exp]im5ts has be5 f\nd v effective 9 produc+ int]e/ 9 ! "w & 9 improvem5t at x4 ,9 :at o!r ari?metical functions may we expect ! same8 7#aj7 ,beside ! g5]al pr9ciples conc]n+ ! nature & causation ( individual diff]5ces1 !re mu/ obvi\sly be1 9 exi/5ce or obta9able z a possible result ( prop] inve/igation1 a great fund ( k ( special diff]5ces relevant to ! le>n+ ( read+1 spell+1 geography1 ari?metic1 & ! l4 ,:at are ! facts z f> z kn[n8 ,:at are ! means ( le>n+ m ( !m8 ,c\rtis f9ds t a *ild may be specially /rong 9 addition & yet be specially weak 9 subtraction 9 comp>ison ) o!rs ( 8 age & grade4 ,x ev5 seems t su* subtle & intricate t5d5cies are inh]it$4 ,h[ f> is su* specialization ! rule8 ,is x1 = example1 ! case t a *ild may h a special gift = spell+ c]ta9 sorts ( words1 = draw+ faces r ?an fl[]s1 = le>n+ anci5t hi/ory r ?an mod]n8 99 99 99 99 99 ,su* are our problems3 ?is volume discusses !m 9 ! case ( ari?metic4 ,! /ud5t :o wi%es to relate ! discussion to ! g5]al p$agogy ( ari?metic may pr(itably read1 9 connection ) ?is volume3 ,! ,tea*+ ( ,elem5t>y ,ma!matics1 by ,d4 ,e4 ,smi? '#ja1 ,! ,tea*+ ( ,prim>y ,ari?metic1 by ,h4 ,suzzallo '#aa1 ,h[ to ,tea* ,ari?metic1 by ,j4 ,c4 ,br[n & ,l4 ,d4 ,c(fman '#ad1 ,! ,tea*+ ( ,ari?metic1 by ,paul ,klapp] '#af1 & ,! ,new ,me?ods 9 ,ari?metic1 by ! au?or '#ba4 ,,! ,,psy*ology ,,( ,,ari?metic ,,*apt] ,i ,,! ,,nature ,,( ,,ari?metical ,,abilities ,accord+ to common s5se1 ! task ( ! elem5t>y s*ool is to tea*3--7#a7 ! mean+s ( numb]s1 7#b7 ! nature ( our sy/em ( decimal notation1 7#c7 ! mean+s ( addition1 subtraction1 multiplication1 & division1 & 7#d7 ! nature & relations ( c]ta9 common measures2 to secure 7#e7 ! ability to add1 subtract1 multiply1 & divide ) integ]s1 common & decimal fractions1 & d5om9ate numb]s1 7#f7 ! ability to apply ! k & p[] repres5t$ by 7#a7 to 7#e7 9 solv+ problems1 & 7#g7 c]ta9 specific abilities to solve problems conc]n+ p]c5tage1 int]e/1 & o!r common occurr5ces 9 bus9ess life4 ,?is /atem5t ( ! functions to be develop$ & improv$ is s\nd & useful s f> z x goes1 b x does n g f> en\gh to make ! task entirely cle>4 ,if tea*]s _h no?+ b ! /atem5t above z a guide to :at *anges !y 7 to make 9 _! pupils1 !y w\ld (t5 leave out important features ( ari?metical tra9+1 & put 9 =ms ( tra9+ t a wise educational plan w\ld n tol]ate4 ,x is also ! case t diff]5t lead]s 9 ari?metical tea*+1 ?\gh !y might all subscribe to ! g5]al /atem5t ( ! previ\s p>agraph1 c]ta9ly d n 9 practice h id5tical notions ( :at ari?metic %\ld be = ! elem5t>y s*ool pupil4 ,! ord9>y view ( ! nature ( ari?metical le>n+ is obscure or inadequate 9 f\r respects4 ,x does n def9e :at 'k ( ! mean+s ( numb]s' is2 x does n take acc\nt ( ! v l>ge am\nt ( tea*+ ( language :i* is done & %\ld be done z a "p ( ! tea*+ ( ari?metic2 x does n di/+ui% betwe5 ! ability to meet c]ta9 quantitative problems z life (f]s !m & ! ability to meet ! problems provid$ by textbooks & c\rses ( /udy2 x leaves '! ability to apply ari?metical k & p[]' z a r my/ical g5]al faculty to be improv$ by "s educational magic4 ,! f\r necess>y am5dm5ts may be discuss$ briefly4 ,,k ,,( ,,! ,,mean+s ,,( ,,numb]s ,k ( ! mean+s ( ! numb]s f "o to t5 may mean k t '"o' means a s+le ?+ ( ! sort nam$1 t two means "o m ?an "o1 t ?ree means "o m ?an two1 & s on4 ,?is we may call ! s]ies mean+4 ,to "k ! mean+ ( 'six' 9 ?is s5se is to "k t x is "o m ?an five & "o less ?an sev5--t x is betwe5 five & sev5 9 ! numb] s]ies4 ,or we may mean by k ( ! mean+s ( numb]s1 k t two fits a collection ( two units1 t ?ree fits a collection ( ?ree units1 & s on1 ea* numb] be+ a "n = a c]ta9 siz$ collection ( discrete ?+s1 su* z apples1 p5nies1 boys1 balls1 f+]s1 & ! o!r cu/om>y objects ( enum]ation 9 ! prim>y s*ool4 ,?is we may call ! collection mean+4 ,to "k ! mean+ ( six 9 ?is s5se is to be able to "n correctly any collection ( six sep>ate1 easily di/+ui%able individual objects4 ,9 ! ?ird place1 k ( ! numb]s f "o to t5 may mean k t two is twice :atev] is call$ "o1 t ?ree is ?ree times :atev] is "o1 & s on4 ,?is is1 ( c\rse1 ! ratio mean+4 ,to "k ! mean+ ( six 9 ?is s5se is to "k t if is "o1 a l9e half a foot long is six1 t if is "o1 is ab\t six1 :ile if is "o1 is ab\t six1 & ! l4 ,9 ! f\r? place1 ! mean+ ( a numb] may be a small] or l>g] fraction ( its implications--its num]ical relations1 facts ab\t x4 ,to "k six 9 ?is s5se is to "k t x is m ?an five or f\r1 less ?an sev5 or eight1 twice ?ree1 ?ree times two1 ! sum ( five & "o1 or ( f\r & two1 or ( ?ree & ?ree1 two less ?an eight--t ) f\r x makes t51 t x is half ( twelve1 & ! l4 ,?is we may call ! 'nucleus ( facts' or relational mean+ ( a numb]4 ,ord9>y s*ool practice has commonly accept$ ! second mean+ z t :i* x is ! task ( ! s*ool to tea* beg9n]s1 b ea* ( ! o!r mean+s has be5 alleg$ to be ! ess5tial "o--! s]ies idea by ,phillips '#ig1 ! ratio idea by ,mc,lellan & ,dewey '#ie & ,spe] '#ig1 & ! relational idea by ,grube & 8 foll[]s4 ,?is div]sity ( views conc]n+ :at ! function is t is to be improv$ 9 ! case ( le>n+ ! mean+s ( ! numb]s "o to t5 is n a trifl+ matt] ( def9ition1 b produces v great diff]5ces 9 s*ool practice4 ,consid]1 = example1 ! pr$om9ant value assign$ to c\nt+ by ,phillips 9 ! passage quot$ bel[1 & ! samples ( ! sort ( "w at :i* *ildr5 7 kept employ$ = mon?s by too ard5t foll[]s ( ,spe] & ,grube4 ,,! ,,s]ies ,,idea ,,ov]emphasiz$ ,?is is ess5tially ! c\nt+ p]iod1 & any words t c be arrang$ into a s]ies furni% all t is necess>y4 ,c\nt+ is fundam5tal1 & c\nt+ t is spontane\s1 free f s5sible obs]vation1 & f ! /ra9 ( reason4 ,a /udy ( ^! orig9al me?ods %[s t multiplication 0 develop$ out ( c\nt+1 & n f addition z ne>ly all textbooks treat x4 ,multiplication is c\nt+4 ,:5 *ildr5 c\nt by #d's1 etc41 !y acc5t ! same z c\nt+ gymna/ics or music4 ,:5 a *ild n[ c\nts on its f+]s x simply reproduces a /age 9 ! gr[? ( ! civilization ( all nations4 ,i w\ld emphasize aga9 t dur+ ! c\nt+ p]iod !re is a some:at spontane\s developm5t ( ! numb] s]ies-idea :i* ,prey] has discuss$ 9 8 ,ari?mog5esis2 t an imm5se mom5tum is giv5 by a sy/ematic s]ies ( names2 & t ^! names are g5]ally fir/ le>n$ & appli$ to objects lat]4 ,a lady tea*] told me t ! ,sup]9t5d5t did n wi% ! tea*]s to all[ ! *ildr5 to c\nt on _! f+]s1 b %e fail$ to see :y c\nt+ ) horse-*e/nuts 0 any bett]4 ,h] *ildr5 c\ld h>dly avoid us+ _! f+]s 9 c\nt+ o!r objects yet !y foll[$ ! s]ies to #ajj )\t hesitation or ref]5ce to _! f+]s4 ,?is spontane\s c\nt+ p]iod1 or nam+ & foll[+ ! s]ies1 %\ld prec$e its application to objects4 ,d4,e4 ,phillips1 '#ig1 p4 #bch4 ,,! ,,ratio ,,idea ,,ov]emphasiz$ ,illu/ration3 ,,fig4 #a4 ,ratios4--#a4 ,select solids hav+ ! relation1 or ratio1 ( a1 b1 c1 d1 o1 e4 #b4 ,"n ! solids1 a1 b1 c1 d1 o1 e4 ,! means ( express+ mu/ be z freely suppli$ z ! means ( discov]y4 ,! pupil is n expect$ to inv5t t]ms4 #c4 ,tell all y c ab\t ! relation ( ^! units4 #d4 ,unite units & tell :at ! sum equals4 #e4 ,make /atem5ts l ?is3 o less e equals b4 #f4 c c be sep>at$ into h[ _m d's8 into h[ _m b's8 #g4 c c be sep>at$ into h[ _m b's8 ,:at is ! "n ( ! l>ge/ unit t c be f\nd 9 bo? c & d an exact numb] ( times8 #h4 ,ea* ( ! o!r units equals :at "p ( c8 #i4 ,if b is #a1 :at is ea* ( ! o!r units8 #aj4 ,if a is #a1 :at is ea* ( ! o!r units8 #aa4 ,if b is #a1 h[ _m #a's are !re 9 ea* ( ! o!r units8 #ab4 ,if d is #a1 h[ _m #a's & p>ts ( #a 9 ea* ( ! o!r units8 #ac4 #b is ! relation ( :at units8 #ad4 #c is ! relation ( :at units8 #ae4 #a_/b is ! relation ( :at units8 #af4 #b_/c is ! relation ( :at units8 #ag4 ,:i* units h ! relation #c_/b8 #ah4 ,:i* unit is #c times z l>ge z #a_/b ( b8 #ai4 c equals #f times #a_/c ( :at unit8 #bj4 #a_/c ( :at unit equals #a_/f ( c8 #ba4 ,:at equals #a_/b ( c8 d equals h[ _m six?s ( c8 #bb4 o equals #e times #a_/c ( :at unit8 #bc4 #a_/c ( :at unit equals #a_/e ( o8 #bd4 #b_/c ( d equals :at unit8 b equals h[ _m ?irds ( d8 #be4 #b is ! ratio ( d to #a_/c ( :at unit8 #c is ! ratio ( d to #a_/b ( :at unit8 #bf4 d equals #c_/d ( :at unit8 #c_/d is ! ratio ( :at units8 ,spe]1 '#ig1 p4 #i;f4 ,,! ,,relational ,,idea ,,ov]emphasiz$ ,an inspection ( books ( ! eighties :i* foll[$ ! ,grube me?od 7= example1 ! ,new ,elem5t>y ,ari?metic by ,e4,e4 ,:ite '#hc7 w %[ undue emphasis on ! relational ideas4 ,!re w be ov] a hundr$ & fifty successive tasks all1 or ne>ly all1 on #g & -#g4 ,!re w be mu* writt5 "w ( ! sort %[n bel[3 ,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 -- -- -- :i* mu/ h sorely tri$ ! eyes ( all conc]n$4 ,pupils are taught to give ! analysis & syn!sis ( ea* ( ! n9e digits4 ,yet ! au?or /ates t he does n c>ry ! pr9ciple ( ! ,grube me?od to ! extreme ( useless repetition & me*anism4 ,x %\ld be obvi\s t all f\r mean+s h claims ^u ! att5tion ( ! elem5t>y s*ool4 ,f\r is ! ?+ betwe5 ?ree & five 9 ! numb] s]ies2 x is ! "n = a c]ta9 siz$ collection ( discrete objects2 x is also ! "n = a cont9u\s magnitude equal to f\r units--= f\r qu>ts ( milk 9 a gallon pail z truly z = f\r sep>ate qu>t-pails ( milk2 x is also1 if we "k x well1 ! ?+ got by add+ "o to ?ree or subtract+ six f t5 or tak+ two two's or half ( eight4 ,to "k ! mean+ ( a numb] means to "k some:at ab\t x 9 all ( ^! respects4 ,! difficulty has be5 ! n>r[ vision ( ! extremi/s4 ,a *ild mu/ n be left int]m9ably c\nt+2 9 fact ! "o-m-ness ( ! numb] s]ies c almo/ be _h z a by-product4 ,a *ild mu/ n be re/rict$ to ex]cises ) collections objectifi$ z 9 ,fig4 #b or /at$ 9 words z s _m apples1 oranges1 hats1 p5s1 etc41 :5 "w ) measurem5t ( cont9u\s quantities ) v>y+ units--in*es1 feet1 y>ds1 glassfuls1 p9ts1 qu>ts1 seconds1 m9utes1 h\rs1 & ! l--is s easy & s significant4 ,on ! o!r h&1 ! elaboration ( artificial problems ) fictiti\s units ( measure j to h relative magnitudes z 9 ! ex]cises on page #e is a wa/eful sacrifice4 ,simil>ly1 special drills emphasiz+ ! fact t eighte5 is elev5 & sev51 twelve & six1 ?ree less ?an tw5ty-"o1 & ! l1 are simply idolatr\s2 ^! facts ab\t eighte51 s f> z !y are ne$$1 are bett] le>n$ 9 ! c\rse ( actual column-addition & -subtraction4 ,illu/ration3 ,,fig4 #b4 ,,ari?metical ,,language ,! second improvem5t to be made 9 ! ord9>y notion ( :at ! functions to be improv$ are 9 ! case ( ari?metic is to include among ^! functions ! k ( c]ta9 words4 ,! und]/&+ ( su* words z bo?1 all1 9 all1 toge!r1 less1 diff]5ce1 sum1 :ole1 "p1 equal1 buy1 sell1 h left1 measure1 is conta9$ 91 & ! l1 is necess>y 9 ari?metic z truly z is ! und]/&+ ( numb]s !mselves4 ,x mu/ be provid$ = by ! s*ool2 = pre-s*ool & extra-s*ool tra9+ does n furni% x1 or furni%es x too late4 ,x c be provid$ = mu* bett] 9 connection ) ! tea*+ ( ari?metic ?an 9 connection ) ! tea*+ ( ,engli%4 ,x has n be5 provid$ =4 ,an exam9ation ( ! fir/ fifty pages ( eight rec5t textbooks = beg9n]s 9 ari?metic reveals v slight att5tion to ?is matt] at ! be/ & no att5tion at all 9 "s cases4 ,?ree ( ! books d n ev5 use ! ^w sum1 & "o uses x only once 9 ! fifty pages4 ,9 all ! f\r hundr$ pages ! ^w diff]5ce occurs only tw5ty times4 ,:5 ! words are us$1 no great +5uity or c>e appe>s 9 ! means ( mak+ sure t _! mean+s are und]/ood4 ,! *ief reason :y x has n be5 provid$ = is precisely t ! common notion ( :at ! functions are t ari?metic is to develop has left out ( acc\nt ?is function ( intellig5t response to quantitative t]ms1 o!r ?an ! names ( ! numb]s & processes4 ,k ( language ov] a mu* wid] range is a necess>y elem5t 9 ari?metical ability 9 s f> z ! latt] includes ability to solve v]bally /at$ problems4 ,z ari?metic is n[ taught1 x does include t ability1 & a l>ge "p ( ! "t ( wise tea*+ is giv5 to improv+ ! function 'kn[+ :at a problem /ates & :at x asks =4' ,s9ce1 h[ev]1 ?is und]/&+ ( v]bally /at$ problems may n be an absolutely necess>y elem5t ( ari?metic1 x is be/ to def] its consid]ation until we h se5 :at ! g5]al function ( problem-solv+ is4 ,,problem-,,solv+ ,! ?ird respect 9 :i* ! function1 'ability 9 ari?metic1' ne$s cle>] def9ition1 is ?is 'problem-solv+4' ,! aim ( ! elem5t>y s*ool is to provide = correct & economical response to g5u9e problems1 su* z kn[+ ! total due = c]ta9 real quantities at c]ta9 real prices1 kn[+ ! correct *ange to give or get1 keep+ h\sehold acc\nts1 calculat+ wages due1 comput+ areas1 p]c5tages1 & disc\nts1 e/imat+ quantities ne$$ ( c]ta9 mat]ials to make c]ta9 h\sehold or %op products1 & ! l4 ,life br+s ^! problems usually ei!r ) a real situation 7z :5 "o buys & c\nts ! co/ & 8 *ange71 or ) a situation t "o imag9es or describes to himself 7z :5 "o figures out h[ mu* money he mu/ save p] week to be able to buy a =ty-doll> bicycle be=e a c]ta9 date74 ,sometimes1 h[ev]1 ! problem is describ$ 9 words to ! p]son :o mu/ solve x by ano!r p]son 7z :5 a life insurance ag5t says1 ',y pay only #be c5ts a week f n[ till--& y get #bej !n'2 or :5 an employ] says1 ',y\r wages w\ld be #i doll>s a week1 ) lun*eon furni%$ & bonuses ( su* & su* am\nts'74 ,sometimes also ! problem is describ$ 9 pr9t$ or writt5 words to ! p]son :o mu/ solve x 7z 9 an adv]tisem5t or 9 ! lett] ( a cu/om] ask+ = an e/imate on ?is or t74 ,! problem may be 9 "p real1 9 "p imag9$ or describ$ to oneself1 & 9 "p describ$ to "o orally or 9 pr9t$ or writt5 words 7z :5 ! propos$ articles = pur*ase lie be=e "o1 ! am\nt ( money "o has 9 ! bank is imag9$1 ! %opkeep] (f]s #aj p]c5t disc\nt1 & ! pr9t$ price li/ is !re to be read74 ,to fit pupils to solve ^! real1 p]sonally imag9$1 or self-describ$ problems1 & 'describ$-by-ano!r' problems1 s*ools h reli$ almo/ exclusively on tra9+ ) problems ( ! la/ sort only4 ,! foll[+ page tak5 almo/ at r&om f "o ( ! be/ rec5t textbooks c\ld be p>allel$ by ?\s&s ( o!rs2 & ! oral problems put by tea*]s h1 z a rule1 no real situation support+ !m4 #a4 ,at #gj c5ts p] #ajj p\nds1 :at w be ! am\nt ( duty on an invoice ( #cfbb /eel rails1 ea* rail be+ #bg feet long & weigh+ #fj p\nds to ! y>d8 #b4 ,a man _h prop]ty valu$ at #fejj4 ,:at w be 8 taxes at ! rate ( #aj4hj p] #ajjj8 #c4 ,multiply sev5ty ?\s& f\rte5 hundr$-?\s&?s by "o hundr$ n9e million?s1 & divide ! product by five hundr$ =ty-five4 #d4 ,:at numb] multipli$ by #dc-c_/d w produce #bfe-e_/h8 #e4 ,:at decimal ( a bu%el is #c qu>ts8 #f4 ,a man sells #e_/h ( an acre ( l& = #ic4ge4 ,:at w\ld be ! value ( 8 f>m ( #aej-c_/d acres at ! same rate8 #g4 ,a coal deal] buys #cge tons coal at #d4be p] ton ( #bbdj p\nds4 ,he sells x at #d4ej p] ton ( #bjjj p\nds4 ,:at is 8 pr(it8 #h4 ,b\ght #fj y>ds ( clo? at ! rate ( #b y>ds = #e1 & #hj y>ds m at ! rate ( #d y>ds = #i4 ,i imm$iately sold ! :ole ( x at ! rate ( #e y>ds = #ab4 ,h[ mu* did ,i ga98 #i4 ,a man pur*as$ #dj bu%els ( apples at #a4ej p] bu%el4 ,tw5ty-five hundr$?s ( !m 7 damag$1 & he sold !m at #bj c5ts p] peck4 ,he sold ! rema9d] at #ej c5ts p] peck4 ,h[ mu* did he ga9 or lose8 #aj4 ,if oranges are #cg-a_/b c5ts p] doz51 h[ _m boxes1 ea* conta9+ #dhj1 c be b\ght = #fj8 #aa4 ,a man c d a piece ( "w 9 #ah-c_/d days4 ,:at "p ( x c he d 9 #f-b_/c days8 #ab4 ,h[ old to-day is a boy t 0 born ,oct4 #bi1 #ahif8 ,wal%1 '#jf1 ,"p ,i1 p4 #afe4 ,z a result1 tea*]s & textbook writ]s h come to ?9k ( ! functions ( solv+ ari?metical problems z id5tical ) ! function ( solv+ ! describ$ problems :i* !y give 9 s*ool 9 books1 exam9ation pap]s1 & ! l4 ,if !y d n ?9k explicitly t ?is is s1 !y /ill act 9 tra9+ & 9 te/+ pupils z if x 7 s4 ,x is n4 ,problems %\ld be solv$ 9 s*ool to ! end t pupils may solve ! problems :i* life (f]s4 ,to "k :at *ange "o %\ld receive aft] a giv5 real pur*ase1 to keep "o's acc\nts accurately1 to adapt a recipe = six s z to make en\gh ( ! article = f\r p]sons1 to e/imate ! am\nt ( se$ requir$ = a plot ( a giv5 size f ! /atem5t ( ! am\nt requir$ p] acre1 to make ) surety ! applications t ! h\sehold1 small /ores1 & ord9>y trades require--su* is ! ability t ! elem5t>y s*ool %\ld develop4 ,o!r ?+s be+ equal1 ! s*ool %\ld set problems 9 ari?metic :i* life !n & lat] w set1 %\ld favor ! situations :i* life itself (f]s & ! responses :i* life itself dem&s4 ,o!r ?+s are n always equal4 ,! same am\nt ( "t & ef=t w (t5 be m productive t[>d ! f9al end if direct$ dur+ s*ool to 'made-up' problems4 ,! keep+ ( p]sonal f9ancial acc\nts z a s*ool ex]cise is usually impracticable1 p>tly because "s ( ! *ildr5 h no e>n+s or all[ance--no acc\nts to keep1 & p>tly because ! task ( sup]vis+ "w :5 ea* *ild has a diff]5t problem is too great = ! tea*]4 ,! use ( real h\sehold & %op problems w be easy only :5 ! s*ool program includes ! h\sehold arts & indu/rial education1 & :5 ^! subjects !mselves are taught s z to improve ! functions us$ by real life4 ,v (t5 ! mo/ effici5t c\rse is to make sure t ari?metical proc$ures are appli$ to ! real & p]sonally initiat$ problems :i* !y fit1 by hav+ a c]ta9 numb] ( su* problems arise & be solv$2 !n to make sure t ! simil>ity betwe5 ^! real problems & c]ta9 describ$ problems ( ! textbook or tea*]'s giv+ is appreciat$2 & !n to give ! ne$$ drill "w ) describ$ problems4 ,9 _m cases ! s*ool practice is fairly well ju/ifi$ 9 assum+ t solv+ describ$ problems w prep>e ! pupil to solve ! correspond+ real problems actually mu* bett] ?an ! same am\nt ( "t sp5t on ! real problems !mselves4 ,all ?is is true1 yet ! g5]al pr9ciple rema9s t1 o!r ?+s be+ equal1 ! s*ool %\ld favor real situations1 %\ld pres5t issues z life w pres5t !m4 ,:]e o!r ?+s make ! use ( v]bally describ$ problems ( ! ord9>y type desirable1 ^! %\ld be *os5 s z to give a maximum ( prep>ation = ! real applications ( ari?metic 9 life4 ,we %\ld n1 = example1 c>elessly use any problem t comes to m9d 9 apply+ a c]ta9 pr9ciple1 b %\ld /op to consid] j :at ! situations ( life really require & %[ cle>ly ! application ( t pr9ciple4 ,= example1 contra/ ^! two problems apply+ cancellation3-- ,a4 ,a man sold #bd lambs at #ah apiece on ea* ( six days1 & b\ght #h p\nds ( metal ) ! proce$s4 ,h[ mu* did he pay p] ounce = ! metal8 ,b4 ,h[ tall mu/ a rectangul> tank #af long by #h wide be to hold z mu* z a rectangul> tank #bd by #ah by #f8 ,! fir/ problem n only pres5ts a situation t w\ld r>ely or nev] occur1 b also takes a way to f9d ! answ] t w\ld n1 9 t situation1 be tak5 s9ce ! price set by ano!r w\ld det]m9e ! am\nt4 ,mu* ?\ght & +5uity %\ld 9 ! future be exp5d$ 9 elim9at+ problems ^: solution does n improve ! real function to be improv$ by appli$ ari?metic1 or improves x at too great co/1 & 9 devis+ problems :i* prep>e directly = life's dem&s & /ill c fit into a curriculum t c be adm9i/]$ by "o tea*] 9 *>ge ( ?irty or =ty pupils1 "u ! limitations ( s*ool life4 ,! foll[+ illu/rations w to "s ext5t %[ concretely :at ! ability to apply ! k & p[] repres5t$ by ab/ract or pure ari?metic--! s-call$ fundam5tals--9 solv+ problems %\ld mean & :at x %\ld n mean4 ,samples ( ,desirable ,applications ( ,ari?metic 9 ,problems :]e ! ,situation is ,actually ,pres5t to ,s5se 9 ,:ole or 9 ,"p ,keep+ ! scores & decid+ :i* side beat & by h[ mu* 9 appropriate classroom games1 spell+ mat*es1 & ! l4 ,comput+ co/s1 mak+ & inspect+ *ange1 tak+ inv5tories1 & ! l ) a real or play /ore4 ,mapp+ ! s*ool g>d51 divid+ x into allotm5ts1 plann+ = ! pur*ase ( se$s1 & ! l4 ,measur+ "o's own a*ievem5t & progress 9 te/s ( ^w-k1 spell+1 addition1 subtraction1 spe$ ( writ+1 & ! l4 ,measur+ ! rate ( improvem5t p] h\r ( practice or p] week ( s*ool life1 & ! l4 ,e/imat+ co/s ( food cook$ 9 ! s*ool kit*51 articles made 9 ! s*ool %ops1 & ! l4 ,comput+ ! co/ ( telegrams1 po/age1 expressage1 = a real message or package1 f ! publi%$ t>iffs4 ,comput+ co/s f mail ord] catalogues & ! l4 ,samples ( ,desirable ,applications ( ,ari?metic :]e ! ,situation is ,n ,pres5t to ,s5se ,! samples giv5 "h all conc]n ! subtraction ( fractions4 ,samples conc]n+ any o!r ari?metical pr9ciple may be f\nd 9 ! appropriate pages ( any text :i* conta9s problem-mat]ial select$ ) consid]ation ( life's ne$s4 ,a #a4 ,dora is mak+ jelly4 ,! recipe calls = #bd cups ( sug> & %e has only #ba-a_/b4 ,%e has no "t to g to ! /ore s %e has to borr[ ! sug> f a neighbor4 ,h[ mu* mu/ %e get8 ,subtract #bd ,?9k #a_/b & #a_/b #a4 ,write #a_/b4 #ba-a_/b ,?9k #b & #b #d4 ,write ! #b4 -------- #b-a_/b #b4 ,a box full ( soap weighs #bi-a_/b lb4 ,! empty box weighs #c-a_/b lb4 ,h[ mu* does ! soap alone weigh8 #c4 ,on ,july #a1 ,mr4 ,lewis b\ght a #ej-lb4 bag ( ice-cream salt4 ,on ,july #ae !re 7 j #aa-a_/b lb4 left4 ,h[ mu* _h he us$ 9 ! two weeks8 #d4 ,grace promis$ to pick #cj qt4 blueb]ries = h] "m4 ,s f> %e has pick$ #ah-a_/b qt4 ,h[ _m m qu>ts mu/ %e pick8 ,b ,?is table ( numb]s tells ,weight ( ,m>y ,adams :at ,nell's baby si/] ,m>y ,:5 born #g-c_/h lb4 weigh$ e two mon?s f #b mon?s old #aa-a_/d lb4 ! "t %e 0 born till %e #d mon?s old #ad-a_/h lb4 0 a ye> old4 #f mon?s old #ae-c_/d lb4 #h mon?s old #ag-e_/h lb4 #aj mon?s old #ai-a_/b lb4 #ab mon?s old #ba-c_/h lb4 #a4 ,h[ mu* did ! ,adams baby ga9 9 ! fir/ two mon?s8 #b4 ,h[ mu* did ! ,adams baby ga9 9 ! second two mon?s8 #c4 ,9 ! ?ird two mon?s8 #d4 ,9 ! f\r? two mon?s8 #e4 ,f ! "t x 0 #h mon?s old till x 0 #aj mon?s old8 #f4 ,9 ! la/ two mon?s8 #g4 ,f ! "t x 0 born till x 0 #f mon?s old8 ,c #a4 ,hel5's exact av]age = ,decemb] 0 #hg-a_/c4 ,kate's 0 #hd-a_/b4 ,h[ mu* high] 0 ,hel5's ?an ,kate's8 #hg-a_/c ,h[ d y ?9k ( #a_/b & #a_/c8 #hd-a_/b ,h[ d y ?9k ( #a-b_/f8 ------ ,h[ d y *ange ! #d8 #b4 ,f9d ! exact av]age = ea* girl 9 ! foll[+ li/4 ,write ! answ]s cle>ly s t y c see !m easily4 ,y w use !m 9 solv+ problems #c1 #d1 #e1 #f1 #g1 & #h4 ,alice ,dora ,emma ,grace ,l\ise ,m>y ,nell ,rebecca ,read+ #ia #hg #hc #ha #gi #gg #gf #gc ,language #hh #gh #hb #gi #gc #gh #gc #ge ,ari?metic #hi #he #gi #ge #hd #hg #hi #hj ,spell+ #ij #gi #ge #hj #hb #ia #fh #ha ,geography #ia #hg #hc #ge #gh #he #gc #gi ,writ+ #ij #hh #ge #gb #ic #ib #ie #gh #c4 ,:i* girl _h ! highe/ av]age8 #d4 ,h[ mu* high] 0 h] av]age ?an ! next highe/8 #e4 ,h[ mu* diff]5ce 0 !re betwe5 ! highe/ & ! l[e/ girl8 #f4 ,0 ,emma's av]age high] or l[] ?an ,l\ise's8 ,h[ mu*8 #g4 ,h[ mu* diff]5ce 0 !re betwe5 ,alice's av]age & ,dora's8 #h4 ,h[ mu* diff]5ce 0 !re betwe5 ,m>y's av]age & ,nell's8 #i4 ,write five o!r problems ab\t ^! av]ages1 & solve ea* ( !m4 ,samples ( ,undesirable ,applications ( ,ari?metic#a ,w has ,,xxi m>bles1 ,,xii jack/ones1 ,,xxxvi pieces ( /r+4 ,h[ _m ?+s _h he8 ,george's kite rose ,,cdxxxv feet & ,tom's w5t ,,lxiii feet high]4 ,h[ high did ,tom's kite rise8 ,if f ,,dciv we take ,,cciv ! result w be a numb] ,,iv times z l>ge z ! numb] ( doll>s ,mr4 ,dane paid = 8 horse4 ,h[ mu* did he pay = 8 horse8 ,hannah has #e_/h ( a doll>1 ,susie #g_/be1 ,nellie #c_/d1 ,norah #ac_/af4 ,h[ mu* money h !y all toge!r8 ,a man saves #c-ag_/hj doll>s a week4 ,h[ mu* does he save 9 a ye>8 ,a tree fell & 0 brok5 into #d pieces1 #ac-a_/f feet1 #aj-c_/g feet1 #h-a_/b feet1 & #g-af_/ba feet long4 ,h[ tall 0 ! tree8 ,annie's fa!r gave h] #bj apples to divide among h] fri5ds4 ,%e gave ea* "o #b-b_/i apples apiece4 ,h[ _m playmates _h %e8 ,john _h #ag-b_/e apples4 ,he divid$ 8 :ole apples into fif?s4 ,h[ _m pieces _h he 9 all8 ,a l&lady has #c-c_/g pies to be divid$ among h] #h bo>d]s4 ,h[ mu* w ea* bo>d] receive8 ,!re are tw5ty columns ( spell+ words 9 ,m>y's lesson & #af words 9 ea* column4 ,h[ _m words are 9 h] lesson8 ,!re are #i nuts 9 a p9t4 ,h[ _m p9ts 9 a pile ( #e1hhh1fgc nuts8 ,! ,adams s*ool conta9s eight rooms2 ea* room conta9s #dh pupils2 if ea* pupil has eight c5ts1 h[ mu* h !y toge!r8 ,a pile ( wood 9 ! =m ( a cube conta9s #ae-a_/b cords4 ,:at are ! dim5sions to ! ne>e/ in*8 ,a man #f ft4 high weighs #age lb4 ,h[ tall is 8 wife :o is ( simil> build1 & weighs #abe lb48 ,a /ick ( timb] is 9 ! %ape ( ! fru/um ( a squ>e pyramid1 ! l[] base be+ #bb 94 squ>e & ! upp] #ad 94 squ>e4 ,h[ _m cubic feet 9 ! log1 if x is #bb ft4 long8 ,mr4 ,ames1 be+ ask$ 8 age1 repli$3 ,if y cube "o half ( my age & add #da1dgb to ! result1 ! sum w be "o half ! cube ( my age4 ,h[ old am ,i8 #a ,! foll[+ & lat] problems are tak5 f actual textbooks or c\rses ( /udy or /ate exam9ations2 to avoid invidi\s comp>isons1 !y are n exact quotations1 b are equival5ts 9 pr9ciple & =m1 z /at$ 9 ! preface4 ,^! samples1 j giv51 ( ! k9d ( problem-solv+ t %\ld n be emphasiz$ 9 s*ool tra9+ ref] 9 "s cases to books ( =ty ye>s back1 b ! foll[+ repres5t ! results ( a collection made 9 #aiaj f books !n 9 excell5t repute4 ,x requir$ only ab\t an h\r to collect !m2 & ,i am confid5t t a ?\s& su* problems describ+ situations t ! pupil w nev] enc\nt] 9 real life1 or putt+ que/ions t he w nev] be ask$ 9 real life1 c\ld easily be f\nd 9 any t5 textbooks ( ! decade f #aijj to #aiaj4 ,if !re are #bej k]nels ( corn on "o e>1 h[ _m are !re on #bd e>s ( corn ! same size8 ,maud is f\r times z old z h] si/]1 :o is #d ye>s old4 ,:at is ! sum ( _! ages8 ,if ! fir/ c5tury began ) ! ye> #a1 ) :at ye> does x end8 ,e spid] has #h comp\nd eyes4 ,h[ _m eyes h #ba spid]s8 ,a nail #d in*es long is driv5 ?r\gh a bo>d s t x projects #a4fie in*es on "o side & #a4dbh on ! o!r4 ,h[ ?ick is ! bo>d8 ,f9d ! p]imet] ( an envelope #e 94 by #c-a_/d 94 ,h[ _m m9utes 9 #e_/i ( #i_/d ( an h\r8 ,mrs4 ,knox is #c_/d z old z ,mr4 ,knox1 :o is #dh ye>s old4 ,_! son ,edw>d is #d_/i z old z 8 "m4 ,h[ old is ,edw>d8 ,suppose a pie to be exactly r\nd & #aj-a_/b miles 9 diamet]4 ,if x 7 cut into #f equal pieces1 h[ long w\ld ! curv$ edge ( ea* piece be8 #h-a_/c ( a class ( #cf boys 7 abs5t on a ra9y day4 #cc-a_/c ( ^? pres5t w5t out ( ! room to ! s*ool y>d4 ,h[ _m 7 left 9 ! room8 ,j aft] a ton ( hay 0 weigh$ 9 m>ket1 a horse ate "o p\nd ( x4 ,:at 0 ! ratio ( :at he ate to :at 0 left8 ,if a fan hav+ #ae rays op5s out s t ! out] rays =m a /raight l9e1 h[ _m degrees are !re betwe5 any two adjac5t rays8 ,"o half ( ! di/ance betwe5 ,/4 ,l\is & ,new ,orleans is #bhj miles m ?an #a_/aj ( ! di/ance2 :at is ! di/ance betwe5 ^! places8 ,if ! pressure ( ! atmosph]e is #ad4g lb4 p] squ>e in* :at is ! pressure on ! top ( a table #a-a_/d yd4 long & #b_/c yd4 wide8 #ac_/bh ( ! total acreage ( b>ley 9 #aijj 0 #ajj1jjj acres2 :at 0 ! total acreage8 ,:at is ! lea/ numb] ( bananas t a "m c exactly divide betwe5 h] #b sons1 or among h] #d daught]s1 or among all h] *ildr58 ,if ,alice 7 two ye>s old] ?an f\r times h] actual age %e w\ld be z old z h] aunt1 :o is #ch ye>s old4 ,h[ old is ,alice8 ,?ree m5 walk ar\nd a circul> isl&1 ! circumf]5ce ( :i* is #cfj miles4 ,a walks #ae miles a day1 ,b #ah miles a day1 & ,c #bd miles a day4 ,if !y />t toge!r & walk 9 ! same direction1 h[ _m days w elapse be=e !y w be toge!r aga98 ,) only ?irty or =ty doll>s a ye> to sp5d on a pupil's education1 ( :i* p]haps eight doll>s are sp5t on improv+ 8 ari?metical abilities1 ! imm$iate guidance ( 8 responses to real situations & p]sonally initiat$ problems has to be supplem5t$ l>gely by guidance ( 8 responses to problems describ$ 9 words1 diagrams1 pictures1 & ! l4 ,( ^! latt]1 words w be us$ mo/ (t54 ,z a consequ5ce ! und]/&+ ( ! words us$ 9 ^! descriptions becomes a "p ( ! ability requir$ 9 ari?metic4 ,su* ^w k is also requir$ 9 s f> z ! problems to be solv$ 9 real life are at times describ$1 z 9 adv]tisem5ts1 bus9ess lett]s1 & ! l4 ,?is is recogniz$ by ev]ybody 9 ! case ( words l rema9d]1 pr(it1 loss1 ga91 int]e/1 cubic capacity1 gross1 net1 & disc\nt1 b holds equally ( let1 suppose1 balance1 av]age1 total1 borr[$1 reta9$1 & _m su* semi-te*nical words1 & may hold also ( hundr$s ( o!r words unless ! textbook & tea*] are c>eful to use only words & s5t5ce /ructures :i* daily life & ! class "w 9 ,engli% h made well kn[n to ! pupils4 ,to apply ari?metic to a problem a pupil mu/ und]/& :at ! problem is2 problem-solv+ dep5ds on problem-read+4 ,9 actual s*ool practice tra9+ 9 problem-read+ w be less & less necess>y z we get rid ( problems to be solv$ simply = ! sake ( solv+ !m1 unnecess>ily unreal problems1 & clumsy descriptions1 b x w rema9 to "s ext5t z an important jo9t task = ! 'ari?metic' & 'read+' ( ! elem5t>y s*ool4 ,,ari?metical ,,reason+ ,! la/ respect 9 :i* ! nature ( ari?metical abilities requires def9ition conc]ns ari?metical reason+4 ,an adequate treatm5t ( ! reason+ t may be expect$ ( pupils 9 ! elem5t>y s*ool & ( ! mo/ effici5t ways to enc\rage & improve x _c be giv5 until we h /udi$ ! =mation ( habits4 ,= reason+ is ess5tially ! organization & control ( habits ( ?\ght4 ,c]ta9 matt]s may1 h[ev]1 be decid$ "h4 ,! fir/ conc]ns ! use ( computation & problems m]ely = discipl9e1--t is1 ! emphasis on tra9+ 9 reason+ reg>dless ( :e!r ! problem is o!rwise wor? reason+ ab\t4 ,x us$ to be ?\ght t ! m9d 0 a set ( faculties or abilities or p[]s :i* grew /rong & compet5t by be+ ex]cis$ 9 a c]ta9 way1 no matt] on :at !y 7 ex]cis$4 ,problems t c\ld n occur 9 life1 & 7 entirely devoid ( any wor?y int]e/1 save ! intellectual int]e/ 9 solv+ !m1 7 suppos$ to be ne>ly or q z useful 9 tra9+ ! m9d to reason z ! g5u9e problems ( ! home1 %op1 or trade4 ,any?+ t gave ! m9d a *ance to reason w\ld d2 & pupils labor$ to f9d :5 ! m9ute h& & h\r h& w\ld be toge!r1 or h[ _m %eep a %eph]d _h if half ( :at he _h plus t5 0 "o ?ird ( twice :at he _h6 ,we n[ "k t ! tra9+ dep5ds l>gely on ! p>ticul> data us$1 s t effici5t discipl9e 9 reason+ requires t ! pupil reason ab\t matt]s ( real importance4 ,!re is no magic ess5ce or faculty ( reason+ t works 9 g5]al & irrespective ( ! p>ticul> facts & relations reason$ ab\t4 ,s we %\ld try to f9d problems :i* n only /imulate ! pupil to reason1 b also direct 8 reason+ 9 useful *annels & rew>d x by results t are ( real significance4 ,we %\ld replace ! purely discipl9>y problems by problems t are also valuable z special tra9+ = important p>ticul> situations ( life4 ,reason+ s\ght = reason+'s sake alone is too wa/eful an exp5diture ( "t & is also likely to be inf]ior z reason+4 ,! second matt] conc]ns ! relative m]its ( 'cat*' problems1 :]e ! pupil has to g aga9/ "s cu/om>y habit ( ?9k+1 & :at we may call 'r\t9e' problems1 :]e ! regul> ways ( ?9k+ t h s]v$ him 9 ! pa/ w1 except = "s blund]1 guide him rightly4 ,consid]1 = example1 ^! f\r problems3 #a4 ,a man b\ght t5 doz5 eggs = #b4ej & sold !m = #cj c5ts a doz54 ,h[ _m c5ts did he lose8 #b4 ,i w5t into ,smi?'s /ore at #i ,a4,m4 & rema9$ until #aj ,a4,m4 ,i b\ght six y>ds ( g+ham at #dj c5ts a y>d & ?ree y>ds ( musl9 at #bj c5ts a y>d & gave a #e4jj bill4 ,h[ long 0 ,i 9 ! /ore8 #c4 ,:at mu/ y divide #dh by to get half ( twice #f8 #d4 ,:at mu/ y add to #ai to get #cj8 ,! 'cat*' problem is n[ 9 disrepute1 ! wise tea*] feel+ by a sort ( intuition t to willfully require a pupil to reason to a result %>ply contr>y to t to :i* previ\s habits lead him is risky4 ,! f\r illu/rations j giv5 %[1 h[ev]1 t m]e 'cat*9ess' or 'contra-previ\s-habit-ness' 9 a problem is n en\gh to condemn x4 ,! f\r? problem is a cat* problem1 b s useful a "o t x has be5 adopt$ 9 _m mod]n books z a r\t9e drill6 ,! fir/ problem1 on ! contr>y1 all1 save ^? :o dem& no high] crit]ion = a problem ?an t x make ! pupil '?9k1' w\ld reject4 ,x dem&s ! rev]sal ( fix$ habits to no valid purpose2 = 9 life ! "q 9 su* case w\ld nev] 7or almo/ nev]7 be ',h[ _m c5ts did he lose8' b ',:at 0 ! result8' or simply ',:at ( x8' ,?is problem weak5s )\t excuse ! *ild's confid5ce 9 ! tra9+ he has _h4 ,problems l 7#b7 are giv5 by tea*]s ( excell5t reputation1 b probably d m h>m ?an good4 ,if a pupil %\ld int]rupt 8 tea*] dur+ ! recitation 9 ari?metic by say+1 ,i got up at #g o'clock to multiply #i by #b-c_/d & got #bd-c_/d = my answ]2 0 t ! "r "t to get up8 ! tea*] w\ld n ?ank =tune = ! /imulus to ?\ght b w\ld ?9k ! *ild a fool4 ,su* cat* que/ions may be fairly useful z an object lesson on ! value ( se>* = ! ess5tial elem5t 9 a situation if a great v>iety ( !m are giv5 "o aft] ano!r ) r\t9e problems int]mix$ & ) w>n+ ( ! g5]al nature ( ! ex]cise at ! beg9n+4 ,ev5 s1 x %\ld be rememb]$ t reason+ %\ld be *iefly a =ce organiz+ habits1 n oppos+ !m2 & also t !re are en\gh bad habits to be oppos$ to give all necess>y tra9+4 ,fabricat$ puzzle situations :]e9 a peculi> hidd5 elem5t ( ! situation makes ! good habits call$ up by ! situation mislead+ are useful !re=e r z a relief & amus+ v>iation 9 ari?metical "w ?an z /imuli to ?\ght4 ,problems l ! ?ird quot$ above we might call puzzl+ r ?an 'cat*' problems4 ,!y h value z drills 9 analysis ( a situation into its elem5ts t w amuse ! gift$ *ildr51 & z te/s ( c]ta9 abilities4 ,!y also require t ( _m confus+ habits1 ! "r "o be *os51 r ?an t ord9>y habits be set aside by "s hidd5 elem5t 9 ! situation4 ,n en\gh is kn[n ab\t _! effect to enable u to decide :e!r or n ! elem5t>y s*ool %\ld include special facility ) !m z "o ( ! ari?metical functions t x specially tra9s4 ,! f\r? 'cat*' quot$ above1 :i* all w\ld admit is a good problem1 is good because x opposes a good habit = ! sake ( ano!r good habit1 =ces ! analysis ( an elem5t ^: analysis life v mu* requires1 & does x ) no obvi\s wa/e4 ,x is n safe to leave a *ild ) ! "o habit ( respond+ to 'add1 #ai1 #cj' by #di1 = 9 life ! 'h #ai1 mu/ get 4444 to h #cj' situation is v frequ5t & important4 ,on ! :ole1 ! ord9>y problems :i* ord9>y life pr(f]s seem to be ! sort t %\ld be reason$ out1 ?\gh ! elem5t>y s*ool may include ! less noxi\s =ms ( pure m5tal gymna/ics = ^? pupils :o l !m4 ,,summ>y ,^! discussions ( ! mean+s ( numb]s1 ! l+ui/ic dem&s ( ari?metic1 ! di/9ction betwe5 s*ola/ic & real applications ( ari?metic1 & ! possible re/rictions ( tra9+ 9 reason+1--may s]ve z illu/rations ( ! significance ( ! "q1 ,:at are ! functions t ! elem5t>y s*ool tries to improve 9 its tea*+ ( ari?metic8 ,o!r matt]s might well be consid]$ 9 ?is connection1 b ! ma9 outl9e ( ! "w ( ! elem5t>y s*ool is n[ fairly cle>4 ,! ari?metical functions or abilities :i* x seeks to improve are1 we may say3-- 7#a7 ,work+ k ( ! mean+s ( numb]s z names = c]ta9 siz$ collections1 = c]ta9 relative magnitudes1 ! magnitude ( unity be+ kn[n1 & = c]ta9 c5t]s or nuclei ( relations to o!r numb]s4 7#b7 ,work+ k ( ! sy/em ( decimal notation4 7#c7 ,work+ k ( ! mean+s ( addition1 subtraction1 multiplication1 & division4 7#d7 ,work+ k ( ! nature & relations ( c]ta9 common measures4 7#e7 ,work+ ability to add1 subtract1 multiply1 & divide ) integ]s1 common & decimal fractions1 & d5om9ate numb]s1 all be+ real positive numb]s4 7#f7 ,work+ k ( words1 symbols1 diagrams1 & ! l z requir$ by life's simpl] ari?metical dem&s or by economical prep>ation !re=4 7#g7 ,! ability to apply all ! above z requir$ by life's simpl] ari?metical dem&s or by economical prep>ation !re=1 includ+ 7#g a7 c]ta9 specific abilities to solve problems conc]n+ areas ( rectangles1 volumes ( rectangul> solids1 p]c5ts1 int]e/1 & c]ta9 o!r common occurr5ces 9 h\sehold1 factory1 & bus9ess life4 ,,! ,,sociology ,,( ,,ari?metic ,! phrase 'life's simpl] ari?metical dem&s' is necess>ily left vague4 ,j :at use is be+ made ( ari?metic 9 ?is c\ntry 9 #aibj by ea* p]son !re91 we "k only v r\ghly4 ,:at may be call$ a 'sociology' ( ari?metic is v mu* ne$$ to inve/igate ?is matt]4 ,= r>e or difficult dem&s ! elem5t>y s*ool %\ld n prep>e2 !re are too _m o!r desirable abilities t x %\ld improve4 ,a mo/ int]e/+ beg9n+ at su* an inv5tory ( ! actual uses ( ari?metic has be5 made by ,wilson '#ai & ,mit*ell4#b ,al?\gh _! /udies ne$ to be mu* ext5d$ & *eck$ by o!r me?ods ( inquiry1 two ma9 facts seem fairly c]ta94 ,fir/1 ! great majority ( p 9 ! great majority ( _! do+s use only v elem5t>y ari?metical processes4 ,9 #agcg cases ( addition report$ by ,wilson1 sev5 eigh?s 7 ( five numb]s or less4 ,ov] half ( ! multipli]s report$ 7 "o-figure numb]s4 ,ov] #ie p] c5t ( ! fractions op]at$ ) 7 includ$ 9 ?is li/3 #a_/b #a_/d #c_/d #a_/c #b_/c #a_/h #c_/h #a_/e #b_/e #d_/e4 ,?ree f\r?s ( all ! cases report$ 7 simple "o-/ep computations ) integ]s or ,unit$ ,/ates money4 ,second1 !y (t5 use ^! v elem5t>y processes1 n because su* are ! quicke/ & mo/ conv5i5t1 b because !y h lo/1 or maybe nev] _h1 ma/]y ( ! m advanc$ processes :i* w\ld d ! "w bett]4 ,! #e & #aj c5t /ores1 ! c\nt] ) ,any?+ on ?is c\nt] = #be;c1 & ! arrangem5ts = paym5ts on ! in/allm5t plan are famili> in/ances ( human avoidance ( ari?metic4 ,wilson f\nd v slight use ( decimals2 & ,mit*ell f\nd m5 comput+ ) #di?s z common fractions :5 ! use ( decimals w\ld h be5 m effici5t4 ,if giv5 #abj seconds to d a te/ l t %[n bel[1 lead+ lawy]s1 physicians1 manufactur]s1 & bus9ess m5 & _! wives w1 accord+ to my exp]i5ce1 get only ab\t half ! "w "r4 ,_m wom51 f9d+ on _! meat bill #g-c_/h lb4 roa/ beef #b4cf1 w sp5d "t & money to telephone ! but*] ask+ h[ mu* roa/ beef 0 p] p\nd1 because !y h no sure p[] 9 divid+ by a mix$ numb]4 #b ,! "w ( ,mit*ell has n be5 publi%$1 b ! au?or has _h ! privilege ( exam9+ x4 ,te/ ,p]=m ! op]ations indicat$4 ,express all fractions 9 answ]s 9 l[e/ t]ms4 ,add3 #c_/d #a_/f 4#be #d yr4 #f mo4 #a yr4 #b mo4 #f yr4 #i mo4 #c yr4 #f mo4 #d yr4 #e mo4 ----------- ,subtract3 #h4f - #f4jejjg #g_/h - #b_/c #e-g_/af - #b-c_/af ,multiply3 #bi ft4 #f 94 #g x #h x #d-a_/b #h ------------ ,divide3 #d-a_/b _/ #g ,x seems probable t ! s*ool tra9+ 9 ari?metic ( ! pa/ has n giv5 en\gh att5tion to p]fect+ ! m elem5t>y abilities4 ,& we %all lat] f9d fur!r evid5ce ( ?is4 ,on ! o!r h&1 ! fact t p 9 g5]al d n at pres5t use a process may n mean t !y ought n to use x4 ,life's simpl] ari?metical dem&s c]ta9ly d n include matt]s l ! rules = f9d+ cube root or true disc\nt1 :i* no s5sible p]son uses4 ,!y %\ld n include matt]s l comput+ ! lat]al surface or volume ( pyramids & cones1 or kn[+ ! cu/oms ( pla/]]s & pap] hang]s1 :i* are us$ only by highly specializ$ trades4 ,!y %\ld n include matt]s l int]e/ on call loans1 usury1 exact int]e/1 & ! r$isc\nt+ ( notes1 :i* conc]n only brok]s1 bank cl]ks1 & ri* m54 ,!y %\ld n include ! te*nique ( cu/oms :i* are vani%+ f effici5t practice1 su* z simple int]e/ on am\nt = times long] ?an a ye>1 days ( grace1 or extremes & means 9 proportions4 ,!y %\ld n include any elaborate practice ) v l>ge numb]s1 or decimals beyond ?\s&?s1 or ! addition & subtraction ( fractions :i* n "o p]son 9 a hundr$ has to add or subtract (t5] ?an once a ye>4 ,:5 we h an adequate sociology ( ari?metic1 /at+ accurately j :o %\ld use ea* ari?metical ability & h[ (t51 we %all be able to def9e ! task ( ! elem5t>y s*ool 9 ?is respect4 ,= ! pres5t1 we may proce$ by common s5se1 guid$ by two limit+ rules4 ,! fir/ is1--,x is no m desirable = ! elem5t>y s*ool to tea* all ! facts ( ari?metic ?an to tea* all ! words 9 ! ,engli% language1 or all ! topography ( ! globe1 or all ! details ( human physiology4 ,! second is1--,x is n desirable to elim9ate any elem5t ( ari?metical tra9+ until y h some?+ bett] to put 9 its place4 ,,*apt] ,,ii ,,! ,,measurem5t ,,( ,,ari?metical ,,abilities ,"o ( ! be/ ways to cle> up notions ( :at ! functions are :i* s*ools %\ld develop & improve is to get measures ( !m4 ,if any giv5 k or skill or p[] or ideal exi/s1 x exi/s 9 "s am\nt4 ,a s]ies ( am\nts ( x1 v>y+ f less to m1 def9es ! ability itself 9 a way t no g5]al v]bal description c d4 ,?us1 a s]ies ( weights1 #a lb41 #b lb41 #c lb41 #d lb41 etc41 helps to tell u :at we mean by weight4 ,by f9d+ a s]ies ( words l only1 smoke1 ano!r1 pretty1 answ]1 tailor1 circus1 telephone1 saucy1 & beg9n+1 :i* are spell$ correctly by kn[n & decreas+ p]c5tages ( *ildr5 ( ! same age1 or ( ! same s*ool grade1 we "k bett] :at we mean by 'spell+-difficulty4' ,inde$1 until we c measure ! effici5cy & improvem5t ( a function1 we are likely to be vague & loose 9 our ideas ( :at ! function is4 ,a ,,sample ,,measurem5t ,,( ,,an ,,ari?metical ,,ability3 ,,! ,,ability ,,to ,,add ,,integ]s ,consid] fir/1 z a sample1 ! measurem5t ( ability to add integ]s4 ,! foll[+ 7 ! examples us$ 9 ! measurem5ts made by ,/one '#jh3 #eif #dfie #dbh #hgb #bcge #id #gidh #djeb #ge #fghf #fced #cjd #efg #bfj #fde #heh #ejda #ihd #iddg #aedc #hig #gdii ---- --- ---- ,! scor+ 0 z foll[s3 ,cr$it ( #a = ea* column add$ correctly4 ,/one comb9$ measures ( o!r abilities ) ?is 9 a total score = am\nt done correctly 9 #ab m9utes4 ,/one also scor$ ! correctness ( ! additions 9 c]ta9 "w 9 multiplication4 ,c\rtis uses a %eet ( tw5ty-f\r tasks or 'examples1' ea* consi/+ ( ! addition ( n9e ?ree-place numb]s z %[n bel[4 ,eight m9utes is all[$4 ,he scores ! am\nt done by ! numb] ( examples1 & also scores ! numb] ( examples done correctly1 b does n sugge/ any comb9ation ( ^! two into a g5]al-effici5cy score4 #ibg #cgi #gef #hcg #ibd #aaj #hed #ife #cdd --- ,! au?or long ago propos$ t pupils be measur$ also ) s]ies l a to g %[n bel[1 9 :i* ! difficulty increases /ep by /ep4 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 con/ruct$ 8 well-kn[n te/s on ?is pr9ciple1 ?\gh he uses only "o example at ea* /ep ( difficulty in/ead ( eight or t5 z sugge/$ above4 ,8 te/1 s f> z addition ( integ]s goes1 is3-- ,,s]ies ,a4 ,,addition ,,scale 79 "p7 ,by ,clif=d ,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 --- ,9 8 orig9al report1 ,woody gives no s*eme = scor+ an individual1 wisely assum+ t1 ) s few samples at ea* degree ( difficulty1 a pupil's score w\ld be too unreliable = individual diagnosis4 ,! te/ is reliable = a class2 & = a class ,woody us$ ! degree ( difficulty su* t a /at$ fraction ( ! class c d ! "w correctly1 if tw5ty m9utes is all[$ = ! ?irty-eight examples ( ! entire te/4 ,! measurem5t ( ev5 s simple a matt] z ! effici5cy ( a pupil's responses to ^! te/s 9 add+ integ]s is really r complex4 ,!re is fir/ ( all ! problem ( comb9+ spe$ & accuracy into "s s+le e/imate4 ,/one gives no cr$it = a column unless x is correctly add$4 ,c\rtis evades ! difficulty by report+ bo? numb] done & numb] correct4 ,! au?or's s*eme1 :i* gives specifi$ weights to spe$ & accuracy at ea* /ep ( ! s]ies1 involves a r intricate computation4 ,?is difficulty ( equat+ spe$ & accuracy 9 add+ means precisely t we h inadequate notions ( :at ! ability is t ! elem5t>y s*ool %\ld improve4 ,until1 = example1 we h decid$ :e!r1 = a giv5 gr\p ( pupils1 fifte5 ,c\rtis attempts ) t5 "r1 is or is n a bett] a*ievem5t ?an t5 ,c\rtis attempts ) n9e "r1 we h n decid$ j :at ! bus9ess ( ! tea*] ( addition is1 9 ! case ( t gr\p ( pupils4 ,!re is also ! difficulty ( comp>+ results :5 %ort & long columns are us$4 ,correctness ) a %ort column1 say ( five figures1 te/ifies to k ( ! process & to ! p[] to d f\r successive s+le additions )\t error4 ,correctness ) a long column1 say ( t5 digits1 te/ifies to k ( ! process & to ! p[] to d n9e successive s+le additions )\t error4 ,n[ if a pupil's precision 0 su* t on ! av]age he made "o mi/ake 9 eight s+le additions1 he w\ld get ab\t half ( 8 five-digit columns "r & almo/ none ( 8 t5-digit columns "r4 7,he w\ld d ?is1 t is1 if he add$ 9 ! cu/om>y way4 ,if he 7 taught to *eck results by repeat$ addition1 by add+ 9 half-columns & ! l1 8 p]c5tages ( accurate answ]s might be greatly increas$ 9 bo? cases & be made approximately equal47 ,l5g? ( column 9 a te/ ( addition "u ord9>y conditions ?us automatically ov]weights precision 9 ! s+le additions z comp>$ ) k ( ! process1 & ability at c>ry+4 ,fur!r1 9 ! case ( a column ( :atev] size1 ! result z ord9>ily scor$ does n di/+ui% betwe5 "o1 two1 ?ree1 or m 7up to ! limit7 errors 9 ! s+le additions4 ,yet1 obvi\sly1 a pupil :o1 add+ ) t5-digit columns1 has half ( 8 answ]-figures wrong1 probably (t5 makes two or m errors )9 a column1 :]eas a pupil :o has only "o column-answ] 9 t5 wrong1 probably almo/ nev] makes m ?an "o error )9 a column4 ,a %ort-column te/ is !n advisable z a means ( int]pret+ ! results ( a long-column te/4 ,f9ally1 ! *oice ( a %ort-column or ( a long-column te/ is indicative ( ! measur]'s notion ( ! k9d ( effici5cy ! _w prop]ly dem&s ( ! s*ool4 ,tw5ty ye>s ago ! au?or w\ld h be5 readi] to accept a long-column te/ ?an he n[ is4 ,9 ! _w at l>ge1 long-column addition is be+ m & m done by ma*9e1 ?\gh x p]si/s /ill 9 great frequ5cy 9 ! bookkeep+ ( weekly & mon?ly acc\nts 9 local groc]ies1 but*] %ops1 & ! l4 ,! se>* = a measure ( ability to add ?us puts ! problem ( spe$ v]sus precision1 & ( %ort-column v]sus long-column additions cle>ly be=e u4 ,! latt] problem has h>dly be5 realiz$ at all by ! ord9>y def9itions ( ability to add4 ,x may be said fur!r t ! measurem5t ( ability to add gives ! sci5tific /ud5t a %ock by ! lack ( precision f\nd ev]y:]e 9 s*ools4 ,( :at value is x to a graduate ( ! elem5t>y s*ool to be able to add ) examples l ^? ( ! ,c\rtis te/1 gett+ only eight out ( t5 "r8 ,nobody w\ld pay a comput] = t ability4 ,! pupil c\ld n keep 8 own acc\nts ) x4 ,! suppos$ discipl9>y value ( habits ( precision runs ! risk ( turn+ negative 9 su* a case4 ,x appe>s1 at lea/ to ! au?or1 imp]ative t *eck+ %\ld be taught & requir$ until a pupil c add s+le columns ( t5 digits ) n ov] "o wrong answ] 9 tw5ty columns4 ,spe$ is useful1 especially indirectly z an indication ( control ( ! sep>ate high]-decade additions1 b ! social dem& = addition bel[ a c]ta9 /&>d ( precision is nil1 & its discipl9>y value is nil or negative4 ,?is w be made a matt] ( fur!r /udy lat]4 ,,measurem5ts ,,( ,,abilities ,,9 ,,computation ,measurem5ts ( ^! abilities may be ( two sorts--7#a7 ( ! spe$ & accuracy %[n 9 do+ "o same sort ( task1 z illu/rat$ by ! ,c\rtis te/ = addition %[n on page #bh2 & 7#b7 ( h[ h>d a task c be done p]fectly 7or ) "s specifi$ precision7 )9 a c]ta9 assign$ "t or less1 z illu/rat$ by ! au?or's r\gh te/ = addition %[n on pages #bh & #bi1 & by ! ,woody te/s1 :5 ext5d$ to include alt]native =ms4 ,! ,c\rtis te/s1 orig9at$ z an improvem5t on ! ,/one te/s & s9ce elaborat$ by ! p]si/5t devotion ( _! au?or1 are a /&>d in/rum5t ( ! fir/ sort = measur+ ! s-call$ 'fundam5tal' ari?metical abilities ) integ]s4 ,!y are %[n on ?is & ! foll[+ page4 ,te/s ( ! second sort are ! ,woody te/s1 :i* include op]ations ) integ]s1 common & decimal fractions1 & d5om9ate numb]s1 ! ,ball\ te/ = common fractions '#af1 & ! ,ladd] ex]cises ( ! ,?orndike ari?metics4 ,"s ( ^! are %[n on pages #cf to #da4 ,c\rtis ,te/ ,ari?metic4 ,te/ ,no4 #a4 ,addition ,s]ies ,b ,y w be giv5 eight m9utes to f9d ! answ]s to z _m ( ^! addition examples z possible4 ,write ! answ]s on ?is pap] directly und]nea? ! examples4 ,y are n expect$ to be able to d !m all4 ,y w be m>k$ = bo? spe$ & accuracy1 b x is m important to h y\r answ]s "r ?an to try a great _m 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 --- --- --- --- --- --- --- --- & sixte5 m addition examples ( n9e ?ree-place numb]s4 ,c\rtis ,te/ ,ari?metic4 ,te/ ,no4 #b4 ,subtraction ,s]ies ,b ,y w be giv5 f\r m9utes to f9d ! answ]s to z _m ( ^! subtraction examples z possible4 ,write ! answ]s on ?is pap] directly und]nea? ! examples4 ,y are n expect$ to be able to d !m all4 ,y w be m>k$ = bo? spe$ & accuracy1 b x is m important to h y\r answ]s "r ?an to try a great _m examples4 #ajggiedia #gejhhhbd #iaejjjec #hgiciihc #ggaigjbi #egdjfcid #aiijaefc #gbbjgcaf --------- -------- -------- -------- & tw5ty m tasks ( ! same sort4 ,c\rtis ,te/ ,ari?metic4 ,te/ ,no4 #c4 ,multiplication ,s]ies ,b ,y w be giv5 six m9utes to "w z _m ( ^! multiplication examples z possible4 ,y are n expect$ to be able to d !m all4 ,d y\r "w directly on ?is pap]2 use no o!r4 ,y w be m>k$ = bo? spe$ & accuracy1 b x is m important to get correct answ]s ?an to try a l>ge numb] ( examples4 #hbdf #ghdc #dhcg #cdgh #fdhb #bi #gjb #hc #ae #df ---- ---- ---- ---- ---- & tw5ty m multiplication examples ( ! same sort4 ,c\rtis ,te/ ,ari?metic4 ,te/ ,no4 #d4 ,division ,s]ies ,b ,y w be giv5 eight m9utes to "w z _m ( ^! division examples z possible4 ,y are n expect$ to be able to d !m all4 ,d y\r "w directly on ?is pap]2 use no o!r4 ,y w be m>k$ = bo? spe$ & accuracy1 b x is m important to get correct answ]s ?an to try a l>ge numb] ( examples4 #be7fgge #id7heceb #cg7iiij #hf7hjjff & tw5ty m division examples ( ! same sort4 ,,s]ies ,b4 ,,multiplication ,,scale ,by ,clif=d ,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 ---- ----- ,,s]ies ,b4 ,,division ,,scale ,by ,clif=d ,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 ( #abh #ej _/ #g 7#ai7 7#bc7 7#bg7 7#bh7 #bdh _/ #g #bc7dfi #g_/h ( #fbd 4#jjc74jicf 7#cj7 7#cd7 7#cf7 #c_/d _/ #e #fb4ej _/ #a-a_/d #i7fi lbs4 #i oz4 ,ball\ ,te/ ,addition ( ,fractions ,te/ #a ,te/ #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 --- ---- --- ---- ,te/ #c ,te/ #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 ----- --- ---- --- ,te/ #e ,te/ #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 ,ladd] ,?orndike1 '#ag1 ,,iii1 #e ,beg9 at ! bottom ( ! ladd]4 ,see if y c climb to ! top )\t mak+ a mi/ake4 ,be sure to copy ! numb]s correctly4 ,/ep #f4 a4 ,add #a-a_/c yd41 #g_/h yd41 #a-a_/d yd41 #c_/d yd41 #g_/h yd41 & #a-a_/b yd4 b4 ,add #fb-a_/b;c1 #ff-b_/c;c1 #ef-a_/d;c1 #fj;c1 & #fb-a_/b;c4 c4 ,add #a-e_/af1 #a-i_/cb1 #a-c_/h1 #a-aa_/cb1 & #a-g_/af4 d4 ,add #a-a_/c yd41 #a-a_/d yd41 #a-a_/b yd41 #b yd41 #c_/d yd41 & #b_/c yd4 ,/ep #e4 a4 ,add #d ft4 #f-a_/b 941 #ec-a_/d 941 #e ft4 #a_/b 941 #ef-c_/d 941 & #e ft4 b4 ,add #g lb41 #f lb4 #aa oz41 #g-a_/b lb41 #f lb4 #d-a_/b oz41 & #h-a_/b lb4 c4 ,add #a hr4 #f m94 #bj sec41 #eh m94 #ae sec41 #a hr4 #d m941 & #ee m94 d4 ,add #g doll>s1 #ac half doll>s1 #ba qu>t]s1 #ag dimes1 & #ai nickels4 ,/ep #d4 a4 ,add 4#je-a_/b1 4#jf1 4#jd-c_/d1 4#jb-c_/d1 & 4#je-a_/d4 b4 ,add 4#cc-a_/c1 4#ab-a_/b1 4#ah1 4#af-b_/c1 4#jh-a_/c & 4#ae4 c4 ,add 4#jh-a_/c1 4#jf-a_/d1 4#ba1 4#jc-c_/d1 & 4#af-b_/c4 d4 ,add 4#fb1 4#fd-a_/b1 4#ff-b_/c1 4#aj-a_/d1 & 4#fh4 ,/ep #c4 a4 ,add #g-a_/d1 #f-a_/b1 #h-c_/h1 #e-c_/d1 #i-e_/h & #c-g_/h4 b4 ,add #d-e_/h1 #ab1 #g-a_/b1 #h-c_/d1 #f & #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 & #i-a_/b4 ,/ep #b4 a4 ,add #ab4jd1 4#if1 #d4g1 #i4fbe1 #c4be1 & #bj4 b4 ,add 4#eh1 #f4jc1 4#jgi1 #d4bjf1 #b4ge1 & #aj4d4 c4 ,add #eb1 #bi4h1 #da4jg1 #a4iac1 #b4f1 & #aaj4 d4 ,add #bi4g1 #cae1 #bf4ge1 #ai4jjd1 #h4gic1 & #bj4je4 ,/ep #a4 a4 ,add #aj-c_/e1 #aa-a_/e1 #aj-d_/e1 #aa1 #aa-b_/e1 #aj-c_/e1 & #aa4 b4 ,add #g-c_/h1 #f-e_/h1 #h1 #i-a_/h1 #g-g_/h1 #e-c_/h1 & #h-a_/h4 c4 ,add #ba-a_/b1 #ah-c_/d1 #ca-a_/b1 #ai-a_/d1 #ag-a_/d1 #bb1 & #af-a_/b4 d4 ,add #ad-e_/ab1 #ab-g_/ab1 #i-aa_/ab1 #f-a_/ab1 & #e4 ,a ,subtraction ,ladd] ,?orndike1 '#ag1 ,,iii1 #aa ,/ep #i4 a4 #b4af mi4 - #a-c_/d mi4 b4 #e4gb ft4 - #e ft4 #c 94 c4 #b m94 #aj-a_/b sec4 - #ic4d sec4 d4 #cj4bh ,a4 - #aj-a_/e ,a4 e4 #aj gal4 #b-a_/b qt4 - #d4fbc gal4 ,/ep #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 ------- ------ ------ ------ ----- ,/ep #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 ------ ------ ------ ------ ------ ,/ep #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 ------ ----- ------ ------ ------ ,/ep #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 ------ ------ ------ ------ ------ ,/ep #d4 a4 #d hr4 - #b hr4 #ag m94 b4 #d lb4 #g oz4 - #b lb4 #aa oz4 c4 #a lb4 #e oz4 - #ac oz4 d4 #g ft4 - #b ft4 #h 94 e4 #a bu4 - #a pk4 ,/ep #c4 a b c d e #ib mi4 #fgce mi4 #c - #hi;c #bh4d mi4 #ejh4dj #hd4ae mi4 #ffhi mi4 #ah4jd mi4 #bjh4fb --------- -------- -------- --------- ------- ,/ep #b4 a b c d e #be4jj #ajj4jj #gej4jj #fabd sq4 mi4 #ghdf sq4 mi4 #i4cf #ga4bh #gcf4ej #bdid sq4 mi4 #bghi sq4 mi4 ------ ------- ------- ------------ ------------ ,/ep #a4 a b c d e #ah4fd #be4ci #ef4gj #hai4d mi4 #fg4ee mi4 #g4dj #ac4cg #de4fj #bji4b mi4 #cf4ad mi4 ------ ------ ------ --------- --------- ,an ,av]age ,ladd] ,?orndike1 '#ag1 ,,iii1 #acb ,f9d ! av]age ( ! quantities on ea* l9e4 ,beg9 ) ,/ep #a4 ,climb to ! top )\t mak+ a mi/ake4 ,be sure to copy ! numb]s correctly4 ,ext5d ! division to two decimal places if necess>y4 ,/ep #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 ,/ep #e4 a4 #e ft4 #c-a_/b 941 #fa-a_/d 941 #eh-c_/d 941 #d ft4 #aa 94 b4 #f lb4 #i oz41 #f lb4 #aa oz41 #g-a_/d lb41 #g-c_/h lb4 c4 #a hr4 #d m94 #dj sec41 #eh m94 #ce sec41 #a-a_/d hr4 d4 #b4h miles1 #c-a_/b miles1 #b4gb miles ,/ep #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 ,/ep #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 ,/ep #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 ,/ep #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 ,z su* te/s are wid5$ to cov] ! :ole task ( ! elem5t>y s*ool 9 respect to ari?metic1 & accept$ by compet5t au?orities z adequate measures ( a*ievem5t 9 comput+1 !y w give1 z has be5 said1 a work+ def9ition ( ! task4 ,! read] w obs]ve1 = example1 t "w su* z ! foll[+1 ?\gh /ill f\nd 9 _m textbooks & classrooms1 does n1 9 g5]al1 appe> 9 ! mod]n te/s & scales4 ,r$uce ! foll[+ improp] fractions to mix$ numb]s3-- #ai_/ac #dc_/ba #agf_/be #aih_/ad ,r$uce to integral or mix$ numb]s3-- #facha_/cg #bacd_/fg #dac_/dac #fig_/bbe ,simplify3-- #c_/d ( #h_/i ( #c_/e ( #ae_/bb ,r$uce to l[e/ t]ms3-- #ceg_/ebg #bfd_/cab #dib_/ggi #dah_/hgd #hed_/agfi #cj_/gce #dd_/bdb #gg_/hdg #ah_/bdc #if_/bbd ,f9d diff]5ces3-- #f-b_/g #h-e_/aa #h-d_/ac #e-a_/d #g-a_/h #c-a_/ad #e-a_/g #c-g_/ac #b-aa_/ad #b-a_/g ------ ------ ------ ------- ------ ,squ>e3-- #b_/c #d_/e #e_/g #f_/i #aj_/aa #ab_/ac #b_/g #ae_/af #ai_/bj #ag_/ah #be_/cj #da_/ec ,multiply3-- #b_/aa x #cc #cb x #c_/ad #ci x #b_/ac #fj x #aa_/bh #gg x #d_/aa #fc x #b_/bg #ed x #h_/de #fe x #c_/ac #cdd-af_/ba #dcb-b_/g ,,measurem5ts ,,( ,,ability ,,9 ,,appli$ ,,ari?metic3 ,,! ,,solution ,,( ,,problems ,/one '#jh measur$ a*ievem5t ) ! foll[+ problems1 fifte5 m9utes be+ ! "t all[$4 ,solve z _m ( ! foll[+ problems z y h "t =2 "w !m 9 ord] z numb]$3 #a4 ,if y buy #b tablets at #g c5ts ea* & a book = #fe c5ts1 h[ mu* *ange %\ld y receive f a two-doll> bill8 #b4 ,john sold #d ,saturday ,ev5+ ,po/s at #e c5ts ea*4 ,he kept #a_/b ! money & ) ! o!r #a_/b he b\ght ,sunday pap]s at #b c5ts ea*4 ,h[ _m did he buy8 #c4 ,if ,james _h #d times z mu* money z ,george1 he w\ld h #af4 ,h[ mu* money has ,george8 #d4 ,h[ _m p5cils c y buy = #ej c5ts at ! rate ( #b = #e c5ts8 #e4 ,! uni=ms = a baseball n9e co/ #b4ej ea*4 ,! %oes co/ #b a pair4 ,:at 0 ! total co/ ( uni=ms & %oes = ! n9e8 #f4 ,9 ! s*ools ( a c]ta9 city !re are #bbjj pupils2 #a_/b are 9 ! prim>y grades1 #a_/d 9 ! gramm> grades1 #a_/h 9 ! high s*ool1 & ! re/ 9 ! night s*ool4 ,h[ _m pupils are !re 9 ! night s*ool8 #g4 ,if #c-a_/b tons ( coal co/ #ba1 :at w #e-a_/b tons co/8 #h4 ,a news deal] b\ght "s magaz9es = #a4 ,he sold !m = #a4bj1 ga9+ #e c5ts on ea* magaz9e4 ,h[ _m magaz9es 7 !re8 #i4 ,a girl sp5t #a_/h ( h] money = c> f>e1 & ?ree times z mu* = clo!s4 ,half ( :at %e _h left 0 #hj c5ts4 ,h[ mu* money did %e h at fir/8 #aj4 ,two girls receive #b4aj = mak+ buttonholes4 ,"o makes #db1 ! o!r #bh4 ,h[ %all !y divide ! money8 #aa4 ,mr4 ,br[n paid "o ?ird ( ! co/ ( a build+2 ,mr4 ,johnson paid #a_/b ! co/4 ,mr4 ,johnson receiv$ #ejj m annual r5t ?an ,mr4 ,br[n4 ,h[ mu* did ea* receive8 #ab4 ,a freight tra9 left ,albany = ,new ,york at #f o'clock4 ,an express left on ! same track at #h o'clock4 ,x w5t at ! rate ( #dj miles an h\r4 ,at :at "t ( day w x ov]take ! freight tra9 if ! freight tra9 /ops aft] x has gone #ef miles8 ,! crit]ia he _h 9 m9d 9 select+ ! problems 7 z foll[s3-- ,! ma9 purpose ( ! reason+ te/ is ! det]m9ation ( ! ability ( ,,vi ,a *ildr5 to reason 9 ari?metic4 ,to ?is end1 ! problems1 z select$ & arrang$1 are meant to embody ! foll[+ conditions3-- #a4 ,situations equally concrete to all ,,vi ,a *ildr54 #b4 ,graduat$ difficulties4 a4 ,z to ari?metical ?9k+4 b4 ,z to famili>ity ) ! situation pres5t$4 #c4 ,! omission ( a4 ,l>ge numb]s4 b4 ,p>ticul> memory requirem5ts4 c4 ,cat* problems4 d4 ,all subject matt] except :ole numb]s1 fractions1 & ,unit$ ,/ates money4 ,! te/ is purposely s long t only v r>ely did any pupil fully complete x 9 ! fifte5 m9ute limit4 ,cr$its 7 giv5 ( #a1 = ea* ( ! fir/ five problems1 #a4d1 #a4b1 & #a4f respectively = problems #f1 #g1 & #h1 & ( #b = ea* ( ! o!rs4 ,c\rtis s\ght to improve ! ,/one te/ ( problem-solv+1 replac+ x by ! two te/s reproduc$ bel[4 ,,ari?metic--,te/ ,no4 #f4 ,spe$ ,te/--,reason+ ,d n "w ! foll[+ examples4 ,read ea* example ?r\gh1 make up y\r m9d :at op]ation y w\ld use if y 7 go+ to "w x1 !n write ! "n ( ! op]ation select$ 9 ! blank space aft] ! example4 ,use ! foll[+ abbreviations3--,add4 = addition1 ,sub4 = subtraction1 ,mul4 = multiplication1 & ,div4 = division4 --------------- ,,op]ation --------------- #a4 ,a girl br\ght a collection ( #cg color$ po/al c>ds to s*ool "o day1 & gave away #ai c>ds to h] fri5ds4 ,h[ _m c>ds did %e h left to take home8 --------------- #b4 ,five boys play$ m>bles4 ,:5 ! game 0 ov]1 ea* boy _h ! same numb] ( m>bles4 ,if !re 7 #de m>bles altoge!r1 h[ _m did ea* boy h8 --------------- #c4 ,a girl1 wat*+ f a w9d[1 saw #bg automobiles pass ! s*ool ! fir/ h\r1 & #cc ! second4 ,h[ _m autos pass$ by ! s*ool 9 ! two h\rs8 --------------- #d4 ,9 a c]ta9 s*ool !re 7 eight rooms & ea* room _h seats = #ej *ildr54 ,:5 all ! places 7 tak51 h[ _m *ildr5 7 !re 9 ! s*ool8 --------------- #e4 ,a club ( boys s5t _! treasur] to buy baseballs4 ,!y gave him #c4ae to sp5d4 ,h[ _m balls did !y expect him to buy1 if ! balls co/ #de;c4 apiece8 --------------- #f4 ,a tea*] weigh$ all ! girls 9 a c]ta9 grade4 ,if "o girl weigh$ #gi p\nds & ano!r #aaj p\nds1 h[ _m p\nds heavi] 0 "o girl ?an ! o!r8 --------------- #g4 ,a girl want$ to buy a #e-p\nd box ( c&y to give z a pres5t to a fri5d4 ,%e decid$ to get ! k9d wor? #ce;c4 a p\nd4 ,:at did %e pay = ! pres5t8 --------------- #h4 ,"o day 9 vacation a boy w5t on a fi%+ trip & caught #ab fi% 9 ! morn+1 & #g 9 ! aft]noon4 ,h[ _m fi% did he cat* altoge!r8 --------------- #i4 ,a boy liv$ #ae blocks ea/ ( a s*ool2 8 *um liv$ on ! same /reet1 b #aa blocks we/ ( ! s*ool4 ,h[ _m blocks ap>t 7 ! two boys' h\ses8 --------------- #aj4 ,a girl 0 #e times z /rong z h] small si/]4 ,if ! little girl c\ld lift a weight ( #bj p\nds1 h[ l>ge a weight c\ld ! old] girl lift8 --------------- #aa4 ,! *ildr5 ( a s*ool gave a sleigh-ride p>ty4 ,!re 7 #bgj *ildr5 to g on ! ride & ea* sleigh held #cj *ildr54 ,h[ _m sleighs 7 ne$$8 --------------- #ab4 ,9 ,septemb] !re 7 #dc *ildr5 9 ! eigh? grade ( a c]ta9 s*ool2 by ,june !re 7 #ei4 ,h[ _m *ildr5 ent]$ ! grade dur+ ! ye>8 --------------- #ac4 ,a girl :o liv$ #ag blocks away walk$ to s*ool & back twice a day4 ,:at 0 ! total numb] ( blocks ! girl walk$ ea* day 9 go+ to & f s*ool8 --------------- #ad4 ,a boy :o made #fg;c4 a day c>ry+ pap]s1 0 hir$ to run on a long err& = :i* he receiv$ #ej;c4 ,:at 0 ! total am\nt ! boy e>n$ t day8 --------------- ,total ,"r --------------- 7,two m simil> problems foll[47 ,te/ #f & ,te/ #h are f ! ,c\rtis ,/&>d ,te/4 ,us$ by p]mission ( ,s4 ,a4 ,c\rtis4 ,,ari?metic--,te/ ,no4 #h4 ,reason+ ,9 ! blank space bel[1 "w z _m ( ! foll[+ examples z possible 9 ! "t all[$4 ,"w !m 9 ord] z numb]$1 ent]+ ea* answ] 9 ! answ] column be=e comm5c+ a new example4 ,d n "w on any o!r pap]4 --------- ,,answ] --------- #a4 ,! *ildr5 9 a c]ta9 s*ool gave a ,*ri/mas p>ty4 ,"o ( ! pres5ts 0 a box ( c&y4 ,9 fill+ ! boxes1 "o grade us$ #af p\nds ( c&y1 ano!r #ag p\nds1 a ?ird #ab p\nds1 & a f\r? #ac p\nds4 ,:at did ! c&y co/ at #bf;c4 a p\nd8 --------- #b4 ,a s*ool 9 a c]ta9 city us$ #beaf pieces ( *alk 9 #cg s*ool days4 ,?ree new rooms 7 op5$1 ea* room hold+ #ej *ildr51 & ! s*ool 0 !n f\nd to use #hd /icks ( *alk p] day4 ,h[ _m m /icks ( *alk 7 us$ p] day ?an at fir/8 --------- #c4 ,sev]al boys w5t on a bicycle trip ( #aejj miles4 ,! fir/ week !y rode #cgd miles1 ! second week #bfd miles1 ! ?ird #dbc miles1 ! f\r? #dja miles4 ,!y f9i%$ ! trip ! next week4 ,h[ _m miles did !y ride ! la/ week8 --------- #d4 ,=ty-five boys 7 hir$ to pick apples f #ae trees 9 an apple or*>d4 ,9 #ej m9utes ea* boy _h pick$ #dh *oice apples4 ,if all ! apples pick$ 7 pack$ away c>efully 9 #h boxes ( equal size1 h[ _m apples 7 put 9 ea* box8 --------- #e4 ,9 a c]ta9 s*ool #baf *ildr5 gave a sleigh-ride p>ty4 ,!y r5t$ #g sleighs at a co/ ( #cj4jj & paid #bd4jj = ! refre%m5ts4 ,! p>ty travell$ #ae miles 9 #b-a_/b h\rs & _h a v pleasant "t4 ,:at 0 ea* *ild's %>e ( ! exp5se8 --------- #f4 ,a girl f\nd1 by c>eful c\nt+1 t !re 7 #bdjj lett]s on "o page ( h] hi/ory1 & only #bbie lett]s on a page ( h] read]4 ,h[ _m m lett]s _h %e read 9 "o book ?an 9 ! o!r if %e _h read #dg pages 9 ea* ( ! books8 --------- #g4 ,ea* ( #ei rooms 9 ! s*ools ( a c]ta9 city contribut$ #be pres5ts to a ,*ri/mas ent]ta9m5t = poor *ildr54 ,! /ores ( ! city gave #aihf o!r articles = pres5ts4 ,:at 0 ! total numb] ( pres5ts giv5 away at ! ent]ta9m5t8 --------- #h4 ,=ty-eight *ildr5 f a c]ta9 s*ool paid #aj;c4 apiece to ride #g miles on ! c>s to a woods4 ,!re 9 a few h\rs !y ga!r$ #bgfe nuts4 #fje ( ^! 7 bad1 b ! re/ 7 %>$ equally among ! *ildr54 ,h[ _m good nuts did ea* "o get8 --------- ,total --------- ,^! propos$ measures ( ability to apply ari?metic illu/rate v nicely ! diff]5ces ( op9ion conc]n+ :at appli$ ari?metic & ari?metical reason+ %\ld be4 ,! ?9k] :o emphasizes ! fact t 9 life out ( s*ool ! situation dem&+ quantitative treatm5t is usually real r ?an describ$1 w condemn a te/ all ( ^: con/itu5ts are describ$ problems4 ,unless we are excessively hopeful conc]n+ ! transf] ( ideas ( me?od & proc$ure f "o m5tal function to ano!r we %all prote/ aga9/ ! artificiality ( ,no4 #c ( ! ,/one s]ies1 & ( ! entire ,c\rtis ,te/ #h except ,no4 #d4 ,! ,c\rtis spe$-reason+ te/ 7,no4 #f7 is a /rik+ example ( ! mixture ( ability to und]/& quantitative relations ) ! ability to und]/& words4 ,consid] ^! five1 = example1 9 comp>ison ) ! revis$ v]sions atta*$4#c #c ,! =m ( ,te/ #f quot$ "h is t giv5 by ,c\rtis '#aa-'ab1 p4 #bj4 ,?is diff]s a little f ! o!r s]ies ( ,te/ #f1 %[n on pages #dc & #dd4 #a4 ,! *ildr5 ( a s*ool gave a sleigh-ride p>ty4 ,!re 7 #i sleighs1 & ea* sleigh held #cj *ildr54 ,h[ _m *ildr5 7 !re 9 ! p>ty8 ,,revision4 ,if "o sleigh holds #cj *ildr51 #i sleighs hold 4444 *ildr54 #b4 ,two s*ool-girls play$ a numb]-game4 ,! score ( ! girl t lo/ 0 #eg po9ts & %e 0 beat5 by #af po9ts4 ,:at 0 ! score ( ! girl t won8 ,,revision4 ,m>y & ,nell play$ a game4 ,m>y _h a score ( #eg4 ,nell beat ,m>y by #af4 ,nell _h a score ( 4444 #c4 ,a girl c\nt$ ! automobiles t pass$ a s*ool4 ,! total 0 #fj 9 two h\rs4 ,if ! girl saw #bg pass ! fir/ h\r h[ _m did %e see ! second8 ,,revision4 ,9 two h\rs a girl saw #fj automobiles4 ,%e saw #bg ! fir/ h\r4 ,%e saw 4444 ! second h\r4 #d4 ,on a playgr\nd !re 7 five equal gr\ps ( *ildr5 ea* play+ a diff]5t game4 ,if !re 7 #ge *ildr5 all toge!r1 h[ _m 7 !re 9 ea* gr\p8 ,,revision4 #ge p\nds ( salt j fill$ five boxes4 ,! boxes 7 exactly alike4 ,!re 7 4444 p\nds 9 a box4 #e4 ,a tea*] weigh$ all ! *ildr5 9 a c]ta9 grade4 ,"o girl weigh$ #gj p\nds4 ,h] old] si/] 0 #di p\nds heavi]4 ,h[ _m p\nds did ! si/] weigh8 ,,revision4 ,m>y weighs #gj lb4 ,jane weighs #di p\nds m ?an ,m>y4 ,jane weighs 4444 p\nds4 ,! di/9ction betwe5 a problem describ$ z cle>ly & simply z possible & ! same problem put awkw>dly or 9 ill-kn[n words or willfully obscur$ %\ld be reg>d$2 & z a rule measurem5ts ( ability to apply ari?metic %\ld es*ew all ne$less obscurity or purely l+ui/ic difficulty4 ,= example1 ,a boy b\ght a two-c5t /amp4 ,he gave ! man 9 ! /ore #aj c5ts4 ,! "r *ange 0 4444 c5ts4 is bett] z a te/ ?an ,if a boy1 pur*as+ a two-c5t /amp1 gave a t5-c5t /amp 9 paym5t1 :at *ange %\ld he be expect$ to receive 9 return8 ,! di/9ction betwe5 ! description ( a bona fide problem t a human be+ might be call$ on to solve out ( s*ool & ! description ( imag9>y possibilities or puzzles %\ld also be consid]$4 ,nos4 #c & #i ( ,/one are bad because to frame ! problems "o mu/ fir/ "k ! answ]s1 s t 9 reality !re c\ld nev] be any po9t 9 solv+ !m4 ,x is probably safe to say t nobody 9 ! _w ev] did or ev] w or ev] %\ld f9d ! numb] ( apples 9 a box by ! task ( ,no4 #d ( ! ,c\rtis ,te/ #h4 ,?is atta*es no blame to ,dr4 ,/one or to ,mr4 ,c\rtis4 ,until v rec5tly we 7 all s us$ to ! artificial problems ( ! traditional sort t we did n expect any?+ bett]2 & s bl9d to ! language dem&s ( describ$ problems t we did n see _! v great influ5ce4 ,c\rtis himself has be5 active 9 re=m & has po9t$ out 7'#ac1 p4 #d f47 ! defects 9 8 ,te/s #f & #h4 ,te/s ,nos4 #f & #h1 ! s-call$ reason+ te/s1 h prov$ ! lea/ satisfactory ( ! s]ies4 ,! judgm5ts ( v>i\s tea*]s & sup]9t5d5ts z to ! inequalities ( ! units 9 any "o te/1 & ( ! diff]5ces betwe5 ! diff]5t editions ( ! same te/1 h prov$ ! ne$ ( inve/igat+ ^! que/ions4 ,te/s ( adults 9 _m l9es ( comm]cial "w h yield$ 9 _m cases l[] scores ?an ^? ( ! av]age eigh? grade *ildr54 ,at ! same "t ! scores ( c]ta9 individuals ( m>k$ ability h be5 high1 & !re appe>s to be a g5]al relation betwe5 ability 9 ^! te/s & accuracy 9 ! ab/ract "w4 ,! mo/ significant facts1 h[ev]1 h be5 ! difficulties exp]i5c$ by tea*]s 9 attempt+ to rem$y ! defects 9 reason+4 ,x is c]ta9 t ! te/s measure abilities ( value b ! abilities are probably n :at !y seem to be4 ,9 an attempt to measure ! value ( diff]5t units1 = in/ance1 z _m problems z possible 7 con/ruct$ bas$ ^u a s+le situation4 ,tw5ty-"o v>ieties 7 secur$ by v>y+ ! relative =m ( ! "q & ! relative position ( ! diff]5t phrases4 ,"o ( ^! prov$ n9ete5 times z h>d z ano!r z measur$ by ! numb] ( mi/akes made by ! *ildr52 yet ! cause ( ! diff]5ce 0 m]ely ! *anges 9 ! phras+4 ,?is & o!r facts ( ! same k9d seem to %[ t ,te/s #f & #h measure ma9ly ! ability to read4 ,! sci5tific measurem5t ( ! abilities & a*ievem5ts conc]n$ ) appli$ ari?metic or problem-solv+ is ?us a matt] = ! future4 ,9 ! case ( describ$ problems a beg9n+ has be5 made 9 ! s]ies :i* =m a "p ( ! ,national ,intellig5ce ,te/s '#bj1 "o ( :i* is %[n on page #di f4 ,9 ! case ( problems ) real situations1 no?+ 9 sy/ematic =m is yet available4 ,sy/ematic te/s & scales1 besides def9+ ! abilities we are to e/abli% & improve1 are ( v great s]vice 9 measur+ ! /atus & improvem5t ( individuals & ( classes1 & ! effects ( v>i\s me?ods ( in/ruction & ( /udy4 ,!y are ?us helpful to pupils1 tea*]s1 sup]visors1 & sci5tific inve/igators2 & are be+ m & m widely us$ e ye>4 ,in=mation conc]n+ ! m]its ( ! diff]5t te/s1 ! proc$ure to foll[ 9 giv+ & scor+ !m1 ! age & grade /&>ds to be us$ 9 int]pret+ results1 & ! l1 is available 9 ! manuals ( ,educational ,measurem5t1 su* z ,c\rtis1 ,manual ( ,in/ructions = ,giv+ & ,scor+ ! ,c\rtis ,/&>d ,te/s 9 ! ,?ree ,r's '#ad2 ,/>*1 ,educational ,measurem5ts '#af2 ,*apman & ,ru%1 ,sci5tific ,measurem5t ( ,classroom ,products '#ag2 ,monroe1 ,de,voss1 & ,kelly1 ,educational ,te/s & ,measurem5ts '#ag2 ,wilson & ,hoke1 ,h[ to ,measure '#bj2 & ,mc,call1 ,h[ to ,measure 9 ,education '#ba4 ,,te/ #a ,national ,intellig5ce ,te/s4 ,scale ,a4 ,=m #a1 ,edition #a ,f9d ! answ]s z quickly z y c4 ,write ! answ]s on ! dott$ l9es4 ,use ! side ( ! page to figure on4 ,beg9 "h #a ,five c5ts make #a nickel4 ,h[ _m nickels make a dime8 ,answ] 44444 #b ,john paid #e doll>s = a wat* & #c doll>s = a *a94 ,h[ _m doll>s did he pay = ! wat* & *a98 ,answ] 44444 #c ,nell is #ac ye>s old4 ,m>y is #i ye>s old4 ,h[ mu* y\ng] is ,m>y ?an ,nell8 ,answ] 44444 #d ,"o qu>t ( ice cream is en\gh = #e p]sons4 ,h[ _m qu>ts ( ice cream are ne$$ = #be p]sons8 ,answ] 44444 #e ,john's gr&mo!r is #hf ye>s old4 ,if %e lives1 9 h[ _m ye>s w %e be #ajj ye>s old8 ,answ] 44444 #f ,if a man gets #b4ej a day1 :at w he be paid = six days' "w8 ,answ] 44444 #g ,h[ _m in*es are !re 9 a foot & a half8 ,answ] 44444 #h ,:at is ! co/ ( #ab cakes at #f = #e c5ts8 ,answ] 44444 #i ,! uni=ms = a baseball team ( n9e boys co/ #b4ej ea*4 ,! %oes co/ #b a pair4 ,:at 0 ! total co/ ( uni=ms & %oes = ! n9e8 ,answ] 44444 #aj ,a tra9 t usually arrives at half-pa/ t5 0 #ag m9utes late4 ,:5 did x arrive8 ,answ] 44444 #aa ,at #aj;c a y>d1 :at is ! co/ ( a piece #aj-a_/b ft4 long8 ,answ] 44444 #ab ,a man e>ns #f a day half ! "t1 #d4ej a day "o f\r? ( ! "t1 & no?+ on ! rema9+ days = a total p]iod ( #dj days4 ,:at did he e>n 9 all 9 ! #dj days8 ,answ] 44444 #ac ,:at p] c5t ( #hjj is #d ( #ajjj8 ,answ] 44444 #ad ,if #fj m5 ne$ #aejj lb4 fl\r p] mon?1 :at is ! requirem5t p] man p] day c\nt+ a mon? z #cj days8 ,answ] 44444 #ae ,a c> goes at ! rate ( a mile a m9ute4 ,a truck goes #bj miles an h\r4 ,h[ _m times z f> w ! c> g z ! truck 9 #aj seconds8 ,answ] 44444 #af ,! area ( ! base 7inside measure7 ( a cyl9drical tank is #ij squ>e feet4 ,h[ tall mu/ x be to hold #ajj cubic y>ds8 ,answ] 44444 ,f ,national ,intellig5ce ,te/s by ,national ,rese>* ,c\ncil4 ,copyright1 #aibj1 by ,! ,_w ,book ,company1 ,yonk]s-on-,hudson1 ,new ,york4 ,us$ by p]mission ( ! publi%]s4 ,,*apt] ,,iii ,,! ,,con/itution ,,( ,,ari?metical ,,abilities ,,! ,,elem5t>y ,,functions ,,( ,,ari?metical ,,le>n+ ,x w\ld be a useful "w = "s "o to try to analyze ari?metical le>n+ into ! unit>y abilities :i* compose x1 %[+ j :at1 9 detail1 ! m9d has to d 9 ord] to be prep>$ to pass a ?or\gh te/ on ! :ole ( ari?metic4 ,^! unit>y abilities w\ld make a v long li/4 ,exam9ation ( a well-plann$ textbook w %[ t su* an ability z multiplication is treat$ z a composite ( ! foll[+3 k ( ! multiplications up to #i x #i2 ability to multiply two 7or m7-place numb]s by #b1 #c1 & #d :5 'c>ry+' is n requir$ & no z]os occur 9 ! multiplic&2 ability to multiply by #b1 #c1 444 #i1 ) c>ry+2 ! ability to h&le z]os 9 ! multiplic&2 ! ability to multiply ) two-place numb]s n end+ 9 z]o2 ! ability to h&le z]o 9 ! multipli] z la/ numb]2 ! ability to multiply ) ?ree 7or m7-place numb]s n includ+ a z]o2 ! ability to multiply ) ?ree- & f\r-place numb]s ) z]o 9 second or ?ird1 or second & ?ird1 z well z 9 la/ place2 ! ability to save "t by annex+ z]os2 & s on & on ?r\gh a long li/ ( fur!r abilities requir$ to multiply ) ,unit$ ,/ates money1 decimal fractions1 common fractions1 mix$ numb]s1 & d5om9ate numb]s4 ,! units or '/eps' ?us recogniz$ by c>eful tea*+ w\ld make a long li/1 b x is probable t a /ill m c>eful /udy ( ari?metical ability z a hi]>*y ( m5tal habits or connections w\ld greatly increase ! li/4 ,consid]1 = example1 ord9>y column addition4 ,! majority ( tea*]s probably treat ?is z a simple application ( ! k ( ! additions to #i #i1 plus und]/&+ ( 'c>ry+4' ,on ! contr>y !re are at lea/ sev5 processes or m9or functions involv$ 9 two-place column addition1 ea* ( :i* is psy*ologically di/9ct & requires di/9ct educational treatm5t4 ,^! are3-- ,a4 ,le>n+ to keep "o's place 9 ! column z "o adds4 ,b4 ,le>n+ to keep 9 m9d ! result ( ea* addition until ! next numb] is add$ to x4 ,c4 ,le>n+ to add a se5 to a ?\ght-( numb]4 ,d4 ,le>n+ to neglect an empty space 9 ! columns4 ,e4 ,le>n+ to neglect #js 9 ! columns4 ,f4 ,le>n+ ! application ( ! comb9ations to high] decades may = ! less gift$ pupils involve z mu* "t & labor z le>n+ all ! orig9al addition tables4 ,& ev5 = ! mo/ gift$ *ild ! =mation ( ! connection '#h & #g #ae' probably nev] q insures ! pres5ce ( ! connections '#ch & #g #de' & '#ah #g #be4' ,g4 ,le>n+ to write ! figure signify+ units r ?an ! total sum ( a column4 ,9 p>ticul>1 le>n+ to write #j 9 ! cases :]e ! sum ( ! column is #aj1 #bj1 etc4 ,le>n+ to 'c>ry' also involves 9 itself at lea/ two di/9ct processes1 by :atev] way x is taught4 ,we f9d evid5ce ( su* specialization ( functions 9 ! results ) su* te/s z ,woody's4 ,= example1 #b #e #a 4444 surely involves abilities 9 "p diff]5t f #b #d #c - because only #gg p]c5t ( *ildr5 9 grade #c d ! =m] correctly1 :]eas #ie p]c5t ( *ildr5 9 t grade d ! latt] correctly4 ,9 grade #b ! diff]5ce is ev5 m m>k$4 ,9 ! case ( subtraction #d #d - involves abilities diff]5t f ^? involv$ 9 #i #c1 - be+ mu* less (t5 solv$ correctly 9 grades #b & #d4 #f #j - is mu* h>d] ?an ei!r ( ! above4 #dc #a #ba #b #cc #ac is mu* h>d] ?an #ce4 -- -- ,x may be said t ^! diff]5ces 9 difficulty are due to diff]5t am\nts ( practice4 ,?is is probably n true1 b if x 71 x w\ld n *ange ! argum5t2 if ! two abilities 7 id5tical1 ! practice ( "o w\ld improve ! o!r equally4 ,i %all n und]take "h ?is task ( li/+ & describ+ ! elem5t>y functions :i* con/itute ari?metical le>n+1 p>tly because :at !y are is n fully kn[n1 p>tly because 9 _m cases a f9al ability may be con/itut$ 9 sev]al diff]5t ways ^: descriptions become necess>ily t$i\s1 & p>tly because an adequate /atem5t ( :at is kn[n w\ld f> outrun ! space limits ( ?is *apt]4 ,in/ead1 ,i %all illu/rate ! results by "s samples4 ,,k ,,( ,,! ,,mean+ ,,( ,a ,,fraction ,z a fir/ sample1 consid] k ( ! mean+ ( a fraction4 ,is ! ability 9 "q simply to und]/& t a fraction is a /atem5t ( ! numb] ( p>ts1 ea* ( a c]ta9 size1 ! upp] numb] or num]ator tell+ h[ _m p>ts are tak5 & ! l[] numb] or d5om9ator tell+ :at fraction ( unity ea* "p is8 ,& is ! educational treatm5t requir$ simply to describe & illu/rate su* a /atem5t & h ! pupils apply x to ! recognition ( fractions & ! int]pretation ( ea* ( !m8 ,& is ! le>n+ process 7#a7 ! =mation ( ! notions ( "p1 size ( "p1 numb] ( "p1 7#b7 relat+ ! la/ two to ! numb]s 9 a fraction1 &1 z a necess>y consequ5ce1 7#c7 apply+ ^! notions adequately :5ev] "o enc\nt]s a fraction 9 op]ation8 ,precisely ?is 0 ! notion a few g5]ations ago4 ,! nature ( fractions 0 taught z "o pr9ciple1 9 "o /ep1 & ! habits ( deal+ ) fractions 7 suppos$ to be d$uc$ f ! g5]al law ( a fraction's nature4 ,z a result ! subject ( fractions _h to be long delay$1 0 /udi$ at great co/ ( "t & ef=t1 &1 ev5 s1 rema9$ a my/]y to all save gift$ pupils4 ,^! gift$ pupils probably ( _! own accord built up ! ability piecemeal out ( con/itu5t insights & habits4 ,at all ev5ts1 sci5tific tea*+ n[ does build up ! total ability z a fusion or organization ( less] abilities4 ,:at ^! are w be se5 be/ by exam9+ ! means tak5 to get !m4 7#a7 ,fir/ comes ! association ( #a_/b ( a pie1 #a_/b ( a cake1 #a_/b ( an apple1 & su* l ) _! concrete mean+s s t a pupil c prop]ly "n a cle>ly designat$ half ( an obvi\s unit l an orange1 pe>1 or piece ( *alk4 ,! same degree ( und]/&+ ( #a_/d1 #a_/h1 #a_/c1 #a_/f1 & #a_/e is secur$4 ,! pupil is taught t #a pie #b #a_/bs1 #c #a_/cs1 #d #a_/ds1 #e #a_/es1 #f #a_/fs1 & #h #a_/hs2 simil>ly = #a cake1 #a apple1 & ! l4 ,s f> he und]/&s #a_/x ( y 9 ! s5se ( c]ta9 simple p>ts ( obvi\sly unit>y ys4 7#b7 ,next comes ! association ) #a_/b ( an in*1 #a_/b ( a foot1 #a_/b ( a glassful & o!r cases :]e y is n s obvi\sly a unit>y object ^: pieces /ill %[ _! d]ivation f x4 ,simil>ly = #a_/d1 #a_/c1 etc4 7#c7 ,next comes ! association ) #a_/b ( a collection ( eight pieces ( c&y1 #a_/c ( a doz5 eggs1 #a_/e ( a squad ( t5 soldi]s1 etc41 until #a_/b1 #a_/c1 #a_/d1 #a_/e1 #a_/f1 & #a_/h are und]/ood z names ( c]ta9 p>ts ( a collection ( objects4 7#d7 ,next comes ! simil> association :5 ! nature ( ! collection is left undef9$1 ! pupil respond+ to #a_/b ( #f is 4441 #a_/d ( #h is 4441 #b is #a_/e ( 4441 #a_/c ( #f is 4441 #a_/c ( #i is 4441 #b is #a_/c ( 4441 & ! l4 ,ea* ( ^! abilities is ju/ifi$ 9 tea*+ by its intr9sic m]its1 irrespective ( its lat] s]vice 9 help+ to con/itute ! g5]al und]/&+ ( ! mean+ ( a fraction4 ,! habits ?us =m$ 9 grades #c or #d are ( con/ant s]vice !n & !reaft] 9 & out ( s*ool4 7#e7 ,) ^! comes ! use ( #a_/e ( #aj1 #ae1 #bj1 etc41 #a_/f ( #ab1 #ah1 #db1 etc41 z a useful v>iety ( drill on ! division tables1 valuable 9 itself1 & a means ( mak+ ! notion ( a unit fraction m g5]al by add+ #a_/g & #a_/i to ! s*eme4 7#f7 ,next comes ! connection ( #c_/d1 #b_/e1 #c_/e1 #d_/e1 #b_/c1 #a_/f1 #e_/f1 #c_/h1 #e_/h1 #g_/h1 #c_/aj1 #g_/aj1 & #i_/aj1 ea* ) its mean+ z a c]ta9 "p ( "s conv5i5tly divisible unit1 &1 7#g7 & 7#h71 connections betwe5 ^! fractions & _! mean+s z p>ts ( c]ta9 magnitudes 7#g7 & collections 7#h7 ( conv5i5t size1 & 7#i7 connections betwe5 ^! fractions & _! mean+s :5 ! nature ( ! magnitude or collection is un/at$1 z 9 #d_/e ( #ae 4441 #e_/h ( #cb 4444 7#aj7 ,t ! relation is g5]al is %[n by us+ x ) numb]s requir+ writt5 division & multiplication1 su* z #g_/h ( #agcf 4441 & ) ,unit$ ,/ates money4 ,elem5ts 7#f7 to 7#aj7 aga9 are useful ev5 if ! pupil nev] goes f>!r 9 ari?metic4 ,"o ( ! commone/ uses ( fractions is 9 calculat+ ! co/ ( fractions ( y>ds ( clo?1 & fractions ( p\nds ( meat1 *eese1 etc4 ,! next /ep 7#aa7 is to und]/& to "s ext5t ! pr9ciple t ! value ( any ( ^! fractions is unalt]$ by multiply+ or divid+ ! num]ator & d5om9ator by ! same numb]4 ,! drills 9 express+ fractions 9 l[] & high] t]ms :i* accompli% ?is are p>allel$ by 7#ab7 & 7#ac7 simple ex]cises 9 add+ & subtract+ fractions to %[ t fractions are quantities t c be op]at$ on l any quantities1 & by 7#ad7 simple "w ) mix$ numb]s 7addition & subtraction & r$uctions71 & 7#ae7 improp] fractions4 ,all t is done ) improp] fractions is 7a7 to h ! pupil use a few ( !m z he w\ld any fractions & 7b7 to note _! equival5t mix$ numb]s4 ,9 7#ab71 7#ac71 & 7#ad7 only fractions ( ! same d5om9ators are add$ or subtract$1 & 9 7#ab7 7#ac71 7#ad71 & 7#ae7 only fractions ) #b1 #c1 #d1 #e1 #f1 #h1 or #aj 9 ! d5om9ator are us$4 ,z hi!rto1 ! "w ( 7#aa7 to 7#ae7 is useful 9 & ( itself4 7#af7 ,def9itions are giv5 ( ! foll[+ type3-- ,numb]s l #b1 #c1 #d1 #g1 #aa1 #bj1 #cf1 #adj1 #iba are call$ :ole numb]s4 ,numb]s l #g_/h1 #a_/e1 #b_/c1 #c_/d1 #aa_/h1 #g_/f1 #a_/c1 #d_/c1 #a_/h1 #a_/f are call$ fractions4 ,numb]s l #e-a_/d1 #g-c_/h1 #i-a_/b1 #af-d_/e1 #cae-g_/h1 #a-a_/c1 #a-b_/c are call$ mix$ numb]s4 7#ag7 ,! t]ms num]ator & d5om9ator are connect$ ) ! upp] & l[] numb]s compos+ a fraction4 ,build+ ?is some:at elaborate s]ies ( m9or abilities seems to be a v r\ndab\t way ( gett+ k ( ! mean+ ( a fraction1 & is1 if we take no acc\nt ( :at is got along ) ?is k4 ,tak+ acc\nt ( ! intr9sically useful habits t are built up1 "o might retort t ! pupil gets 8 k ( ! mean+ ( a fraction at z]o co/4 ,,k ,,( ,,! ,,subtraction ,,& ,,division ,,tables ,consid] next ! k ( ! subtraction & division ',tables4' ,! usual treatm5t presupposes t le>n+ !m consi/s ( =m+ indep5d5tly ! 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 ,9 fact1 h[ev]1 ^! #abf bonds are n =m$ indep5d5tly4 ,except p]haps 9 ! case ( ! dulle/ tw5tie? ( pupils1 !y are some:at facilitat$ by ! already le>n$ additions & multiplications4 ,& by prop] arrangem5t ( ! le>n+ !y may be enorm\sly facilitat$ !reby4 ,inde$1 we may replace ! indep5d5t memoriz+ ( ^! facts by a set ( in/ructive ex]cises :]e9 ! pupil d]ives ! subtractions f ! correspond+ additions by simple acts ( reason+ or selective ?9k+4 ,z soon z ! additions giv+ sums ( #i or less are le>n$1 let ! pupil attack an ex]cise l ! foll[+3-- ,write ! miss+ numb]s3-- ,a ,b ,c ,d #c & 444 are #e4 #e & 444 are #h4 #d & 444 are #e4 #d & 444 are #h4 #c & 444 are #i4 #c & 444 are #f4 #e & 444 are #f4 #a & 444 are #g4 #d & 444 are #g4 #d & 444 are #i4 #f & 444 are #i4 #f & 444 are #g4 #e & 444 are #g4 #b & 444 #f4 #a & 444 are #h4 #h & 444 are #i4 #f & 444 are #h4 #e & 444 #i4 #c & 444 are #g4 #c 444 are #d4 #d & 444 are #f4 #b & 444 #g4 #a 444 are #c4 #g 444 are #h4 #b & 444 are #e4 #c & 444 #h4 #a 444 are #e4 #d 444 are #i4 #b & 444 #h4 #a & 444 #d4 #d 444 are #h4 #b 444 are #c4 #c & 444 #f4 #b & 444 #d4 #g 444 are #i4 #a 444 are #i4 #f & 444 #i4 #c & 444 #h4 #b 444 #d4 #c 444 #f4 #d & 444 #f4 #f & 444 #g4 #c 444 #h4 #e 444 #i4 #d & 444 #g4 #b & 444 #e4 #d 444 #e4 #a 444 #c4 ,! task = reason+ is only to try1 "o aft] ano!r1 numb]s t seem promis+ & to select ! "r "o :5 f\nd4 ,) a little /imulus & direction *ildr5 c ?us d]ive ! subtractions up to ^? ) #i z ! l>g] numb]4 ,let !m !n be taught to d ! same ) ! pr9t$ =ms3-- ,subtract #i #g #h #e #h #f #c #e #f #b #b #d etc4 - - - - - - & #i - #g 4441 #i - #e 4441 #g - #e 4441 etc4 ,9 ! case ( ! divisions1 suppose t ! pupil has le>n$ 8 fir/ table & ga9$ surety 9 su* ex]cises z3-- #d #es 4444 #f x #e 4444 #i nickels 4444 c5ts4 #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 "o ball co/s #e c5ts1 two balls co/ 4444 c5ts1 ?ree balls co/ 4444 c5ts1 etc4 ,he may !n be set at once to "w at ! answ]s to ex]cises l ! foll[+3-- ,write ! answ]s & ! miss+ numb]s3-- ,a ,b ,c ,d 4444 #es #ae #dj 4444 #es 4444 x #e #be #bj c5ts 4444 nickels4 4444 #es #bj #bj 4444 #es 4444 x #e #ej #cj c5ts 4444 nickels4 4444 #es #dj #ae 4444 #es 4444 x #e #ce #ae c5ts 4444 nickels4 4444 #es #be #de 4444 #es 4444 x #e #aj #dj c5ts 4444 nickels4 4444 #es #cj #ej 4444 #es 4444 x #e #dj 4444 #es #ce #be 4444 #es 4444 x #e #de ,e ,= #e c5ts y c buy #a small loaf ( bread4 ,= #aj c5ts y c buy #b small loaves ( bread4 ,= #be c5ts y c buy 4444 small loaves ( bread4 ,= #de c5ts y c buy 4444 small loaves ( bread4 ,= #ce c5ts y c buy 4444 small loaves ( bread4 ,f #e c5ts pays #a c> f>e4 #ae c5ts pays 4444 c> f>es4 #aj c5ts pays 4444 c> f>es4 #bj c5ts pays 4444 c> f>es4 ,g ,h[ _m #e c5t balls c y buy ) #cj c5ts8 4444 ,h[ _m #e c5t balls c y buy ) #ce c5ts8 4444 ,h[ _m #e c5t balls c y buy ) #be c5ts8 4444 ,h[ _m #e c5t balls c y buy ) #ae c5ts8 4444 ,9 ! case ( ! mean+ ( a fraction1 ! ability1 & s ! le>n+1 is mu* m elaborate ?an common practice has assum$2 9 ! case ( ! subtraction & division tables ! le>n+ is mu* less s4 ,9 nei!r case is ! le>n+ ei!r m]e memoriz+ ( facts or ! m]e und]/&+ ( a pr9ciple 9 ab/racto foll[$ by its application to concrete cases4 ,x is 7& ?is we %all f9d true ( almo/ all effici5t le>n+ 9 ari?metic7 ! =mation ( connections & _! use 9 su* an ord] t ea* helps ! o!rs to ! maximum degree1 & s t ea* w d ! maximum am\nt = ari?metical abilities o!r ?an ! "o specially conc]n$1 & = ! g5]al compet5ce ( ! le>n]4 ,,le>n+ ,,! ,,processes ,,( ,,computation ,z ano!r in/ructive topic 9 ! con/itution ( ari?metical abilities1 we may take ! case ( ! reason+ involv$ 9 und]/&+ ! manipulations ( figures 9 two 7or m7-place addition & subtraction1 multiplication & division involv+ a two 7or m7-place numb]1 & ! manipulations ( decimals 9 all f\r op]ations4 ,! psy*ology ( ^! is ( special int]e/ & importance4 ,= !re are two opposite explanations possible "h1 lead+ to two opposite !ories ( tea*+4 ,! common explanation is t ^! me?ods ( manipulation1 if und]/ood at all1 are und]/ood z d$uctions f ! prop]ties ( our sy/em ( decimal notation4 ,! o!r is t !y are und]/ood p>tly z inductions f ! exp]i5ce t !y always give ! "r answ]4 ,! fir/ explanation leads to ! common prelim9>y d$uctive explanations ( ! textbooks4 ,! o!r leads to explanations by v]ification2 e4g41 ( addition by c\nt+1 ( subtraction by addition1 ( multiplication by addition1 ( division by multiplication4 ,samples ( ^! two sorts ( explanation are giv5 bel[4 ,,%ort ,,multiplication ,,)\t ,,c>ry+3 ,,d$uctive ,,explanation ,,multiplication is ! process ( tak+ "o numb] z _m times z !re are units 9 ano!r numb]4 ,! ,,product is ! result ( ! multiplication4 ,! ,,multiplic& is ! numb] to be tak54 ,! ,,multipli] is ! numb] d5ot+ h[ _m times ! multiplic& is to be tak54 ,! multipli] & multiplic& are ! ,,factors4 ,multiply #fbc by #c ,,op]ation ,multiplic& #fbc ,multipli] #c ---- ,product #ahfi ,,explanation4--,= conv5i5ce we write ! multipli] "u ! multiplic&1 & beg9 ) units to multiply4 #c times #c units are #i units4 ,we write ! n9e units 9 units' place 9 ! product4 #c times #b t5s are #f t5s4 ,we write ! #f t5s 9 t5s' place 9 ! product4 #c times #f hundr$s are #ah hundr$s1 or #a ?\s& & #h hundr$s4 ,! #a ?\s& we write 9 ?\s&s' place & ! #h hundr$s 9 hundr$s' place 9 ! product4 ,!re=e1 ! product is #a ?\s& #h hundr$s1 #f t5s & #i units1 or #ahfi4 ,,%ort ,,multiplication ,,)\t ,,c>ry+3 ,,inductive ,,explanation #a4 ,! *ildr5 ( ! ?ird grade are to h a picnic4 #cb are go+4 ,h[ _m s&wi*es w !y ne$ if ea* ( ! #cb *ildr5 has f\r s&wi*es8 ,"h is a quick way to f9d out3-- #cb ,?9k #d x #b1 write #h "u ! #b 9 ! ones column4 #d ,?9k #d x #c1 write #ab "u ! #c 9 ! t5s column4 -- #b4 ,h[ _m bananas w !y ne$ if ea* ( ! #cb *ildr5 has two bananas8 #cb x #b or #b x #cb w give ! answ]4 #c4 ,h[ _m little cakes w !y ne$ if ea* *ild has ?ree cakes8 #cb x #c or #c x #cb w give ! answ]4 #cb #c x #b 4444 ,:]e d y write ! #f8 #c #c x #c 4444 ,:]e d y write ! #i8 -- #d4 ,prove t #abh1 #fd1 & #if are "r by add+ f\r #cbs1 two #cbs1 & ?ree #cbs4 #cb #cb #cb #cb #cb #cb #cb #cb #cb -- -- -- ,multiplication ,y multiply :5 y f9d ! answ]s to que/ions l ,h[ _m are #i x #c8 ,h[ _m are #c x #cb8 ,h[ _m are #h x #e8 ,h[ _m are #d x #db8 #a4 ,read ^! l9es4 ,say ! "r numb]s :]e ! dots are3 ,if y add #c to #cb1 y h 4444 #ce is ! sum4 ,if y subtract #c f #cb1 ! result is 4444 #bi is ! diff]5ce or rema9d]4 ,if y multiply #c by #cb or #cb by #c1 y h 4444 #if is ! product4 ,f9d ! products4 ,*eck y\r answ]s to ! fir/ l9e by add+4 #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 ! #i 9 ! ones column4 #c ,write ! #c 9 ! t5s column4 --- ,write ! #f 9 ! hundr$s column4 ,*eck y\r answ] by add+4 ,add #bac #bac #bac --- #ah4 #ai4 #bj4 #ba4 #bb4 #bc4 #bd4 #bad #cab #dcb #bca #acb #cad #bdc #b #c #b #c #c #b #b --- --- --- --- --- --- --- ,,%ort ,,division3 ,,d$uctive ,,explanation ,divide #ahbe by #d ,divisor #d #ahbe ,divid5d -------- #def-a_/d ,quoti5t ,,explanation4--,= conv5i5ce we write ! divisor at ! left ( ! divid5d1 & ! quoti5t bel[ x1 & beg9 at ! left to divide4 #d is n conta9$ 9 #a ?\s& any ?\s& times1 !re=e ! quoti5t conta9s no unit ( any ord] high] ?an hundr$s4 ,consequ5tly we f9d h[ _m times #d is conta9$ 9 ! hundr$s ( ! divid5d4 #a ?\s& & #h hundr$s are #ah hundr$s4 #d is conta9$ 9 #ah hundr$s #d hundr$ times & #b hundr$s rema9+4 ,we write ! #d hundr$s 9 ! quoti5t4 ,! #b hundr$s we consid] z unit$ ) ! #b t5s1 mak+ #bb t5s4 #d is conta9$ 9 #bb t5s #e t5s times1 & #b t5s rema9+4 ,we write ! #e t5s 9 ! quoti5t1 & ! rema9+ #b t5s we consid] z unit$ ) ! #e units1 mak+ #be units4 #d is conta9$ 9 #be units #f units times & #a unit rema9+4 ,we write ! #f units 9 ! quoti5t & indicate ! division ( ! rema9d]1 #a unit1 by ! divisor #d4 ,!re=e ! quoti5t ( #ahbe divid$ by #d is #def-a_/d1 or #def & #a rema9d]4 ,,%ort ,,division3 ,,inductive ,,explanation ,divid+ ,l>ge ,numb]s #a4 ,tom1 ,dick1 ,w1 & ,fr$ put 9 #b c5ts ea* to buy an eight-c5t bag ( m>bles4 ,!re are #abh m>bles 9 x4 ,h[ _m %\ld ea* boy h1 if !y divide ! m>bles equally among ! f\r boys8 ----- #d #abh ,?9k #ab ?ree #ds4 ,write ! #c ov] ! #b 9 ! t5s column4 ,?9k #h two #ds4 ,write ! #b ov] ! #h 9 ! ones column4 #cb is "r1 because #d x #cb #abh4 #b4 ,m>y1 ,nell1 & ,alice are go+ to buy a book z a pres5t = _! ,sunday-s*ool tea*]4 ,! pres5t co/s #fi c5ts4 ,h[ mu* %\ld ea* girl pay1 if !y divide ! co/ equally among ! ?ree girls8 ---- #cfi ,?9k #f 4444 #cs4 ,write ! #b ov] ! #f 9 ! t5s column4 ,?9k #i 4444 #cs4 ,write ! #c ov] ! #i 9 ! ones column4 #bc is "r1 = #c x #bc #fi4 #c4 ,divide ! co/ ( a #if-;c5t pres5t equally among ?ree girls4 ,h[ mu* %\ld ea* girl pay8 ------ #cif #d4 ,divide ! co/ ( an #hd-;c5t pres5t equally among #d girls4 ,h[ mu* %\ld ea* girl pay8 #e4 ,le>n ?is3 7,read _/ z divid$ by47 #ab #d #af4 #af is ! sum4 #ab - #d #h4 #h is ! diff]5ce or rema9d]4 #ab x #d #dh4 #dh is ! product4 #ab _/ #d #c4 #c is ! quoti5t4 #f4 ,f9d ! quoti5ts4 ,*eck y\r answ]s by multiply+4 ---- ---- ---- ----- ----- ----- #cii #bhf #eaee #fbdf #dafh #cbai ,unev5 division is taught by ! same g5]al plan1 ext5d$4 ,,long ,,division3 ,,d$uctive ,,explanation ,to ,divide by ,long ,division #a4 ,let x be requir$ to divide #cdeca by #ae4 ,op]ation ,divid$ ,divisor #ae7cdeca7bcjb-a_/ae ,quoti5t #cj -- #de #de -- #ca #cj -- #a ,rema9d] ,= conv5i5ce we write ! divisor at ! left & ! quoti5t at ! "r ( ! divid5d1 & beg9 to divide z 9 ,%ort ,division4 #ae is conta9$ 9 #c t5-?\s&s #j t5-?\s&s times2 !re=e1 !re w be #j t5-?\s&s 9 ! quoti5t4 ,take #cd ?\s&s2 #ae is conta9$ 9 #cd ?\s&s #b ?\s&s times2 we write ! #b ?\s&s 9 ! quoti5t4 #ae x #b ?\s&s #cj ?\s&s1 :i*1 subtract$ f #cd ?\s&s1 leaves #d ?\s&s #dj hundr$s4 ,add+ ! #e hundr$s1 we h #de hundr$s4 #ae 9 #de hundr$s #c hundr$s times2 we write ! #c hundr$s 9 ! quoti5t4 #ae x #c hundr$s #de hundr$s1 :i* subtract$ f #de hundr$s1 leaves no?+4 ,add+ ! #c t5s1 we h #c t5s4 #ae 9 #c t5s #j t5s times2 we write #j t5s 9 ! quoti5t4 ,add+ to ! ?ree t5s1 :i* equal #cj units1 ! #a unit1 we h #ca units4 #ae 9 #ca units #b units times2 we write ! #b units 9 ! quoti5t4 #ae x #b units #cj units1 :i*1 subtract$ f #ca units1 leaves #a unit z a rema9d]4 ,indicat+ ! division ( ! #a unit1 we annex ! fractional expression1 #a_/ae unit1 to ! integral "p ( ! quoti5t4 ,!re=e1 #cdeca divid$ by #ae is equal to #bcjb-a_/ae4 ,b4 ,gre5leaf1 ,practical ,ari?metic1 '#gc1 p4 #di4 ,,long ,,division3 ,,inductive ,,explanation ,divid+ by ,l>ge ,numb]s #a4 ,j be=e ,*ri/mas ,frank's fa!r s5t #cfj oranges to be divid$ among ! *ildr5 9 ,frank's class4 ,!re are #bi *ildr54 ,h[ _m oranges %\ld ea* *ild receive8 ,h[ _m oranges w be left ov]8 ,"h is ! be/ way to f9d out3 #ab & #ab ,?9k h[ _m #bis !re are 9 #cf4 #a is "r4 rema9d] ,write #a ov] ! #f ( #cf4 ,multiply #bi by #a4 #bi 7#cfj ,write ! #bi "u ! #cf4 ,subtract #bi f #cf4 #bi ,write ! #j ( #cfj aft] ! #g4 --- ,?9k h[ _m #bis !re are 9 #gj4 #b is "r4 #gj ,write #b ov] ! #j ( #cfj4 ,multiply #bi by #b4 #eh ,write ! #eh "u #gj4 ,subtract #eh f #gj4 -- ,!re is #ab rema9d]4 #ab ,ea* *ild gets #ab oranges1 & !re are #ab left ov]4 ,?is is "r1 = #ab multipli$ by #bi #cdh1 & #cdh #ab #cfj4 99 99 99 99 99 #h4 ,9 ,no4 #h1 keep on divid+ by #ca until y h us$ ! #e1 ! #h1 & ! #g1 & h f\r #ca7ii1ehg figures 9 ! quoti5t4 #i4 #aj4 #aa4 #ab4 #ac4 #bb7bec #bb7bhie #ba7hhia #bb7bij #cb7af1cfh ,*eck y\r results = #i1 #aj1 #aa1 #ab1 & #ac4 #a4 ,! boys & girls ( ! ,welf>e ,club plan to e>n money to buy a victrola4 ,!re are #bc boys & girls4 ,!y c get a good second-h& victrola = #e4ge4 ,h[ mu* mu/ ea* e>n if !y divide ! co/ equally8 ,"h is ! be/ way to f9d out3 4#be ,?9k h[ _m #bcs !re are 9 #eg4 #b is "r4 ----- ,write #b ov] ! #g ( #eg4 ,multiply #bc by #b4 #bce4ge ,write #df "u #eg & subtract4 ,write ! #e ( #ege #df aft] ! #aa4 ---- ,?9k h[ _m #bcs !re are 9 #aae4 #e is "r4 #aae ,write #e ov] ! #e ( #ege4 ,multiply #bc by #e4 #aae ,write ! #aae "u ! #aae t is !re & subtract4 ---- ,!re is no rema9d]4 ,put & ! decimal po9t :]e !y belong4 ,ea* *ild mu/ e>n #be c5ts4 ,?is is "r1 = 4#be multipli$ by #bc #e4ge4 #b4 ,divide #ga4gf equally among #bc p]sons4 ,h[ mu* is ea* p]son's %>e8 #c4 ,*eck y\r result = ,no4 #b by multiply+ ! quoti5t by ! divisor4 ,f9d ! quoti5ts4 ,*eck ea* quoti5t by multiply+ x by ! divisor4 #d4 #e4 #f4 #g4 #h4 #bc7ii4ac #be7ah4ej #ba7abi4ae #ac7bi4be #cb7gc4ib #a bu%el #cb qt4 #i4 ,h[ _m bu%els are !re 9 #bhh qt48 #aj4 ,9 #aib qt48 #aa4 ,9 #daf qt48 ,crucial exp]im5ts are lack+1 b !re are sev]al l9es ( well-atte/$ evid5ce4 ,fir/ ( all1 !re c be no d\bt t ! great majority ( pupils le>n ^! manipulations at ! />t f ! plac+ ( units "u units1 t5s "u t5s1 etc41 9 add+1 to ! plac+ ( ! decimal po9t 9 division ) decimals1 by imitation & bl9d foll[+ ( specific in/ructions1 & t a v l>ge proportion ( ! pupils d n to ! end1 t is to ! fif? s*ool-ye>1 und]/& !m z necess>y d$uctions f decimal notation4 ,x also seems probable t ?is proportion w\ld n be mu* r$uc$ no matt] h[ +5i\sly & c>efully ! d$uctions 7 expla9$ by textbooks & tea*]s4 ,evid5ce ( ?is fact w appe> abundantly to any "o :o w obs]ve s*oolroom life4 ,x also appe>s 9 ! fact t aft] ! prop]ties ( ! decimal notation h be5 ?us us$ aga9 & aga92 e4g41 = d$uc+ 'c>ry+' 9 addition1 'borr[+' 9 subtraction1 'c>ry+' 9 multiplication1 ! value ( ! digits 9 ! p>tial product1 ! value ( ea* rema9d] 9 %ort division1 ! value ( ! quoti5t figures 9 division1 ! addition1 subtraction1 multiplication1 & division ( ,unit$ ,/ates money1 & ! plac+ ( ! decimal po9t 9 multiplication1 no compet5t tea*] d>es to rely ^u ! pupil1 ev5 ?\gh he n[ has f\r or m ye>s' exp]i5ce ) decimal notation1 to d$uce ! plac+ ( ! decimal po9t 9 division ) decimals4 ,x may be an illusion1 b "o seems to s5se 9 ! bett] textbooks a recognition ( ! futility ( ! attempt to secure d$uctive d]ivations ( ^? manipulations4 ,i ref] to ! brevity ( ! explanations & _! ins]tion 9 su* a =m t !y w influ5ce ! pupils' ?9k+ z little z possible4 ,at any rate ! fact is sure t mo/ pupils d n le>n ! manipulations by d$uctive reason+1 or und]/& !m z necess>y consequ5ces ( ab/ract pr9ciples4 ,x is a common op9ion t ! only alt]native is kn[+ !m by rote4 ,?is1 ( c\rse1 is "o common alt]native1 b ! o!r explanation sugge/s t und]/&+ ! manipulations by inductive reason+ f _! results is ano!r & an important alt]native4 ,! manipulations ( 'long' multiplication1 = in/ance1 le>n$ by imitation or me*anical drill1 are f\nd to give = #be x ,a a result ab\t twice z l>ge z = #ac x ,a1 = #ch or #ci x ,a a result ab\t ?ree times z l>ge2 = #aae x ,a a result ab\t t5 times z l>ge z = #aa x ,a4 ,) ev5 ! v dull pupils ! proc$ure is v]ifi$ at lea/ to ! ext5t t x gives a result :i* ! sci5tific exp]t 9 ! case--! tea*]--calls "r4 ,) ev5 ! v bright pupils1 :o c appreciate ! relation ( ! proc$ure to decimal notation1 ?is relation may be us$ n z ! sole d$uction ( ! proc$ure be=eh&1 b z "o p>tial means ( v]ify+ x aft]w>d4 ,or !re may be ! condition ( half-appreciation ( ! relation 9 :i* ! pupil uses k ( ! decimal notation to conv9ce himself t ! proc$ure does1 b n t x mu/ give ! "r answ]1 ! answ] be+ '"r' because ! tea*]1 ! answ]-li/1 & collat]al evid5ce assure him ( x4 ,i h tak5 ! manipulation ( ! p>tial products z an illu/ration because x is "o ( ! lea/ favor$ cases = ! explanation ,i am pres5t+4 ,if we take ! fir/ case :]e a manipulation may be d$uc$ f decimal notation1 kn[n m]ely by rote1 or v]ifi$ inductively1 namely1 ! addition ( two-place numb]s1 x seems sure t ! m5tal processes j describ$ are almo/ ! univ]sal rule4 ,surely 9 our s*ools at pres5t *ildr5 add ! #c ( #bc to ! #c ( #ec & ! #b ( #bc to ! #e ( #ec at ! />t1 9 n9e cases out ( t5 because !y see ! tea*] d s & are told to d s4 ,!y are protect$ f add+ #c #c #b #e n by any d$uction ( any sort b because !y d n "k h[ to add #h & #e1 because !y h be5 taught ! habit ( add+ figures t /& "o above ! o!r1 or ) a betwe5 !m2 & because !y are %[n or told :at !y are to d4 ,!y are protect$ f add+ #c #e & #b #c1 aga91 by no d$uctive reason+ b = ! second & ?ird reasons j giv54 ,9 n9e cases out ( t5 !y d n ev5 ?9k ( ! possibility ( add+ 9 any o!r way ?an ! '#c #c1 #b #e' way1 mu* less d !y select t way on acc\nt ( ! facts t #ec #ej #c & #bc #bj #c1 t #ej #bj #gj1 t #c #c #f1 & t 7a b7 7c d7 7a c7 7b d76 ,j z surely all b ! v dulle/ tw5tie? or s ( *ildr5 come 9 ! end to some?+ m ?an rote k1--to und]/&1 to "k t ! proc$ure 9 "q is "r4 ,:e!r !y "k :y #gf is "r dep5ds ^u :at is meant by :y4 ,if x means t #gf is ! result :i* compet5t p agree ^u1 !y d4 ,if x means t #gf is ! result :i* w\ld come f accurate c\nt+ !y p]haps "k :y z well z !y w\ld h1 _h !y be5 giv5 full explanations ( ! relation ( ! proc$ure 9 two-place addition to decimal notation4 ,if :y means because #ec #ej #c1 #bc #bj #c1 #ej #bj #gj1 & 7a b7 7c d7 7a c7 7b d71 !y d n4 ,nor1 ,i am tempt$ to add1 w\ld mo/ ( !m by any sort ( tea*+ :atev]4 ,i conclude1 !re=e1 t s*ool *ildr5 may & d reason ab\t & und]/& ! manipulations ( numb]s 9 ?is inductive1 v]ify+ way )\t be+ able to1 or at lea/ )\t1 "u pres5t conditions1 f9d+ x pr(itable to d]ive !m d$uctively4 ,i believe1 9 fact1 t pure ari?metic z x is le>n$ & kn[n is l>gely an inductive sci5ce4 ,at "o extreme is a m9ority to :om x is a s]ies ( d$uctions f pr9ciples2 at ! o!r extreme is a m9ority to :om x is a s]ies ( bl9d habits2 betwe5 ! two is ! great majority1 repres5t+ e gradation b c5t]+ ab\t ! type ( ! inductive ?9k]4 ,,*apt] ,,iv ,,! ,,con/itution ,,( ,,ari?metical ,,abilities 7,,cont9u$73 ,,! ,,selection ,,( ,,! ,,bonds ,,to ,,be ,,=m$ ,:5 ! analysis ( ! m5tal functions involv$ 9 ari?metical le>n+ is made ?or\gh x turns into ! "q1 ',:at are ! elem5t>y bonds or connections t con/itute ^! functions8' & :5 ! problem ( tea*+ ari?metic is reg>d$1 z x %\ld be 9 ! light ( pres5t psy*ology1 z a problem 9 ! developm5t ( a hi]>*y ( intellectual habits1 x becomes 9 l>ge measure a problem ( ! *oice ( ! bonds to be =m$ & ( ! discov]y ( ! be/ ord] 9 :i* to =m !m & ! be/ means ( =m+ ea* 9 t ord]4 ,,! ,,importance ,,( ,,habit-,,=mation ,! importance ( habit-=mation or connection-mak+ has be5 grossly und]e/imat$ by ! majority ( tea*]s & writ]s ( textbooks4 ,=1 9 ! fir/ place1 ma/]y by d$uctive reason+ ( su* matt]s z 'c>ry+' 9 addition1 'borr[+' 9 subtraction1 ! value ( ! digits 9 ! p>tial products 9 multiplication1 ! manipulation ( ! figures 9 division1 ! plac+ ( ! decimal po9t aft] multiplication or division ) decimals1 or ! manipulation ( ! figures 9 ! multiplication & division ( fractions1 is impossible or extremely unlikely 9 ! case ( *ildr5 ( ! ages & exp]i5ce 9 "q4 ,!y d n z a rule d$uce ! me?od ( manipulation f _! k ( decimal notation4 ,r !y le>n ab\t decimal notation by c>ry+1 borr[+1 writ+ ! la/ figure ( ea* p>tial product "u ! multipli] :i* gives t product1 etc4 ,!y le>n ! me?od ( manipulat+ numb]s by see+ !m employ$1 & by m or less bl9dly acquir+ !m z associative habits4 ,9 ! second place1 we1 :o h already =m$ & long us$ ! "r habits & are !reby protect$ aga9/ ! casual mislead+s ( un=tunate m5tal connections1 c h>dly realize ! =ce ( m]e association4 ,:5 a *ild writes sixte5 z #fa1 or f9ds #dbh z ! sum ( #ae #ai #af #ah -- or gives #fdb z an answ] to #bg x #cf1 or says t #d divid$ by #a_/d #a1 we are tempt$ to consid] him m5tally p]v]se1 =gett+ or p]haps nev] hav+ und]/ood t he goes wrong = exactly ! same g5]al reason t we g "r2 namely1 ! g5]al law ( habit-=mation4 ,if we /udy ! cases ( #fa = #af1 we %all f9d !m occurr+ 9 ! "w ( pupils :o aft] hav+ be5 drill$ 9 writ+ #bf1 #cf1 #df1 #fb1 #fc1 & s on1 9 :i* ! ord] ( ! six 9 writ+ is ! same z x is 9 spee*1 return to writ+ ! 'te5 numb]s4 ,if our language said onety-"o = elev5 & onety-six = sixte51 we %\ld probably nev] f9d su* errors except z 'lapses' or z ! results ( misp]ception or lack ( memory4 ,!y w\ld !n be m frequ5t be=e ! #bjs1 #cjs1 etc41 7 le>n$4 ,if pupils are giv5 mu* drill on writt5 s+le column addition involv+ ! high] decades 7ea* "t writ+ ! two-figure sum71 !y are =m+ a habit ( writ+ #bh aft] ! sum ( #h1 #f1 #i1 & #e is rea*$2 & x %\ld n surprise u if ! pupil /ill occasionally writes ! two-figure sum = ! fir/ column ?\gh a second column is to be add$ also4 ,on ! contr>y1 unless "s c\nt] =ce influ5ces him1 he is absolutely sure to make ?is mi/ake4 ,! la/ mi/ake quot$ 7#d _/ #a_/d #a7 is int]e/+ because "h we h possibly "o ( ! cases :]e d$uction f psy*ology alone c give con/ructive aid to tea*+4 ,multiplication & division by fractions h be5 notori\s = _! difficulty4 ,! =m] is n[ alleviat$ by us+ ( in/ead ( x until ! new habit is fix$4 ,! latt] is /ill approa*$ ) elaborate caution & ) v>i\s means ( %[+ :y "o mu/ 'inv]t & multiply' or 'multiply by ! reciprocal4' ,b 9 ! au?or's op9ion x seems cle> t ! difficulty 9 multiply+ & divid+ by a fraction 0 n t *ildr5 felt any logical objections to cancel+ or inv]t+4 ,i fancy t ! majority ( !m w\ld *e]fully inv]t any fraction ?ree times ov] or cancel numb]s at r&om 9 a column if !y 7 %[n h[ to d s4 ,b if y are a y\ng/] inexp]i5c$ 9 num]ical ab/ractions & if y h _h divide connect$ ) 'make small]' ?ree ?\s& times & nev] once connect$ ) 'make bigg]1' y are sure to be some:at impell$ to make ! numb] small] ! ?ree ?\s& & fir/ "t y are ask$ to divide x4 ,"s ( my read]s w probably confess t ev5 n[ !y feel a slight irritation or d\bt 9 say+ or writ+ t #af_/a _/ #a_/h #abh4 ,! habits t h be5 confirm$ by e multiplication & division by integ]s are1 9 ?is p>ticul> ( '! ratio ( result to numb] op]at$ ^u1' directly oppos$ to ! =mation ( ! habits requir$ ) fractions4 ,& t is1 ,i believe1 ! ma9 cause ( ! difficulty4 ,its treatm5t !n becomes easy1 z w be %[n lat]4 ,^! illu/rations c\ld be add$ to almo/ indef9itely1 especially 9 ! case ( ! responses made to ! s-call$ 'cat*' problems4 ,! fact is t ! le>n] r>ely c1 & almo/ nev] does1 survey & analyze an ari?metical situation & ju/ify :at he is go+ to d by articulate d$uctions f pr9ciples4 ,he usually feels ! situation m or less vaguely & responds to x z he has respond$ to x or "s situation l x 9 ! pa/4 ,ari?metic is to him n a logical doctr9e :i* he applies to v>i\s special in/ances1 b a set ( r specializ$ habits ( behavior t[>d c]ta9 sorts ( quantities & relations4 ,& 9 s f> z he does come to "k ! doctr9e x is *iefly by do+ ! w ( ! ma/]4 ,?is is true ev5 ) ! cle>e/ expositions1 ! wise/ use ( objective aids1 & full enc\ragem5t ( orig9ality on ! pupil's "p4 ,le/ ! la/ few p>agraphs be misund]/ood1 ,i ha/5 to add t ! psy*ologi/s ( to-day d n wi% to make ! le>n+ ( ari?metic a m]e matt] ( acquir+ ?\s&s ( disconnect$ habits1 nor to decrease by "o jot ! pupil's g5u9e compreh5sion ( its g5]al tru?s4 ,!y wi% him to reason n less ?an he has 9 ! pa/1 b m4 ,!y f9d1 h[ev]1 t y d n secure reason+ 9 a pupil by dem&+ x1 & t 8 le>n+ ( a g5]al tru? )\t ! prop] developm5t ( organiz$ habits back ( x is likely to be1 n a rational le>n+ ( t g5]al tru?1 b only a me*anical memoriz+ ( a v]bal /atem5t ( x4 ,!y h come to "k t reason+ is n a magic =ce work+ 9 indep5d5ce ( ord9>y habits ( ?\ght1 b an organization & cooep]ation ( ^? v habits on a high] level4 ,! old] p$agogy ( ari?metic /at$ a g5]al law or tru? or pr9ciple1 ord]$ ! pupil to le>n x1 & gave him tasks to d :i* he c\ld n d pr(itably unless he und]/ood ! pr9ciple4 ,x left him to build up himself ! p>ticul> habits ne$$ to give him und]/&+ & ma/]y ( ! pr9ciple4 ,! new] p$agogy is c>eful to help him build up ^! connections or bonds ahead ( & along ) ! g5]al tru? or pr9ciple1 s t he c und]/& x bett]4 ,! old] p$agogy comm&$ ! pupil to reason & let him suff] ! p5alty ( small pr(it f ! "w if he did n4 ,! new] provides in/ructive exp]i5ces ) numb]s :i* w /imulate ! pupil to reason s f> z he has ! capacity1 b w /ill be pr(itable to him 9 concrete k & skill1 ev5 if he lacks ! ability to develop ! exp]i5ces into a g5]al und]/&+ ( ! pr9ciples ( numb]s4 ,! new] p$agogy secures m reason+ 9 reality by n pret5d+ to secure s mu*4 ,! new] p$agogy ( ari?metic1 !n1 scrut9izes e elem5t ( k1 e connection made 9 ! m9d ( ! le>n]1 s z to *oose ^? :i* provide ! mo/ in/ructive exp]i5ces1 ^? :i* w gr[ toge!r into an ord]ly1 rational sy/em ( ?9k+ ab\t numb]s & quantitative facts4 ,x is n en\gh = a problem to be a te/ ( und]/&+ ( a pr9ciple2 x mu/ also be helpful 9 & ( itself4 ,x is n en\gh = an example to be a case ( "s rule2 x mu/ help review & consolidate habits already acquir$ or lead up to & facilitate habits to be acquir$4 ,e detail ( ! pupil's "w mu/ d ! maximum s]vice 9 ari?metical le>n+4 ,,desirable ,,bonds ,,n[ ,,(t5 ,,neglect$ ,z hi!rto1 ,i %all n try to li/ completely ! elem5t>y bonds t ! c\rse ( /udy 9 ari?metic %\ld provide =4 ,! be/ means ( prep>+ ! /ud5t ( ?is topic = s\nd criticism & helpful inv5tion is to let him exam9e repres5tative cases ( bonds n[ (t5 neglect$ :i* %\ld be =m$ & repres5tative cases ( useless1 or ev5 h>mful1 bonds n[ (t5 =m$ at consid]able wa/e ( "t & ef=t4 7#a7 ,numb]s z measures ( cont9u\s quantities4--,! numb]s "o1 two1 ?ree1 #a1 #b1 #c1 etc41 %\ld be connect$ soon aft] ! beg9n+ ( ari?metic ea* ) ! appropriate am\nt ( "s cont9u\s quantity l l5g? or volume or weight1 z well z ) ! appropriate siz$ collection ( apples1 c\nt]s1 blocks1 & ! l4 ,l9es %\ld be label$ #a foot1 #b feet1 #c feet1 etc42 "o in*1 two in*es1 ?ree in*es1 etc42 weights %\ld be lift$ & call$ "o p\nd1 two p\nds1 etc42 ?+s %\ld be measur$ 9 glassfuls1 h&fuls1 p9ts1 & qu>ts4 ,o!rwise ! pupil is likely to limit ! mean+ (1 say1 f\r to f\r s5sibly discrete ?+s & to h difficulty 9 multiplication & division4 ,measur+1 or c\nt+ by ins5sibly m>k$ (f repetitions ( a unit1 b9ds ea* numb] "n to its mean+ z ---- times :atev] #a is1 m surely ?an m]e c\nt+ ( ! units 9 a collection c1 & %\ld re5=ce ! latt]4 7#b7 ,additions 9 ! high] decades4--,9 ! case ( all save ! v gift$ *ildr51 ! additions ) high] decades--t is1 ! bonds1 #af #g #bc1 #bf #g #cc1 #cf #g #dc1 #ad #h #bb1 #bd #h #cb1 & ! l--ne$ to be specifically practic$ until ! t5d5cy becomes g5]aliz$4 ',c\nt+' by #bs beg9n+ ) #a1 & ) #b1 c\nt+ by #cs beg9n+ ) #a1 ) #b1 & ) #c1 c\nt+ by #ds beg9n+ ) #a1 ) #b1 ) #c1 & ) #d1 & s on1 make easy beg9n+s 9 ! =mation ( ! decade connections4 ,practice ) isolat$ bonds %\ld soon be add$ to get fre] use ( ! bonds4 ,! "w ( column addition %\ld be *eck$ = accuracy s t a pupil w cont9ually get b5eficial practice r ?an 'practice 9 error4' 7#c7 ,! unev5 divisions4--,! quoti5ts ) rema9d]s = ! divisions ( e numb] to #ai by #b1 e numb] to #bi by #c1 e numb] to #ci by #d1 & s on %\ld be taught z well z ! ev5 divisions4 ,a table l ! foll[+ w be f\nd a conv5i5t means ( mak+ ^! connections3-- #aj 4444 #bs #aj 4444 #cs & 4444 rem4 #aj 4444 #ds & 4444 rem4 #aj 4444 #es #aa 4444 #bs & 4444 rem4 #aa 4444 #cs & 4444 rem4 4 4 4 #hi 4444 #is & 4444 rem4 ,^! bonds mu/ be =m$ be=e %ort division c be effici5t1 are useful z a p>tial help t[>d selection ( ! prop] quoti5t figures 9 long division1 & are ! *ief in/rum5ts = "o ( ! important problem s]ies 9 appli$ ari?metic1--,h[ _m xs c ,i buy = y c5ts at z c5ts p] x & h[ mu* w ,i h left8 ,t ^! bonds are at pres5t sadly neglect$ is %[n by ,kirby '#ac1 :o f\nd t pupils 9 ! la/ half ( grade #c & ! fir/ half ( grade #d c\ld d only ab\t f\r su* examples p] m9ute 79 a t5-m9ute te/71 & ev5 at t rate made f> f p]fect records1 ?\gh !y _h be5 taught ! regul> division tables4 ,sixty m9utes ( practice result$ 9 a ga9 ( ne>ly #ge p]c5t 9 numb] done p] m9ute1 ) an increase 9 accuracy z well4 7#d7 ,! equation =m4--,! equation =m ) an unkn[n quantity to be det]m9$1 or a miss+ numb] to be f\nd1 %\ld be connect$ ) its mean+ & ) ! problem attitude long be=e a pupil beg9s algebra1 & 9 ! m9ds ( pupils :o nev] w /udy algebra4 ,*ildr5 :o h j b>ely le>n$ to add & subtract le>n easily to d su* "w z ! foll[+3-- ,write ! miss+ numb]s3-- #d #h 4444 #e 4444 #ad 4444 #c #aa 4444 #e #b #af #g 4444 #ab 4444 #e ,! equation =m is ! simple/ uni=m way yet devis$ to /ate a quantitative issue4 ,x is capable ( indef9ite ext5sion if c]ta9 easily und]/ood conv5tions ab\t p>5!ses & fraction signs are le>n$4 ,x %\ld be employ$ widely 9 acc\nt+ & ! treatm5t ( comm]cial problems1 & w\ld be except = outworn conv5tions4 ,x is a lead+ contribution ( algebra to bus9ess & indu/rial life4 ,ari?metic c make x ne>ly z well4 ,x saves m "t 9 ! case ( drills on r$uc+ fractions to high] & l[] t]ms alone ?an is requir$ to le>n its mean+ & use4 ,to rewrite a quantitative problem z an equation & !n make ! easy selection ( ! necess>y te*nique to solve ! equation is "o ( ! mo/ univ]sally useful intellectual devices kn[n to man4 ,! words 'equals1' 'equal1' 'is1' 'are1' 'makes1' 'make1' 'gives1' 'give1' & _! r>] equival5ts %\ld !re=e e>ly give way on _m occasions to ! '' :i* s f> surpasses !m 9 ultimate conv5i5ce & simplicity4 7#e7 ,addition & subtraction facts 9 ! case ( fractions4--,9 ! case ( add+ & subtract+ fractions1 c]ta9 specific bonds--betwe5 ! situation ( halves & ?irds to be add$ & ! responses ( ?9k+ ( ! numb]s z equal to s _m six?s1 betwe5 ! situation ?irds & f\r?s to be add$ & ?9k+ ( !m z s _m twelf?s1 betwe5 f\r?s & eigh?s to be add$ & ?9k+ ( !m z eigh?s1 & ! l--%\ld be =m$ sep>ately4 ,! g5]al rule ( ?9k+ ( fractions z _! equival5ts ) "s conv5i5t d5om9ator %\ld come z an organization & ext5sion ( su* special habits1 n z an edict f ! textbook or tea*]4 7#f7 ,fractional equival5ts4--,effici5cy requires t 9 ! end ! mu* us$ r$uctions %\ld be firmly connect$ ) ! situations :]e !y are ne$$4 ,!y may z well1 !re=e1 be s connect$ f ! beg9n+1 ) ! ga9 ( mak+ ! g5]al process f> easi] = ! dull pupils to ma/]4 ,we %all see lat] t1 = all save ! v gift$ pupils1 ! economical way to get an und]/&+ ( ari?metical pr9ciples is n1 usually1 to le>n a rule & !n apply x1 b to p]=m in/ructive op]ations &1 9 ! c\rse ( p]=m+ !m1 to get insight into ! pr9ciples4 7#g7 ,protective habits 9 multiply+ & divid+ ) fractions4--,9 multiply+ & divid+ ) fractions special bonds %\ld be =m$ to c\nt]act ! n[ h>mful influ5ce ( ! 'multiply get a l>g] numb]' & 'divide get a small] numb]' bonds :i* all "w ) integ]s has be5 re5=c+4 ,= example1 at ! beg9n+ ( ! sy/ematic "w ) multiplication by a fraction1 let ! foll[+ be pr9t$ cle>ly at ! top ( e relevant page ( ! textbook & display$ on ! blackbo>d3-- ,:5 y multiply a numb] by any?+ m ?an #a ! result is l>g] ?an ! numb]4 ,:5 y multiply a numb] by #a ! result is ! same z ! numb]4 ,:5 y multiply a numb] by any?+ less ?an #a ! result is small] ?an ! numb]4 ,let ! pupils e/abli% ! new habit by _m su* ex]cises z3-- #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 ,9 ! case ( division by a fraction ! old h>mful habit %\ld be c\nt]act$ & ref9$ by simil> rules & ex]cises z foll[s3-- ,:5 y divide a numb] by any?+ m ?an #a ! result is small] ?an ! numb]4 ,:5 y divide a numb] by #a ! result is ! same z ! numb]4 ,:5 y divide a numb] by any?+ less ?an #a ! result is l>g] ?an ! numb]4 ,/ate ! miss+ numb]s3-- #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 ' (' means 'hundr$?s times'4--,9 ! case ( p]c5tage a s]ies ( bonds l ! foll[+ %\ld be =m$3-- #e p]c5t ( 4#je times #bj 4#bj #f 4#jf #be 4#be x #ab 4#ab x #c 4#jc x ,f\r five-m9ute drills on su* connections betwe5 'x p]c5t (' & 'its decimal equival5t times' are wor? an h\r's /udy ( v]bal def9itions ( ! mean+ ( p]c5t z p] hundr$ or ! l4 ,! only use ( ! /udy ( su* def9itions is to facilitate ! lat] =mation ( ! bonds1 &1 ) all save ! bright] pupils1 ! bonds are m ne$$ = an und]/&+ ( ! def9itions ?an ! def9itions are ne$$ = ! =mation ( ! bonds4 7#i7 ,habits ( v]ify+ results4--,bonds %\ld e>ly be =m$ betwe5 c]ta9 manipulations ( numb]s & c]ta9 means ( *eck+1 or v]ify+ ! correctness (1 ! manipulation 9 "q4 ,! additions to #i #i & ! subtractions to #ah - #i %\ld be v]ifi$ by objective addition & subtraction & c\nt+ until ! pupil has sure comm&2 ! multiplications to #i x #i %\ld be v]ifi$ by objective multiplication & c\nt+ ( ! result 79 piles ( t5s & a pile ( ones7 eight or t5 times1#d & by addition eight or t5 times2#d ! divisions to #ha _/ #i %\ld be v]ifi$ by multiplication & occasionally objectively until ! pupil has sure comm&2 column addition %\ld be *eck$ by add+ ! columns sep>ately & add+ ! sums s obta9$1 & by mak+ two %ort] tasks ( ! giv5 task & add+ ! two sums2 '%ort' multiplication %\ld be v]ifi$ eight or t5 times by addition2 'long' multiplication %\ld be *eck$ by rev]s+ multipli] & multiplic& & 9 o!r ways2 '%ort' & 'long' division %\ld be v]ifi$ by multiplication4 #d ,eight or t5 times 9 all1 n eight or t5 times = ea* fact ( ! tables4 ,^! habits ( te/+ an obta9$ result are ( ?reefold value4 ,!y enable ! pupil to f9d 8 own errors1 & to ma9ta9 a /&>d ( accuracy by himself4 ,!y give him a s5se ( ! relations ( ! processes & ! reasons :y ! "r ways ( add+1 subtract+1 multiply+1 & divid+ are "r1 su* z only ! v bright pupils c get f v]bal explanations4 ,!y put 8 acquisition ( a c]ta9 p[]1 say multiplication1 to a real & intelligible use1 9 *eck+ ! results ( 8 practice ( a new p[]1 & s in/ill a respect = ari?metical p[] & skill 9 g5]al4 ,! "t sp5t 9 su* v]ification produces ^! results at little co/2 = ! practice 9 add+ to v]ify multiplications1 9 multiply+ to v]ify divisions1 & ! l is ne>ly z good = g5]al drill & review ( ! addition & multiplication !mselves z practice devis$ = t special purpose4 ,e>ly "w 9 add+1 subtract+1 & r$uc+ fractions %\ld be v]ifi$ by objective aids 9 ! %ape ( l9es & areas divid$ 9 suitable fractional p>ts4 ,e>ly "w ) decimal fractions %\ld be v]ifi$ by ! use ( ! equival5t common fractions = 4#be1 4#ge1 4#abe1 4#cge1 & ! l4 ,multiplication & division ) fractions1 bo? common & decimal1 %\ld 9 ! e>ly /ages be v]ifi$ by objective aids4 ,! plac+ ( ! decimal po9t 9 multiplication & division ) decimal fractions %\ld be v]ifi$ by su* ex]cises z3-- #bj ,x _c be #bjj2 = #bjj x #a4bc is mu* m ?an #bd4f4 ,x _c be #b2 = #b x #a4bc is mu* less ?an #bd4f4 #a4bc 7#bd4fj #bdf ---- ,! e/abli%m5t ( habits ( v]ify+ results & _! use is v greatly ne$$4 ,! p]c5tage ( wrong answ]s 9 ari?metical "w 9 s*ools is n[ s high t ! pupils are (t5 be+ practic$ 9 error4 ,9 _m cases !y c feel no g5u9e & effective confid5ce 9 ! processes1 s9ce _! own use ( ! processes br+s wrong answ]s z (t5 z "r4 ,9 solv+ problems !y (t5 _c decide :e!r !y h done ! "r ?+ or ! wrong1 s9ce ev5 if !y h done ! "r ?+1 !y may h done x inaccurately4 ,a wrong answ] to a problem is !re=e too (t5 ambigu\s & un9/ructive to !m4#e #e ,! facts conc]n+ ! pres5t inaccuracy ( s*ool "w 9 ari?metic w be f\nd on pages #ajb to #aje4 ,^! illu/rations ( ! la/ few pages are samples ( ! proc$ures recomm5d$ by a consid]ation ( all ! bonds t "o might =m & ( ! contribution t ea* w\ld make t[>d ! abilities t ! /udy ( ari?metic %\ld develop & improve4 ,x is by do+ m or less at haphaz>d :at psy*ology tea*es u to d delib]ately & sy/ematically 9 ?is respect t _m ( ! pa/ advances 9 ! tea*+ ( ari?metic h be5 made4 ,,wa/eful ,,& ,,h>mful ,,bonds ,a scrut9y ( ! bonds n[ =m$ 9 ! tea*+ ( ari?metic ) que/ions conc]n+ ! exact s]vice ( ea*1 results 9 a li/ ( bonds ( small value or ev5 no value1 s f> z a psy*ologi/ c det]m9e4 ,i pres5t "h samples ( su* psy*ologically unju/ifiable bonds ) "s ( ! reasons = _! defici5cies4 7#a7 ,arbitr>y units4--,9 drills int5d$ to improve ! ability to see & use ! mean+s ( numb]s z names = ratios or relative magnitudes1 x is unwise to employ entirely arbitr>y units4 ,! proc$ure 9 ,,ii 7on page #hd7 is bett] ?an t 9 ,i4 ,in*es1 half-in*es1 feet1 & c5timet]s are bett] z units ( l5g? ?an arbitr>y ,z4 ,squ>e in*es1 squ>e c5timet]s1 & squ>e feet are bett] = areas4 ,ounces & p\nds %\ld be lift$ r ?an arbitr>y weights4 ,p9ts1 qu>ts1 glassfuls1 cupfuls1 h&fuls1 & cubic in*es are bett] = volume4 ,all ! real m]it 9 ! drills on relative magnitude advocat$ by ,spe]1 ,mc,lellan & ,dewey1 & o!rs c be secur$ )\t sp5d+ "t 9 relat+ magnitudes = ! sake ( relative magnitude alone4 ,! use ( units ( measure 9 drills :i* w nev] be us$ 9 bona fide measur+ is l ! use ( fractions l sev5?s1 elev5?s1 & ?irte5?s4 ,a v little ( x is p]haps desirable to te/ ! appreciation ( c]ta9 g5]al pr9ciples1 b = regul> tra9+ x %\ld give place to ! use ( units ( practical significance4 ,illu/ration3 ,,fig4 #c4 ,a ---------- ,b ------------------------------ ,c -------------------- ,d ---------------------------------------- ,i4 ,if ,a is #a :i* l9e is #b8 ,:i* l9e is #d8 ,:i* l9e is #c8 ,a & ,c toge!r equal :at l9e8 ,a & ,b toge!r equal :at l9e8 ,h[ mu* long] is ,b ?an ,a8 ,h[ mu* long] is ,b ?an ,c8 ,h[ mu* long] is ,d ?an ,a8 ,illu/ration3 ,,fig4 #d4 ,a ---------- ,b ------------------------------ ,c -------------------- ,d ---------------------------------------- ,,ii4 ,a is #a in* long4 ,:i* l9e is #b in*es long8 ,:i* l9e is #d in*es long8 ,:i* l9e is #c in*es long8 ,a & ,c toge!r make 444 in*es8 ,a & ,b toge!r make 444 in*es8 ,b is 444 444 long] ?an ,a8 ,b is 444 444 long] ?an ,c8 ,d is 444 444 long] ?an ,a8 7#b7 ,multiples ( #aa4--,! multiplications ( #b to #ab by #aa & #ab z s+le connections %\ld be left = ! pupil to acquire by himself z he ne$s !m4 ,^! connections int]f]e ) ! process ( le>n+ two-place multiplication4 ,! manipulations ( numb]s !re requir$ c be le>n$ mu* m easily if #aa & #ab are us$ z multipli]s 9 j ! same way t #gh or #if w\ld be4 ,lat] ! #ab x #b1 #ab x #c1 etc41 may be taught4 ,!re is less reason = kn[+ ! multiples ( #aa ?an = kn[+ ! multiples ( #ae1 #af1 or #be4 7#c7 ,ab/ract & concrete numb]s4--,! elaborate emphasis ( ! suppos$ fact t we _c multiply #gbf by #h doll>s & ! /ill m elaborate explanations ( :y nev]!less we f9d ! co/ ( #gbf articles at #h ea* by multiply+ #gbf by #h & call+ ! answ] doll>s are wa/eful4 ,! same holds ( ! correspond+ p$antry ab\t division4 ,^! imag9>y difficulties %\ld n be rais$ at all4 ,! pupil %\ld n ?9k ( multiply+ or divid+ m5 or doll>s1 b simply ( ! necess>y equation & ( ! sort ( ?+ t ! miss+ numb] repres5ts4 #h x #gbf 4444 ,answ] is doll>s1 or #h1 #gbf1 multiply4 ,answ] is doll>s1 is all t he ne$s to ?9k1 & is 9 ! be/ =m = 8 ?\ght4 ,conc]n+ ! di/9ction betwe5 ab/ract & concrete numb]s1 bo? logic & common s5se z well z psy*ology support ! cont5tion ( ,mc,dougle '#ad1 p4 #bjf;f41 :o writes3-- ,! mo/ elem5t>y c\nt+1 ev5 t /age :5 ! c\nts 7 n c>ri$ 9 ! m9d1 b m]ely 9 not*es on a /ick or by ,de,morgan's /ones 9 a pot1 requires "s ?\ght2 & ! mo/ advanc$ c\nt+ implies memory ( ?+s4 ,! t]ms1 !re=e1 ab/ract & concrete numb]1 h long s9ce ceas$ to be us$ by ?9k+ p4 ,rec5tly ! writ] visit$ an ari?metic class 9 a ,/ate ,normal ,s*ool & saw a gr\p ( practically adult /ud5ts confus$ ab\t ?is v "q conc]n+ ab/ract & concrete numb]s1 accord+ to _! previ\s tra9+ 9 ! conv5tionalities ( ! textbook4 ,_! tea*] div]t$ ! "w ( ! h\r & %e & ! class sp5t almo/ ! :ole p]iod 9 ree/abli%+ ! requirem5ts 't ! product mu/ always be ! same k9d ( unit z ! multiplic&1' & 'add5ds mu/ all be alike to be add$4' ,?is is n an exceptional case4 ,?r\gh\t ! :ole range ( tea*+ ari?metic 9 ! public s*ools pupils are obfuscat$ by ! philosophical encumbrances :i* h be5 impos$ ^u ! simple/ processes ( num]ical "w4 ,! "t is surely ripe1 n[ t we are readju/+ our ideas ( ! subject ( ari?metic1 to revise "s ( ^! wa/eful & di%e>t5+ practices4 ,algebra hi/orically grew out ( ari?metic1 yet x has n be5 lad5 ) ?is di/9ction4 ,no pupil 9 algebra lets x equal ! horses2 he lets x equal ! numb] ( horses1 & proce$s to drop ! idea ( horses out ( 8 consid]ation4 ,he multiplies1 divides1 & extracts ! root ( ! numb]1 sometimes h&l+ fractions 9 ! process1 & f9ally int]prets ! result accord+ to ! conditions ( 8 problem4 ,( c\rse1 9 ! e>ly numb] "w !re h be5 ! s5se-objects f :i* numb] has be5 p]ceiv$1 b ! m9d retreats naturally f objectivity to ! pure conception ( numb]1 & !n to ! numb] symbol4 ,! foll[+ is tak5 f ! app5dix to ,horn's !sis1 :]e a sev5? grade girl gets ! population ( ! ,unit$ ,/ates 9 #ahbj3-- #g1hfb1aff :ites #bcc1fcd free negroes #a1ech1jbb slaves --------- #i1fcc1hbb ,9 ?is problem ?ree diff]5t k9ds ( add5ds are comb9$1 if we accept ! usual di/9ction4 ,"s may say t ?is is a mi/ake1--t ! pupil trans=m$ ! ':ites1' 'free negroes1' & 'slaves' into a common unit1 su* z 'p' ( 'population' & !n add$ ^! common units4 ,b ?is 'explanation' is entirely gratuit\s1 z "o w f9d if he que/ions ! pupil ab\t ! process4 ,x w be f\nd t ! *ild simply add$ ! figures z numb]s only & !n int]pret$ ! result1 accord+ to ! /atem5t ( ! problem1 )\t s mu* m5tal gymna/ics4 ,! writ] has que/ion$ hundr$s ( /ud5ts 9 ,normal ,s*ool "w on ?is po9t1 & he believes t ! ord9>y m9d-movem5t is correctly set =? "h1 no matt] h[ well "o may ma9ta9 z an academic proposition t ?is is n logical4 ,_m classes 9 ! ,ea/]n ,k5tucky ,/ate ,normal h be5 giv5 ?is problem to solve1 & !y inv>iably get ! same result3-- ',9 a g>d5 on ! ,summit are z _m cabbage-heads z ! total numb] ( ladies & g5tlem5 9 ?is class4 ,h[ _m cabbage-heads 9 ! g>d58' ,& ! blackbo>d solution looks l ?is ea* "t3-- #bi ladies #ae g5tlem5 -- #dd cabbage-heads ,s1 also1 "o may say3 ,i h #f times z _m %eep z y h c[s4 ,if y h #e c[s1 h[ _m %eep h ,i8 ,"h we w\ld multiply ! numb] ( c[s1 :i* is #e1 by #f & call ! result #cj1 :i* mu/ be l9k$ ) ! idea ( %eep because ! conditions impos$ by ! problem dem& x4 ,! m9d naturally 9 ?is "w sep>ates ! pure numb] f its situation1 z 9 algebra1 h&les x accord+ to ! laws gov]n+ ari?metical comb9ations1 & labels ! result z ! /atem5t ( ! problem dem&s4 ,?is is express$ 9 ! foll[+1 :i* is tacitly accept$ 9 algebra1 & %\ld be accept$ equally 9 ari?metic3 ',9 all computations & op]ations 9 ari?metic1 all numb]s are ess5tially ab/ract & %\ld be s treat$4 ,!y are concrete only 9 ! ?\ght process t att5ds ! op]ation & int]prets ! result4' 7#d7 ,lea/ common multiple4--,! :ole set ( bonds involv$ 9 le>n+ 'lea/ common multiple' %\ld be left out4 ,9 add+ & subtract+ fractions ! pupil %\ld n f9d ! lea/ common multiple ( _! d5om9ators b %\ld f9d any common multiple t he c f9d quickly & correctly4 ,no intellig5t p]son w\ld ev] wa/e "t 9 se>*+ = ! lea/ common multiple ( six?s1 ?irds1 & halves except = ! un=tunate traditions ( an ov]sy/ematiz$ ari?metic1 b w\ld ?9k ( _! equival5ts 9 six?s or twelf?s or tw5ty-f\r?s or any o!r conv5i5t common multiple4 ,! process ( f9d+ ! lea/ common multiple is ( su* exce$+ly r>e application 9 sci5ce or bus9ess or life g5]ally t ! textbooks h to resort to purely fanta/ic problems to give drill 9 its use4 7#e7 ,greate/ common divisor4--,! :ole set ( bonds involv$ 9 le>n+ 'greate/ common divisor' %\ld also be left out4 ,9 r$uc+ fractions to l[e/ t]ms ! pupil %\ld divide by any?+ t he sees t he c divide by1 favor+ l>ge divisors1 & cont9ue do+ s until he gets ! fraction 9 t]ms suitable = ! purpose 9 h&4 ,! read] probably nev] has _h occasion to compute a greate/ common divisor s9ce he left s*ool4 ,if he has comput$ any1 ! *ances are t he w\ld h sav$ "t by solv+ ! problem 9 "s o!r way6 ,! foll[+ problems are tak5 at r&om f ^? giv5 by "o ( ! be/ ( ! textbooks t make ! attempt to apply ! facts ( ,greate/ ,common ,divisor & ,lea/ ,common ,multiple to problems4#f ,mo/ ( ^! problems are fanta/ic4 ,! o!rs are trivial1 or are bett] solv$ by trial & adaptation4 #a4 ,a c]ta9 s*ool consi/s ( #acb pupils 9 ! high s*ool1 #aed 9 ! gramm>1 & #aih 9 ! prim>y grades4 ,if ea* gr\p is divid$ into sections ( ! same numb] conta9+ z _m pupils z possible1 h[ _m pupils w !re be 9 ea* section8 #b4 ,a f>m] has #bdj bu4 ( :eat & #ibj bu4 ( oats1 :i* he desires to put into ! lea/ numb] ( boxes ( ! same capacity1 )\t mix+ ! two k9ds ( gra94 ,f9d h[ _m bu%els ea* box mu/ hold4 #c4 ,f\r bells toll at int]vals ( #c1 #g1 #ab1 & #ad seconds respectively1 & beg9 to toll at ! same in/ant4 ,:5 w !y next toll toge!r8 #d4 ,a1 ,b1 ,c1 & ,d />t toge!r1 & travel ! same way ar\nd an isl& :i* is #fjj mi4 9 circuit4 ,a goes #bj mi4 p] day1 ,b #cj1 ,c #be1 & ,d #dj4 ,h[ long mu/ _! j\rney+ cont9ue1 9 ord] t !y may all come toge!r aga98 #e4 ,! p]iods ( ?ree planets :i* move uni=mly 9 circul> orbits r\nd ! sun1 are respectively #bjj1 #bej1 & #cjj da4 ,suppos+ _! positions relatively to ea* o!r & ! sun to be giv5 at any mom5t1 det]m9e h[ _m da4 mu/ elapse be=e !y aga9 h exactly ! same relative positions4 #f ,mc,lellan & ,ames1 ,public ,s*ool ,ari?metic #aijj4 7#f7 ,r>e & unimportant words4--,! bonds betwe5 r>e or unimportant words & _! mean+s %\ld n be =m$ = ! m]e sake ( v]bal v>iety 9 ! problems ( ! textbook4 ,a pupil %\ld n be expect$ to solve a problem t he _c read4 ,he %\ld n be expect$ 9 grades #b & #c1 or ev5 9 grade #d1 to read words t he has r>ely or nev] se5 be=e4 ,he %\ld n be giv5 elaborate drill 9 read+ dur+ ! "t devot$ to ! treatm5t ( quantitative facts & relations4 ,all ?is is s obvi\s t x may seem ne$less to relate4 ,x is n4 ,) _m textbooks x is n[ necess>y to give def9ite drill 9 read+ ! words 9 ! pr9t$ problems int5d$ = grades #b1 #c1 & #d1 or to replace !m by oral /atem5ts1 or to leave ! pupils 9 confusion conc]n+ :at ! problems are t !y are to solve4 ,_m good tea*]s make a regul> read+-lesson out ( e page ( problems be=e hav+ !m solv$4 ,!re %\ld be no su* necessity4 ,to def9e r>e & unimportant concretely1 ,i w say t = pupils up to ! middle ( grade #c1 su* words z ! foll[+ are r>e & unimportant 7?\gh ea* ( !m occurs 9 ! v fir/ fifty pages ( "s well-kn[n beg9n]'s book 9 ari?metic74 abs5tees acc\nt ,adele admitt$ ,agnes agre$ ,albany ,all5 all[$ alt]nate ,&rew ,arkansas arriv$ assembly automobile bak+ p[d] balance b>ley begg> ,b]tie ,bessie b9 ,bo/on b\quet bronze buck:eat ,byron camphor ,c>l ,c>rie ,cecil ,*>lotte *>ity ,*icago c9namon ,cl>a clo!sp9s collect comma committee conc]t confection] cranb]ries crane currants dairyman ,daniel ,david deal] debt deliv]$ ,d5v] dep>tm5t deposit$ dictation dis*>g$ discov] discov]y di%-wat] drug due ,edg> ,eddie ,edw9 election electric ,ella ,emily enroll$ ent]ta9m5t envelope ,e/h] ,e!l exce$s explanation expression g5]ally g5tlem5 ,gilb]t ,grace grad+ ,graham gramm> ,h>old hat*et ,h]alds hesitation ,horace ,mann impossible income indicat$ inmo/ ins]ts in/allm5ts in/antly insurance ,i[a ,jack ,j5nie ,johnny ,joseph j\rney ,julia ,ka!r9e lettuce-plant libr>y ,lottie ,lula m>g9 ,m>?a ,mat!w ,maud mead[ m5tally m]cury m9]al ,miss\ri molasses ,morton movem5ts musl9 ,nellie nieces ,oakl& obs]v+ obta9$ (f]$ (fice onions opposite orig9al package packet palm ,patrick ,paul paym5ts peep ,pet] p]* phaeton photograph piano pigeons ,pilgrims pres]v+ proprietor pur*as$ ,ra*el ,ralph rapidity r readily receipts regi/] rem&$ respectively ,rob]t ,rog] ,ru? rye ,samuel ,san ,francisco seldom %e>$ %+les skyrockets sloop solve speckl$ sponges spr\t /ack ,/eph5 /rap successfully sugge/$ sunny supply ,susan ,susie's syllable talcum t]m te/ !rmomet] ,?omas torp$oes trad] transaction treasury tricycle tube two-seat$ unit$ usually vacant v>i\s vase velocip$e votes walnuts ,walt] ,wa%+ton wat*$ :i/le woodl& wor/$ 7#g7 ,mislead+ facts & proc$ures4--,bonds %\ld n be =m$ betwe5 articles ( comm]ce & grossly inaccurate prices !re=1 betwe5 ev5ts & grossly improbable consequ5ces1 or causes or accompanim5ts !re(1 nor betwe5 ?+s1 qualities1 & ev5ts :i* h no important connections "o ) ano!r 9 ! real _w4 ,9 g5]al1 ?+s %\ld n be put toge!r 9 ! pupil's m9d t d n belong toge!r4 ,if ! read] d\bts ! ne$ ( ?is w>n+ let him exam9e problems #a to #e1 all f reputable books t are 9 common use1 or h be5 )9 a few ye>s1 & consid] h[ addition1 subtraction1 & ! habits belong+ ) ea* are confus$ by ex]cise #f4 #a4 ,if a duck fly+ #c_/e z fa/ z a hawk flies #ij miles 9 an h\r1 h[ fa/ does ! hawk fly8 #b4 ,at #e_/h ( a c5t apiece h[ _m eggs c ,i buy = #fj8 #c4 ,at 4#fh a pair h[ _m pairs ( ov]%oes c y buy = #haf8 #d4 ,at 4#ac a doz5 h[ _m doz5 bananas c y buy = #c4ab8 #e4 ,h[ _m pecks ( beans c be put into a box t w hold j #ba bu%els8 #f4 ,write answ]s3 #ecg ,beg9n+ at ! bottom say #aa1 #ah1 & #b 7writ+ x 9 #cfe its place7 are #bj4 #e1 #aa1 #ad1 & #f 7writ+ x7 are #bj1 8 #e1 #aj4 ,! numb]1 omitt$1 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 & absurdities4--,bonds %\ld n be =m$ betwe5 insignificant or fooli% que/ions & ! labor ( answ]+ !m1 nor betwe5 ! g5]al ari?metical "w ( ! s*ool & su* insignificant or fooli% que/ions4 ,! foll[+ are samples f rec5t textbooks ( excell5t /&+3-- ,on "o side ( ,george's slate !re are #cb words1 & on ! o!r side #bf words4 ,if he erases #f words f "o side1 & #h f ! o!r1 h[ _m words rema9 on 8 slate8 ,a c]ta9 s*ool has #ad rooms1 & an av]age ( #dj *ildr5 9 a room4 ,if e "o 9 ! s*ool %\ld make #ejj /raight m>ks on ea* side ( 8 slate1 h[ _m w\ld be made 9 all8 #h times ! numb] ( /ripes 9 our flag is ! numb] ( ye>s f #ahjj until ,roosevelt 0 elect$ ,presid5t4 ,9 :at ye> 0 he elect$ ,presid5t8 ,f ! ,decl>ation ( ,indep5d5ce to ! ,_w's ,fair 9 ,*icago 0 #i times z _m ye>s z !re are /ripes 9 ! flag4 ,h[ _m ye>s 0 x8 7#i7 ,useless me?ods4--,bonds %\ld n be =m$ betwe5 a describ$ situation & a me?od ( treat+ ! situation :i* w\ld n be a useful "o to foll[ 9 ! case ( ! real situation4 ,= example1 ,if ,i set #if trees 9 r[s1 sixte5 trees 9 a r[1 h[ _m r[s w ,i h8 =ms ! habit ( treat+ by division a problem t 9 reality w\ld be solv$ by c\nt+ ! r[s4 ,s also ,i wi% to give #be c5ts to ea* ( a gr\p ( boys & f9d t x w require #b4ge4 ,h[ _m boys are 9 ! gr\p8 =ms ! habit ( answ]+ a "q by division ^: answ] mu/ already h be5 pres5t to give ! data ( ! problem4 7#aj7 ,problems ^: answ]s w\ld1 9 real life1 be already kn[n4--,! cu/om ( giv+ problems 9 textbooks :i* c\ld n occur 9 reality because ! answ] has to be kn[n to frame ! problem is a natural result ( ! lazy au?or's t5d5cy to "w out a problem to fit a c]ta9 process & a c]ta9 answ]4 ,su* bogus problems are v1 v common4 ,9 a r&om sampl+ ( a doz5 pages ( ,g5]al ,review problems 9 "o ( ! mo/ widely us$ ( rec5t textbooks1 ,i f9d t ab\t #f p]c5t ( ! problems are ( ?is sort4 ,among ! problems extemporiz$ by tea*]s ^! bogus problems are probably /ill m frequ5t4 ,su* are3-- ,a cl]k 9 an (fice address$ lett]s accord+ to a giv5 li/4 ,aft] %e _h address$ #bejj1 #d_/i ( ! names on ! li/ _h n be5 us$2 h[ _m names 7 9 ! entire li/8 ,! ,canadian p[] canal at ,sault ,/e4 ,m>ie furni%$ #bj1jjj horse p[]4 ,! canal on ! ,mi*igan side furni%$ #b-a_/b times z mu*4 ,h[ _m horse p[] does ! latt] furni%8 ,x may be ass]t$ t ! ideal ( giv+ z describ$ problems only problems t might occur & dem& ! same sort ( process = solution ) a real situation1 is too exact+4 ,if a problem is compreh5sible & s]ves to illu/rate a pr9ciple or give useful drill1 t is en\gh1 tea*]s may say4 ,= really sci5tific tea*+ x is n en\gh4 ,moreov]1 if problems are giv5 m]ely z te/s ( k ( a pr9ciple or z means to make "s fact or pr9ciple cle> or emphatic1 & are n expect$ to be ( direct s]vice 9 ! quantitative "w ( life1 x is bett] to let ! fact be kn[n4 ,= example1 ,i am ?9k+ ( a numb]4 ,half ( ?is numb] is twice six4 ,:at is ! numb]8 is bett] ?an ,a man left 8 wife a c]ta9 sum ( money4 ,half ( :at he left h] 0 twice z mu* z he left to 8 son1 :o receives #fjjj4 ,h[ mu* did he leave 8 wife8 ,! =m] is bett] because x makes no false pret5ses4 7#aa7 ,ne$less l+ui/ic difficulties4--,x %\ld be unnecess>y to add t bonds %\ld n be =m$ betwe5 ! pupil's g5]al attitude t[>d ari?metic & ne$less1 useless difficulty 9 language or ne$less1 useless1 wrong reason+4 ,our tea*+ is1 h[ev]1 /ill ta9t$ by bo? ( ^! un=tunate connections1 :i* dispose ! pupil to ?9k ( ari?metic z a my/]y & folly4 ,consid]1 = example1 ! pr(itless l+ui/ic difficulty ( problems #a-f1 ^: quantitative difficulties are simply ^? (3-- #a4 #e #h #c #g #b4 #fd _/ #h1 & k t #a peck #h qu>ts #c4 #ab _/ #d #d4 #f _/ #b #e4 #c x #b #f4 #d x #d #a4 ,:at am\nt %\ld y obta9 by putt+ toge!r #e c5ts1 #h c5ts1 #c c5ts1 & #g c5ts8 ,did y f9d ?is result by add+ or multiply+8 #b4 ,h[ _m times mu/ y empty a peck measure to fill a basket hold+ #fd qu>ts ( beans8 #c4 ,if a girl commits to memory #d pages ( hi/ory 9 "o day1 9 h[ _m days w %e commit to memory #ab pages8 #d4 ,if ,fr$ _h #f *ick5s h[ _m times c\ld he give away #b *ick5s to 8 companions8 #e4 ,if a croquet-play] drove a ball ?r\gh #b ar*es at ea* /roke1 ?r\gh h[ _m ar*es w he drive x by #c /rokes8 #f4 ,if mamma cut ! pie into #d pieces & gave ea* p]son a piece1 h[ _m p]sons did %e h = d9n] if %e us$ #d :ole pies = dess]t8 ,ari?metically ?is "w belongs 9 ! fir/ or second ye>s ( le>n+4 ,b *ildr5 ( grades #b & #c1 save a few1 w\ld be utt]ly at a loss to und]/& ! language4 ,we are n yet free f ! follies illu/rat$ 9 ! lessons ( pages #if to #ii1 :i* my/ifi$ our p>5ts4 ,,lesson ,i ,illu/ration3 ,,fig4 #e4 #a4 ,9 ?is picture1 h[ _m girls are 9 ! sw+8 #b4 ,h[ _m girls are pull+ ! sw+8 #c4 ,if y c\nt bo? girls toge!r1 h[ _m are !y8 ,"o girl & "o o!r girl are h[ _m8 #d4 ,h[ _m kitt5s d y see on ! /ump8 #e4 ,h[ _m on ! gr\nd8 #f4 ,h[ _m kitt5s are 9 ! picture8 ,"o kitt5 & "o o!r kitt5 are h[ _m8 #g4 ,if y %\ld ask me h[ _m girls are 9 ! sw+1 or h[ _m kitt5s are on ! /ump1 ,i c\ld answ] al\d1 ,"o2 or ,i c\ld write ,"o2 or ?us1 #a4 #h4 ,if ,i write ,"o1 ?is is call$ ! ^w ,"o4 #i4 ,?is1 #a1 is nam$ a figure ,"o1 because x means ! same z ! ^w ,"o1 & /&s = ,"o4 #aj4 ,write #a4 ,:at is ?is nam$8 ,:y8 #aa4 ,a figure #a may /& = "o girl1 "o kitt51 or "o any?+4 #ab4 ,:5 *ildr5 fir/ att5d s*ool1 :at d !y beg9 to le>n8 ,ans4 ,lett]s & words4 #ac4 ,c\ld y read or write be=e y _h le>n$ ei!r lett]s or words8 #ad4 ,if we h all ! lett]s toge!r1 !y are nam$ ! ,alphabet4 #ae4 ,if we write or speak words1 !y are nam$ ,language4 #af4 ,y are comm5c+ to /udy ,ari?metic2 & y c read & write 9 ,ari?metic only z y le>n ! ,alphabet & ,language ( ,ari?metic4 ,b little "t w be requir$ = ?is purpose4 ,,lesson ,,ii ,illu/ration3 ,,fig4 #f4 #a4 ,if we speak or write words1 :at d we "n !m1 :5 tak5 toge!r8 #b4 ,:at are y comm5c+ to /udy8 ,ans4 ,ari?metic4 #c4 ,:at ,language mu/ y n[ le>n8 #d4 ,:at d we "n ?is1 #a8 ,:y8 #e4 ,?is figure1 #a1 is "p ( ! ,language ( ,ari?metic4 #f4 ,if ,i %\ld write some?+ to /& = ,two--two girls1 two kitt5s1 or two ?+s ( any k9d--:at d y ?9k we w\ld "n x8 #g4 ,a figure ,two is writt5 ?us3 #b4 ,make a figure two4 #h4 ,:y d we "n ?is a figure two8 #i4 ,?is figure two 7#b7 is "p ( ! ,language ( ,ari?metic4 #aj4 ,9 ?is picture "o boy is sitt+1 play+ a flageolet4 ,:at is ! o!r boy do+8 ,if ! boy /&+ %\ld sit d[n by ! o!r1 h[ _m boys w\ld be sitt+ toge!r8 ,"o boy & "o o!r boy are h[ _m boys8 #aa4 ,y see a flageolet & a viol94 ,!y are musical in/rum5ts4 ,"o musical in/rum5t & "o o!r musical in/rum5t are h[ _m8 #ab4 ,i w write ?us3 #a #a #b4 ,we say t #a boy & #a o!r boy1 c\nt$ toge!r1 are #b boys2 or are equal to #b boys4 ,we w n[ write some?+ to %[ t ! fir/ #a & ! o!r #a are to be c\nt$ toge!r4 #ac4 ,we "n a l9e drawn ?us1 -1 a horizontal l9e4 ,draw su* a l9e4 ,"n x4 #ad4 ,a l9e drawn ?us1 1 we "n a v]tical l9e4 ,draw su* a l9e4 ,"n x4 #ae4 ,n[ ,i w put two su* l9es toge!r2 ?us1 4 ,:at k9d ( a l9e d we "n ! fir/ 7-78 ,& :at d we "n ! la/8 778 ,are ^! l9es long or %ort8 ,:]e d !y cross ea* o!r8 #af4 ,ea* ( y write ?us3 -1 1 4 #ag4 ,?is1 1 is nam$ ,plus4 ,plus means m2 & also means m4 #ah4 ,i w write4 ,"o & ,"o ,m ,equal ,two4 #ai4 ,n[ ,i w write "p ( ?is 9 ! ,language ( ,ari?metic4 ,i write ! fir/ ,"o ?us1 #a2 !n ! o!r ,"o ?us1 #a4 ,aft]w>d ,i write1 = ! ^w ,m1 ?us1 1 plac+ ! betwe5 #a & #a1 s t ! :ole /&s ?us3 #a #a4 ,z ,i write1 ,i say1 ,"o & ,"o m4 #bj4 ,ea* ( y write #a #a4 ,read :at y h writt54 #ba4 ,?is 1 :5 writt5 betwe5 ! #as1 %[s t !y are to be put toge!r1 or c\nt$ toge!r1 s z to make #b4 #bb4 ,because %[s :at is to be done1 x is call$ a ,sign4 ,if we take its "n1 ,plus1 & ! ^w ,sign1 & put bo? words toge!r1 we h ,sign ,plus1 or ,plus ,sign4 ,9 speak+ ( ?is we may call x ,sign ,plus1 or ,plus ,sign1 or ,plus4 #bc4 #a1 #b1 1 are "p ( ! ,language ( ,ari?metic4 ,write ! foll[+ 9 ! ,language ( ,ari?metic3 #bd4 ,"o & "o m4 #be4 ,"o & two m4 #bf4 ,two & "o m4 7#ab7 ,ambiguities & falsities4--,consid] ! ambiguities & false reason+ ( ^! problems4 #a4 ,if y c e>n #d c5ts a day1 h[ mu* c y e>n 9 #f weeks8 7,are ,sundays c\nt$8 ,%\ld a *ild :o e>ns #d c5ts "s day expect to repeat ! feat daily87 #b4 ,h[ _m l9es mu/ y make to draw t5 triangles & five squ>es8 7,i c d ?is ) #h l9es1 ?\gh ! answ] ! book requires is #ej47 #c4 ,a runn] ran twice ar\nd an #a_/h mile track 9 two m9utes4 ,:at di/ance did he run 9 #b_/c ( a m9ute8 7,i d n "k1 b ,i d "k t1 save by *ance1 he did n run exactly #b_/c ( #a_/h mile47 #d4 ,john e>n$ #d4ce 9 a week1 & ,h5ry e>n$ #a4ic4 ,!y put _! money toge!r & b\ght a gun4 ,:at did x co/8 7,maybe #e1 maybe #aj4 ,did !y pay = ! :ole ( x8 ,did !y use all _! e>n+s1 or less1 or m87 #e4 ,ri*>d has #ab nickels 9 8 purse4 ,h[ mu* m ?an #ej c5ts w\ld y give him = !m8 7,w\ld a wise *ild give #fj c5ts to a boy :o want$ to swap #ab nickels !re=1 or w\ld he suspect a trick & hold on to 8 own co9s87 #f4 ,if a horse trots #aj miles 9 "o h\r h[ f> w he travel 9 #i h\rs8 #g4 ,if a girl c pick #c qu>ts ( b]ries 9 #a h\r h[ _m qu>ts c %e pick 9 #c h\rs8 7,^! la/ two1 ) a tea*] insi/+ on ! #ij & #i1 might well deprive a matt]-(-fact boy ( respect = ari?metic = weeks !reaft]47 ,! economics & physics ( ! next f\r problems speak = !mselves4 #h4 ,i lo/ #ae by sell+ a horse = #he4 ,:at 0 ! value ( ! horse8 #i4 ,if float+ ice has #g times z mu* ( x "u ! surface ( ! wat] z above x1 :at "p is above wat]8 ,if an iceb]g is #ej ft4 above wat]1 :at is ! entire height ( ! iceb]g8 ,h[ high above wat] w\ld an iceb]g #cjj ft4 high h to be8 #aj4 ,a man's sal>y is #ajjj a ye> & 8 exp5ses #fbe4 ,h[ _m ye>s w elapse be=e he is wor? #aj1jjj if he is wor? #bejj at ! pres5t "t8 #aa4 ,s\nd travels #aabj ft4 a second4 ,h[ long aft] a cannon is fir$ 9 ,new ,york w ! report be he>d 9 ,philadelphia1 a di/ance ( #ij miles8 ,,guid+ ,,pr9ciples ,! read] may be we>i$ ( ^! special details conc]n+ bonds n[ neglect$ t %\ld be =m$ & useless or h>mful bonds =m$ = no valid reason4 ,any "o ( !m by itself is p]haps a m9or matt]1 b :5 we h cur$ all our faults 9 ?is respect & f\nd all ! possibilities = wis] selection ( bonds1 we %all h enorm\sly improv$ ! tea*+ ( ari?metic4 ,! ideal is su* *oice ( bonds 7&1 z w be %[n lat]1 su* arrangem5t ( !m7 z w mo/ improve ! functions 9 "q at ! lea/ co/ ( "t & ef=t4 ,! guid+ pr9ciples may be kept 9 m9d 9 ! =m ( sev5 simple b gold5 rules3-- #a4 ,consid] ! situation ! pupil faces4 #b4 ,consid] ! response y wi% to connect ) x4 #c4 ,=m ! bond2 d n expect x to come by a miracle4 #d4 ,o!r ?+s be+ equal1 =m no bond t w h to be brok54 #e4 ,o!r ?+s be+ equal1 d n =m two or ?ree bonds :5 "o w s]ve4 #f4 ,o!r ?+s be+ equal1 =m bonds 9 ! way t !y are requir$ lat] to act4 #g4 ,favor1 !re=e1 ! situations :i* life itself w (f]1 & ! responses :i* life itself w dem&4 ,,*apt] ,v ,,! ,,psy*ology ,,( ,,drill ,,9 ,,ari?metic3 ,,! ,,/r5g? ,,( ,,bonds ,an inv5tory ( ! bonds to be =m$ 9 le>n+ ari?metic %\ld be accompani$ by a /atem5t ( h[ /rong ea* bond is to be made & kept ye> by ye>4 ,s9ce1 h[ev]1 ! inv5tory itself has be5 pres5t$ "h only 9 samples1 ! detail$ /atem5t ( desir$ /r5g? = ea* bond _c be made4 ,only c]ta9 g5]al facts w be not$ "h4 ,,! ,,ne$ ,,( ,,/rong] ,,elem5t>y ,,bonds ,! con/itu5t bonds involv$ 9 ! fundam5tal op]ations ) numb]s ne$ to be mu* /rong] ?an !y n[ are4 ,inaccuracy 9 ^! op]ations means weakness ( ! con/itu5t bonds4 ,inaccuracy exi/s1 & to a degree t deprives ! subject ( mu* ( its possible discipl9>y value1 makes ! pupil's a*ievem5ts ( slight value = use 9 bus9ess or indu/ry1 & prev5ts ! pupil f v]ify+ 8 "w ) new processes by "s previ\sly acquir$ process4 ,! inaccuracy t exi/s may be se5 9 ! measurem5ts made by ! _m inve/igators :o h us$ ari?metical tasks z te/s ( fatigue1 practice1 individual diff]5ces & ! l1 & 9 ! special /udies ( ari?metical a*ievem5ts = _! own sake made by ,c\rtis & o!rs4 ,burg]/e9 '#ia1 us+ su* examples z #bhgjdeafichbgfedfcig #cehfidbgceiafchbgbfc ---------------------- & simil> long numb]s to be multipli$ by #b or by #c or by #d or by #e or by #f1 f\nd #hea errors 9 #bh1bfg answ]-figures1 or #c p] hundr$ answ]-figures1 or #c_/e ( an error p] example4 ,! *ildr5 7 #i-a_/b to #ae ye>s old4 ,las] '#id1 us+ ! same sort ( addition & multiplication1 f\nd some:at ov] #c errors p] hundr$ answ]-figures 9 ! case ( boys & girls av]ag+ #aa-a_/b ye>s1 dur+ ! p]iod ( _! mo/ accurate "w4 ,holmes '#ie1 us+ addition ( ! sort j describ$1 f\nd #cdf errors 9 #bc1gac answ]-figures or ab\t #a-a_/b p] hundr$4 ,! *ildr5 7 f all grades f ! ?ird to ! eigh?4 ,9 ,las]'s "w1 #ba1 #ai1 #ac1 & #aj answ]-figures 7 obta9$ p] m9ute4 ,fri$ri* '#ig ) simil> examples1 giv+ ! v long "t ( #bj m9utes = obta9+ ab\t #bjj answ]-figures1 f\nd f #a to #b p] hundr$ wrong4 ,k+ '#jg _h *ildr5 9 grade #e d sums1 ea* consi/+ ( #e two-place numb]s4 ,9 ! mo/ accurate "w-p]iod1 !y made #a error p] #bj columns4 ,9 multiply+ a f\r-place by a f\r-place numb] !y _h less ?an "o total answ] "r out ( ?ree4 ,9 ,new ,york ,city ,c\rtis f\nd '#aa-'ab ) 8 ,te/ #g t 9 #ab m9utes ! av]age a*ievem5t ( f\r?-grade *ildr5 is #h4h units attempt$ ) #d4b "r4 ,9 grade #e ! facts are #aj4i attempts ) #e4h "r2 9 grade #f1 #ab4e attempts ) #g4j "r2 9 grade #g1 #ae attempts ) #h4e "r2 9 grade #h1 #ae4g attempts ) #aj4a "r4 ,^! results are ne> en\gh to ^? obta9$ f ! c\ntry at l>ge to s]ve z a text "h4 ,! foll[+ 7 set z (ficial /&>ds1 9 an excell5t s*ool sy/em1 ,c\rtis ,s]ies ,b be+ us$3-- ,,spe$ ,,p]c5t ,,( ,,grade4 ,,attempts4 ,,correct ,,answ]s4 ,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 & #ee ff4 f\nd t1 9 add+ columns l ^? pr9t$ bel[1 *ildr5 9 grade #d got on ! av]age less ?an #hj p]c5t ( correct answ]s4 ,_! av]age spe$ 0 ab\t #b columns p] m9ute4 ,9 do+ division ( ! sort pr9t$ bel[ *ildr5 ( grades #c ,b & #d ,a got less ?an #ie p]c5t ( correct answ]s1 ! av]age spe$ be+ #d divisions p] m9ute4 ,9 bo? cases ! sl[] comput]s 7 no m accurate ?an ! fa/] ones4 ,practice improv$ ! spe$ v rapidly1 b ! accuracy rema9$ sub/antially un*ang$4 ,br[n '#aa & '#ab f\nd a simil> l[ /atus ( ability & notable improvem5t f a mod]ate am\nt ( 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 & 4444 r4 #cj 4444 #gs & 4444 r4 #hi 4444 #is & 4444 r4 #bj 4444 #hs & 4444 r4 #ef 4444 #fs & 4444 r4 #ca 4444 #ds & 4444 r4 #hf 4444 #is & 4444 r4 ,x is cle> t num]ical "w z inaccurate z ?is has little or no comm]cial or indu/rial value4 ,if cl]ks got only six answ]s out ( t5 "r z 9 ! ,c\rtis te/s1 "o w\ld ne$ to h at lea/ f\r cl]ks make ea* computation & w\ld ev5 !n h to *eck _m ( _! discrepancies by ! "w ( /ill o!r cl]ks1 if he want$ 8 acc\nts to %[ less ?an "o error p] hundr$ acc\nt+ units ( ! ,c\rtis size4 ,x is also cle> t ! habits ( 444 absolute accuracy1 & satisfaction 9 tru? z a result :i* ari?metic is suppos$ to fur!r mu/ be l>gely my?ical 9 pupils :o get "r answ]s only f ?ree to n9e times out ( t56 ,,e>ly ,,ma/]y ,! bonds 9 "q cle>ly mu/ be made f> /rong] ?an !y n[ are4 ,!y %\ld 9 fact be /rong en\gh to aboli% errors 9 computation1 except = ^? due to tempor>y lapses4 ,x is mu* bett] = a *ild to "k half ( ! multiplication tables1 & to "k t he does n "k ! re/1 ?an to half-"k !m all2 & ?is holds good ( all ! elem5t>y bonds requir$ = computation4 ,any bond %\ld be made to "w p]fectly1 ?\gh sl[ly1 v soon aft] its =mation is begun4 ,spe$ c easily be add$ by prop] practice4 ,! *ief reasons :y ?is is n done n[ seem to be ! foll[+3 7#a7 ,c]ta9 important bonds 7l ! additions ) high] decades7 are n giv5 en\gh att5tion :5 !y are fir/ us$4 7#b7 ,! special tra9+ necess>y :5 a bond is us$ 9 a diff]5t connection 7z :5 ! multiplications to #i x #i are us$ 9 examples l #gbi #h --- :]e ! pupil has also to *oose ! "r numb] to multiply1 keep 9 m9d :at is c>ri$1 use x prop]ly1 & write ! "r figure 9 ! "r place1 & c>ry a figure1 or rememb] t he c>ries none7 is neglect$4 7#c7 ,! pupil is n taught to *eck 8 "w4 7#d7 ,he is n made responsible = sub/antially accurate results4 ,fur!rmore1 ! requirem5t ( 7#d7 )\t ! tra9+ ( 7#a71 7#b71 & 7#c7 w involve ei!r a fruitless failure on ! "p ( _m pupils1 or an utt]ly unju/ requirem5t ( "t4 ,! common error ( suppos+ t ! task ( computation ) integ]s consi/s m]ely 9 le>n+ ! additions to #i #i1 ! subtractions to #ah - #i1 ! multiplications to #h x #i1 & ! divisions to #ha _/ #i1 & 9 apply+ ?is k 9 connection ) ! pr9ciples ( decimal notation1 has _h a l>ge %>e 9 p]mitt+ ! gross inaccuracy ( ari?metical "w4 ,! bonds involv$ 9 'kn[+ ! tables' d n make up "o f\r? ( ! bonds involv$ 9 real add+1 subtract+1 multiply+1 & divid+ 7) integ]s alone74 ,x %\ld be not$ t if ! tra9+ m5tion$ 9 7#a7 & 7#b7 is well c>$ =1 ! *eck+ ( results z recomm5d$ 9 7#c7 becomes enorm\sly m valuable ?an x is "u pres5t conditions1 ?\gh ev5 n[ x is "o ( our s\nde/ practices4 ,if a *ild kn[s ! additions to high] decades s t he c add a se5 "o-place numb] to a ?\ght-( two-place numb] 9 ?ree seconds or less ) a correct answ] #aii times out ( #bjj1 !re is only an inf9itesimal *ance t a t5-figure column twice add$ 7once up1 once d[n7 a few m9utes ap>t ) id5tical answ]s w be wrong4 ,suppose t1 9 long multiplication1 a pupil c multiply to #i x #i :ile keep+ 8 place & keep+ track ( :at he is 'c>ry+' & ( :]e to write ! figure he writes1 & c add :at he c>ries )\t los+ track ( :at he is to add x to1 :]e he is to write ! unit figure1 :at he is to multiply next & by :at1 & :at he w !n h to c>ry1 9 ea* case to a surety ( #ii p]c5t ( correct responses4 ,!n two id5tical answ]s got by multiply+ "o ?ree-place numb] by ano!r a few m9utes ap>t1 & ) rev]sal ( ! numb]s1 w n be wrong m ?an twice 9 8 entire s*ool c>e]4 ,*ecks approa* pro(s :5 ! con/itu5t bonds are /rong4 ,if1 on ! contr>y1 ! fundam5tal bonds are s weak t !y d n "w accurately1 *eck+ becomes mu* less tru/wor?y & also v mu* m labori\s4 ,9 fact1 x is possible to %[ t bel[ a c]ta9 po9t ( /r5g? ( ! fundam5tal bonds1 ! "t requir$ = *eck+ is s great t "p ( x might bett] be sp5t 9 improv+ ! fundam5tal bonds4 ,= example1 suppose t a pupil has to f9d ! sum ( five numb]s l #b4di1 #e4be1 #f4ej1 #g4hi1 & #c4ge4 ,c\nt+ ea* act ( hold+ 9 m9d ! numb] to be c>ri$ & ea* writ+ ( a column's result z equival5t 9 difficulty to "o addition1 su* a sum equals n9ete5 s+le additions4 ,on ?is basis & ) c]ta9 additional e/imates#g we c compute ! practical consequ5ces = a pupil's use ( addition 9 life accord+ to ! ma/]y ( x t he has ga9$ 9 s*ool4 #g ,^! conc]n all[ances = two errors occurr+ 9 ! same example & = ! same wrong answ] be+ obta9$ 9 bo? orig9al "w & *eck "w4 ,i h s comput$ ! am\nt ( *eck+ a pupil w h to d to rea* two agree+ numb]s 7out ( two1 or ?ree1 or f\r1 or five1 or :atev] ! numb] be=e he gets two t are alike71 accord+ to 8 ma/]y ( ! elem5t>y processes4 ,! facts appe> 9 ,table #a4 ,x is obvi\s t a pupil ^: ma/]y ( ! elem5ts is t d5ot$ by gett+ !m "r #if times out ( #ajj w require s mu* "t = *eck+ t1 ev5 if he 7 nev] to use ?is ability = any?+ save a few ?\s& sums 9 addition1 he w\ld d well to improve ?is ability be=e he tri$ to d ! sums4 ,an ability ( #aii out ( #bjj1 or #iie out ( #ajjj1 seems likely to save mu* m "t ?an w\ld be tak5 to acquire x1 & a reasonable def5se c\ld be made = requir+ #iif or #iig out ( #ajjj4 ,a precision ( f #iie to #iig out ( #ajjj be+ requir$1 & ord9>y sagacity be+ us$ 9 ! tea*+1 spe$ w sub/antially take c>e ( itself4 ,c\nt+ on ! f+]s or 9 words w n give t precision4 ,sl[ rec\rse to memory ( s]ial addition tables w n give t precision4 ,no?+ save sure memory ( ! facts op]at+ "u ! conditions ( actual examples w give x4 ,& su* memories w op]ate ) suffici5t spe$4 ,,table #a ,,! ,,effect ,,( ,,ma/]y ,,( ,,! ,,elem5t>y ,,facts ,,( ,,addition ,,^u ,,! ,,labor ,,requir$ ,,to ,,secure ,,two ,,agree+ ,,answ]s ,,:5 ,,add+ ,,five ,,?ree-,,figure ,,numb]s ,,ma/]y ,,( ,,approximate ,,approximate ,,approximate ,,approximate ,,! ,,numb] ,,( ,,numb] ,,( ,,numb] ,,( ,,numb] ,,( ,,elem5t>y ,,wrong ,,answ]s,,agree+ ,,agree+ ,,*eck+s ,,additions ,,9 ,,sums ,,( #e ,,answ]s1 ,,answ]s1 ,,requir$ 7,,ov] ,,times ,,"r ,,?ree-,,place ,,aft] ,,"o ,,aft] ,a ,,& ,,above ,,! ,,9 #ajjj ,,numb]s ,,p] ,,*eck+1 ,,*eck+ ,,( ,,fir/ ,,g5]al #ajjj ,,p] #ajjj ,,! ,,fir/ ,,*eck+ ,,( ,,discrepancies,,! #ajjj ,,sums7 ,,to ,,secure ,,two ,,agree+ ,,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 ---------------------------------------- -------------------------- ,!re is "o intellig5t objection to ! special practice necess>y to e/abli% ari?metical connections s fully z to give ! accuracy :i* bo? utilit>ian & discipl9>y aims require4 ,x may be said t ! pupils 9 grades #c1 #d1 & #e _c appreciate ! ne$ & t consequ5tly ! "w w be dull1 b>r51 & ali51 )\t close p]sonal appropriation by ! pupil's nature4 ,x is true t no vehem5t life-purpose is directly involv$ by ! problem ( p]fect+ "o's p[] to add #g to #bh 9 grade #b1 or by ! problem ( multiply+ #bec by #h accurately 9 grade #c or by precise subtraction 9 long division 9 grade #d4 ,x is also true1 h[ev]1 t ! mo/ humanly int]e/+ ( problems--"o t ! pupil attacks mo/ :ole-he>t$ly--w n be solv$ correctly unless ! pupil has ! necess>y associative me*anisms 9 ord]2 & ! sur] he is ( !m1 ! fre] he is to ?9k out ! problem z su*4 ,fur!r1 computation is n dull if ! pupil c compute4 ,he does n himself object to its b>r5ness ( vital mean+1 s long z ! b>r5ness ( failure is prev5t$4 ,we mu/ n =get t pupils l to le>n4 ,9 tea*+ excessively dull individuals1 :o has n (t5 obs]v$ ! great int]e/ :i* !y display 9 any?+ t !y are enabl$ to ma/]8 ,!re is pa?os 9 _! joy 9 le>n+ to recognize p>ts ( spee*1 p]=m algebraic simplifications1 or translate ,lat9 s5t5ces1 & 9 o!r accompli%m5ts equally mean+less to all _! int]e/s save ! univ]sal human int]e/ 9 success & recognition4 ,/ill fur!r1 x is n v h>d to %[ to pupils ! imp]ative ne$ ( accuracy 9 scor+ games1 9 ! %op1 9 ! /ore1 & 9 ! (fice4 ,f9ally1 ! argum5t t accurate "w ( ?is sort is ali5 to ! pupil 9 ^! grades is /ill /rong] aga9/ inaccurate "w ( ! same sort4 ,if we are to tea* computation ) two- & ?ree- & f\r-place numb]s at all1 x %\ld be taught z a reliable in/rum5t1 n z a comb9ation ( vague memories & fai?4 ,! au?or is ready to cut computation ) numb]s above #aj out ( ! curriculum ( grades #a-f z soon z m valuable educational in/rum5ts are (f]$ 9 its place1 b he is conv9c$ t no?+ 9 *ild-nature makes a l>ge v>iety ( inaccurate comput+ m int]e/+ or educative or g]mane to felt ne$s1 ?an a small] v>iety ( accurate comput+6 ,,! ,,/r5g? ,,( ,,bonds ,,= ,,tempor>y ,,s]vice ,! second g5]al fact is t c]ta9 bonds are ( s]vice = only a limit$ "t & s ne$ to be =m$ only to a limit$ & slight degree ( /r5g?4 ,! data ( problems set to illu/rate a pr9ciple or improve "s habit ( computation are1 ( c\rse1 ! cle>e/ cases4 ,! pupil ne$s to rememb] t ,john b\ght #c loaves ( bread & t !y 7 #e-;c5t loaves & t he gave #be c5ts to ! bak] only long en\gh to use ! data to decide :at *ange ,john %\ld receive4 ,! connections betwe5 ! total describ$ situation & ! answ] obta9$1 suppos+ "s consid]able computation to int]v5e1 is a bond t we let expire almo/ z soon z x is born4 ,x is sometimes assum$ t ! bond betwe5 a c]ta9 gr\p ( features :i* make a problem a ',buy a ?+s at b p] ?+1 f9d total co/' problem or a ',buy a ?+s at b p] ?+1 :at *ange f c' problem or a ',:at ga9 on buy+ = a & sell+ = b' problem or a ',h[ _m ?+s at a ea* c ,i buy = b c5ts' problem--x is assum$ t ! bond betwe5 ^! ess5tial def9+ features & ! op]ation or op]ations requir$ = solution is z tempor>y z ! bonds ) ! "n ( ! buy] or ! price ( ! ?+4 ,x is assum$ t all problems are & %\ld be solv$ by "s pure act ( reason+ )\t help or h9drance f bonds ) ! p>ticul> v]bal /ructure & vocabul>y ( ! problems4 ,:e!r or n !y %\ld be1 !y are n4 ,e "t t a pupil solves a 'b\ght-sold' problem by subtraction he /r5g!ns ! t5d5cy to respond to any problem :atsoev] t conta9s ! words 'b\ght =' & 'sold =' by subtraction2 & he w by no means surely /op & survey e su* problem 9 all its elem5ts to make sure t no o!r feature makes inapplicable ! t5d5cy to subtract :i* ! 'b\ght sold' evokes4 ,to prev5t pupils f respond+ to ! =m ( /atem5t r ?an ! ess5tial facts1 we %\ld !n n tea* !m to =get ! =m ( /atem5t1 b r give !m all ! common =ms ( /atem5t to :i* ! response 9 "q is an appropriate response1 & only su*4 ,if a c]ta9 =m ( /atem5t does 9 life always signify a c]ta9 ari?metical proc$ure1 ! bond betwe5 x & t proc$ure may prop]ly be made v /rong4 ,ano!r case ( ! =mation ( bonds to only a slight degree ( /r5g? conc]ns ! use ( s-call$ 'crut*es' su* z writ+ 1 -1 & x 9 copy+ problems l ^? bel[3-- ,add ,subtract ,multiply #bc #gi #cb #fa #bd #c -- -- -- or alt]+ ! figures :5 'borr[+' 9 subtraction1 & ! l4 ,s9ce x is undesirable t ! pupil %\ld reg>d ! 'crut*' response z ess5tial to ! total proc$ure1 or become s us$ to hav+ x t he w be di/urb$ by its abs5ce lat]1 x is suppos$ t ! bond betwe5 ! situation & ! crut* %\ld n be fully =m$4 ,!re is a bett] way out ( ! difficulty1 9 case crut*es are us$ at all4 ,?is is to associate ! crut* ) a special 'set1' & its non-use ) ! g5]al set :i* is to be ! p]man5t "o4 ,= example1 *ildr5 may be taught f ! />t nev] to write ! crut* sign or crut* figure unless ! "w is accompani$ by ,write 444 to help y to4444 ,write - to help y to ,f9d ! diff]5ces3-- rememb] t y mu/ #ci #fg #gh #ef #de subtract 9 ?is r[4 #bc #dd #cf #bf #bd -- -- -- -- -- ,rememb] t y mu/ ,f9d ! diff]5ces3-- subtract 9 ?is r[4 #he #bg #if #ch #gh #fc #ad #ea #de #cb -- -- -- -- -- ,! bond evok+ ! use ( ! crut* may !n be =m$ ?or\ghly en\gh s t !re is no hesitation1 insecurity1 or error1 )\t int]f]+ to any h>mful ext5t ) ! m g5]al bond f ! situation to "w )\t ! crut*4 ,,! ,,/r5g? ,,( ,,bonds ,,) ,,te*nical ,,facts ,,& ,,t]ms ,ano!r in/ructive case conc]ns ! bonds betwe5 c]ta9 words & _! mean+s1 & betwe5 c]ta9 situations ( comm]ce1 indu/ry1 or agriculture & useful facts ab\t ^! situations4 ,illu/rations ( ! =m] are ! bonds betwe5 cube root1 hect>e1 brok]age1 commission1 indorsem5t1 v]tex1 adjac5t1 nonagon1 sector1 draft1 bill ( ex*ange1 & _! mean+s4 ,illu/rations ( ! latt] are ! bonds f ,money be+ l5t ') int]e/' at no specifi$ rate1 :at rate is *>g$8 to ,! legal rate ( ! /ate1 f ,x p] ,m z a rate = lumb] to ,means ,x p] ?\s& bo>d feet1 a bo>d foot be+ #a ft4 by #a ft4 by #a 94 ,x is argu$ by _m t su* bonds are valuable = a %ort "t2 namely1 :ile ari?metical proc$ures 9 connection ) :i* !y s]ve are le>n$1 b t _! value is only to s]ve z a means = le>n+ ^! proc$ures & t !reaft] !y may be =gott54 ,!y are =m$ only z accessory means to c]ta9 m purely ari?metical k or discipl9e2 aft] ?is is acquir$ !y may be =gott54 ,ev]ybody does 9 fact =get !m1 rele>n+ !m lat] if life requires4 ,s runs ! argum5t4 ,9 "s cases le>n+ su* words & facts only to use !m 9 solv+ a c]ta9 sort ( problems & !n =get !m may be pr(itable4 ,! practice is1 h[ev]1 exce$+ly risky4 ,x is true t ev]ybody does 9 fact =get _m su* mean+s & facts1 b ?is commonly means ei!r t !y %\ld n h be5 le>n$ at all at ! "t t !y 7 le>n$1 or t !y %\ld h be5 le>n$ m p]man5tly1 or t details %\ld h be5 le>n$ ) ! expectation t !y !mselves w\ld be =gott5 b t a g5]al fact or attitude w\ld rema94 ,= example1 duodecagon %\ld n be le>n$ at all 9 ! elem5t>y s*ool2 indorsem5t %\ld ei!r n be le>n$ at all !re1 or be le>n$ = p]man5ce ( a ye> or m2 ! details ( ! metric sy/em %\ld be s taught z to leave = sev]al ye>s at lea/ k ( ! facts t !re is a sy/em s nam$ t is important1 ^: tables g by t5s1 hundr$s1 or ?\s&s1 & a t5d5cy 7n necess>ily /rong7 to connect met]1 kilogram1 & lit] ) measurem5t by ! metric sy/em & ) approximate e/imates ( _! sev]al magnitudes4 ,if an ari?metical proc$ure seems to require accessory bonds :i* are to be =gott51 once ! proc$ure is ma/]$1 we %\ld be suspici\s ( ! value ( ! proc$ure itself4 ,if pupils =get :at comp\nd int]e/ is1 we may be sure t !y w usually also h =gott5 h[ to compute x4 ,surely !re is wa/e if !y h le>n$ :at x is only to le>n h[ to compute x only to =get h[ to compute x6 ,,! ,,/r5g? ,,( ,,bonds ,,conc]n+ ,,! ,,reasons ,,= ,,ari?metical ,,processes ,! next case ( ! =mation ( bonds to slight /r5g? is ! problematic "o ( =m+ ! bonds involv$ 9 und]/&+ ! reasons = c]ta9 processes only to =get !m aft] ! process has become a habit4 ,%\ld a pupil1 t is1 le>n :y he inv]ts & multiplies1 only to =get x z soon z he c be tru/$ to divide by a fraction8 ,%\ld he le>n :y he puts ! units figure ( ea* p>tial product 9 multiplication "u ! figure t he multiplies by1 only to =get ! reason z soon z he has comm& ( ! process8 ,%\ld he le>n :y he gets ! numb] ( squ>e in*es 9 a rectangle by multiply+ ! l5g? by ! wid?1 bo? be+ express$ 9 l9e> in*es1 & =get :y z soon z he is compet5t to make computations ( ! areas ( rectangles8 ,on g5]al psy*ological gr\nds we %\ld be suspici\s ( =m+ bonds only to let !m die ( />vation lat]1 & t5d to expect t elaborate explanations le>n$ only to be =gott5 ei!r %\ld n be le>n$ at all1 or %\ld be le>n$ at su* a "t & 9 su* a way t !y w\ld n be =gott54 ,especially we %\ld expect t ! g5]al pr9ciples ( ari?metic1 ! :ys & :]e=es ( its fundam5tal ways ( manipulat+ numb]s1 ought to be ! la/ bonds ( all to be =gott54 ,details ( h[ y arrang$ numb]s to multiply might vani%1 b ! g5]al reasons = ! plac+ w\ld be expect$ to p]si/ & enable "o to inv5t ! detail$ manipulations t _h be5 =gott54 ,?is suspicion is1 ,i ?9k1 ju/ifi$ by facts4 ,! doctr9e t ! cu/om>y d$uctive explanations ( :y we inv]t & multiply1 or place ! p>tial products z we d be=e add+1 may be all[$ to be =gott5 once ! actual habits are 9 work+ ord]1 has a suspici\s s\rce4 ,x arose to meet ! criticism t s mu* "t & ef=t 7 requir$ to keep ^! d$uctive explanations 9 memory4 ,! fact 0 t ! pupil le>n$ to compute correctly irrespective ( ! d$uctive explanations4 ,!y 7 only an add$ burd54 ,8 inductive le>n+ t ! proc$ure gave ! "r answ] really taught him4 ,s he wisely %uffl$ (f ! extra burd5 ( facts ab\t ! consequ5ces ( ! nature ( a fraction or ! place values ( our decimal notation4 ,! bonds weak5$ because !y 7 n us$4 ,!y 7 n us$ because !y 7 n useful 9 ! %ape & at ! "t t !y 7 =m$1 or because ! pupil 0 unable to und]/& ! explanations s z to =m !m at all4 ,! criticism 0 valid & %\ld h be5 met 9 "p by replac+ ! d$uctive explanations by inductive v]ifications1 & 9 "p by us+ ! d$uctive reason+ z a *eck aft] ! process itself is ma/]$4 ,! v same discussions ( place-value :i* are futile z pro( t y mu/ d a c]ta9 ?+ be=e y h done x1 (t5 become in/ructive z an explanation ( :y ! ?+ t y h le>n$ to d & are famili> ) & h v]ifi$ by o!r te/s works z well z x does4 ,! g5]al d$uctive !ory ( ari?metic %\ld n be le>n$ only to be =gott54 ,mu* ( x %\ld1 by mo/ pupils1 n be le>n$ at all4 ,:at is le>n$ %\ld be le>n$ mu* lat] ?an n[1 z a syn!sis & rationale ( habits1 n z _! creator4 ,:at is le>n$ ( su* d$uctive !ory %\ld rank among ! mo/ r ?an lea/ p]man5t ( a pupil's /ock ( ari?metical k & p[]4 ,!re are bonds :i* are =m$ only to be lo/1 & bonds =m$ only to be lo/ 9 _! fir/ =m1 be+ us$ 9 a new organization z mat]ial = bonds ( a high] ord]2 b ! bonds involv$ 9 d$uctive explanations ( :y c]ta9 processes are "r are n su*3 !y are n to be =m$ j to be =gott51 nor z m]e propa$eutics to r\t9e manipulations4 ,,propa$eutic ,,bonds ,! =mation ( bonds to a limit$ /r5g? because !y are to be lo/ 9 _! fir/ =m1 be+ work$ ov] 9 diff]5t ways 9 o!r bonds to :i* !y are propa$eutic or contribut+ is ! mo/ important case ( l[ /r5g?1 or r l[ p]man5ce1 9 bonds4 ,! bond betwe5 f\r #es 9 a column to be add$ & ! response ( ?9k+ '#aj1 #ae1 #bj' is wor? =m+1 b x is displac$ lat] by ! multiplication bond or direct connection ( 'f\r #es to be add$' ) '#bj4' ,c\nt+ by #bs f #b1 #cs f #c1 #ds f #d1 #es f #e1 etc41 =ms s]ial bonds :i* z s]ies might well be left to disappe>4 ,_! sep>ate /eps are kept z p]man5t bonds = use 9 column addition1 b _! s]ial nature is *ang$ f #b 7& #b7 #d1 7& #b7 #f1 7& #b7 #h1 etc41 to two #bs #d1 ?ree #bs #f1 f\r #bs #h1 etc42 aft] play+ _! "p 9 produc+ ! bonds :]eby any multiple ( #b by #b to #i1 c be got1 ! orig9al s]ial bonds are1 z s]ies1 ne$$ no long]4 ,! v]bal response ( say+ '&' 9 add+1 aft] help+ to e/abli% ! bonds :]eby ! g5]al set ( ! m9d t[>d add+ cooep]ates ) ! numb]s se5 or ?\ght ( to produce _! sum1 %\ld disappe>2 or rema9 s slurr$ 9 inn] spee* z to (f] no b> to spe$4 ,! rule = su* bonds is1 ( c\rse1 to =m !m /rongly en\gh s t !y "w quickly & accurately = ! "t be+ & facilitate ! bonds t are to replace !m1 b n to ov]le>n !m4 ,!re is a diff]5ce betwe5 le>n+ some?+ to be held = a %ort "t1 & ! same am\nt ( en]gy sp5t 9 le>n+ = long ret5tion4 ,! =m] sort ( le>n+ is1 ( c\rse1 appropriate ) _m ( ^! propa$eutic bonds4 ,! bonds m5tion$ z illu/rations are n purely propa$eutic1 nor =m$ only to be transmut$ into some?+ else4 ,ev5 ! say+ ( '&' 9 addition has "s g5u9e1 intr9sic value 9 di/+ui%+ ! process ( addition1 & may p]haps be usefully review$ = a brief space dur+ ! fir/ /eps 9 add+ common fractions4 ,"s su* propa$eutic bonds may be wor? :ile ap>t f _! value 9 prep>+ = o!r bonds4 ,consid]1 = example1 ex]cises l ^? %[n bel[ :i* are propa$eutic to long division1 giv+ ! pupil "s basis 9 exp]i5ce = 8 selection ( ! quoti5t figures4 ,^! multiplications are intr9sically wor? do+1 especially ! #abs & #bes4 ,:atev] ! pupil rememb]s ( !m w be to 8 advantage4 #a4 ,c\nt by #aas to #acb1 beg9n+ #aa1 #bb1 #cc4 #b4 ,c\nt by #abs to #add1 beg9n+ #ab1 #bd1 #cf4 #c4 ,c\nt by #bes to #cjj1 beg9n+ #be1 #ej1 #ge4 #d4 ,/ate ! miss+ numb]s3-- ,a4 ,b4 ,c4 ,d4 #c #aas #e #aas #h ft4 4444 94 #b doz5 #d #abs #c #abs #aj ft4 4444 94 #d doz5 #e #abs #f #abs #g ft4 4444 94 #aj doz5 #f #aas #ab #aas #d ft4 4444 94 #e doz5 #i #aas #b #abs #f ft4 4444 94 #g doz5 #g #abs #i #abs #i ft4 4444 94 #ab doz5 #h #abs #g #aas #aa ft4 4444 94 #i doz5 #aa #aas #ab #abs #e ft4 4444 94 #f doz5 #e4 ,c\nt by #bes to #b4ej1 say+1 #be c5ts1 #ej c5ts1 #ge c5ts1 "o doll>1 & s on4 #f4 ,c\nt by #aes to #a4ej4 #g4 ,f9d ! products4 ,d n use p5cil4 ,?9k :at !y 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 ,/ate ! miss+ numb]s3-- ,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 ,f9d ! quoti5ts & rema9d]s4 ,if y ne$ to use pap] & p5cil to f9d !m1 y may4 ,b f9d z _m z y c )\t p5cil & pap]4 ,d ,r[ ,a fir/4 ,!n d ,r[ ,b4 ,!n ,r[ ,c1 etc4 ,r[ ,a4 #aade #abde #bede #aede #bade #bbde ,r[ ,b4 #beee #aaee #abee #aeee #bbee #cjee ,r[ ,c4 #abfj #befj #aefj #aafj #cjfj #bafj ,r[ ,d4 #abge #aage #aege #bege #cjge #cege ,r[ ,e4 #aaajj #abajj #beajj #aeajj #cjajj #bbajj ,r[ ,f4 #aaif #abif #beif #aeif #cjif #bbif ,r[ ,g4 #beaje #aaaje #aeaje #abaje #bbaje #ceaje ,r[ ,h4 #abfd #aefd #befd #aafd #bbfd #bafd ,r[ ,i4 #aahj #abhj #aehj #behj #cehj #bahj ,r[ ,j4 #bebjj #cjbjj #gebjj #fcbjj #febjj #ffbjj ,d ?is section aga94 ,d all ! fir/ column fir/4 ,!n d ! second column1 !n ! ?ird1 & s on4 ,consid]1 f ! same po9t ( view1 ex]cises l 7#c x #d7 #b1 7#g x #f7 #e1 7#i x #d7 #f1 giv5 z a prep>ation = writt5 multiplication4 ,! "w ( #dh #fh #dg #c #g #i -- -- -- & ! l is facilitat$ if ! pupil has easy control ( ! process ( gett+ a product1 & keep+ x 9 m9d :ile he adds a "o-place numb] to x4 ,! practice ) 7#c x #d7 #b & ! l is also good practice intr9sically4 ,s "s tea*]s provide sy/ematic prep>atory drills ( ?is type j be=e or along ) ! beg9n+ ( %ort multiplication4 ,9 "s cases ! bonds are purely propa$eutic or are =m$ only = lat] recon/ruction4 ,!y !n diff] little f 'crut*es4' ,! typical crut* =ms a habit :i* has actually to be brok51 :]eas ! purely propa$eutic bond =ms a habit :i* is left to ru/ out f disuse4 ,= example1 z an introduction to long division1 a pupil may be giv5 ex]cises us+ "o-figure divisors 9 ! long =m1 z3-- #ggc & #e rema9d] #g7edaf #di -- #ea #di -- #bf #ba -- #e ,! important recomm5dation conc]n+ ^! purely propa$eutic bonds1 & bonds =m$ only = lat] recon/ruction1 is to be v critical ( !m1 & n indulge 9 !m :51 by ! ex]cise ( en\gh +5uity1 "s bond wor?y ( a p]man5t place 9 ! individual's equipm5t c be devis$ :i* w d ! "w z well4 ,ari?metical tea*+ has done v well 9 ?is respect1 t5d+ to err by leav+ out really valuable prep>atory drills r ?an by ins]t+ uneconomical ones4 ,x is 9 ! tea*+ ( read+ t we f9d ! =mation ( propa$eutic bonds ( dubi\s value 7) lett]s1 phonograms1 diacritical m>ks1 & ! l7 (t5 c>ri$ to demon/rably wa/eful extremes4 ,,*apt] ,,vi ,,! ,,psy*ology ,,( ,,drill ,,9 ,,ari?metic3 ,,! ,,am\nt ,,( ,,practice ,,& ,,! ,,organization ,,( ,,abilities ,,! ,,am\nt ,,( ,,practice ,x w be in/ructive if ! read] w p]=m ! foll[+ exp]im5t z an introduction to ! discussion ( ?is *apt]1 be=e read+ any ( ! discussion4 ,suppose t a pupil does all ! "w1 oral & writt51 computation & problem-solv+1 pres5t$ = grades #a to #f inclusive 7t is1 9 ! fir/ two books ( a ?ree-book s]ies7 9 ! av]age textbook n[ us$ 9 ! elem5t>y s*ool4 ,h[ _m times w he h ex]cis$ ea* ( ! v>i\s bonds involv$ 9 ! f\r op]ations ) integ]s %[n bel[8 ,t is1 h[ _m times w he h ?\ght1 #a & #a are #b1 #a & #b are #c1 etc48 ,e case ( ! action ( ea* bond is to be c\nt$4 ,,! ,,fundam5tal ,,bonds #a #a #b - #a #a x #a #b _/ #a #a #b #b - #b #b x #a #b _/ #b #a #c #c x #a #a #d #d x #a #a #e #c - #a #e x #a #c _/ #a #a #f #c - #b #f x #a #c _/ #b #a #g #c - #c #g x #a #c _/ #c #a #h #h x #a #a #i #i x #a #d - #a #d _/ #a #d - #b #d _/ #b #aa 7or #ba or #ca1 etc47 #a #d - #c #a x #b #d _/ #c #aa #b #d - #d #b x #b #d _/ #d #aa #c #c x #b #aa #d #d x #b #aa #e #e - #a #e x #b #e _/ #a #aa #f #e - #b #f x #b #e _/ #b #aa #g #e - #c #g x #b #e _/ #c #aa #h #e - #d #h x #b #e _/ #d #aa #i #e - #e #i x #b #e _/ #e #f - #a #a x #c #f _/ #a #b #a #f - #b #b x #c #f _/ #b #b #b #f - #c #c x #c #f _/ #c #b #c #f - #d #d x #c #f _/ #d #b #d #f - #e #e x #c #f _/ #e #b #e #f - #f #f x #c #f _/ #f #b #f #g x #c #b #g #h x #c #b #h #g - #a #i x #c #g _/ #a #b #i #g - #b #g _/ #b #g - #c #g _/ #c #g - #d #a x #d #g _/ #d #ab 7or #bb or #cb1 etc47 #a #g - #e #b x #d #g _/ #e #ab #b #g - #f & s on #g _/ #f #g - #g to #i x #i #g _/ #g & s on to & s on & s on to #i #i to #ah - #i #hb _/ #i #ai 7or #bi or #ci1 etc47 #i #hc _/ #i1 etc4 ,if e/imat+ = ! entire s]ies is too long a task1 x w be suffici5t to use eight or t5 f ea*1 say3-- #c #b #ac1 #bc1 etc4 #b #g #b #ag1 #bg1 etc4 #b #c #c #c #c #d #d #d #d #e #e #e #e #f #f #f #f #g #g #g #g #h #h #h #h #i #i #i #i #c - #c #g - #g #i x #g #fc _/ #i #d #h #g x #i #fd #e #i #h x #f #fe #f #aj #f x #h #ff #g #aa #fg #h #ab #fh #i #ac #fi #aj #ad #gj #aa #ae #ga #ab #af ,,table #b ,,e/imates ,,( ,,! ,,am\nt ,,( ,,practice ,,provid$ ,,9 ,,books ,i ,,& ,,ii ,,( ,,! ,,av]age ,,?ree-,,book ,,text ,,9 ,,ari?metic2 ,,by #ej ,,exp]i5c$ ,,tea*]s ,,l[e/ ,,m$ian ,,highe/ ,,range ,,requir$ ,,to ,,ari?metical ,,fact ,,e/imate,,e/imate,,e/imate ,,include ,,half ,,( ,,! ,,e/imates ---------------------------------------- -------------------------- #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 ,hav+ made 8 e/imates ! read] %\ld comp>e !m fir/ ) simil> e/imates made by exp]i5c$ tea*]s 7%[n on page #abd f471 & !n ) ! results ( actual c\nts = repres5tative textbooks 9 ari?metic 7%[n on pages #abf to #acb74 ,x w be obs]v$ 9 ,table #b t ev5 exp]i5c$ tea*]s v>y enorm\sly 9 _! e/imates ( ! am\nt ( practice giv5 by an av]age textbook 9 ari?metic1 & t mo/ ( !m are 9 s]i\s error by ov]e/imat+ ! am\nt ( practice4 ,9 g5]al x is ! fact t we use textbooks 9 ari?metic ) v vague & errone\s ideas ( :at is 9 !m1 & ?9k !y give mu* m practice ?an !y d4 ,! au?ors ( ! textbooks z a rule also probably _h only v vague & errone\s ideas ( :at 0 9 !m4 ,if !y _h kn[n1 !y w\ld almo/ c]ta9ly h revis$ _! books4 ,surely no au?or w\ld int5tionally provide ne>ly f\r times z mu* practice on #b #b z on #h #h1 or eight times z mu* practice on #b x #b z on #i x #h1 or elev5 times z mu* practice on #b - #b z on #ag - #h1 or ov] =ty times z mu* practice on #b _/ #b z on #ge _/ #h & #ge _/ #i1 bo? toge!r4 ,surely no au?or w\ld h provid$ int5tionally only tw5ty to ?irty occurr5ces ea* ( #af - #g1 #af - #h1 #af - #i1 #ag - #h1 #ag - #i1 & #ah - #i = ! entire c\rse ?r\gh grade #f2 or h left ! practice on #fj _/ #g1 #fj _/ #h1 #fj _/ #i1 #fa _/ #g1 #fa _/ #h1 #fa _/ #i1 & ! l to occur only ab\t once a ye>6 ,,table #c ,,am\nt ,,( ,,practice3 ,,addition ,,bonds ,,9 ,a ,,rec5t ,,textbook 7,a7 ,,( ,,excell5t ,,repute4 ,,books ,i ,,& ,,ii1 ,,all ,,save ,,f\r ,,sections ,,( ,,supplem5t>y ,,mat]ial1 ,,to ,,be ,,us$ ,,at ,,! ,,tea*]',s ,,discretion ,! ,table reads3 #b #b 0 us$ #bbf times1 #ab #b 0 us$ #gd times1 #bb #b1 #cb #b1 #db #b1 & s on 7 us$ #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 ,,am\nt ,,( ,,practice3 ,,subtraction ,,bonds ,,9 ,a ,,rec5t ,,textbook 7,a7 ,,( ,,excell5t ,,repute4 ,,books ,i ,,& ,,ii1 ,,all ,,save ,,f\r ,,sections ,,( ,,supplem5t>y ,,mat]ial1 ,,to ,,be ,,us$ ,,at ,,! ,,tea*]',s ,,discretion ,,subtrah5ds ,,m9u5ds ---------------------------------------- ------------- #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 exclud+ #a-a1 #b-b1 etc4 #abeh #gee #efe #gac #ega #eeh #cbg #efi #cja ,,table #e ,,frequ5cies ,,( ,,subtractions ,,n ,,includ$ ,,9 ,,table #d ,^! are cases :]e ! pupil w\ld1 by reason ( 8 /age ( advancem5t1 probably op]ate #ce-cj1 #df-df1 etc41 z "o bond4 ,,subtrah5ds ---------------------------------------- #a #b #c #d #e #f #g #h #i ,,m9u5ds #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 ,,am\nt ,,( ,,practice3 ,,multiplication ,,bonds ,,9 ,,ano!r ,,rec5t ,,textbook 7,b7 ,,( ,,excell5t ,,repute4 ,,books ,i ,,& ,,ii ,,multiplic&s ,,multipli]s ---------------------------------------- ----------------- #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 ,,am\nt ,,( ,,practice3 ,,divisions ,,)\t ,,rema9d] ,,9 ,,textbook ,b1 ,,p>ts ,i ,,& ,,ii ,,divisors ,,divid5ds ---------------------------------------- ------ #b #c #d #e #f #g #h #i ,totals ---------------------------------------- --------------------- ,integral #cig #bbd #bej #acj #ic #dd #ih #bc #abei multiples ( #b to #i #bef #abd #aeb #gi #bh #dc #fa #be #gfh 9 sequ5ce2 i4e41 #d _/ #b #cah #abc #acj #fe #ej #ai #ci #ai #gfc occurr$ #cig times1 #beh #ih #hf #aje #be #bd #cd #bj #fej #f _/ #b occurr$ #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 ,,) ,,& ,,)\t ,,rema9d]s4 ,,book ,b ,all "w ?r\gh grade #f1 except e/imates ( quoti5t figures 9 long division4 ,divid5d #b #c #d #e ,divisor #a #b #a #b #c #a #b #c #d #a #b #c #d #e ,numb] ( ,occurr5ces #da #chf #bg #ahi #bdj #bf #cig #ff #ahe #bc #acf #dc #ec #ace ,divid5d #f #g ,divisor #a #b #c #d #e #f #a #b #c #d #e #f #g ,numb] ( ,occurr5ces #ba #bef #bbd #fh #dc #hc #bc #gb #ee #ch #df #cb #ed ,divid5d #h #i ,divisor #a #b #c #d #e #f #g #h #a #b #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #ag #cah #cj #bej #bb #bh #ci #ia #ai #ej #abd #di #be #ae #ah #cj #ch ,divid5d #aj #aa ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #beh #ch #df #abj #ai #i #bd #bd #cb #ba #af #c #g #aa #ad #c ,divid5d #ab #ac ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #aih #abc #aeb #bi #ic #i #af #g #de #af #ae #aa #g #d #e #c ,divid5d #ad #ae ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #gg #bj #ac #e #h #dd #h #f #fi #ih #af #gi #h #h #d #f ,divid5d #af #ag ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #ahj #ai #acj #ad #f #i #ih #c #fa #i #ae #ad #f #f #ab #c ,divid5d #ah #ai ,divisor #b #c #d #e #f #g #h #i #b #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #fi #di #ac #f #bh #g #g #bc #ba #f #aj #e #c #d #aj #d ,divid5d #bj #ba ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #bd #hf #fe #aa #c #bc #e #ed #ab #h #e #dc #aj #e ,divid5d #bb #bc ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #ag #af #ae #h #ac #f #ae #g #h #aa #h #f #c #b ,divid5d #bd #be ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #ia #gf #ah #ej #e #fa #a #aa #ac #aje #e #f #e #c ,divid5d #bf #bg ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #e #f #c #c #d #f #c #df #h #aj #d #b #f #be ,divid5d #bh #bi ,divisor #c #d #e #f #g #h #i #c #d #e #f #g #h #i ,numb] ( ,occurr5ces #d #cf #h #c #ai #c #g #f #h #j #e #aa #b #c ,divid5d #cj #ca #cb ,divisor #d #e #f #g #h #i #d #e #f #g #h #i #d #e #f #g #h #i ,numb] ( ,occurr5ces #ba #bg #be #f #g #ac #d #c #a #a #d #b #ej #aa #c #f #ci #e ,divid5d #cc #cd #ce ,divisor #d #e #f #g #h #i #d #e #f #g #h #i #d #e #f #g #h #i ,numb] ( ,occurr5ces #h #g #g #b #f #a #h #c #e #b #a #a #aj #ca #e #bd #e #c ,divid5d #cf #cg #ch ,divisor #d #e #f #g #h #i #d #e #f #g #h #i #d #e #f #g #h #i ,numb] ( ,occurr5ces #cg #af #bb #b #f #ai #ab #h #g #e #c #i #g #h #g #a #a #e ,divid5d #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 ,numb] ( ,occurr5ces #d #c #g #d #c #a #ch #i #b #cd #b #f #f #c #g #e #g #bh #cj #aj #c ,divid5d #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 ,numb] ( ,occurr5ces #g #e #aj #c #b #g #f #d #e #j #bd #f #g #aj #bj #c #c #b #b #b ,divid5d #dg #dh #di #ej ,divisor #e #f #g #h #i #e #f #g #h #i #e #f #g #h #i #f #g #h #i ,numb] ( ,occurr5ces #f #b #b #j #c #g #ag #d #cc #b #d #g #bg #i #b #d #f #c #h ,divid5d #ea #eb #ec #ed ,divisor #f #g #h #i #f #g #h #i #f #g #h #i #f #g #h #i ,numb] ( ,occurr5ces #b #c #a #b #e #e #e #c #d #c #b #b #ab #e #a #af ,divid5d #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 ,numb] ( ,occurr5ces #e #c #d #b #j #ac #af #h #j #c #a #c #b #b #c #a #b #c #j #c ,divid5d #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 ,numb] ( ,occurr5ces #c #i #a #a #b #e #d #f #a #ag #e #i #e #bb #j #a #aj #a ,divid5d #ff #fg #fh #fi #gj #ga ,divisor #g #h #i #g #h #i #g #h #i #g #h #i #h #i #h #i ,numb] ( ,occurr5ces #b #a #d #j #a #a #a #c #b #j #f #a #f #b #a #j ,divid5d #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 ,numb] ( ,occurr5ces #af #aj #g #e #c #c #e #c #c #b #c #j #d #a #j #b ,divid5d #hj #ha #hb #hc #hd #he #hf #hg #hh #hi ,divisor #i #i #i #i #i #i #i #i #i #i ,numb] ( ,occurr5ces #d #ae #b #d #a #b #j #c #b #g ,tables #c to #h %[ t ev5 gift$ au?ors make in/rum5ts = in/ruction 9 ari?metic :i* conta9 mu* less practice on c]ta9 elem5t>y facts ?an tea*]s suppose2 & :i* conta9 relatively mu* m practice on ! m easily le>n$ facts ?an on ^? :i* are h>d] to le>n4 ,h[ mu* practice %\ld be giv5 9 ari?metic8 ,h[ %\ld x be divid$ among ! diff]5t bonds to be =m$8 ,bel[ a c]ta9 am\nt !re is wa/e because1 z has be5 %[n 9 ,*apt] ,,vi1 ! pupil w ne$ m "t to detect & correct 8 errors ?an w\ld h be5 requir$ to give him ma/]y4 ,above a c]ta9 am\nt !re is wa/e because ( unproductive ov]le>n+4 ,if #ffh is j en\gh = #b x #b1 #hb is n en\gh = #i x #h4 ,if #hb is j en\gh = #i x #h1 #ffh is too mu* = #b x #b4 ,x is possible to f9d ! answ]s to ^! que/ions = ! pupil ( m$ian ability 7or any /at$ ability7 by suitable exp]im5ts4 ,! am\nt ( practice w1 ( c\rse1 v>y accord+ to ! ability ( ! pupil4 ,x w also v>y accord+ to ! int]e/ ar\s$ 9 him & ! satisfaction he feels 9 progress & ma/]y4 ,x w also v>y accord+ to ! am\nt ( practice ( o!r relat$ bonds2 #g #g #ad & #fj _/ #g #h & #d rema9d] w help ! =mation ( #g #h #ae & #fa _/ #g #h & #e rema9d]4 ,x w also1 ( c\rse1 v>y ) ! g5]al difficulty ( ! bond1 #ag - #h #i be+ "u ord9>y conditions ( tea*+ h>d] to =m ?an #g - #b #e4 ,until suitable exp]im5ts are at h& we may e/imate = ! fundam5tal bonds z foll[s1 assum+ t by ! end ( grade #f a /r5g? ( #aii correct out ( #bjj is to be _h1 & t ! tea*+ is by an intellig5t p]son work+ 9 accord ) psy*ological pr9ciples z to bo? ability & int]e/4 ,= "o ( ! easi] bonds1 mo/ facilitat$ by o!r bonds 7su* z #b x #e #aj1 or #aj - #b #h1 or ! d\ble bond #g two #cs & #a rema9d]7 9 ! case ( ! m$ian or av]age pupil1 twelve practices 9 ! week ( fir/ le>n+1 support$ by tw5ty-five practices dur+ ! two mon?s foll[+1 & ma9ta9$ by ?irty practices well spread ov] ! lat] p]iods %\ld be en\gh4 ,= ! m gift$ pupils less] am\nts d[n to six1 twelve1 & fifte5 may suffice4 ,= ! less gift$ pupils m may be requir$ up to ?irty1 fifty1 & a hundr$4 ,x is to be d\bt$1 h[ev]1 :e!r pupils requir+ ne>ly two hundr$ repetitions ( ea* ( ^! easy bonds %\ld be taught ari?metic beyond a few matt]s ( practical necessity4 ,= bonds ( ord9>y difficulty1 ) av]age facilitation f o!r bonds 7su* z #aa - #c1 #d x #g1 or #dh _/ #h #f7 9 ! case ( ! m$ian or av]age pupil1 we may e/imate tw5ty practices 9 ! week ( fir/ le>n+1 support$ by ?irty1 & ma9ta9$ by fifty practices well spread ov] ! lat] p]iods4 ,gift$ pupils may ga9 & keep ma/]y ) twelve1 fifte51 & tw5ty practices respectively4 ,pupils dull at ari?metic may ne$ up to tw5ty1 sixty1 & two hundr$4 ,"h1 aga91 x is to be d\bt$ :e!r a pupil = :om ari?metical facts1 well taught & made int]e/+1 are s h>d to acquire z ?is1 %\ld le>n _m ( !m4 ,= bonds ( great] difficulty1 less facilitat$ by o!r bonds 7su* z #ag - #i1 #h x #g1 or #ab-a_/b ( #a_/h (71 ! practice may be f t5 to a hundr$ p]c5t m ?an ! above4 ,,und]le>n+ ,,& ,,ov]le>n+ ,if we accept ! above provisional e/imates z reasonable1 we may consid] ! h>m done by giv+ less & by giv+ m ?an ^! reasonable am\nts4 ,giv+ less is indef5sible4 ,! pupil's "t is wa/$ 9 excessive *eck+ to f9d 8 errors4 ,he is 9 dang] ( be+ practic$ 9 error4 ,8 att5tion is div]t$ f ! le>n+ ( new facts & processes by ! necessity ( ?9k+ out ^! suppos$ly ma/]$ facts4 ,all new bonds are h>d] to le>n ?an !y %\ld be because ! bonds :i* %\ld facilitate !m are n /rong en\gh to d s4 ,giv+ m does h>m to "s ext5t by us+ up "t t c\ld be sp5t bett] = o!r purposes1 & 7?\gh n necess>ily7 by detract+ f ! pupil's int]e/ 9 ari?metic4 ,9 c]ta9 cases1 h[ev]1 su* excess practice & ov]le>n+ are actually desirable4 ,?ree cases are ( special importance4 ,! fir/ is ! case ( a bond op]at+ "u a *ang$ m5tal set or adju/m5t4 ,a pupil may "k #g x #h adequately z a ?+ by itself1 b ne$ m practice to op]ate x 9 #bhe #g --- :]e he has to rememb] t #c is to be add$ to ! #ef :5 he obta9s x1 & t only ! #i is to be writt5 d[n1 ! #e to be held 9 m9d = lat] use4 ,! practice requir$ to op]ate ! bond effici5tly 9 ?is new set is desirable1 ev5 ?\gh x is excess f a n>r[] po9t ( view1 & causes ! /raight=w>d 'sev5 eights are fifty-six' to be ov]le>n$4 ,s also a pupil's "w ) #bd1 #cd1 #dd1 etc41 #i may react to give :at w\ld be excess practice f ! po9t ( view ( #d #i alone2 8 "w 9 e/imat+ approximate quoti5t figures 9 long division may give excess practice on ! division tables4 ,!re are _m su* cases4 ,ev5 add+ ! #e & #g 9 #e_/ab #g_/ab is n q ! same task z add+ #e & #g undi/urb$ by ! fact t !y are twelf?s4 ,we "k f> too little ab\t ! am\nt ( practice ne$$ to adapt ari?metical bonds to effici5t op]ation 9 ^! m complicat$ conditions to e/imate ev5 approximately ! all[ances to be made4 ,b "s all[ance1 & (t5 a r l>ge all[ance1 mu/ be made4 ,! second is ! case :]e ! computation 9 g5]al %\ld be made v easy & sure = ! pupil except = "s "o new elem5t t is be+ le>n$4 ,= example1 9 tea*+ ! mean+ & uses ( ',av]ages' & ( unev5 division1 we may delib]ately use #b1 #c1 & #d z divisors r ?an #g & #i1 s z to let all ! pupil's en]gy be sp5t 9 le>n+ ! new facts1 & s t ! fraction 9 ! quoti5t may be some?+ easily und]/ood1 real1 & significant4 ,9 tea*+ ! addition ( mix$ numb]s1 we may use1 9 ! e>ly /eps1 #aa-a_/b #ac-a_/b #bd ------ r ?an #gi-a_/b #ih-a_/b #fg ------ s z to save att5tion = ! new process itself4 ,9 cancellation1 we may give excess practice to divisions by #b1 #c1 #d1 & #e 9 ord] to make ! transf] to ! new habits ( consid]+ two numb]s toge!r f ! po9t ( view ( _! divisibility by "s numb]4 ,9 introduc+ trade disc\nt1 we may give excess practice on '#e (' & '#aj (' delib]ately1 s t ! mean+ ( disc\nt may n be obscur$ by difficulties 9 ! computation itself4 ,excess practice on1 & ov]le>n+ (1 c]ta9 bonds is ?us v (t5 ju/ifiable4 ,! ?ird case conc]ns bonds ^: importance = practical uses 9 life or z notable facilitators ( o!r bonds is s great t !y may pr(itably be br\ght to a great] /r5g? ?an #aii correct out ( #bjj at a spe$ ( #b sec4 or less1 or be br\ght to t degree ( /r5g? v e>ly4 ,examples ( bonds ( su* special practical use are ! subtractions f #aj1 #a_/b #a_/b1 #a_/b #a_/d1 #a_/b ( #fj1 #a_/d ( #fj1 & ! fractional p>ts ( #ab & ( #a4jj4 ,examples ( notable facilitat+ bonds are t5 #ajs #ajj1 t5 #ajjs #ajjj1 additions l #b #b1 #c #c1 & #d #d1 & all ! multiplication tables to #i x #i4 ,9 consid]ation ( ^! ?ree modify+ cases or pr9ciples1 a volume c\ld well be writt5 conc]n+ j h[ mu* practice to give to ea* bond1 9 ea* ( ! types ( complex situations :]e x has to op]ate4 ,!re is evid5tly ne$ = mu* exp]im5tation to expose ! facts1 & = mu* sagacity & inv5tiv5ess 9 mak+ sure ( effective le>n+ )\t wa/eful ov]le>n+4 ,! facts ( prim>y importance are3-- 7#a7 ,! textbook or o!r in/rum5t ( in/ruction :i* is a tea*]'s g5]al guide may give f> too little practice on c]ta9 bonds4 7#b7 ,x may divide ! practice giv5 9 ways t are app>5tly unju/ifiable4 7#c7 ,! tea*] ne$s !re=e to "k h[ mu* practice x does give1 :]e to supplem5t x1 & :at to omit4 7#d7 ,! omissions1 on gr\nds ( app>5t excess practice1 %\ld be made only aft] c>eful consid]ation ( ! ?ird pr9ciple describ$ above4 7#e7 ,! am\nt ( practice %\ld always be consid]$ 9 ! light ( its int]e/ & appeal to ! pupil's t5d5cy to "w ) full p[] & zeal4 ,m]e repetition ( bonds :5 ! le>n] does n c>e :e!r he is improv+ is r>ely ju/ifiable on any gr\nds4 7#f7 ,practice t is actually 9 excess is n a v grave defect if x is enjoy$ & improves ! pupil's attitude t[>d ari?metic4 ,n mu* "t is lo/2 a hundr$ practices = ea* ( a ?\s& bonds aft] ma/]y to #aii 9 #bjj at #b seconds w use up less ?an #fj h\rs1 or #ae h\rs p] ye> 9 grades #c to #f4 7#g7 ,by ! prop] division ( practice among bonds1 ! arrangem5t ( le>n+ s t ea* bond helps ! o!rs1 ! adroit %ift+ ( practice ( a bond to ea* new type ( situation requir+ x to op]ate "u *ang$ conditions1 & ! elim9ation ( excess practice :]e no?+ sub/antial is ga9$1 notable improvem5ts ov] ! pa/ hit-&-miss cu/oms may be expect$4 7#h7 ,unless ! mat]ial = practice is adequate1 well balanc$ & suffici5tly motivat$1 ! tea*] mu/ keep close acc\nt ( ! le>n+ ( pupils4 ,o!rwise disa/r\s und]le>n+ ( _m bonds is almo/ sure to occur & ret>d ! pupil's developm5t4 ,,! ,,organization ,,( ,,abilities ,!re is dang] t ! ne$ ( brevity & simplicity :i* has made u speak s (t5 ( a bond or an ability1 & ( ! am\nt ( practice x requires1 may mislead ! read] into ?9k+ t ^! bonds & abilities are to be =m$ ea* by itself alone & kept s4 ,!y %\ld r>ely be =m$ s & nev] kept s4 ,?is we h indicat$ f "t to "t by ref]5ces to ! importance ( =m+ a bond 9 ! way 9 :i* x is to be us$1 to ! action ( bonds 9 *ang$ situations1 to facilitation ( "o bond by o!rs1 to ! cooep]ation ( abilities1 & to _! integration into a total ari?metical ability4 ,z a matt] ( fact1 only a small "p ( drill "w 9 ari?metic %\ld be ! =mation ( isolat$ bonds4 ,ev5 ! v "y pupil le>n+ #e & #c are #h %\ld le>n x ) '#e & #e #aj1' '#e & #b #g1' at ! back ( 8 m9d1 s to speak4 ,ev5 s e>ly1 #e #c #h %\ld be "p ( an organiz$1 cooep]at+ sy/em ( bonds4 ,lat] #ej #cj #hj %\ld become alli$ to x4 ,ea* bond %\ld be consid]$1 n simply z a sep>ate tool to be put 9 a comp>tm5t until ne$$1 b also z an improvem5t ( "o total tool or ma*9e1 ari?metical ability4 ,!re are diff]5ces ( c\rse4 ,k ( squ>e root c be reg>d$ some:at z a sep>ate tool to be %>p5$1 poli%$1 & us$ by itself1 :]eas k ( ! multiplication tables _c4 ,yet ev5 squ>e root is probably be/ made m closely a "p ( ! total ability1 be+ taught z a special case ( divid+ :]e divisor is to be ! same z quoti5t1 ! process be+ "o ( e/imat+ & correct+4 ,9 g5]al we d n wi% ! pupil to be a repository ( sep>at$ abilities1 ea* ( :i* may op]ate only if y ask him ! sort ( que/ions :i* ! tea*] us$ to ask him1 or o!rwise indicate to him :i* p>ticul> ari?metical tool he is to use4 ,r he is to be an effective organization ( abilities1 cooep]at+ 9 useful ways to meet ! quantitative problems life (f]s4 ,he %\ld n z a rule h to ?9k 9 su* fa%ion z3 ,is ?is int]e/ or disc\nt8 ,is x simple int]e/ or comp\nd int]e/8 ,:at did ,i d 9 comp\nd int]e/8 ,h[ d ,i multiply by #b p]c5t8 ,! situation t calls up int]e/ %\ld also call up ! k9d ( int]e/ t is appropriate1 & ! te*nique ( op]at+ ) p]c5ts %\ld be s weld$ toge!r ) int]e/ 9 8 m9d t ! "r cooep]ation w occur almo/ )\t sup]vision by him4 ,z ea* new ability is acquir$1 !n1 we seek to h x take its place z an improvem5t ( a ?9k+ be+1 z a cooep]ative memb] ( a total organization1 z a soldi] fight+ toge!r ) o!rs1 z an elem5t 9 an educat$ p]sonality4 ,su* an organization ( bonds w n =m itself any m ?an any "o bond w create itself4 ,if ! elem5ts ( ari?metical ability are to act toge!r z a total organiz$ unifi$ =ce !y mu/ be made to act toge!r 9 ! c\rse ( le>n+4 ,:at we wi% to h "w toge!r we mu/ put toge!r & give practice 9 teamwork4 ,we c d mu* to secure su* cooep]ative action :5 & :]e & z x is ne$$ by a v simple exp$i5t2 namely1 to give practice ) computation & problems su* z life provides1 in/ead ( mak+ up drills & problems m]ely to apply ea* fact or pr9ciple by itself4 ,?\gh a pupil has solv$ scores ( problems read+1 ,a triangle has a base ( a feet & an altitude ( b feet1 :at is its area8 he may /ill be practically helpless 9 f9d+ ! area ( a triangul> plot ( gr\nd2 /ill m helpless 9 us+ ! =mula = a triangle :i* is "o ( two into :i* a trapezoid is divid$4 ,?\gh a pupil has le>n$ to solve problems 9 trade disc\nt1 simple int]e/1 comp\nd int]e/1 & bank disc\nt "o at a "t1 /at$ 9 a few set =ms1 he may be practically helpless be=e ! actual s]ies ( problems confront+ him 9 />t+ 9 bus9ess1 & may take money out ( ! sav+s bank :5 he ought to borr[ on a "t loan1 or delay paym5t on 8 bills :5 by pay+ ca% he c\ld save money z well z improve 8 /&+ ) ! :olesal]4 ,in/ead ( mak+ up problems to fit ! abilities giv5 by s*ool in/ruction1 we %\ld pref]ably modify s*ool in/ruction s t ari?metical abilities w be organiz$ into an effective total ability to meet ! problems t life w (f]4 ,/ill m g5]ally1 e bond =m$ %\ld be =m$ ) due consid]ation ( e o!r bond t has be5 or w be =m$2 e ability %\ld be practic$ 9 ! mo/ effective possible relations ) o!r abilities4 ,,*apt] ,,vii ,,! ,,sequ5ce ,,( ,,topics3 ,,! ,,ord] ,,( ,,=mation ,,( ,,bonds ,! bonds to be =m$ hav+ be5 *os51 ! next /ep is to arrange = _! mo/ economical ord] ( =mation--to arrange to h ea* help ! o!rs z mu* z possible--to arrange = ! maximum ( facilitation & ! m9imum ( inhibition4 ,! pr9ciple is obvi\s en\gh & w\ld probably be admitt$ 9 !ory by any intellig5t tea*]1 b 9 practice we are /ill w$d$ to conv5tional usages :i* arose long be=e ! psy*ology ( ari?metic 0 /udi$4 ,= example1 we inh]it ! conv5tion ( /udy+ addition ( integ]s ?or\ghly1 & !n subtraction1 & !n multiplication1 & !n division1 & _m ( u foll[ x ?\gh nobody has ev] giv5 a pro( t ?is is ! be/ ord] = ari?metical le>n+4 ,we inh]it also ! opposite conv5tion ( /udy+ 9 a s-call$ spiral plan1 a little addition1 subtraction1 multiplication1 & division1 & !n "s m ( ea*1 & !n "s m1 & _m ( u foll[ ?is cu/om1 ) an unreason$ fai? t *ang+ ab\t f "o process to ano!r is p] se helpful4 ,su* conv5tions are v /rong1 illu/rat+ our common t5d5cy to *]i% mo/ ^? cu/oms :i* we _c ju/ify6 ,! r$uctions ( d5om9ate numb]s asc5d+ & desc5d+ 71 until rec5tly1 9 mo/ c\rses ( /udy1 kept until grade #d or grade #e 0 rea*$1 al?\gh ?is mat]ial is ( f> great] value = drills on ! multiplication & division tables ?an ! cu/om>y problems ab\t apples1 eggs1 oranges1 tablets1 & p5hold]s4 ,by "s hi/orical accid5t or = good reasons ! g5]al treatm5t ( d5om9ate numb]s 0 put late2 by our naive notions ( ord] & sy/em we felt t any use ( d5om9ate numb]s be=e ?is "t 0 h]etical2 we ?us became bl9d to ! advantages ( qu>ts & p9ts = ! tables ( #bs2 y>ds & feet = ! tables ( #cs2 gallons & qu>ts = ! tables ( #ds2 nickels & c5ts = ! #es2 weeks & days = ! #gs2 pecks & qu>ts = ! #hs2 & squ>e y>ds & squ>e feet = ! #is4 ,problems l #e y>ds feet or #ae feet y>ds h n only ! advantages ( brevity1 cle>ness1 practical use1 real ref]5ce1 & ready v>iation1 b also ! v great advantage t "p ( ! data h to be ?\ght ( 9 a useful way in/ead ( read (f f ! page4 ,9 life1 :5 a p]son has tw5ty c5ts ) :i* to buy tablets ( a c]ta9 sort1 he ?9ks ( ! price 9 mak+ 8 pur*ase1 ask+ x ( ! cl]k only 9 case he does n "k x1 & 9 plann+ 8 pur*ases be=eh& he ?9ks ( prices z a rule4 ,9 spite ( ^! & o!r advantages1 n "o textbook 9 t5 up to #aijj made e>ly use ( ^! ex]cises ) d5om9ate numb]s4 ,s /rong is m]e use & wont4 ,besides ^! conv5tional cu/oms1 !re has be51 9 ^? responsible = ari?metical in/ruction1 an admiration = an arrangem5t ( topics t is easy = a p]son1 aft] he kn[s ! subject1 to use 9 ?9k+ ( its con/itu5t p>ts & _! relations4 ,su* arrangem5ts are (t5 call$ 'logical' arrangem5ts ( subject matt]1 ?\gh !y are (t5 f> f logical 9 any useful s5se4 ,n[ ! easie/ ord] 9 :i* to ?9k ( a hi]>*y ( habits aft] y h =m$ !m all may be an extremely difficult ord] 9 :i* to =m !m4 ,! criticism ( o!r ord]s z 'scrappy1' or 'unsy/ematic1' valid en\gh if ! c\rse ( /udy is ?\ght ( z an object ( contemplation1 may be fooli% if ! c\rse ( /udy is reg>d$ z a work+ in/rum5t = fur!r+ ari?metical le>n+4 ,we mu/ rememb] t all our sy/ematiz+ & label+ is l>gely )\t mean+ to ! pupils4 ,!y _c at any po9t appreciate ! sy/em z a progression f t po9t t[>d ?is & t1 s9ce !y h no k ( ! '?is or t4' ,!y d n z a rule ?9k ( _! "w 9 grade #d z an outcome ( _! "w 9 grade #c ) ext5sions ( a to a#a1 & additions ( b#b & b#c to b & b#a1 & ref9em5ts ( c & d by c#d & d#e4 ,!y c\ld give only ! vague/ acc\nt ( :at !y did 9 grade #c1 mu* less ( :y x %\ld h be5 done !n4 ,!y are n mu* di/urb$ by a lack ( s-call$ 'sy/em' & 'logical' progression = ! same reason t !y are n mu* help$ by _! pres5ce4 ,:at !y ne$ & c use is a dynamically effective sy/em or ord]1 "o t !y c le>n easily & reta9 long by1 reg>dless ( h[ x w\ld look 9 a museum ( ari?metical sy/ems4 ,unless _! actual ari?metical habits are usefully relat$ x does no good to see ! s-call$ logical relations2 & if _! habits are usefully relat$1 x does n v mu* matt] :e!r or n !y d see ^!2 f9ally1 !y c be br\ght to see !m be/ by fir/ acquir+ ! "r habits 9 a dynamically effective ord]4 ,,decreas+ ,,int]f]5ce ,,& ,,increas+ ,,facilitation ,psy*ology (f]s no s+le1 easy1 royal road to discov]+ ?is dynamically be/ ord]4 ,x c only survey ! bonds1 ?9k :at ea* dem&s z pr]equisite & (f]s z future help1 recomm5d c]ta9 ord]s = trial1 & measure ! effici5cy ( ea* ord] z a means ( atta9+ ! ends desir$4 ,! +5i\s ?\ght & c>eful exp]im5tation ( _m able work]s w be requir$ = _m ye>s to come4 ,psy*ology c1 h[ev]1 ev5 n[1 give solid con/ructive help 9 _m in/ances1 ei!r by recomm5d+ ord]s t seem almo/ c]ta9ly bett] ?an ^? 9 vogue1 or by propos+ ord]s = trial :i* c be ju/ifi$ or reject$ by crucial te/s4 ,consid]1 = example1 ! situation1 'a column ( "o-place numb]s to be add$1 ^: sum is ov] #i1' & ! response 'writ+ d[n ! sum4' ,?is bond is commonly firmly fix$ be=e addition ) two-place numb]s is und]tak54 ,z a result ! pupil has fix$ a habit t he has to break :5 he le>ns two-place addition4 ,if oral answ]s only are giv5 ) su* s+le columns until two-place addition is well "u way1 ! int]f]5ce is avoid$4 ,9 _m c\rses ( /udy ! ord] ( sy/ematic =mation ( ! 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 ,?is is probably wrong 9 two respects4 ,!re is abundant reason to believe t ! x #es %\ld be le>n$ fir/1 s9ce !y are easi] to le>n ?an ! #as or ! #bs1 & give ! idea ( multiply+ m emphatically & cle>ly4 ,!re is also abundant reason to believe t ! #a x #e1 #a x #b1 #a x #c1 etc41 %\ld be put v late--aft] at lea/ ?ree or f\r tables are le>n$1 s9ce ! "q ,:at is #a times #b8 7or #c or #e7 is unnecess>y until we come to multiplication ( two- & ?ree-place numb]s1 seems a fooli% "q until !n1 & obscures ! notion ( multiplication if put e>ly4 ,also ! facts are be/ le>n$ once = all z ! habits #a times k is ! same z k1 & k times #a is ! same z k4#h #h ,! v e>ly le>n+ ( #b x #b1 #b x #c1 #c x #b1 #b x #d1 #d x #b1 #c x #c1 & p]haps a few m multiplications is n consid]$ "h4 ,x is advisable4 ,! treatm5t ( #j x #j1 #j x #a1 #a x #j1 etc41 is n consid]$ "h4 ,x is probably be/ to def] ! 'x #j' bonds until aft] all ! o!rs are =m$ & are be+ us$ 9 %ort multiplication1 & to =m !m 9 close connection ) _! use 9 %ort multiplication4 ,! '#j x' bonds may well be def]r$ until !y are ne$$ 9 'long' multiplication1 #j x #j com+ la/ ( all4 ,9 ano!r connection x 0 recomm5d$ t ! divisions to #ha _/ #i be le>n$ by selective ?9k+ or reason+ f ! multiplications4 ,?is det]m9es ! ord] ( bonds s f> z to place ! =mation ( ! division bonds soon aft] ! le>n+ ( ! multiplications4 ,= o!r reasons x is well to make ! proximity close4 ,"o ( ! arbitr>y sy/ematizations ( ! ord] ( =mation ( bonds re/ricts op]ations at fir/ to ! numb]s #a to #aj1 !n to numb]s "u #ajj1 !n to numb]s "u #ajjj1 !n to numb]s "u #aj1jjj4 ,ap>t f ! avoidance ( unreal & p$antic problems 9 appli$ ari?metic to :i* "w ) l>ge numb]s 9 l[ grades does some:at pr$ispose a tea*]1 !re is little m]it 9 ?is re/riction ( ! ord] ( =mation ( bonds4 ,its dem]its are _m4 ,= example1 :5 ! pupil is le>n+ to 'c>ry' 9 addition he c be giv5 bett] practice by soon includ+ tasks ) sums above #ajj1 & c get a valuable s5se ( ! g5]al use ( ! process by be+ giv5 a few examples ) ?ree- & f\r-place numb]s to be add$4 ,! same holds = subtraction4 ,inde$1 !re is some?+ to be said 9 favor ( us+ six- or sev5-place numb]s 9 subtraction1 en=c+ ! 'borr[+' process by hav+ x done aga9 & aga9 9 ! same example1 & putt+ x "u control by hav+ ! decision betwe5 'borr[+' & 'n borr[+' made aga9 & aga9 9 ! same example4 ,:5 ! multiplication tables are le>n$ ! mo/ important use = !m is n 9 t$i\s reviews or trivial problems ) answ]s "u #ajj1 b 9 regul> '%ort' multiplication ( two- & ?ree- & ev5 f\r-place numb]s4 ,j z ! addition comb9ations function ma9ly 9 ! high]-decade modifications ( !m1 s ! multiplication comb9ations function *iefly 9 ! cases :]e ! bond has to op]ate :ile ! add$ tasks ( keep+ "o's place1 add+ :at has be5 c>ri$1 writ+ d[n ! "r figure 9 ! "r place1 & hold+ ! "r numb] = lat] addition1 are also tak5 c>e (4 ,x seems be/ to introduce su* %ort multiplication z soon z ! x #es1 x #bs1 x #cs1 & x #ds are le>n$ & to put ! x #fs1 x #gs1 & ! re/ to "w 9 su* %ort multiplication z soon z ea* is le>n$4 ,/ill sur] is ! ne$ = f\r-1 five-1 & six-place numb]s :5 two-place numb]s are us$ 9 multiply+4 ,:5 ! process ) a two-place multipli] is le>n$1 multiplications by ?ree-place numb]s %\ld soon foll[4 ,!y are n m difficult !n ?an lat]4 ,on ! contr>y1 if ! pupil gets us$ to multiply+ only z "o does ) two-place multipli]s1 he w suff] m by ! result+ int]f]5ce ?an he does f gett+ six- or sev5-place answ]s ^: mean+ he _c exactly realize4 ,!y tea* ! rationale & ! manipulations ( long multiplication ) especial economy because ! pr9ciples & ! proc$ures are us$ two or ?ree times ov] & ! contra/s betwe5 ! values :i* ! p>tial products h 9 add+ become ?ree in/ead ( "o4 ,! entire matt] ( long multiplication ) integ]s & ,unit$ ,/ates money %\ld be treat$ z a tea*+ unit & ! bonds =m$ 9 close organization1 ev5 ?\gh numb]s z l>ge z #ijj1jjj are occasionally involv$4 ,! reason is n t x is m logical1 or less scrappy1 b t ea* ( ! bonds 9 "q ?us gets mu* help f1 & gives mu* help to1 ! o!rs4 ,9 %>p contra/ to a topic l 'long multiplication' /&s a topic l d5om9ate numb]s4 ,x mo/ c]ta9ly %\ld n be treat$ z a l>ge tea*+ unit1 & all ! bonds involv$ 9 add+1 subtract+1 multiply+1 & divid+ ) all ! ord9>y sorts ( measures %\ld c]ta9ly n be =m$ 9 close sequ5ce4 ,! r$uctions asc5d+ & desc5d+ = _m ( ! measures %\ld be taught z drills on ! appropriate multiplication & division tables4 ,! r$uction ( feet & in*es to in*es1 y>ds & feet to y>ds1 gallons & qu>ts to qu>ts1 & ! l are admirable ex]cises 9 connection ) ! 7a x b7 c 4444 problems1--! ',b\ght #c lbs4 ( sug> at #g c5ts & #e c5ts wor? ( mat*es' problems4 ,! r$uctions ( in*es to feet & in*es & ! l are admirable ex]cises 9 ! d 74444 x b7 c or 'mak+ *ange' problem1 :i* 9 its small-numb] =ms is an excell5t prep>atory /ep = %ort division4 ,!y are also ( great s]vice 9 e>ly "w ) fractions4 ,! feet-mile1 squ>e-foot-squ>e-in*1 & o!r simple relations give a g5u9e & intelligible dem& = multiplication ) l>ge numb]s4 ,k ( ! metric sy/em = l9e> & squ>e measure w\ld p]haps1 z an introduction to decimal fractions1 m ?an save ! "t sp5t to le>n x4 ,x w\ld ev5 p]haps be wor? :ile to inv5t a measure 7call x ! twoqua7 midway betwe5 ! qu>t & gallon & tea* c>ry+ 9 addition & borr[+ 9 subtraction by tea*+ fir/ ! addition & subtraction ( 'gallon1 twoqua1 qu>t1 & p9t' s]ies6 ,_m ( ! bonds :i* a sy/em-made tradition huddl$ toge!r uselessly 9 a *apt] on d5om9ate numb]s %\ld ?us be =m$ z helpful prep>ations = & applications ( o!r bonds all ! way f ! fir/ to ! eigh? half-ye> ( in/ruction 9 ari?metic4 ,! bonds involv$ 9 ! ability to respond correctly to ! s]ies3-- #e 4444 #bs & 4444 rema9d] #e 4444 #cs & 4444 rema9d] #hh 4444 #is & 4444 rema9d] %\ld be =m$ be=e1 n dur+1 ! tra9+ 9 %ort division4 ,!y are admirable at t po9t z practice on ! division tables2 are ( practical s]vice 9 ! mak+-*ange problems ( ! small pur*ase & ! l2 & simplify ! o!rwise intricate task ( keep+ "o's place1 *oos+ ! quoti5t figure1 multiply+ by x1 subtract+ & hold+ 9 m9d ! new numb] to be divid$1 :i* is compos$ half ( ! rema9d] & half ( a figure 9 ! writt5 divid5d4 ,?is *ange ( ord] is a good illu/ration ( ! ne>ly g5]al rule t ,:5 ! practice or review requir$ to p]fect or hold c]ta9 bonds c1 by an inexp5sive modification1 be turn$ into a useful prep>ation = new bonds1 t modification %\ld be made4 ,! bonds involv$ 9 ! f\r op]ations ) ,unit$ ,/ates money %\ld be =m$ 9 grades #c & #d along ) or v soon aft] ! correspond+ bonds ) ?ree-place & f\r-place integ]s4 ,?is /atem5t w\ld h seem$ prepo/]\s to ! p$agogues ( fifty ye>s ago4 ,unit$ ,/ates money1 !y w\ld h said1 is an application ( decimals4 ,h[ c x be le>n$ until ! ess5tials ( decimal fractions are kn[n8 ,h[ w ! *ild und]/& :5 multiply+ 4#ge by #c t #c times #e c5ts is #a dime & #e c5ts1 or t #c times #gj c5ts is #b doll>s & #a dime8 ,:y p]plex ! "y pupils ) ! difficulties ( plac+ ! decimal po9t8 ,:y di/urb ! le>n+ ( ! f\r op]ations ) integ]s by add+ at ea* /ep a second 'proc$ure ) ,unit$ ,/ates money'8 ,! case illu/rates v well ! error ( ! old] ov]sy/ematic treatm5t ( ! ord] ( topics & ! /ill m important error ( confus+ ! logic ( pro( ) ! psy*ology ( le>n+4 ,to prove t #c x 4#ge #b4be to ! satisfaction ( c]ta9 ari?meticians1 y may ne$ to "k ! !ory ( decimal fractions2 b to d su* multiplication all a *ild ne$s is to d j :at he has be5 do+ ) integ]s & !n ,put a be=e ! answ] to %[ t x means doll>s & c5ts1 & put a decimal po9t 9 ! answ] to %[ :i* figures mean doll>s & :i* figures mean c5ts4 ,& ?is is g5]al4 ,! ability to op]ate ) integ]s plus ! two habits ( prefix+ & sep>at+ doll>s f c5ts 9 ! result w enable him to op]ate ) ,unit$ ,/ates money4 ,consequ5tly good practice came to use ,unit$ ,/ates money n z a consequ5ce ( decimal fractions1 le>n$ by _! aid1 b z an introduction to decimal fractions :i* aids ! pupil to le>n !m4 ,s x has gradually pu%$ "w ) ,unit$ ,/ates money fur!r & fur!r back1 ?\gh some:at timidly4 ,we ne$ n be timid4 ,! pupil w h no difficulty 9 add+1 subtract+1 multiply+1 & divid+ ) ,unit$ ,/ates money--unless we create x by our explanations6 ,if we simply =m ! two bonds describ$ above & %[ by prop] v]ification t ! proc$ure always gives ! "r answ]1 ! e>ly tea*+ ( ! f\r op]ations ) ,unit$ ,/ates money w 9 fact actually %[ a le>n+ pr(it6 ,x w save m "t 9 ! "w ) integ]s ?an 0 sp5t 9 tea*+ x6 ,=1 9 ! fir/ place1 x w help to make "w ) f\r-place & five-place numb]s m intelligible & vital4 ,a pupil c und]/& #af4ge or #bh4gi m easily ?an #afge or #bhgi4 ,! =m] may be ! prices ( a suit or sew+ ma*9e or bicycle4 ,9 ! second place1 x p]mits ! use ( a l>ge /ock ( g5u9e problems ab\t sp5d+1 sav+1 %>+1 & ! l ) adv]tisem5ts & catalogues & s*ool ent]prises4 ,9 ! ?ird place1 x p]mits ! use ( common-s5se *ecks4 ,a boy may f9d "o f\r? ( #cjjj z #gjej or #ge & n be di/urb$1 b he w mu* m easily realize t "o f\r? ( #cj4jj is n ov] #gj or less ?an #a4 ,ev5 ! decimal po9t ( :i* we us$ to be s afraid may actually help ! eye to keep its place 9 add+4 ,,int]e/ ,s f>1 ! illu/rations ( improvem5ts 9 ! ord] ( bonds s z to get less int]f]5ce & m facilitation ?an ! cu/om>y ord]s secure h s\ght *iefly to improve ! me*anical organization ( ! bonds4 ,any ga9 9 int]e/ :i* ! *anges describ$ effect$ w\ld be l>gely due to ! great] a*ievem5t itself4 ,dewey & o!rs h emphasiz$ a v diff]5t pr9ciple ( improv+ ! ord] ( =mation ( bonds--! pr9ciple ( det]m9ation ( ! bonds to be =m$ by "s vital1 engag+ problem :i* ar\ses int]e/ en\gh to light5 ! labor & :i* goes beyond or ev5 aga9/ cut-&-dri$ plans = sequ5ces 9 ord] to get effective problems4 ,= example1 ! "w ( ! fir/ mon? 9 grade #b;,b might sacrifice facilitations ( ! me*anical sort 9 ord] to put ari?metic to use 9 decid+ :at dim5sions a rabbit's cage %\ld h to give him #ab squ>e feet ( floor space1 h[ mu* bread he %\ld h p] meal to get #f ounces a day1 h[ long a t5-c5t loaf w\ld la/1 h[ _m loaves %\ld be b\ght p] week1 h[ mu* x co/s to fe$ ! rabbit1 h[ mu* he has ga9$ 9 weight s9ce he 0 br\ght to ! s*ool1 & s on4 ,su* sacrifices ( ! optimal ord] if int]e/ 7 equal1 9 ord] to get great] int]e/ or a heal?i] int]e/1 are ju/ifiable4 ,vital problems z nuclei ar\nd :i* to organize ari?metical le>n+ are ( prime importance4 ,x is ev5 safe probably to insi/ t "s g5u9e problem-situation requir+ a new process1 su* z addition ) c>ry+1 multiplication by two-place numb]s1 or division ) decimals1 be provid$ 9 e case z a "p ( ! introduction to t process4 ,! sacrifice %\ld n be too great1 h[ev]2 ! se>* = vital problems t fit an economical ord] ( subject matt] is z mu* ne$$ z ! am5dm5t ( t ord] to fit kn[n int]e/s2 & ! assurance t a problem helps ! pupil to le>n ari?metic is z important z ! assurance t ari?metic is us$ to help ! pupil solve 8 p]sonal problems4 ,mu* +5uity & exp]im5tation w be requir$ to f9d ! ord] t is satisfactory 9 bo? quality & quantity ( int]e/ or motive & helpfulness ( ! bonds "o to ano!r4 ,! difficulty ( organiz+ ari?metic ar\nd attractive problems is mu* increas$ by ! fact ( class in/ruction4 ,= any "o pupil vital1 p]sonal problems or projects c\ld be f\nd to provide = _m ari?metical abilities2 & any necess>y k & te*nique :i* ^! projects did n develop c\ld be someh[ fitt$ 9 along ) !m4 ,b ?irty *ildr51 half boys & half girls1 v>y+ by five ye>s 9 age1 com+ f diff]5t homes1 ) diff]5t native capacities1 w n1 9 ,septemb]1 #aibj1 unanim\sly feel a vital ne$ to solve any "o problem1 & !n conv5i5tly feel ano!r on1 say1 ,octob] #ae6 ,9 ! me*anical laws ( le>n+ *ildr5 are mu* alike1 & ! ga9 we may hope to make f r$uc+ inhibitions & increas+ facilitations is1 = ord9>y class-tea*+1 probably great] ?an t to be made f ! discov]y ( attractive c5tral problems4 ,we %\ld1 h[ev]1 get z mu* z possible ( bo?4 ,,g5]al ,,pr9ciples ,! read] may by n[ feel r helpless be=e ! problem ( ! arrangem5t ( ari?metical subject matt]4 ,sometimes y complete a topic1 sometimes y take x piecemeal mon?s or ye>s ap>t1 (t5 y make que] twi/s & %ifts to get a /rategic advantage ov] ! enemy1 he may ?9k1 b are !re no guid+ pr9ciples1 no g5]al rules8 ,!re is only "o t is absolutely g5]al1 to take ! ord] t works be/ = ari?metical le>n+4 ,!re are p>ticul> rules1 b !re are s _m & !y are s limit$ by an 'o!r ?+s be+ equal' clause1 t probably a g5]al eag]ness to ?9k out ! pros & cons = any giv5 proposal is bett] ?an a /iff attempt to adh]e to ^! rules4 ,i w /ate & illu/rate "s ( !m1 & let ! read] judge4 ,o!r ?+s be+ equal1 "o new sort ( bonds %\ld n be />t$ until ! previ\s set is fairly e/abli%$1 & two diff]5t sets %\ld n be />t$ at once4 ,?us1 multiplication ( two- & ?ree-place numb]s by #b1 #c1 #d1 & #e w fir/ use numb]s su* t no c>ry+ is requir$1 & no z]o difficulties are enc\nt]$1 !n introduce c>ry+1 !n introduce multiplic&s l #bjf & #cbj4 ,if o!r ?+s 7 equal1 ! c>ry+ w\ld be split into two /eps--fir/ drills ) 7#d x #f7 #b1 7#c x #g7 #c1 7#e x #d7 #a1 & ! l1 & second ! actual use ( ^! habits 9 ! multiplication4 ,! objection to ?is sep>ation ( ! d\ble habit is t ! fir/ "p ( x 9 isolation is too artificial--t x may be bett] to suff] ! extra difficulty ( =m+ ! two toge!r ?an to tea* s r>ely us$ habits z ! 7a x b7 c s]ies4 ,exp]im5tal te/s are ne$$ to decide ?is po9t4 ,o!r ?+s be+ equal1 bonds %\ld be =m$ 9 su* ord] t none w h to be brok5 lat]4 ,= example1 !re is a /rong argum5t = tea*+ long division fir/1 or v e>ly1 ) rema9d]s1 lett+ ! case ( z]o rema9d] come 9 z "o ( _m4 ,if ! pupils h be5 famili>iz$ ) ! rema9d] notion by ! drills recomm5d$ z prep>ation = %ort division1#i ! use ( rema9d]s 9 long division w (f] little difficulty4 ,! exclusive use ( examples )\t rema9d]s may =m ! habit ( n be+ exact 9 computation1 ( tru/+ to 'com+ out ev5' z a sole *eck1 & ev5 ( writ+ d[n a numb] to fit ! f9al numb] to be divid$ in/ead ( obta9+ x by hone/ multiplication4 #i ,see page #gf4 ,= simil> reasons additions ) #b & #c z well z #a to be 'c>ri$' h mu* to recomm5d !m 9 ! v fir/ /ages ( column addition ) c>ry+4 ,!re is "h ! add$ advantage t a pupil w be m likely to rememb] to c>ry if he has to ?9k :at to c>ry4 ,! pres5t common practice ( us+ small numb]s = ease 9 ! addition itself tea*es _m *ildr5 to ?9k ( c>ry+ z add+ "o4 ,o!r ?+s be+ equal1 arrange to h v>iety4 ,?us x is probably1 ?\gh n surely1 wise to int]rupt ! monotony ( le>n+ ! multiplication & division tables1 by tea*+ ! fundam5tals ( '%ort' multiplication & p]haps ( division aft] ! #es1 #bs1 #cs1 & #ds are le>n$4 ,?is makes a break ( sev]al weeks4 ,! facts = ! #fs1 #gs1 #hs1 & #is c !n be put to v>i$ use z fa/ z le>n$4 ,x is almo/ c]ta9ly wise to int]rupt ! fir/ half-ye>'s "w ) addition & subtraction1 by tea*+ #b x #b1 #b x #c1 #c x #b1 #b x #d1 #d x #b1 #b x #e1 lat] by #b x #aj1 #c x #aj1 #d x #aj1 #e x #aj1 lat] by #a_/b #a_/b1 #a-a_/b #a_/b1 #a_/b ( #b1 #a_/b ( #d1 #a_/b ( #f1 & at "s "t by c]ta9 pr(itable ex]cises :]e9 a pupil tells all he kn[s ab\t c]ta9 numb]s :i* may be made nuclei ( important facts 7say1 #e1 #h1 #aj1 #ab1 #ae1 & #bj74 ,o!r ?+s be+ equal1 use objective aids to v]ify an ari?metical process or inf]5ce aft] x is made1 z well z to provoke x4 ,x is well at times to let pupils d ev]y?+ t !y c ) relations ab/ractly conceiv$1 te/+ _! results by objective c\nt+1 measur+1 add+1 & ! l4 ,= example1 an e>ly /ep 9 add+ %\ld be to %[ ?ree ?+s1 put !m "u a book1 %[ two m1 put ^! "u ! book1 & !n ask h[ _m !re are "u ! book1 lett+ ! objective c\nt+ come lat] z ! te/ ( ! correctness ( ! addition4 ,o!r ?+s be+ equal1 res]ve all explanations ( :y a process mu/ be "r until ! pupils c use ! process accurately1 & h v]ifi$ ! fact t x is "r4 ,except = ! v gift$ pupils1 ! ord9>y prelim9>y d$uctive explanations ( :at mu/ be done are probably useless z means ( tea*+ ! pupils :at to d4 ,!y use up mu* "t & are ( s little p]man5t effect t1 z we h se51 ! v ari?meticians :o advocate mak+ !m1 admit t aft] a pupil has ma/]$ ! process he may be all[$ to =get ! reasons = x4 ,i am n sure t ! d$uctive pro(s ( :y we place ! decimal po9t z we d 9 division by a decimal1 or inv]t & multiply 9 divid+ by a fraction1 & ! l1 are wor? tea*+ at all4 ,if !y are to be taught at all1 ! "t to tea* !m is 7except = ! v gift$7 aft] ! pupil has ma/]$ ! process & has confid5ce 9 x4 ,he !n at lea/ kn[s :at process he is to prove is "r1 & t x is "r1 & has _h "s *ance ( see+ :y x is "r f 8 exp]i5ce ) x4 ,"o m pr9ciple may be m5tion$ )\t illu/ration4 ,arrange ! ord] ( bonds ) due reg>d = ! aims ( ! o!r /udies ( ! curriculum & ! practical ne$s ( ! pupil outside ( s*ool4 ,ari?metic is n a book or a clos$ sy/em ( ex]cises4 ,x is ! quantitative "w ( ! pupils 9 ! elem5t>y s*ool4 ,no n>r[] view ( x is adequate4 ,,*apt] ,,viii ,,! ,,di/ribution ,,( ,,practice ,,! ,,problem ,! same am\nt ( practice may be di/ribut$ 9 v>i\s ways4 ,figures #g to #aj1 = example1 %[ #bjj practices ) division by a fraction di/ribut$ ov] ?ree & a half ye>s ( #aj mon?s 9 f\r diff]5t ways4 ,9 ,fig4 #g1 practice is some:at equally di/ribut$ ov] ! :ole p]iod4 ,9 ,fig4 #h ! practice is di/ribut$ at haphaz>d4 ,9 ,fig4 #i !re is a fir/ ma9 le>n+ p]iod1 a review aft] ab\t t5 weeks1 a review at ! beg9n+ ( ! sev5? grade1 ano!r review at ! beg9n+ ( ! eigh? grade1 & "s casual practice r at r&om4 ,9 ,fig4 #aj !re is a ma9 le>n+ p]iod1 ) reviews dim9i%+ 9 l5g? & sep>at$ by wid] & wid] int]vals1 ) occasional practice !reaft] to keep ! ability alive & heal?y4 ,plans ,i & ,,ii are obvi\sly inf]ior to ,plans ,,iii & ,,iv2 & ,plan ,,iv gives promise ( be+ m effective ?an ,plan ,,iii1 s9ce !re seems dang] t ! pupil work+ by ,plan ,,iii might 9 ! t5 weeks lose too mu* ( :at he _h ga9$ 9 ! initial practice1 & s aga9 9 ! next t5 weeks4 ,x is n wise1 h[ev]1 to try n[ to make close decisions 9 ! case ( practice ) division by a fraction2 or to det]m9e :at ! be/ di/ribution ( practice is = t or any o!r ability to be improv$4 ,! facts ( psy*ology are z yet n adequate = v close decisions1 nor are ! types ( di/ribution ( practice t are be/ adapt$ to diff]5t abilities ev5 approximately work$ out4 ,illu/ration3 ,,fig4 #g4--,plan ,i4 #bjj practices di/ribut$ some:at ev5ly ov] #c-a_/b ye>s ( #aj mon?s4 ,9 ,figs4 #g1 #h1 #i1 & #aj1 ea* t5? ( an in* along ! base l9e repres5ts "o mon?4 ,ea* hundr$? ( a squ>e in* repres5ts f\r practices1 a little squ>e #a_/bj ( an in* wide & #a_/bj in* high repres5t+ "o practice4 ,illu/ration3 ,,fig4 #h4--,plan ,,ii4 #bjj practices di/ribut$ haphaz>d ov] #c-a_/b ye>s ( #aj mon?s4 ,illu/ration3 ,,fig4 #i4--,plan ,,iii4 ,a le>n+ p]iod1 ?ree reviews1 & incid5tal practice4 ,illu/ration3 ,,fig4 #aj4--,plan ,,iv4 ,a le>n+ p]iod ) reviews ( decreas+ l5g? at increas+ int]vals4 ,,sample ,,di/ributions ,let u r exam9e "s actual cases ( di/ribution ( practice f\nd 9 s*ool "w & consid]1 n ! atta9m5t ( ! be/ possible di/ribution1 b simply ! avoidance ( gross blund]s & ! atta9m5t ( reasonable1 def5sible proc$ures 9 ?is reg>d4 ,figures #aa to #ah %[ ! di/ribution ( examples 9 multiplication ) multipli]s ( v>i\s sorts4 ,x /&s = any digit except z]o4 ,o /&s = #j4 ,,xxo ?us means a multipli] l #cej or #bgj or #afj2 ,,xox means multipli]s l #djg1 #ije1 or #bjf2 ,,xx means multipli]s l #be1 #ag1 #ch4 ,ea* ( ^! diagrams cov]s approximately #c-a_/b ye>s ( s*ool "w1 or f ab\t ! middle ( grade #c to ! end ( grade #f4 ,!y are made f c\nts ( f\r textbooks 7,a1 ,b1 ,c1 & ,d71 ! c\nt be+ tak5 = ea* successive #h pages4#aj ,ea* t5? ( an in* along ! base l9e equals #h pages ( ! text 9 "q4 ,ea* 4#ja sq4 94 equals "o example4 ,! books1 x w be obs]v$1 diff] 9 ! am\nt ( practice giv51 z well z 9 ! way 9 :i* x is di/ribut$4 #aj ,at ! end ( a volume or "p1 ! c\nt may be f z few z #e or z _m z #ab pages4 ,^! di/ributions are wor?y ( c>eful /udy2 we %all note only a few sali5t facts ab\t !m "h4 ,( ! di/ributions ( multiplications ) multipli]s ( ! ,,xx type1 t ( book ,d 7,fig4 #ad7 is p]haps ! be/4 ,a 7,fig4 #aa7 has too mu* ( ! practice too late2 ,b 7,fig4 #ab7 gives too little practice 9 ! fir/ le>n+2 ,c 7,fig4 #ac7 gives too mu* 9 ! fir/ le>n+ & 9 grade #f4 ,among ! di/ributions ( multiplication ) multipli]s ( ! ,,xox type1 t ( book ,d 7,fig4 #ah7 is aga9 probably ! be/4 ,a1 ,b1 & ,c 7,figs4 #ae1 #af1 & #ag7 h too mu* practice e>ly & too long int]vals betwe5 reviews4 ,book ,c 7,fig4 #ag7 by a c>eless ov]sight has "o case ( ?is v difficult process1 )\t any explanation1 weeks be=e ! process is taught6 ,illu/ration3 ,,fig4 #aa4--;,di/ribution ( practise ) multipli]s ( ! ,,xx type 9 ! fir/ two books ( ! ?ree-book text ,a4 ,illu/ration3 ,,fig4 #ab4--,same z ,fig4 #aa1 b = text ,b4 ,foll[+ ?is p]iod come c]ta9 pages ( computation to be us$ by ! tea*] at h] discretion1 conta9+ #bd ,,xx multiplications4 ,illu/ration3 ,,fig4 #ac4--,same z ,fig4 #aa1 b = text ,c4 ,illu/ration3 ,,fig4 #ad4--,same z ,fig4 #aa1 b = text ,d4 ,illu/ration3 ,,fig4 #ae4--;,di/ribution ( practice ) multipli]s ( ! ,,xox type 9 ! fir/ two books ( ! ?ree-book text ,a4 ,illu/ration3 ,,fig4 #af4--,same z ,fig4 #ae1 b = text ,b4 ,foll[+ ?is p]iod come c]ta9 pages ( computation to be us$ by ! tea*] at h] discretion1 conta9+ #ag ,,xox multiplications4 ,illu/ration3 ,,fig4 #ag4--,same z ,fig4 #af1 b = text ,c4 ,illu/ration3 ,,fig4 #ah4--,same z ,fig4 #af1 b = text ,d4 ,figures #ai1 #bj1 #ba1 #bb1 & #bc all conc]n ! fir/ two books ( ! ?ree-book text ,e4 ,figure #ai %[s ! di/ribution ( practice on #e x #e 9 ! fir/ two books ( text ,e4 ,! plan is ! same z 9 ,figs4 #aa to #ah1 except t ea* t5? ( an in* along ! base l9e repres5ts t5 pages4 ,figure #bj %[s ! di/ribution ( practice on #g x #g2 ,fig4 #ba %[s x = #f x #g & #g x #f toge!r4 ,9 ,figs4 #bj & #ba also1 #j4a in* along ! base l9e equals t5 pages4 ,figures #bb & #bc %[ ! di/ribution ( practice on ! divisions ( #gb1 #gc1 #gd1 #ge1 #gf1 #gg1 #gh1 & #gi by ei!r #h or #i1 & on ! divisions ( #ha1 #hb 444 #hi by #i4 ,ea* t5? ( an in* along ! base l9e repres5ts t5 pages "h also4 ,figures #ai to #bc %[ no consi/5t plan = di/ribut+ practice4 ,) #e x #e 7,fig4 #ai7 ! am\nt ( practice increases f ! fir/ treatm5t 9 grade #c to ! end ( grade #f1 s t ! di/ribution w\ld be bett] if ! pupil began at ! end & w5t backw>d6 ,) #g x #g 7,fig4 #bj7 ! practice is di/ribut$ r ev5ly & 9 small doses4 ,) #f x #g & #g x #f 7,fig4 #ba7 mu* ( x is 9 v l>ge doses4 ,) ! divisions 7,figs4 #bb & #bc7 ! practice is di/ribut$ m suitably1 ?\gh 9 ,fig4 #bc !re is too mu* ( x giv5 at "o "t 9 ! middle ( ! p]iod4 ,illu/ration3 ,,fig4 #ai4--;,di/ribution ( practice ) #e x #e 9 ! fir/ two books ( ! ?ree-book text ,e4 ,illu/ration3 ,,fig4 #bj4--;,di/ribution ( practice ) #g x #g 9 ! fir/ two books ( text ,e4 ,illu/ration3 ,,fig4 #ba4--;,di/ribution ( practice ) #f x #g or #g x #f 9 ! fir/ two books ( text ,e4 ,illu/ration3 ,,fig4 #bb4--;,di/ribution ( practice ) #gb1 #gc 444 #gi _/ #h or #i 9 ! fir/ two books ( text ,e4 ,illu/ration3 ,,fig4 #bc4--;,di/ribution ( practice ) #ha1 #hb 444 #hi _/ #i 9 ! fir/ two books ( text ,e4 ,,possible ,,improvem5ts ,ev5 if we knew :at ! be/ di/ribution ( practice 0 = ea* ability ( ! _m to be inculcat$ by ari?metical in/ruction1 we c\ld p]haps n provide x = all ( !m4 ,=1 9 ! fir/ place1 ! allotm5ts = "s ( !m might int]f]e ) ^? = o!rs4 ,9 ! second place1 !re are _m o!r consid]ations ( importance 9 ! ord]+ ( topics besides giv+ ! optimal di/ribution ( practice to ea* ability4 ,su* are consid]ations ( int]e/1 ( weld+ sep>ate abilities into an integrat$ total ability1 & ( ! limitations due to ! s*ool s*$ule ) its ,saturdays1 ,sundays1 holidays1 & vacations4 ,improvem5t c1 h[ev]1 be made ov] pres5t practice 9 _m respects4 ,a sci5tific exam9ation ( ! tea*+ ( almo/ any class = a ye>1 or ( _m ( our /&>d in/rum5ts ( in/ruction1 w reveal opportunities = improv+ ! di/ribution ( practice ) no sacrifice ( int]e/1 & ) an actual ga9 9 integrat$ function+ ari?metical p[]4 ,9 p>ticul> x w reveal cases :]e an ability is giv5 practice & !n1 nev] be+ us$ aga91 left to die ( inactivity4 ,x w reveal cases :]e an ability is giv5 practice & !n left s long )\t practice t ! fir/ effect is ne>ly lo/4 ,!re w be cases :]e practice is giv5 & reviews are giv51 b all 9 su* isolation f ev]y?+ else 9 ari?metic t ! ability1 ?\gh exi/5t1 does n become a "p ( ! pupil's g5]al work+ equipm5t4 ,!re w be cases :]e m practice is giv5 9 ! late ?an ! e>li] p]iods = no app>5t extr9sic advantage2 & cases :]e ! practice is put :]e x is = no reason t is obs]vable save t ! tea*] or au?or 9 "q has decid$ to h "s drill "w at t "t6 ,ea* ability has its peculi> ne$s 9 ?is matt]1 & no set rules are at pres5t ( mu* value4 ,x w be en\gh = ! pres5t if we are ar\s$ to ! problem ( di/ribution1 avoid obvi\s follies l ^? j not$1 & ex]cise :at +5uity we h4 ,,*apt] ,,ix ,,! ,,psy*ology ,,( ,,?9k+3 ,,ab/ract ,,ideas ,,& ,,g5]al ,,notions ,,9 ,,ari?metic#aa #aa ,c]ta9 p>agraphs 9 ?is & ! foll[+ *apt] are tak5 f ! au?or's ,educational ,psy*ology1 ) slight modifications4 ,,responses ,,to ,,elem5ts ,,& ,,classes ,! plate :i* y see1 ! egg be=e y at ! breakfa/ table1 & ?is page are concrete ?+s1 b :it5ess1 :e!r ( plate1 egg1 or pap]1 is1 we say1 an ab/ract quality4 ,to be able to ?9k ( :it5ess irrespective ( any concrete :ite object is to be able to h an ab/ract idea or notion ( :ite2 to be able to respond to :it5ess1 irrespective ( :e!r x is a "p ( *9a1 egg%ell1 pap] or :atev] object1 is to be able to respond to ! ab/ract elem5t ( :it5ess4 ,le>n+ ari?metic involves ! =mation ( v _m su* ideas1 ! acquisition ( v _m su* p[]s ( response to elem5ts reg>dless ( ! gross total situations 9 :i* !y appe>4 ,to appreciate ! fiv5ess ( five boys1 five p5cils1 five in*es1 five r+s ( a bell2 to und]/& ! division into eight equal p>ts ( #dj c5ts1 #cb feet1 #fd m9utes1 or #af ones2 to respond correctly to ! fraction relation 9 #b_/c1 #e_/f1 #c_/d1 #g_/ab1 #a_/h1 or any o!r2 to be s5sitive to ! common elem5t ( #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--^! are obvi\s illu/rations4 ,all ! numb]s :i* ! pupil le>ns to und]/& & manipulate are 9 fact ab/ractions2 all ! op]ations are ab/ractions2 p]c5t1 disc\nt1 int]e/1 height1 l5g?1 area1 volume1 are ab/ractions2 sum1 diff]5ce1 product1 quoti5t1 rema9d]1 av]age1 are facts t conc]n elem5ts or aspects :i* may appe> ) c\ntless diff]5t concrete surr\nd+s or concomitants4 ,t[s] is a p>ticul> dog2 y\r h\se lot on ,elm ,/reet is a p>ticul> rectangle2 ,mr4 & ,mrs4 ,i4,s4 ,pet]son & _! daught] ,l\ise are a p>ticul> family ( ?ree4 ,9 contra/ to ^! p>ticul>s1 we mean by a dog1 a rectangle1 & a family ( ?ree1 any specim5s ( ^! classes ( facts4 ,! idea ( a dog1 ( rectangles 9 g5]al1 ( any family ( ?ree is a g5]al notion1 a concept or idea ( a class or species4 ,! ability to respond to any dog1 or rectangle1 or family ( ?ree1 reg>dless ( :i* p>ticul> "o x may be1 is ! g5]al notion 9 action4 ,le>n+ ari?metic involves ! =mation ( v _m su* g5]al notions1 su* p[]s ( response to any memb] ( a c]ta9 class4 ,?us a hundr$ diff]5t siz$ lots may all be respond$ to z rectangles2 #i_/ah1 #ab_/bg1 #ae_/bd1 & #bg_/cf may all be respond$ to z memb]s ( ! class1 'bo? memb]s divisible by #c4' ,! same fact may be respond$ to 9 diff]5t ways accord+ to ! class to :i* x is assign$4 ,?us #d 9 #c_/d1 #d_/e1 #de1 #ed1 & #dje is class$ respectively z 'a c]ta9 siz$ "p ( unity1' 'a c]ta9 numb] ( p>ts ( ! size %[n by ! #e1' 'a c]ta9 numb] ( t5s1' 'a c]ta9 numb] ( ones1' & 'a c]ta9 numb] ( hundr$s4' ,ea* ab/ract quality may become ! basis ( a class ( facts4 ,s f\rness z a quality corresponds to ! class '?+s f\r 9 numb] or size'2 ! fractional quality or relation corresponds to ! class 'fractions4' ,! bonds =m$ ) classes ( facts & ) elem5ts or features by :i* "o :ole class ( facts is di/+ui%$ f ano!r1 are 9 fact1 a *ief conc]n ( ari?metical le>n+4#ab #ab ,x %\ld be not$ t j z concretes give rise to ab/ractions1 s ^! 9 turn give rise to /ill m ab/ract ab/ractions4 ,?us f\rness1 fiv5ess1 tw5tyness1 & ! l give rise to 'integral-numb]-ness4' ,simil>ly j z individuals are gr\p$ into g5]al classes1 s classes are gr\p$ into /ill m g5]al classes4 ,half1 qu>t]1 six?1 & t5? are g5]al notions1 b '"o 444?' is m g5]al2 & 'fraction' is /ill m g5]al4 ,,facilitat+ ,,! ,,analysis ,,( ,,elem5ts ,ab/ractions & g5]alizations !n dep5d ^u analysis & ^u bonds =m$ ) m or less subtle elem5ts r ?an ) gross total concrete situations4 ,! process involv$ is mo/ easily und]/ood by consid]+ ! means employ$ to facilitate x4 ,! fir/ ( ^! is hav+ ! le>n] respond to ! total situations conta9+ ! elem5t 9 "q ) ! attitude ( piecemeal exam9ation1 & ) att5tiv5ess to "o elem5t aft] ano!r1 especially to s ne> an approximation to ! elem5t 9 "q z he c already select = att5tive exam9ation4 ,?is att5tiv5ess to "o elem5t aft] ano!r s]ves to emphasize :atev] appropriate m9or bonds f ! elem5t 9 "q ! le>n] already possesses4 ,?us1 9 tea*+ *ildr5 to respond to ! 'fiv5ess' ( v>i\s collections1 we %[ five boys or five girls or five p5cils1 & say1 ,see h[ _m boys are /&+ up4 ,is ,jack ! only boy t is /&+ "h8 ,are !re m ?an two boys /&+8 ,"n ! boys :ile ,i po9t at !m & c\nt !m4 7,jack7 is "o1 & 7,fr$7 is "o m1 & 7,h5ry7 is "o m4 ,jack & ,fr$ make 7two7 boys4 ,jack & ,fr$ & ,h5ry make 7?ree7 boys4 7,& s on ) ! att5tive c\nt+47 ,! m5tal set or attitude is direct$ t[>d favor+ ! p>tial & pr$om9ant activity ( 'h[-_m-ness' z f> z may be2 & ! useful bonds t ! 'fiv5ess1' ! '"o & "o & "o & "o & "o-ness1' already h1 are emphasiz$ z f> z may be4 ,! second ( ! means us$ to facilitate analysis is hav+ ! le>n] respond to _m situations ea* conta9+ ! elem5t 9 "q 7call x ,a71 b ) v>y+ concomitants 7call ^! ,v4 ,c47 8 response be+ s direct$ z1 s f> z may be1 to sep>ate ea* total response into an elem5t b\nd to ! ,a & an elem5t b\nd to ! ,v4 ,c4 ,?us ! *ild is l$ to associate ! responses--',five boys1' ',five girls1' ',five p5cils1' ',five in*es1' ',five feet1' ',five books1' ',he walk$ five /eps1' ',i hit my desk five times1' & ! l--ea* ) its appropriate situation4 ,! ',five' elem5t ( ! response is ?us b\nd ov] & ov] aga9 to ! 'fiv5ess' elem5t ( ! situation1 ! m5tal set be+ ',h[ _m81' b is b\nd only once to any "o ( ! concomitants4 ,^! concomitants are also su* z h pref]r$ m9or bonds ( _! own 7! sight ( a r[ ( boys p] se t5ds /rongly to call up ! ',boys' elem5t ( ! response74 ,! o!r elem5ts ( ! responses 7boys1 girls1 p5cils1 etc47 h ea* only a slight connection ) ! 'fiv5ess' elem5t ( ! situations4 ,^! slight connections also 9 l>ge "p#ac c\nt]act ea* o!r1 leav+ ! field cle> = :atev] un9hibit$ bond ! 'fiv5ess' has4 #ac ,!y may1 ( c\rse1 also result 9 a fusion or an alt]nation ( responses1 b only r>ely4 ,! ?ird means us$ to facilitate analysis is hav+ ! le>n] respond to situations :i*1 pair by pair1 pres5t ! elem5t 9 a c]ta9 context & pres5t t same context ) ! opposite ( ! elem5t 9 "q1 or ) some?+ at lea/ v unlike ! elem5t4 ,?us1 a *ild :o is be+ taught to respond to '"o fif?' is n only l$ to respond to '"o fif? ( a cake1' '"o fif? ( a pie1' '"o fif? ( an apple1' '"o fif? ( t5 in*es1' '"o fif? ( an army ( tw5ty soldi]s1' & ! l2 he is also l$ to respond to ea* ( ^! 9 contra/ ) 'five cakes1' 'five pies1' 'five apples1' 'five times t5 in*es1' 'five armies ( tw5ty soldi]s4' ,simil>ly ! 'place values' ( t5?s1 hundr$?s1 & ! re/ are taught by contra/ ) ! t5s1 hundr$s1 & ?\s&s4 ,^! means utilize ! laws ( connection-=m+ to dis5gage a response elem5t f gross total responses & atta* x to "s situation elem5t4 ,! =ces ( use1 disuse1 satisfaction1 & discom=t are s maneuv]$ t an elem5t :i* nev] exi/s by itself 9 nature c influ5ce man almo/ z if x did s exi/1 bonds be+ =m$ ) x t act almo/ or q irrespective ( ! gross total situation 9 :i* x inh]es4 ,:at happ5s c be mo/ conv5i5tly put 9 a g5]al /atem5t by us+ symbols4 ,d5ote by a b1 a g1 a l1 a q1 a v1 & a ,b c]ta9 situations alike 9 ! elem5t a & diff]5t 9 all else4 ,suppose t1 by orig9al nature or tra9+1 a *ild responds to ^! 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 t man's neurones are capable ( su* action t r#a1 r#b1 r#g1 r#ab1 r#bb1 & r#bg1 c ea* be made s+ly4 ,case ,i4 ,v>y+ ,concomitants ,suppose t a b1 a g1 a l1 etc41 occur once ea*4 ,we h a b respond$ 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 & a ,b r#a r#bg1 z %[n 9 ,s*eme ,i4 ,s*eme ,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 ?us respond$ to by r#a 7t is1 connect$ ) r#a7 ea* "t1 or six 9 all1 b only once ea* ) b1 g1 l1 q1 v1 & ,b4 b1 g1 l1 q1 v1 & ,b are connect$ once ea* ) r#a & once respectively ) r#b1 r#g1 r#ab1 etc4 ,! bond f a to r#a1 has _h six times z mu* ex]cise z ! bond f a to r#b1 or f a to r#g1 etc4 ,9 any new gross situation1 a #j1 a w be m pr$om9ant 9 det]m9+ response ?an x w\ld o!rwise h be52 & r#a w be m likely to be made ?an r#b1 r#g1 r#ab1 etc41 ! o!r previ\s associates 9 ! response to a situation conta9+ a4 ,t is1 ! bond f ! elem5t a to ! response r#a has be5 notably /r5g!n$4 ,case ,,ii4 ,contra/+ ,concomitants ,n[ suppose t b & g are v dissimil> elem5ts 7e4g41 :ite & black71 t l & q are v dissimil> 7e4g41 long & %ort71 & t v & ,b are also v dissimil>4 ,to be v dissimil> means to be respond$ to v diff]5tly1 s t r#g1 ! response to g1 w be v unlike r#b1 ! response to b4 ,s r#g may be ?\ght ( z rn #b or r-#b4 ,9 ! same way r#ab may be ?\ght ( z rn #ab or r-#ab1 & r#bg may be call$ rn #bb or r-#bb4 ,!n1 if ! situations a b1 a g1 a l1 a q1 a v1 & a ,b are respond$ to1 ea* once1 we h3-- a b respond$ to by r#a r#b1 a g r#a rn #b1 a l r#a r#ab1 a q r#a rn #ab1 a v r#a r#bb1 & a ,b r#a rn #bb1 z %[n 9 ,s*eme ,,ii4 ,s*eme ,,ii a b g l q v ,b 7opp4 ( b7 7opp4 ( l7 7opp4 ( v7 r#a #f #a #a #a #a #a #a rn #a r#b #a #a rn #b #a #a r#ab #a #a rn #ab #a #a r#bb #a #a rn #bb #a #a r#a is connect$ to a by #f repetitions4 r#b & rn #b are ea* connect$ to a by #a repetition1 b s9ce !y int]f]e1 cancel+ ea* o!r s to speak1 ! net result is = a to h z]o t5d5cy to call up r#b or rn #b4 r#ab & rn #ab are ea* connect$ to a by #a repetition1 b !y int]f]e ) or cancel ea* o!r ) ! net result t a has z]o t5d5cy to call up r#ab or rn #ab4 ,s ) r#bb & rn #bb4 ,"h !n ! net result ( ! six connections ( a b1 a g1 a l1 a q1 a v1 & a ,b is to connect a ) r1 & ) no?+ else4 ,case ,,iii4 ,contra/+ ,concomitants & ,contra/+ ,elem5t ,suppose n[ t ! facts are z 9 ,case ,,ii1 b ) ! addition ( six exp]i5ces :]e a c]ta9 elem5t :i* is ! opposite (1 or v dissimil> to1 a is connect$ ) ! response rn #a1 or r-#a :i* is opposite to1 or v dissimil> to r#a4 ,call ?is opposite ( a1 - a4 ,t is1 we h n only a b respond$ to by r#a r#b1 a g r#a rn #b1 a l r#a r#ab1 a q r#a rn #ab1 a v r#a r#bb1 & a ,b r#a rn #bb1 b also - a b respond$ to by rn #a r#b1 - a g rn #a rn #b1 - a l rn #a r#ab1 - a q rn #a rn #ab1 - a v rn #a r#bb1 & - a ,b rn #a rn #bb1 z %[n 9 ,s*eme ,,iii4 ,s*eme ,,iii a opp4 b g l q v ,b ( a 7opp4 ( b7 7opp4 ( l7 7opp4 ( v7 r#a #f #a #a #a #a #a #a rn #a #f #a #a #a #a #a #a r#b #a #a #b rn #b #a #a #b r#ab #a #a #b rn #ab #a #a #b r#bb #a #a #b rn #bb #a #a #b ,9 ?is s]ies ( twelve exp]i5ces a connects ) r#a six times & ! opposite ( a connects ) rn #a six times4 a connects equally (t5 ) ?ree pairs ( mutual de/ructives r#b & rn #b1 r#ab & rn #ab1 r#bb & rn #bb1 & s has z]o t5d5cy to call !m up4 - a has also z]o t5d5cy to call up any ( ^! responses except its opposite1 rn #a4 b1 g1 l1 q1 v1 & ,b are made to connect equally (t5 ) r#a & rn #a4 ,s1 ( ^! elem5ts1 a is ! only "o left ) a t5d5cy to call up r#a4 ,?us1 by ! m]e action ( frequ5cy ( connection1 r#a is connect$ ) a2 ! bonds f a to any?+ except r#a are be+ c\nt]act$1 & ! slight bonds f any?+ except a to r#a are be+ c\nt]act$4 ,! elem5t a becomes pr$om9ant 9 situations conta9+ x2 & its bond t[>d r#a becomes relatively enorm\sly /r5g!n$ & fre$ f competition4 ,^! ?ree processes occur 9 a simil>1 b m complicat$1 =m if ! situations a b1 a g1 etc41 are replac$ by a b c d e f1 a g h i j k1 etc41 & ! responses r#a r#b1 r#a r#g1 r#a r#ab1 etc41 are replac$ 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--provid$ ! r#a1 r#b1 r#c1 r#d1 etc41 c be made s+ly4 ,9 s f> z any "o ( ! responses is necess>ily co-active ) any "o ( ! o!rs 7s t1 = example1 r#ac always br+s r#bf ) x & vice v]sa71 ! exact relations ( ! numb]s record$ 9 s*emes l s*emes ,i1 ,,ii1 & ,,iii on pages #agb to #agd w *ange2 b1 unless r#a has su* an inevitable co-actor1 ! g5]al results ( s*emes ,i1 ,,ii1 & ,,iii w hold good4 ,if r#a does h su* an insep>able co-actor1 say r#b1 !n1 ( c\rse1 a c nev] acquire bonds ) r#a alone1 b ev]y:]e t r#a or r#b appe>s 9 ! prec$+ s*emes ! o!r elem5t mu/ appe> also4 r#a r#b w\ld !n h to be us$ z a unit 9 analysis4 ,! 'a b1' 'a g1' 'a l1' 444 'a ,b' situations may occur unequal numb]s ( times1 alt]+ ! exact num]ical relations ( ! connections =m$ & pres5t$ 9 s*emes ,i1 ,,ii1 & ,,iii2 b ! process 9 g5]al rema9s ! same4 ,s mu* = ! effect ( use & disuse 9 atta*+ appropriate response elem5ts to c]ta9 subtle elem5ts ( situations4 ,!re are ?ree ma9 s]ies ( effects ( satisfaction & discom=t4 ,!y s]ve1 fir/1 to emphasize1 f ! />t1 ! desir$ bonds lead+ to ! responses r#a r#b1 r#a r#g1 etc41 to ! total situations1 & to we$ out ! undesirable ones4 ,!y also act to emphasize1 9 su* comp>isons & contra/s z h be5 describ$1 e action ( ! bond f a to r#a2 & to elim9ate e t5d5cy ( a to connect ) aught save r#a1 & ( aught save a to connect ) r#a4 ,_! ?ird s]vice is to /r5g!n ! bonds produc$ ( appropriate responses to a :]ev] x occurs1 :e!r or n any =mal comp>isons & contra/s take place4 ,! process ( le>n+ to respond to ! diff]5ce ( pit* 9 tones f :atev] in/rum5t1 to ! 'squ>e-root-ness' ( :atev] numb]1 to triangul>ity 9 :atev] size or comb9ation ( l9es1 to equality ( :atev] pairs1 or to hone/y 9 :atev] p]son or in/ance1 is ?us a consequ5ce ( associative le>n+1 requir+ no o!r =ces ?an ^? ( use1 disuse1 satisfaction1 & discom=t4 ,:at happ5s 9 su* cases is t ! response1 by be+ connect$ ) _m situations alike 9 ! pres5ce ( ! elem5t 9 "q & diff]5t 9 o!r respects1 is b\nd firmly to t elem5t & loosely to ea* ( its concomitants4 ,conv]sely any elem5t is b\nd firmly to any "o response t is made to all situations conta9+ x & v1 v loosely to ea* ( ^? responses t are made to only a few ( ! situations conta9+ x4 ,! elem5t ( triangul>ity1 = example1 is b\nd firmly to ! response ( say+ or ?9k+ 'triangle' b only v loosely to ! response ( say+ or ?9k+ :ite1 r$1 blue1 l>ge1 small1 iron1 /eel1 wood1 pap]1 & ! l4 ,a situation ?us acquires bonds n only ) "s response to x z a gross total1 b also ) responses to any ( its elem5ts t h appe>$ 9 any o!r gross totals4 ,appropriate response to an elem5t reg>dless ( its concomitants is a necess>y consequ5ce ( ! laws ( ex]cise & effect if an animal le>ns to make t response to ! gross total situations t conta9 ! elem5t & n to make x to ^? t d n4 ,su* prepot5t det]m9ation ( ! response by "o or ano!r elem5t ( ! situation is no transc5d5tal my/]y1 b1 giv5 ! circum/ances1 a g5]al rule ( all le>n+4 ,su* are at bottom only extreme cases ( ! same le>n+ z a cat exhibits t depresses a plat=m 9 a c]ta9 box :e!r x faces nor? or s\?1 :e!r ! temp]ature is #ej or #hj degrees1 :e!r "o or two p]sons are 9 sight1 :e!r %e is exce$+ly or mod]ately hungry1 :e!r fi% or milk is outside ! box4 ,all le>n+ is analytic1 repres5t+ ! activity ( elem5ts )9 a total situation4 ,9 man1 by virtue ( c]ta9 in/9cts & ! c\rse ( 8 tra9+1 v subtle elem5ts ( situations c s op]ate4 99 99 99 99 99 ,le>n+ by analysis does n (t5 proce$ 9 ! c>efully organiz$ way repres5t$ by ! mo/ +5i\s m>%al+ ( comp>+ & contra/+ activities4 ,! associations ) gross totals1 :]eby 9 ! end an elem5t is elevat$ to indep5d5t p[] to det]m9e response1 may come 9 a haphaz>d ord] ov] a long int]val ( "t4 ,?us a gift$ ?ree-ye>-old boy w h ! response elem5t ( 'say+ or ?9k+ two1' b\nd to ! 'two-ness' elem5t ( v _m situations 9 connection ) ! 'h[-_m' m5tal set2 & he w h made ?is analysis )\t any =mal1 sy/ematic tra9+4 ,an imp]fect & inadequate analysis already made is inde$ usually ! />t+ po9t = :atev] sy/ematic ab/raction ! s*ools direct4 ,?us ! k9d]g>t5 ex]cises 9 analyz+ out numb]1 color1 size1 & %ape commonly assume t '"o-ness' v]sus 'm-?an-"o-ness1' black & :ite1 big & little1 r\nd & n r\nd are1 at lea/ vaguely1 active z elem5ts respond$ to 9 "s indep5d5ce ( _! contexts4 ,moreov]1 ! te/s ( actual trial & success 9 fur!r undirect$ ex]cises usually cooep]ate to confirm & ext5d & ref9e :at ! sy/ematic drills h giv54 ,?us ! ord9>y *ild 9 s*ool is left1 by ! drills on decimal notation1 ) only imp]fect p[] ( response to ! 'place-values4' ,he cont9ues to le>n to respond prop]ly to !m by f9d+ t #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 & ! l1 are satisfy+2 :ile #d x #dj #af1 #bc x #dh #hcb1 #hjj - #h #j1 & ! l1 are n4 ,! process ( analysis is ! same 9 su* casual1 unsy/ematiz$ =mation ( connections ) elem5ts z 9 ! delib]ately manag$1 piecemeal inspection1 comp>ison1 & contra/ describ$ above4 ,,sy/ematic ,,& ,,opportuni/ic ,,/imuli ,,to ,,analysis ,! arrangem5t ( a pupil's exp]i5ces s z to direct 8 att5tion to an elem5t1 v>y its concomitants in/ructively1 /imulate comp>ison1 & ?r[ ! elem5t into relief by contra/ may be by fix$1 =mal1 sy/ematic ex]cises4 ,or x may be by mu* less =mal ex]cises1 spread ov] a long] "t1 & done m or less incid5tally 9 o!r connections4 ,we may call ^! two extremes ! 'sy/ematic' & 'opportuni/ic1' s9ce ! *ief feature ( ! =m] is t x sy/ematically provides exp]i5ces design$ to build up ! p[] ( correct response to ! elem5t1 :]eas ! *ief feature ( ! latt] is t x uses especially su* opportunities z occur by reason ( ! pupil's activities & int]e/s4 ,ea* me?od has its advantages & disadvantages4 ,! sy/ematic me?od *ooses exp]i5ces t are specially design$ to /imulate ! analysis2 x provides ^! at a c]ta9 fix$ "t s t !y may "w toge!r2 x c !n & !re te/ ! pupils to asc]ta9 :e!r !y really h ! p[] to respond to ! elem5t or aspect or feature 9 "q4 ,its disadvantages are1 fir/1 t _m ( ! pupils w feel no ne$ = & atta* no int]e/ or motive to ^! =mal ex]cises2 second1 t "s ( ! pupils may memorize ! answ]s z a v]bal task in/ead ( acquir+ insight into ! facts2 ?ird1 t ! ability to respond to ! elem5t may rema9 re/rict$ to ! special cases devis$ = ! sy/ematic tra9+1 & n be available = ! g5u9e uses ( ari?metic4 ,! opportuni/ic me?od is /rong j :]e ! sy/ematic is weak4 ,s9ce x seizes ^u opportunities creat$ by ! pupil's abilities & int]e/s1 x has ! attitude ( int]e/ m (t54 ,s9ce x builds up ! exp]i5ces less =mally & ov] a wid] space ( "t1 ! pupils are less likely to le>n v]bal answ]s4 ,s9ce its mat]ial comes m f ! g5u9e uses ( life1 ! p[] acquir$ is m likely to be applicable to life4 ,its disadvantage is t x is h>d] to manage4 ,m ?\ght & exp]im5tation are requir$ to f9d ! be/ exp]i5ces2 great] c>e is requir$ to keep track ( ! developm5t ( an ab/raction :i* is taught n 9 two days1 b ov] two mon?s2 & "o may =get to te/ ! pupils at ! end4 ,9 s f> z ! textbook & tea*] are able to ov]come ^! disadvantages by +5uity & c>e1 ! opportuni/ic me?od is bett]4 ,,adaptations ,,to ,,elem5t>y ,,s*ool ,,pupils ,we may expect mu* improvem5t 9 ! =mation ( ab/ract & g5]al ideas 9 ari?metic f ! application ( ?ree pr9ciples 9 addition to ^? already describ$4 ,!y are3 7#a7 ,provide en\gh actual exp]i5ces be=e ask+ ! pupil to und]/& & use an ab/ract or g5]al idea4 7#b7 ,develop su* ideas gradually1 n attempt+ to give complete & p]fect ideas all at once4 7#c7 ,develop su* ideas s f> z possible f exp]i5ces :i* w be valuable to ! pupil 9 & ( !mselves1 q ap>t f _! m]it z aids 9 develop+ ! ab/raction or g5]al notion4 ,consid] ^! ?ree pr9ciples 9 ord]4 ,*ildr51 especially ! less gift$ intellectually1 ne$ m exp]i5ces z a basis = & z applications ( an ari?metical ab/raction or concept ?an are usually giv5 !m4 ,= example1 9 pav+ ! way = ! pr9ciple1 ,any numb] times #j equals #j1 x is n safe to say1 ,john work$ #h days = #j m9utes p] day4 ,h[ _m m9utes did he "w8 & ,h[ mu* is #j times #d c5ts8 ,x w be mu* bett] to sp5d t5 or fifte5 m9utes z foll[s3#ad ,:at does z]o mean8 7,n any4 ,no47 ,h[ _m feet are !re 9 eight y>ds8 ,9 #e y>ds8 ,9 #c y>ds8 ,9 #b y>ds8 ,9 #a y>d8 ,9 #j y>d8 ,h[ _m in*es are !re 9 #d ft48 ,9 #b ft48 ,9 #j ft48 #g pk4 4444 qt4 #e pk4 4444 qt4 #j pk4 4444 qt4 ,a boy receives #fj c5ts an h\r :5 he works4 ,h[ mu* does he receive :5 he works #c hr48 #h hr48 #f hr48 #j hr48 ,a boy receiv$ #fj c5ts a day = #j days4 ,h[ mu* did he receive8 ,h[ mu* is #j times #fjj8 ,h[ mu* is #j times #ejjj8 ,h[ mu* is #j times a million doll>s8 #j times any numb] equals4444 #bcb 7,at ! blackbo>d47 #j "t #bcb equals :at8 #cj ,i write #j "u ! #j4ae #c times #bcb equals :at8 ---- #fifj ,cont9ue at ! blackbo>d ) #gcd #cba #cab #da #bj #dj #cj #fj etc4 --- --- --- -- #ad ,! m gift$ *ildr5 may be put to "w us+ ! pr9ciple aft] ! fir/ m9ute or two4 #ae #bcb #cj ,if desir$ ?is =m may be us$1 ) ! appropriate --- diff]5ce 9 ! =m ( ! que/ions & /atem5ts4 #jjj #fif ---- #fifj ,pupils 9 ! elem5t>y s*ool1 except ! mo/ gift$1 %\ld n be expect$ to ga9 ma/]y ov] su* concepts z common fraction1 decimal fraction1 factor1 & root quickly4 ,!y c le>n a def9ition quickly & le>n to use x 9 v easy cases1 :]e ev5 a vague & imp]fect und]/&+ ( x w guide response correctly4 ,b complete & exact und]/&+ commonly requires !m to take1 n "o intellectual /ep1 b _m2 & ma/]y 9 use commonly comes only z a sl[ gr[?4 ,= example1 suppose t pupils are taught t 4#a1 4#b1 4#c1 etc41 mean #a_/aj1 #b_/aj1 #c_/aj1 etc41 t 4#ja1 4#jb1 4#jc1 etc41 mean #a_/ajj1 #b_/ajj1 #c_/ajj1 etc41 t 4#jja1 4#jjb1 4#jjc1 etc41 mean #a_/ajjj1 #b_/ajjj1 #c_/ajjj1 etc41 & t 4#a1 4#jb1 4#jja1 etc41 are decimal fractions4 ,!y may !n respond correctly :5 ask$ to write a decimal fraction1 or to /ate :i* ( ^!1--#a_/d1 4#d1 #c_/h1 4#jg1 4#jjb1 #e_/f1--;are common fractions & :i* are decimal fractions4 ,!y may be able1 ?\gh by no means all ( !m w be1 to write decimal fractions :i* equal #a_/b & #a_/e1 & ! common fractions :i* equal 4#a & 4#ji4 ,mo/ ( !m w n1 h[ev]1 be able to respond correctly to ,write a decimal mix$ numb]2 or to /ate :i* ( ^!1--#a_/ajj1 4#d-a_/b1 4#jjg_/cej1 4#be1--;are common fractions1 & :i* are decimals2 or to write ! decimal fractions :i* equal #c_/d & #a_/c4 ,if n[ ! tea*] _h giv5 all at once ! additional exp]i5ces ne$$ to provide ! ability to h&le ^! m intricate & subtle features ( decimal-fraction-ness1 ! result w\ld h be5 confusion = mo/ pupils4 ,! g5]al mean+ ( 4#cb1 4#ad1 4#ii1 & ! l requires "s und]/&+ ( 4#cj1 4#aj1 4#ij1 & 4#jb1 4#jd1 4#jh2 b x is n desirable to di/urb ! *ild ) 4#cj :ile he is try+ to ma/] #b4c1 #d4c1 #f4c1 & ! l4 ,decimals 9 g5]al require connection ) place value & ! contra/s ( 4#da ) #da1 #daj1 #d4a1 & ! l1 b if ! relation to place values 9 g5]al is taught 9 ! same lesson ) ! relation to _/#ajs1 _/#ajjs1 _/#ajjjs1 ! m9d w suff] f viol5t indige/ion4 ,a wise p$agogy 9 fact w break up ! process ( le>n+ ! mean+ & use ( decimal fractions into _m tea*+ units1 = example1 z foll[s3-- 7#a7 ,su* famili>ity ) fractions ) l>ge d5om9ators z is desirable = pupils to h1 z by an ex]cise 9 r$uc+ to l[e/ t]ms1 #h_/aj1 #cf_/fd1 #bj_/be1 #ah_/bd1 #bd_/cb1 #ba_/cj1 #be_/ajj1 #dj_/ajj1 & ! l4 ,?is is good z a review ( cancellation1 & z an ext5sion ( ! idea ( a fraction4 7#b7 ,objective "w1 %[+ #a_/aj sq4 ft41 #a_/ej sq4 ft41 #a_/ajj sq4 ft41 & #a_/ajjj sq4 ft41 & hav+ ^! id5tifi$ & ! =ms #a_/aj sq4 ft41 #a_/ajj sq4 ft41 & #a_/ajjj sq4 ft4 le>n$4 ,f9d+ h[ _m feet #a_/aj mile & #a_/ajj mile4 7#c7 ,famili>ity ) _/#ajjs & _/#ajjjs by r$uctions ( #gej_/ajjj1 #ej_/ajj1 etc41 to l[e/ t]ms & by writ+ ! miss+ num]ators 9 #ejj_/ajjj _/#ajj _/#aj & ! l1 & by f9d+ #a_/aj1 #a_/ajj1 & #a_/ajjj ( #cjjj1 #fjjj1 #ijjj1 etc4 7#d7 ,writ+ #a_/aj z 4#a & #a_/ajj z 4#ja1 #aa_/ajj1 #ab_/ajj1 #ac_/ajj1 etc41 z 4#aa1 4#ab1 4#ac4 ,unit$ ,/ates money is us$ z ! introduction4 ,application is made to miles4 7#e7 ,mix$ numb]s ) a fir/ decimal place4 ,! cyclomet] or spe$omet]4 ,add+ numb]s l #i4a1 #ad4g1 #aa4d1 etc4 7#f7 ,place value 9 g5]al f ?\s&s to hundr$?s4 7#g7 ,review ( 7#a7 to 7#f74 7#h7 ,t5?s & hundr$?s ( a mile1 subtraction :5 bo? numb]s ext5d to hundr$?s1 us+ a railroad table ( di/ances4 7#i7 ,?\s&?s4 ,! names 'decimal fractions or decimals1' & 'decimal mix$ numb]s or decimals4' ,drill 9 read+ any numb] to ?\s&?s4 ,! "w w cont9ue ) gradual ext5sion & ref9em5t ( ! und]/&+ ( decimals by le>n+ h[ to op]ate ) !m 9 v>i\s ways4 ,su* may seem a sl[ progress1 b 9 fact x is n1 & _m ( ^! ex]cises :]eby ! pupil acquires 8 ma/]y ( decimals are useful z organizations & applications ( o!r ari?metical facts4 ,t1 x w be rememb]$1 0 ! ?ird pr9ciple3--,develop ab/ract & g5]al ideas by exp]i5ces :i* w be intr9sically valuable4 ,! reason is t1 ev5 ) ! be/ ( tea*+1 "s pupils w n1 )9 any reasonable limits ( "t exp5d$1 acquire ideas t are fully complete1 rigor\s :5 !y %\ld be1 flexible :5 !y %\ld be1 & absolutely exact4 ,_m *ildr5 7& adults1 = t matt]7 c\ld n )9 any reasonable limits ( "t be s taught ! nature ( a fraction t !y c\ld decide un]r+ly 9 orig9al ex]cises l3-- ,is #b4ge_/be a common fraction8 ,is 4#be a decimal fraction8 ,is "o x? ( y a fraction8 ,c ! same words mean bo? a common fraction & a decimal fraction8 ,express #a z a common fraction4 ,express #a z a decimal fraction4 ,^! same *ildr5 c1 h[ev]1 be taught to op]ate correctly ) fractions 9 ! ord9>y uses !re(4 ,& t is ! *ief value ( ari?metic to !m4 ,!y %\ld n be depriv$ ( x because !y _c ma/] its subtl] pr9ciples4 ,s we seek to provide exp]i5ces t w tea* all pupils some?+ ( value1 :ile /imulat+ 9 ^? :o h ! ability ! gr[? ( ab/ract ideas & g5]al pr9ciples4 ,f9ally1 we %\ld be> 9 m9d t work+ ) qualities & relations t are only p>tly und]/ood or ev5 misund]/ood does "u c]ta9 conditions give control ov] !m4 ,! g5]al process ( analytic le>n+ 9 life is to respond z well z "o c2 to get a cle>] idea !reby2 to respond bett] ! next "t2 & s on4 ,= in/ance1 "o gets "s sort ( notion ( :at #a_/e means2 he !n answ]s su* que/ions z #a_/e ( #aj 8 #a_/e ( #e 8 #a_/e ( #bj 82 by be+ told :5 he is "r & :5 he is wrong1 he gets f ^! exp]i5ces a bett] idea ( #a_/e2 aga9 he does 8 be/ ) #a_/e _/#aj1 #a_/e _/#ae1 etc41 & z be=e ref9es & enl>ges 8 concept ( #a_/e4 ,he adds #a_/e to #b_/e1 etc41 #a_/e to #c_/aj1 etc41 #a_/e to #a_/b1 etc41 & !reby ga9s /ill fur!r1 & s on4 ,:at beg9s z a bl9d habit ( manipulation />t$ by imitation may ?us gr[ into ! p[] ( correct response to ! ess5tial elem5t4 ,! pupil :o has at ! />t no notion at all ( 'multiply+' may le>n :at multiply+ is by 8 exp]i5ce t '#d #f multiply+ gives #bd'2 '#c #i multiply+ gives #bg1' etc4 ,if ! pupil keeps on do+ some?+ ) numb]s & diff]5tiates "r results1 he w (t5 rea* 9 ! end ! ab/ractions :i* he is suppos$ to ne$ 9 ! beg9n+4 ,x may ev5 be ! case ) "s ( ! ab/ractions requir$ 9 ari?metic t elaborate provision = compreh5sion be=eh& is n s effici5t z ! same am\nt ( en]gy devot$ p>tly to provision = analysis itself be=eh& & p>tly to practice 9 response to ! elem5t 9 "q )\t full compreh5sion4 ,x c]ta9ly is n ! be/ psy*ology & n ! be/ educational !ory to ?9k t ! pupil fir/ ma/]s a pr9ciple & !n m]ely applies x--fir/ does "s ?9k+ & !n computes by m]e r\t9e4 ,on ! contr>y1 ! applications %\ld help to e/abli%1 ext5d1 & ref9e ! pr9ciple--! "w a pupil does ) numb]s %\ld be a ma9 means ( increas+ 8 und]/&+ ( ! pr9ciples ( ari?metic z a sci5ce4 ,,*apt] ,x ,,! ,,psy*ology ,,( ,,?9k+3 ,,reason+ ,,9 ,,ari?metic ,,! ,,ess5tials ,,( ,,ari?metical ,,reason+ ,we di/+ui% aimless rev]ie1 z :5 a *ild dreams ( a vacation trip1 f purposive ?9k+1 z :5 he tries to "w out ! answ] to ,h[ _m weeks ( vacation c a family h = #abj if ! co/ is #bb a week = bo>d1 #b4be a week = laundry1 & #a4ge a week = incid5tal exp5ses1 & if ! railroad f>es = ! r\nd trip are #ab8 ,we di/+ui% ! process ( response to famili> situations1 su* z five integral numb]s to be add$1 f ! process ( response to novel situations1 su* z 7= a *ild :o has n be5 tra9$ ) simil> problems73--,a man has f\r pieces ( wire4 ,! l5g?s are #abj yd41 #acb met]s1 #afj feet1 & #a_/h mile4 ,h[ mu* m does he ne$ to h #ajjj yd4 9 all8 ,we di/+ui% '?9k+ ?+s toge!r1' z :5 a diagram or problem or pro( is und]/ood1 f ?9k+ ( "o ?+ aft] ano!r z :5 a numb] ( words are spell$ or a poem 9 an unkn[n tongue is le>n$4 ,9 proportion z ?9k+ is purposive1 ) selection f ! ideas t come up1 & 9 proportion z x deals ) novel problems = :i* no ready-made habitual response is available1 & 9 proportion z _m bonds act toge!r 9 an organiz$ way to produce response1 we call x reason+4 ,:5 ! conclusion is rea*$ z ! effect ( _m p>ticul> exp]i5ces1 ! reason+ is call$ inductive4 ,:5 "s pr9ciple already e/abli%$ leads to ano!r pr9ciple or to a conclusion ab\t "s p>ticul> fact1 ! reason+ is call$ d$uctive4 ,9 bo? cases ! process involves ! analysis ( facts into _! elem5ts1 ! selection ( ! elem5ts t are deem$ significant = ! "q at h&1 ! atta*m5t ( a c]ta9 am\nt ( importance or weight to ea* ( !m1 & _! use 9 ! "r relations4 ,?\ght may fail because x has n suitable facts1 or does n select f !m ! "r ones1 or does n atta* ! "r am\nt ( weight to ea*1 or does n put !m toge!r prop]ly4 ,9 ! _w at l>ge1 _m ( our failures 9 ?9k+ are due to n hav+ suitable facts4 ,"s ( my read]s1 = example1 _c solve ! problem--,:at are ! *ances t 9 draw+ a c>d f an ord9>y pack ( play+-c>ds f\r times 9 succession1 ! same c>d w be drawn ea* "t8 ,& x w be probably because !y d n "k c]ta9 facts ab\t ! !ory ( probabilities4 ,! good ?9k]s among su* w\ld look ! matt] up 9 a suitable book4 ,simil>ly1 if a p]son did n happ5 to "k t !re 7 fifty-two c>ds 9 all & t no two 7 alike1 he c\ld n reason out ! answ]1 no matt] :at 8 ma/]y ( ! !ory ( probabilities4 ,if a compet5t ?9k]1 he w\ld fir/ ask ab\t ! size & nature ( ! pack4 ,9 ! actual practice ( reason+1 t is1 we h to survey our facts to see if we lack any t are necess>y4 ,if we d1 ! fir/ task ( reason+ is to acquire ^? facts4 ,?is is specially true ( ! reason+ ab\t ari?metical facts 9 life4 ,w #c-a_/b y>ds ( ?is be en\gh = a dress8 ,reason directs y to le>n h[ wide x is1 :at /yle ( dress y int5d to make ( x1 h[ mu* mat]ial t /yle normally calls =1 :e!r y are a c>eful or a wa/eful cutt]1 & h[ big ! p]son is = :om ! dress is to be made4 ,h[ mu* *eap] z a diet is bread alone1 ?an bread ) butt] add$ to ! ext5t ( #aj ( ! weight ( ! bread8 ,reason directs y to le>n ! co/ ( bread1 ! co/ ( butt]1 ! nutritive value ( bread1 & ! nutritive value ( butt]4 ,9 ! ari?metic ( ! s*ool ?is feature ( reason+ appe>s 9 cases :]e "s fact ab\t common measures mu/ be br\ght to be>1 or "s table ( prices or disc\nts mu/ be consult$1 or "s bus9ess cu/om mu/ be rememb]$ or look$ up4 ,?us ,h[ _m badges1 ea* #i in*es long1 c be made f #b-a_/b yd4 ribbon8 _c be solv$ )\t gett+ into m9d #a yd4 #cf in*es4 ,at ,jones' prices1 :i* co/s m1 #c-c_/d lb4 butt] or #f-a_/b lb4 l>d8 ,h[ mu* m8 is a problem :i* directs ! ?9k] to asc]ta9 ,jones' prices4 ,x may be not$ t su* problems are1 o!r ?+s be+ equal1 some:at bett] tra9+ 9 ?9k+ ?an problems :]e all ! data are giv5 9 ! problem itself 7e4g41 ,:i* co/s m1 #c-c_/d lb4 butt] at #dh;c p] lb4 or #f-a_/b lb4 l>d at #bg;c p] lb48 ,h[ mu* m874 ,at lea/ x is unwise to h s _m problems ( ! latt] sort t ! pupil may come to ?9k ( a problem 9 appli$ ari?metic z a problem :]e ev]y?+ is giv5 & he has only to manipulate ! data4 ,life does n pres5t its problems s4 ,! process ( select+ ! "r elem5ts & atta*+ prop] weight to !m may be illu/rat$ by ! foll[+ problem3--,:i* ( ^! (f]s w\ld y take1 suppos+ t y wi% a ,d4,c4,k4 upright piano1 h #ej sav$1 c save a little ov] #bj p] mon?1 & c borr[ f y\r fa!r at #f int]e/8 ,a ,a ,reliable ,piano4 ,! ,fam\s ,d4,c4,k4 ,upright4 ,y pay #ej ca% d[n & #ba a mon? = only a ye> & a half4 ,no int]e/ to pay4 ,we ask y to pay only = ! piano & all[ y pl5ty ( "t4 ,b ,we (f] ! well-kn[n ,d4,c4,k4 ,piano = #cij4 #ej ca% & #bj a mon? !reaft]4 ,regul> int]e/ at #f4 ,! int]e/ soon is r$uc$ to less ?an #a a mon?4 ,c ,! ,d4,c4,k4 ,piano4 ,special ,(f]1 #cge1 ca%4 ,comp>e our prices ) ^? ( any reliable firm4 ,if y consid] *iefly ! only1 ,no int]e/ to pay1 only1 & pl5ty ( "t 9 (f] ,a1 atta*+ mu* weight to !m & little to ! ?\ght1 ,h[ mu* w #ej plus 7#ah x #ba7 be81 y w probably decide wrongly4 ,! situations ( life are (t5 complicat$ by _m elem5ts ( little or ev5 ( no relevance to ! correct solution4 ,! (f]] ( ,a may belong to y\r *ur*2 y\r de>e/ fri5d may urge y to accept (f] ,b2 y may dislike to talk ) ! deal] :o makes (f] ,c2 y may h a prejudice aga9/ ow+ money to a relative2 t prejudice may be wise or fooli%2 y may h a suspicion t ! ,b piano is %opworn2 t suspicion may be well-f\nd$ or gr\ndless2 ! salesman = ,c says1 ,y don't want y\r fri5ds to say t y b\ght on ! in/allm5t plan4 ,only l[-class p]sons d t1 etc4 ,! /atem5t ( ari?metical problems 9 s*ool usually assi/s ! pupil to ! ext5t ( rul+ out all save def9itely quantitative elem5ts1 & ( rul+ out all quantitative elem5ts except ^? :i* %\ld be consid]$4 ,! fir/ ( ! two simplifications is v b5eficial1 on ! :ole1 s9ce o!rwise !re might be diff]5t correct solutions to a problem accord+ to ! nature & circum/ances ( ! p]sons involv$4 ,! second simplification is (t5 desirable1 s9ce x w (t5 produce great] improvem5t 9 ! pupils1 p] h\r ( "t sp5t1 ?an w\ld be produc$ by ! problems requir+ m selection4 ,x %\ld n1 h[ev]1 be a univ]sal cu/om2 = 9 t case ! pupils are tempt$ to ?9k t 9 e problem !y mu/ use all ! quantities giv51 z "o mu/ use all ! pieces 9 a puzzle picture4 ,x is obvi\s t ! elem5ts select$ mu/ n only be "r b also be 9 ! "r relations to "o ano!r4 ,= example1 9 ! problems bel[1 ! #f mu/ be ?\ght ( 9 relation to a doz5 & z be+ half ( a doz51 & also z be+ #f times #a4 #a mu/ be m5tally ti$ to ea*4 ,! #f z half ( a doz5 mu/ be relat$ to ! #a4jj1 #a4fj1 etc4 ,! #f z #f times #a mu/ be relat$ to ! 4#ji1 4#ad1 etc4 ,buy+ 9 ,quantity ,^! are a groc]'s prices = c]ta9 ?+s by ! doz5 & = a s+le "o4 ,he sells a half doz5 at half ! price ( a doz54 ,f9d out h[ mu* y save by buy+ #f all at "o "t in/ead ( buy+ !m "o at a "t4 ,doz4 ,ea* #a4 ,evaporat$ ,milk #a4jj 4#ji #b4 ,puff$ ,rice #a4fj 4#ad #c4 ,puff$ ,:eat #a4aj 4#aj #d4 ,cann$ ,s\p #a4ij 4#ag #e4 ,s>d9es #a4hj 4#af #f4 ,beans 7,no4 #b cans7 #a4ej 4#ac #g4 ,pork & ,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 ,x is obvi\s also t 9 su* ari?metical "w z we h be5 describ+1 ! pupil1 to be successful1 mu/ '?9k ?+s toge!r4' ,_m bonds mu/ cooep]ate to det]m9e 8 f9al response4 ,z a preface to reason+ ab\t a problem we (t5 h ! discov]y ( ! problem & ! classification ( j :at x is1 & z a po/script we h ! critical inspection ( ! answ] obta9$ to make sure t x is v]ifi$ by exp]im5t or is consi/5t ) kn[n facts4 ,dur+ ! process ( se>*+ =1 select+1 & weight+ facts1 !re may be simil> inspection & validation1 item by item4 ,,reason+ ,,z ,,! ,c,o,oe,p,e,r,a,t,i,o,n ,,( ,,organiz$ ,,habits ,! p$agogy ( ! pa/ made two notable errors 9 practice bas$ on two errors ab\t ! psy*ology ( reason+4 ,x consid]$ reason+ z a some:at magical p[] or ess5ce :i* act$ to c\nt]act & ov]rule ! ord9>y laws ( habit 9 man2 & x sep>at$ too %>ply ! 'und]/&+ ( pr9ciples' by reason+ f ! 'me*anical' "w ( computation1 read+ problems1 rememb]+ facts & ! l1 done by 'm]e' habit & memory4 ,reason+ or selective1 inf]5tial ?9k+ is n at all oppos$ to1 or indep5d5t (1 ! laws ( habit1 b really is _! necess>y result "u ! conditions impos$ by man's nature & tra9+4 ,a clos] exam9ation ( selective ?9k+ w %[ t no pr9ciples beyond ! laws ( read9ess1 ex]cise1 & effect are ne$$ to expla9 x2 t x is only an extreme case ( :at goes on 9 associative le>n+ z describ$ "u ! 'piecemeal' activity ( situations2 & t attribut+ c]ta9 features ( le>n+ to my/]i\s faculties ( ab/raction or reason+ gives no real help t[>d und]/&+ or controll+ !m4 ,x is true t man's behavior 9 meet+ novel problems goes beyond1 or ev5 aga9/1 ! habits repres5t$ by bonds lead+ f gross total situations & cu/om>ily ab/ract$ elem5ts !re(4 ,"o ( ! two reasons !re=1 h[ev]1 is simply t ! f9]1 subtle1 pref]5tial bonds ) subtl] & less (t5 ab/ract$ elem5ts g beyond1 & at times aga9/1 ! gross] & m usual bonds4 ,"o set is z mu* due to ex]cise & effect z ! o!r4 ,! o!r reason is t 9 meet+ novel problems ! m5tal set or attitude is likely to be "o :i* rejects "o aft] ano!r response z _! unfitness to satisfy a c]ta9 desid]atum appe>s4 ,:at rema9s z ! app>5t c\rse ( ?\ght includes only a few ( ! _m bonds :i* did op]ate1 b :i*1 = ! mo/ "p1 7 unsatisfy+ to ! rul+ attitude or adju/m5t4 ,successful responses to novel data1 associations by simil>ity & purposive behavior are 9 only app>5t opposition to ! fundam5tal laws ( associative le>n+4 ,really !y are beautiful examples ( x4 ,man's successful responses to novel data--z :5 he argues t ! diagonal on a "r triangle ( #gif4bgh mm4 base & #acg4bid mm4 altitude w be #hjh4jbb mm41 or t ,m>y ,jones1 born ?is morn+1 w sometime die--are due to habits1 notably ! habits ( response to c]ta9 elem5ts or features1 "u ! laws ( piecemeal activity & assimilation4 ,no?+ is less l ! my/]i\s op]ations ( a faculty ( reason+ transc5d+ ! laws ( connection-=m+1 ?an ! behavior ( m5 9 response to novel situations4 ,let *ildr5 :o h hi!rto confront$ only su* ari?metical tasks1 9 addition & subtraction ) "o- & two-place numb]s & multiplication ) "o-place numb]s1 z ^? exemplifi$ 9 ! fir/ l9e bel[1 be told to d ! examples %[n 9 ! second l9e4 ,,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 -- -- -- ,!y w add ! numb]s1 or subtract ! l[] f ! upp] numb]1 or multiply #c x #b & #b x #c1 etc41 gett+ #ff1 #hf1 & #fbd1 or respond to ! elem5t ( ',multiply' atta*$ to ! two-place numb]s by ,i c't or ,i don't "k :at to d1 or ! l2 or1 if "o is a *ild ( great ability1 he may consid] ! ',multiply' elem5t & ! bigness ( ! numb]s1 be rem9d$ by ^! two aspects ( ! situation ( ! fact t '#i #i multiply' -- gave only #ha1 & t '#aj #aj multiply' -- gave only #ajj1 or ! l2 & s may report an intellig5t & ju/ifi$ ,i c't1 or reject ! plan ( #c x #b & #b x #c1 ) #ff1 #hf1 & #fbd = answ]s1 z unsatisfactory4 ,:at ! *ildr5 w d w1 9 e case1 be a product ( ! elem5ts 9 ! situation t are pot5t ) !m1 ! responses :i* ^! evoke1 & ! fur!r associates :i* ^! responses 9 turn evoke4 ,if ! *ild 7 "o ( suffici5t g5ius1 he might inf] ! proc$ure to be foll[$ z a result ( 8 k ( ! pr9ciples ( decimal notation & ! mean+ ( ',multiply1' respond+ correctly to ! 'place-value' elem5t ( ea* digit & add+ 8 #f t5s & #i t5s1 #bj twos & #c ?irties2 b if he did ?us inv5t ! %or?& addition ( a collection ( tw5ty-?ree collections1 ea* ( #cb units1 he w\ld /ill d x by ! op]ation ( bonds1 subtle b real4 ,association by simil>ity is1 z ,james %[$ long ago1 simply ! t5d5cy ( an elem5t to provoke ! responses :i* h be5 b\nd to x4 abcde leads to vwxyz because a has be5 b\nd to vwxyz by orig9al nature1 ex]cise1 or effect4 ,purposive behavior is ! mo/ important case ( ! influ5ce ( ! attitude or set or adju/m5t ( an organism 9 det]m9+ 7#a7 :i* bonds %all act1 & 7#b7 :i* results %all satisfy4 ,james e>ly describ$ ! =m] fact1 %[+ t ! me*anism ( habit c give ! direct$ness or purposefulness 9 ?\ght's products1 provid$ t me*anism includes some?+ p>allel+ ! problem1 ! aim1 or ne$1 9 "q4 ,! second fact1 t ! set or attitude ( ! man helps to det]m9e :i* bonds %all satisfy1 & :i* %all annoy1 has commonly be5 some:at obscur$ by vague ass]tions t ! selection & ret5tion is ( :at is 9 po9t1 or is ! "r "o1 or is appropriate1 or ! l4 ,x is ?us ass]t$1 or at lea/ h9t$1 t ! w1 ! volunt>y att5tion1 ! consci\sness ( ! problem1 & o!r su* entities are end[$ ) magic p[] to decide :at is ! "r or useful bond & to kill (f ! o!rs4 ,! facts are t 9 purposive ?9k+ & action1 z ev]y:]e else1 bonds are select$ & reta9$ by ! satisfy+ness1 & are kill$ (f by ! discom=t1 :i* !y produce2 & t ! pot5cy ( ! man's set or attitude to make ?is satisfy & t annoy--to put c]ta9 conduction-units 9 read9ess to act & o!rs 9 unread9ess--is 9 e way z important z its pot5cy to set c]ta9 conduction-units 9 actual op]ation4 ,reason+ is n a radically diff]5t sort ( =ce op]at+ aga9/ habit b ! organization & cooep]ation ( _m habits1 ?9k+ facts toge!r4 ,reason+ is n ! negation ( ord9>y bonds1 b ! action ( _m ( !m1 especially ( bonds ) subtle elem5ts ( ! situation4 ,"s outside p[] does n ent] to select & criticize2 ! pupil's own total rep]tory ( bonds relevant to ! problem is :at selects & rejects4 ,an unsuitable idea is n kill$ (f by "s actus purus ( intellect1 b by ! ideas :i* x itself calls up1 9 connection ) ! total set ( m9d ( ! pupil1 & :i* %[ x to be inadequate4 ,almo/ no?+ 9 ari?metic ne$ be taught z a matt] ( m]e unreason+ habit or memory1 nor ne$ any?+1 fir/ taught z a pr9ciple1 ev] become a matt] ( m]e habit or memory4 #e x #d #bj %\ld n be le>n$ z an isolat$ fact1 nor rememb]$ z we rememb] t ,jones' telephone numb] is #fdh ,j #b4 ,almo/ ev]y?+ 9 ari?metic %\ld be taught z a habit t has connections ) habits already acquir$ & w "w 9 an organization ) o!r habits to come4 ,! use ( ?is organiz$ hi]>*y ( habits to solve novel problems is reason+4 ,,*apt] ,,xi ,,orig9al ,,t5d5cies ,,& ,,acquisitions ,,be=e ,,s*ool ,,! ,,utilization ,,( ,,in/9ctive ,,int]e/s ,! activities ess5tial to acquir+ ability 9 ari?metic c rely on little 9 man's in/9ctive equipm5t beyond ! purely intellectual t5d5cies ( curiosity & ! satisfy+ness ( ?\ght = ?\ght's sake1 & ! g5]al enjoym5t ( success r ?an failure 9 an ent]prise to :i* "o sets oneself4 ,x is only by a c]ta9 am\nt ( artifice t we c enli/ o!r vehem5t inborn int]e/s ( *ildhood 9 ! s]vice ( ari?metical k & skill4 ,:5 ?is c be done at no co/ ! ga9 is great4 ,= example1 m>*+ 9 files ( two1 9 files ( ?ree1 9 files ( f\r1 etc41 rais+ ! arms once1 two times1 ?ree times1 %[+ a foot1 a y>d1 an in* ) ! h&s1 & ! l are admirable because le>n+ ! mean+s ( numb]s ?us acquires "s ( ! ze/ ( ! passion = physical action4 ,ev5 9 late grades *ances to make pictures %[+ ! relations ( fractional p>ts1 to cut /rips1 to fold pap]1 & ! l w be useful4 ,v>i\s social in/9cts c be utiliz$ 9 mat*es aft] ! patt]n ( ! spell+ mat*1 conte/s betwe5 r[s1 c]ta9 numb] games1 & ! l4 ,! scor+ ( bo? ! play & ! "w ( ! classroom is a useful field = control by ! tea*] ( ari?metic4 ,hunt '#ab has not$ ! m important games :i* h "s consid]able am\nt ( ari?metical tra9+ z a by-product & :i* are m or less suitable = class use4 ,flynn '#ab has describ$ games1 mo/ ( !m = home use1 :i* give v def9ite ari?metical drill1 ?\gh 9 _m cases ! drills are r beh9d ! ne$s ( *ildr5 old en\gh to und]/& & l ! game itself4 ,x is possible to utilize ! int]e/s 9 my/]y1 tricks1 & puzzles s z to ar\se a c]ta9 =m ( respect = ari?metic & also to get computational "w done4 ,i quote "o simple case f ,miss ,selk9's admirable collection '#ab1 p4 #fi f43-- ,i4 ,,addition ,we mu/ admit t !re is no?+ p>ticul>ly int]e/+ 9 a long column ( numb]s to be add$4 ,let ! tea*]1 h[ev]1 sugge/ t he c write ! answ] at sight1 & ! task w assume a totally diff]5t aspect4 ,a v simple numb] trick ( ?is k9d c be p]=m$ by mak+ use ( ! pr9ciple ( complem5t>y addition4 ,! ari?metical complem5t ( a numb] ) respect to a l>g] numb] is ! diff]5ce betwe5 ^! two numb]s4 ,mo/ int]e/+ results c be obta9$ by us+ complem5ts ) respect to #i4 ,! *ildr5 may be call$ ^u to sugge/ sev]al numb]s ( two1 ?ree1 or m digits4 ,bel[ ^! write an equal numb] ( add5ds & imm$iately ann\nce ! answ]4 ,! *ildr51 impress$ by ?is app>5tly rapid addition1 w set to "w en?usia/ically to te/ ! results ( ?is lightn+ calculation4 ,example3-- #ceg #iii #fhb ,a x #c #gic ---- #biig #fdb #cag ,b #bjf ,explanation3--,! add5ds 9 gr\p ,a are writt5 d[n at r&om or sugge/$ by ! class4 ,^? 9 gr\p ,b are _! complem5ts4 ,to write ! fir/ numb] 9 gr\p ,b we look at ! fir/ numb] 9 gr\p ,a &1 />t+ at ! left write #f1 ! complem5t ( #c ) respect to #i2 #d1 ! complem5t ( #e2 #b1 ! complem5t ( #g4 ,! second & ?ird add5ds 9 gr\p ,b are d]iv$ 9 ! same way4 ,s9ce we h ?ree add5ds 9 ea* gr\p1 ! problem r$uces itself to multiply+ #iii by #c1 or to tak+ #cjjj - #c4 ,any numb] ( add5ds may be us$ & ea* add5d may consi/ ( any numb] ( digits4 ,respect = ari?metic z a s\rce ( tricks & magic is v mu* less important ?an respect = its ev]yday s]vices2 & computation to te/ su* tricks is likely to be und]tak5 zeal\sly only by ! abl] pupils4 ,consequ5tly ?is s\rce ( int]e/ %\ld probably be us$ only sp>+ly1 & p]haps ! tea*] %\ld give su* exhibitions only z a rew>d = effici5cy 9 ! regul> "w4 ,= example1 if ! "w = a week is well done 9 f\r days ! fif? day might be giv5 up to "s semi-ari?metical ent]ta9m5t1 su* z ! demon/ration ( an add+ ma*9e1 ! /ory ( primitive me?ods ( c\nt+1 team races 9 computation1 an exhibition ( lightn+ calculation & intellectual sleight-(-h& by ! tea*]1 or ! volunt>y /udy ( ari?metical puzzles4 ,! int]e/ 9 a*ievem5t1 9 success1 m5tion$ above is /rong] 9 *ildr5 ?an is (t5 realiz$ & makes advisable ! sy/ematic use ( ! practice exp]im5t z a me?od ( tea*+ mu* ( ari?metic4 ,*ildr5 :o ?us compete ) _! own pa/ records1 keep+ an exact score f week to week1 make notable progress & enjoy h>d "w 9 mak+ x4 ,,! ,,ord] ,,( ,,developm5t ,,( ,,orig9al ,,t5d5cies ,negatively ! difficulty ( ! "w t pupils %\ld be expect$ to d is condition$ by ! gradual matur+ ( _! capacities4 ,o!r ?+s be+ equal1 ! common cu/om ( res]v+ h>d ?+s = late 9 ! elem5t>y s*ool c\rse is1 ( c\rse1 s\nd4 ,x seems probable t little is ga9$ by us+ any ( ! *ild's "t = ari?metic be=e grade #b1 ?\gh !re are _m ari?metical facts t he c le>n 9 grade #a4 ,po/ponem5t ( sy/ematic "w 9 ari?metic to grade #c or ev5 grade #d is all[able if bett] ?+s are (f]$4 ,) prop] textbooks & oral & writt5 ex]cises1 h[ev]1 a *ild 9 grades #b & #c c sp5d "t pr(itably on ari?metical "w4 ,:5 all *ildr5 c be held 9 s*ool ?r\gh ! eigh? grade x does n mu* matt] :e!r ari?metic is begun e>ly or late4 ,if1 h[ev]1 _m *ildr5 are to leave 9 grades #e & #f z n[1 we may ?9k x wise to provide someh[ t c]ta9 m9ima ( ari?metical ability be giv5 !m4 ,!re are1 s f> z is kn[n1 no special times & seasons at :i* ! human animal by inn] gr[? is specially ripe = "o or ano!r section or aspect ( ari?metic1 except 9 s f> z ! g5]al inn] gr[? ( intellectual p[]s makes ! m ab/ruse & complex tasks suitable to lat] & lat] ye>s4 ,inde$1 v few ( ev5 ! mo/ en?usia/ic devotees ( ! recapitulation !ory or culture-epo* !ory h attempt$ to apply ei!r to ! le>n+ ( ari?metic1 & ,bran=d is ! only ma!matician1 s f> z ,i "k1 :o has advocat$ su* application1 ev5 temp]$ by elaborate %ift+s & rev]sals ( ! racial ord]4 ,he says3-- ,?us1 = ea* age ( ! individual life--infancy1 *ildhood1 s*ool1 college--may be select$ f ! racial hi/ory ! mo/ appropriate =m 9 :i* ma!matical exp]i5ce c be assimilat$4 ,?us ! capacity ( ! infant & e>ly *ildhood is comp>able ) ! capacity ( animal consci\sness & primitive man4 ,! ma!matics suitable to lat] *ildhood & boyhood 7&1 ( c\rse1 girlhood7 is comp>able ) ,ar*aean ma!matics pass+ on ?r\gh ,greek & ,h9du to m$iaeval ,european ma!matics2 :ile ! /ud5t is become suffici5tly mature to beg9 ! assimilation ( mod]n & highly ab/ract ,european ?\ght4 ,! fill+ 9 ( details mu/ necess>ily be left to ! individual tea*]1 & also1 )9 "s su* broadly m>k$ limits1 ! precise ord] ( ! m>%all+ ( ! mat]ial = ea* age4 ,=1 ?\gh1 on ! :ole1 ma!matical developm5t has gone =w>d1 yet !re h be5 lapses f advances already made4 ,witness ! practical _w-loss ( mu* valuable ,h9du ?\ght1 &1 = long c5turies1 ! neglect ( ,greek ?\ght3 witness ! _w-loss ( ! inv5tion by ! ,babylonians ( ! ,z]o1 until re-inv5t$ by ! ,h9dus1 pass$ on by !m to ! ,arabs1 & by ^! to ,europe4 ,moreov]1 _m blund]s & false />ts & false pr9ciples h m>k$ ! :ole c\rse ( developm5t4 ,9 a phrase1 riv]s h _! backwat]s4 ,b x is precisely ! tea*]'s function to avoid su* racial mi/akes1 to take %ort cuts ultimately discov]$1 & to guide ! "y along ! road ultimately f\nd mo/ accessible ) su* halts & retrac+s--returns up side-cuts--z ! m5tal peculi>ities ( ! pupils dem&4 ,all ?is1 ! practical realization ( ! _s ( ! pr9ciple1 is to be wisely left to ! ma!matical tea*]1 famili> ) ! hi/ory ( ma!matical sci5ce & ) ! p>ticul> limitations ( 8 pupils & himself4 '#jh1 p4 #bde4 ,! latitude ( modification sugge/$ by ,bran=d r$uces ! guidance to be d]iv$ f racial hi/ory to almo/ nil4 ,also x is app>5t t ! racial hi/ory 9 ! case ( ari?metical a*ievem5t is entirely a matt] ( acquisition & social transmission4 ,man's orig9al nature is de/itute ( all ari?metical ideas4 ,! human g]ms d n "k ev5 t "o & "o make two6 ,,inv5tories ,,( ,,ari?metical ,,k ,,& ,,skill ,a sci5tific plan = tea*+ ari?metic w\ld beg9 ) an exact inv5tory ( ! k & skill :i* ! pupils already possess$4 ,our ord9>y notions ( :at a *ild kn[s at entrance to grade #a1 or grade #b1 or grade #c1 & ( :at a fir/-grade *ild or second-grade *ild c d1 are n adequate4 ,if !y 71 we %\ld n f9d reputable textbooks arrang+ to tea* elaborately facts already suffici5tly well kn[n to ov] ?ree qu>t]s ( ! pupils :5 !y ent] s*ool4 ,nor %\ld we f9d o!r textbooks presuppos+ 9 _! fir/ fifty pages a k ( words :i* n half ( ! *ildr5 c read ev5 at ! end ( ! #b ,b grade4 ,we d f9d j su* evid5ce t ord9>y ideas ab\t ! abilities ( *ildr5 at ! beg9n+ ( sy/ematic s*ool tra9+ 9 ari?metic may be 9 gross error4 ,= example1 a reputable & 9 _m ways admirable rec5t book has f\rte5 pages ( ex]cises to tea* ! mean+ ( two & ! fact t "o & "o make two6 ,z an example ( ! rev]se error1 consid] putt+ all ^! words 9 ! fir/ tw5ty-five pages ( a beg9n]'s book3--abs5tees1 att5dance1 blanks1 cont9ue1 copy1 dur+1 examples1 gr\p$1 memorize1 p]fect1 simil>1 spl9ts1 !re=e1 total6 ,little1 almo/ no?+1 has be5 done t[>d provid+ an exact inv5tory comp>$ ) :at ne$s to be done4 ,we may note "h 7#a7 ! facts relevant to ari?metic f\nd by ,/anley ,hall1 ,h>tmann1 & o!rs 9 _! g5]al inve/igations ( ! k possess$ by *ildr5 at entrance to s*ool1 7#b7 ! facts conc]n+ ! p[] ( *ildr5 to p]ceive diff]5ces 9 l5g?1 area1 size ( collection1 & organization )9 a collection su* z is %[n 9 ,fig4 #bd1 & c]ta9 facts & !ories ab\t e>ly aw>5ess ( numb]4 ,9 ! ,b]l9 inquiry ( #ahfi1 k ( ! mean+ ( two1 ?ree1 & f\r appe>$ 9 #gd1 #gd1 & #gc p]c5t ( ! *ildr5 ^u entrance to s*ool4 ,"s ( ^? record$ z ignorant probably really knew1 b fail$ to und]/& t !y 7 expect$ to reply or 7 %y4 ,only #he p]c5t 7 record$ z kn[+ _! fa!rs' names4 ,sev5 eigh?s z _m *ildr5 knew ! mean+s ( two1 ?ree1 & f\r z knew _! fa!rs' names4 ,9 a simil> b m c>eful exp]im5t ) ,bo/on *ildr5 9 ,septemb]1 #ahhj1 ,/anley ,hall f\nd t #ib p]c5t knew ?ree1 #hc p]c5t knew f\r1 & #ga-a_/b p]c5t knew five4 ,?ree 0 kn[n ab\t z well z ! color r$2 f\r 0 kn[n ab\t z well z ! color blue or yell[ or gre54 ,h>tmann '#ij f\nd t two ?irds ( ! *ildr5 ent]+ s*ool 9 ,annab]g c\ld c\nt f "o to t54 ,?is is ab\t z _m z knew money1 or ! famili> objects ( ! t[n1 or c\ld repeat words spok5 to !m4 ,illu/ration3 ,,fig4 #bd4--,objective pres5tation4 ,9 ! ,/an=d =m ( ! ,b9et te/s c\nt+ f\r p5nies is giv5 z an ability ( ! typical f\r-ye>-old4 ,c\nt+ #ac p5nies correctly 9 at lea/ "o out ( two trials1 & kn[+ ?ree ( ! f\r co9s1--p5ny1 nickel1 dime1 & qu>t]1--are giv5 z abilities ( ! typical six-ye>-old4 ,,! ,,p]ception ,,( ,,numb] ,,& ,,quantity ,we "k t educat$ adults c tell h[ _m l9es or dots1 etc41 !y see 9 a s+le glance 7) an exposure too %ort = ! eye to move7 up to f\r or m1 accord+ to ! cle>ness ( ! objects & _! gr\p+4 ,= example1 ,nanu '#jd reports t :5 a numb] ( bright circles on a d>k backgr\nd are %[n to educat$ adults = only 4#jcc second1 t5 c be c\nt$ :5 arrang$ to =m a p>allelogram1 b only five :5 arrang$ 9 a r[4 ,) c]ta9 gr\p+s1 ( c\rse1 _! 'p]ception' involves mu* inf]5ce1 ev5 consci\s addition & multiplication4 ,simil>ly !y c tell1 up to tw5ty & beyond1 ! numb] ( taps1 notes1 or o!r s\nds 9 a s]ies too rapid = s+le c\nt+ if ! s\nds are gr\p$ 9 a conv5i5t rhy?m4 ,^! abilities are1 h[ev]1 ! product ( a long & elaborate le>n+1 includ+ ! le>n+ ( ari?metic itself4 ,elem5t>y psy*ology & common exp]i5ce tea* u t ! m]e obs]vation ( gr\ps or quantities1 no matt] h[ cle> _! numb] quality appe>s to ! p]son :o already kn[s ! mean+s ( numb]s1 does n ( itself create ! k ( ! mean+s ( numb]s 9 "o :o does n4 ,! exp]im5ts ( ,mess5g] '#jc & ,burnett '#jf %[$ t !re is no direct intuitive appreh5sion ev5 ( two z di/9ct f "o4 ,we h to le>n to feel ! two t\*es or see ! two dots or l9es z two4 ,we d n "k by exact measurem5ts ! gr[? 9 *ildr5 ( ?is ability to c\nt or inf] ! numb] ( elem5ts 9 a collection se5 or s]ies he>d4 ,/ill less d we "k :at ! gr[? w\ld be )\t ! influ5ce ( s*ool tra9+ 9 c\nt+1 gr\p+1 add+1 & multiply+4 ,_m textbooks & tea*]s seem to ov]e/imate x greatly4 ,n all educat$ adults c1 ap>t f measurem5t1 decide ) surety :i* ( ^! l9es is ! long]1 or :i* ( ^! areas is ! l>g]1 or :e!r ?is is a n9? or a t5? or an elev5? ( a circle4 ,illu/ration ,*ildr5 ^u ent]+ s*ool h n be5 te/$ c>efully 9 respect to judgm5ts ( l5g? & area1 b we "k f su* /udies z ,gilb]t's '#id t ! diff]5ce requir$ 9 _! case is probably ov] twice t requir$ = *ildr5 ( #ac or #ad4 ,9 judg+ weights1 = example1 a diff]5ce ( #f is p]ceiv$ z easily by *ildr5 #ac to #ae ye>s ( age z a diff]5ce ( #ae by six-ye>-olds4 ,a tea*] :o has adult p[]s ( e/imat+ l5g? or area or weight & :o also kn[s already :i* ( ! two is long] or l>g] or heavi]1 may use two l9es to illu/rate a diff]5ce :i* !y really hide f ! *ild4 ,x is unlikely1 = example1 t ! fir/ ( ^! l9es w\ld be recogniz$ z %ort] ?an ! second by e *ild 9 a f\r?-grade class1 & x is extremely unlikely t x w\ld be recogniz$ z be+ #g_/h ( ! l5g? ( ! latt]1 r ?an #c_/d ( x or #e_/f ( x or #i_/aj ( x or #aa_/ab ( x4 ,if ! two 7 %[n to a second grade1 ) ! "q1 ,! fir/ l9e is #g4 ,h[ long is ! o!r l9e8 !re w\ld be v _m answ]s ( #g or #i2 & ^! might be entirely correct ari?metically1 ! pupils' errors be+ all due to _! inability to comp>e ! l5g?s accurately4 ,a ,b ,c ------- -------- ,d ,e 4'_/4 4'_/4 _/_/ _/_/ ----------- ----------- _/_/ _/_/ 4_/4' 4_/4' ,! quantities us$ %\ld be su* t _! m]e discrim9ation (f]s no difficulty ev5 to a *ild ( blunt$ s5se p[]s4 ,if #g_/h & #a are to be comp>$1 ,a & ,b are n all[able4 ,c1 ,d1 & ,e are mu* bett]4 ,tea*]s probably (t5 und]e/imate or neglect ! s5sory difficulties ( ! tasks !y assign & ( ! mat]ial !y use to illu/rate absolute & relative magnitudes4 ,! result may be m p]nici\s :5 ! pupils answ] correctly ?an :5 !y fail4 ,= _! correct answ]+ may be due to _! div9ation ( :at ! tea*] wants2 & !y may call a ?+ an in* l>g] to suit h] :i* does n really seem l>g] to !m at all4 ,?is1 ( c\rse1 is utt]ly de/ructive ( _! respect = ari?metic z an exact & matt]-(-fact in/rum5t4 ,= example1 if a tea*] drew a s]ies ( l9es #bj1 #ba1 #bb1 #bc1 #bd1 & #be in*es long on ! blackbo>d 9 ?is =m-- & ask$1 ,?is is #bj in*es long1 h[ long is ?is8 %e might1 aft] "s errors & correction !re(1 f9ally secure successful response to all ! l9es by all ! *ildr54 ,b _! appreciation ( ! numb]s #bj1 #ba1 #bb1 #bc1 #bd1 & #be w\ld be actually damag$ by ! ex]cise4 ,,! ,,e>ly ,,aw>5ess ,,( ,,numb] ,!re has be5 "s disagreem5t conc]n+ ! orig9 ( aw>5ess ( numb] 9 ! individual1 9 p>ticul> conc]n+ ! relative importance ( ! p]ception ( h[-_m-ness & t ( h[-mu*-ness1 ( ! p]ception ( a def9$ aggregate & ! p]ception ( a def9$ ratio4 7,see ,mc,lellan & ,dewey '#ie1 ,phillips '#ig & '#ih1 & ,decroly & ,deg& '#ab47 ,! *ief facts ( significance = practice seem to be ^!3 7#a7 ,*ildr5 ) r>e exceptions he> ! names "o1 two1 ?ree1 f\r1 half1 twice1 two times1 m1 less1 z _m z1 aga91 fir/1 second1 & ?ird1 long be=e !y h analyz$ out ! qualities & relations to :i* ^! words ref] s z to feel !m at all cle>ly4 7#b7 ,_! k ( ! qualities & relations is develop$ 9 ! ma9 9 close association ) ! use ( ^! words to ! *ild & by ! *ild4 7#c7 ,! ord9>y exp]i5ces ( ! fir/ five ye>s s develop 9 ! *ild aw>5ess ( ! 'h[ _m some?+s' 9 v>i\s gr\ps1 ( ! relative magnitudes ( two gr\ps or quantities ( any sort1 & ( gr\ps & magnitudes z relat$ to o!rs 9 a s]ies4 ,= in/ance1 if fairly gift$1 a *ild comes1 by ! age ( five1 to see t a r[ ( f\r cakes is an aggregate ( f\r1 see+ ea* cake z a "p ( ! f\r & ! f\r z ! sum ( its p>ts1 to "k t two ( !m are z _m z ! o!r two1 t half ( !m w\ld be two1 & to ?9k1 :5 x is useful = him to d s1 ( f\r z a /ep beyond ?ree on ! way to five1 or to ?9k ( hot z a /ep f w>m on ! way to v hot4 ,! degree ( developm5t ( ^! abilities dep5ds ^u ! activity ( ! law ( analysis 9 ! individual & ! "* ( 8 exp]i5ces4 7#d7 ,he gets c]ta9 bad habits ( response f ! ambiguity ( common usage ( #b1 #c1 #d1 etc41 = second1 ?ird1 f\r?4 ,?us he sees or he>s 8 p>5ts or old] *ildr5 or o!rs c\nt p5nies or rolls or eggs by say+ "o1 two1 ?ree1 f\r1 & s on4 ,he himself is p]haps misl$ into s c\nt+4 ,?us ! names prop]ly belong+ to a s]ies ( aggregations v>y+ 9 am\nt come to be to him ! names ( ! positions ( ! p>ts 9 a c\nt$ :ole4 ,?is happ5s especially ) numb]s above #c or #d1 :]e ! correct exp]i5ce ( ! numb] z a "n = ! gr\p has r>ely be5 pres5t4 ,?is atta*+ to ! c>d9al numb]s above ?ree or f\r ! mean+s ( ! ord9al numb]s seems to affect _m *ildr5 on entrance to s*ool4 ,! numb]+ ( pages 9 books1 h\ses1 /reets1 etc41 & bad tea*+ ( c\nt+ (t5 prolong ?is error4 7#e7 ,he also gets ! habit1 n necess>ily bad1 b (t5 indirectly s1 ( us+ _m names su* z eight1 n9e1 t51 elev51 fifte51 a hundr$1 a million1 )\t any mean+4 7#f7 ,! exp]i5ces ( half1 twice1 ?ree times z _m1 ?ree times z long1 etc41 are r>]2 ev5 if !y 7 n1 !y w\ld /ill be less easily productive ( ! analysis ( ! prop] ab/ract elem5t ?an are ! exp]i5ces ( two1 ?ree1 f\r1 etc41 9 connection ) aggregates ( ?+s ea* ( :i* is usually call$ "o1 su* z boys1 girls1 balls1 apples4 ,exp]i5ces ( ! names1 two1 ?ree1 & f\r1 9 connection ) two twos1 two ?rees1 two f\rs1 are v r>e4 ,h5ce1 ! names1 two1 ?ree1 etc41 mean to ^! *ildr5 9 ! ma91 "o some?+ & "o some?+1 "o some?+ usually call$ "o1 & "o some?+ usually call$ "o1 & ano!r some?+ usually call$ "o1 & m r>ely & imp]fectly two times any?+1 ?ree times any?+1 etc4 ,) respect to ,mr4 ,phillips' emphasis ( ! importance ( ! s]ies-idea 9 *ildr5's m9ds1 ! matt]s ( importance are3 fir/1 t ! k ( a s]ies ( numb] names 9 ord] is ( v little consequ5ce to ! tea*+ ( ari?metic & ( /ill less to ! orig9 ( aw>5ess ( numb]4 ,second1 ! habit ( apply+ ?is s]ies ( words 9 c\nt+ 9 su* a way t #h is associat$ ) ! eigh? ?+1 #i ) ! n9? ?+1 etc41 is ( consequ5ce because x does s mu* mis*ief4 ,?ird1 ! really valuable idea ( ! numb] s]ies1 ! idea ( a s]ies ( gr\ps or ( magnitudes v>y+ by /eps1 is acquir$ lat]1 z a result1 n a cause1 ( aw>5ess ( numb]s4 ,) respect to ! ,mc,lellan-,dewey doctr9e1 ! ratio aspect ( numb]s %\ld be emphasiz$ 9 s*ools1 n because x is ! ma9 orig9 ( ! *ild's aw>5ess ( numb]1 b because x is n1 & because ! ord9>y practical issues ( *ild life d n adequately /imulate its action4 ,x also seems bo? m economical & m sci5tific to introduce x ?r\gh multiplication1 division1 & fractions r ?an to insi/ t #d & #e %all f ! />t mean #d or #e times any?+ t is call$ #a1 = in/ance1 t #h in*es %all be call$ #d two-in*es1 or #aj c5ts1 #e two-c5ts4 ,if ,i int]pret ,pr(essor ,dewey's writ+s correctly1 he w\ld agree t ! use ( in*1 foot1 y>d1 p9t1 qu>t1 ounce1 p\nd1 glassful1 cupful1 h&ful1 spoonful1 c5t1 nickel1 dime1 & doll> gives a suffici5t range ( units = ! fir/ two s*ool ye>s4 ,tea*+ ! mean+s ( #a_/b ( #d1 #a_/b ( #f1 #a_/b ( #h1 #a_/b ( #aj1 #a_/b ( #bj1 #a_/c ( #f1 #a_/c ( #i1 #a_/c ( #cj1 #a_/d ( #h1 two #bs1 five #bs1 & ! l1 9 e>ly grades1 ea* 9 connection ) _m diff]5t units ( measure1 provides a suffici5t assurance t numb]s w connect ) relation%ips z well z ) collections4 ,,*apt] ,,xii ,,int]e/ ,,9 ,,ari?metic ,,c5suses ,,( ,,pupils' ,,int]e/s ,ari?metic1 al?\gh x makes little or no appeal to collect+1 muscul> manipulation1 s5sory curiosity1 or ! pot5t orig9al int]e/s 9 ?+s & _! me*anisms & p & _! passions1 is fairly well lik$ by *ildr54 ,! c5suses ( pupils' likes & dislikes t h be5 made are n models ( sci5tific inve/igation1 & ! result+ p]c5tages %\ld n be us$ uncritically4 ,!y are1 h[ev]1 probably n on ! av]age ov]-favorable to ari?metic 9 any unfair way4 ,"s ( _! results are summ>iz$ bel[4 ,9 g5]al !y %[ ari?metic to be surpass$ 9 int]e/ cle>ly by only ! manual arts 7%opwork & manual tra9+ = boys1 cook+ & sew+ = girls71 draw+1 c]ta9 =ms ( gymna/ics1 & hi/ory4 ,x is ab\t on a level ) read+ & sci5ce4 ,x cle>ly surpasses gramm>1 language1 spell+1 geography1 & religion4 ,lobsi5 '#jc1 :o ask$ "o hundr$ *ildr5 9 ea* ( ! fir/ five grades 7,/uf57 ( ! elem5t>y s*ools ( ,kiel1 ,:i* "p ( ! s*ool "w 7lit]ally1 ':i* in/ruction p]iod'7 d y l be/8 f\nd ari?metic l$ only by draw+ & gymna/ics 9 ! case ( ! boys1 & only by h&work 9 ! case ( ! girls4 ,?is is an exagg]at$ picture ( ! facts1 s9ce no c\nt is made ( ^? :o especially dislike ari?metic4 ,ari?metic is z unpopul> ) "s z x is popul> ) o!rs4 ,:5 full all[ance is made = ?is1 ari?metic /ill has popul>ity above ! av]age4 ,/]n '#je ask$1 ,:i* subject d y l mo/8 & ,:i* subject d y l lea/8 ,! balance 0 greatly 9 favor ( gymna/ics = boys 7#bh--a71 h&work = girls 7#cb--a-a_/b71 & draw+ = bo? 7#af-a_/b--f74 ,writ+ 7#f-a_/b--d71 ari?metic 7#ad-a_/b--ac71 hi/ory 7#i--f-a_/b71 read+ 7#h-a_/b--h71 & s++ 7#f--g-a_/b7 come next4 ,religion1 nature /udy1 physiology1 geography1 geometry1 *emi/ry1 language1 & gramm> are l[4 ,mc,knight '#jg f\nd ) boys & girls 9 grades #g & #h ( c]ta9 ,am]ican cities t ari?metic 0 lik$ bett] ?an any ( ! s*ool subjects except gymna/ics & manual tra9+4 ,! vote z comp>$ ) hi/ory 03-- ,ari?metic #cbg lik$ greatly1 #if dislik$ greatly4 ,hi/ory #afd lik$ greatly1 #aac dislik$ greatly4 ,9 a lat] /udy ,lobsi5 '#ji _h #fbdh pupils f #i to #ae ye>s old repres5t+ all grades ( ! elem5t>y s*ool report1 s f> z !y c\ld1 ! subject mo/ dislik$1 ! subject mo/ lik$1 ! subject next mo/ lik$1 & ! subject next 9 ord]4 ,no *ild 0 =c$ to report all ( ^! f\r judgm5ts1 or ev5 any ( !m4 ,lobsi5 c\nts ! likes & ! dislikes = ea* subject4 ,gymna/ics1 h&work1 & cook+ are by f> ! mo/ popul>4 ,hi/ory & draw+ are next1 foll[$ by ari?metic & read+4 ,bel[ ^! are geography1 writ+1 s++1 nature /udy1 biblical hi/ory1 cate*ism1 & ?ree m9or subjects4 ,lewis '#ac secur$ records f ,engli% *ildr5 9 elem5t>y s*ools ( ! ord] ( pref]5ce ( all ! /udies li/$ bel[4 ,he reports ! results 9 ! foll[+ table ( p]c5ts3 ,,top ,,?ird ,,( ,,middle ,,?ird ,,( ,,l[e/ ,,?ird ,,( ,,/udies ,,= ,,/udies ,,= ,,/udies ,,= ,,int]e/ ,,int]e/ ,,int]e/ ---------------------------------------- ------------------------ ,draw+ #gh #bj #b ,manual ,subjects #ff #bf #h ,hi/ory #fd #bd #ab ,read+ #ec #ch #i ,s++ #cb #dh #bj ,drill #bj #ee #be ,ari?metic #af #ec #ca ,sci5ce #bc #cg #dj ,nature ,/udy #af #cf #dh ,dictation #d #eg #ci ,composition #ah #bh #ed ,scripture #d #ch #eh ,recitation #i #bc #fh ,geography #d #bd #gb ,gramm> -- #f #id ,br&ell '#ac obta9$ data f #bacg ,sw$i% *ildr5 9 ,/ockholm 7#cbg71 ,norrkoep+ 7#hgj71 & ,go!nburg 7#idj74 ,9 g5]al he f\nd1 z o!rs h1 t h&work1 %opwork = boys & h\sehold "w = girls1 & draw+ 7 report$ z mu* bett] lik$ ?an ari?metic4 ,s also 0 hi/ory1 & 79 ?is he diff]s f mo/ /ud5ts ( ?is matt]7 s 7 read+ & nature /udy4 ,gymna/ics he f9ds less lik$ ?an ari?metic4 ,religion1 geography1 language1 spell+1 & writ+ are1 z 9 o!r /udies1 mu* less popul> ?an ari?metic4 ,o!r /udies are by ,lilius '#aa 9 ,f9l&1 ,walsemann '#jg1 ,wi$]kehr '#jg1 ,pomm] '#ad1 ,seekel '#ad1 & ,/]n '#ac & '#ad1 9 ,g]many4 ,!y confirm ! g5]al results /at$4 ,! reasons = ! good %[+ t ari?metic makes are probably ! /r5g? ( its appeal to ! int]e/ 9 def9ite a*ievem5t1 success1 do+ :at "o attempts to d2 & ( its appeal1 9 grades #e to #h1 to ! practical int]e/ ( gett+ on 9 ! _w1 acquir+ abilities t ! _w pays =4 ,( ^!1 ! =m] is 9 my op9ion mu* ! m pot5t int]e/4 ,ari?metic satisfies x especially well1 because1 m ?an any o!r ( ! 'intellectual' /udies ( ! elem5t>y s*ool1 x p]mits ! pupil to see 8 own progress & det]m9e 8 own success or failure4 ,! mo/ important applications ( ! psy*ology ( satisfi]s & annoy]s to ari?metic w !re=e be 9 ! direction ( utiliz+ /ill m effectively ?is int]e/ 9 a*ievem5t4 ,next 9 importance come ! plans to atta* to ari?metical le>n+ ! satisfy+ness ( bodily action1 play1 sociability1 *e]fulness1 & ! l1 & ( significance z a means ( secur+ o!r desir$ ends ?an ari?metical abilities !mselves4 ,next come plans to relieve ari?metical le>n+ f c]ta9 discom=ts su* z ! eye/ra9 ( "s computations & excessive copy+ ( figures4 ,^! w be discuss$ "h 9 ! inv]se ord]4 ,,reliev+ ,,eye/ra9 ,at pres5t ari?metical "w is1 h\r = h\r1 probably m ( a tax ^u ! eyes ?an read+4 ,! task ( copy+ numb]s f a book to a %eet ( pap] is "o ( ! v h>de/ tasks t ! eyes ( a pupil 9 ! elem5t>y s*ools h to p]=m4 ,a c]ta9 am\nt ( su* "w is desirable to tea* a *ild to write numb]s1 to copy exactly1 & to organize mat]ial 9 %ape = computation4 ,b beyond t1 !re is no m reason = a pupil to copy e numb] ) :i* he is to compute ?an = him to copy e ^w he is to read4 ,! mean+less drudg]y ( copy+ figures %\ld be mitigat$ by arrang+ mu* "w 9 ! =m ( ex]cises l ^? %[n on pages #baf1 #bag1 & #bah1 & by hav+ _m ( ! textbook examples 9 addition1 subtraction1 & multiplication done ) a slip ( pap] laid bel[ ! numb]s1 ! answ]s be+ writt5 on x4 ,!re is n only a result+ ga9 9 int]e/1 b also a v great sav+ ( "t = ! pupil 7v (t5 copy+ an example m ?an quadruples ! "t requir$ to get its answ]71 & a mu* great] effici5cy 9 sup]vision4 ,ari?metical errors are n confus$ ) errors ( copy+1#af & ! tea*]'s task ( foll[+ a pupil's "w on ! page is r$uc$ to a m9imum1 ea* pupil hav+ put ! same "p ( ! day's "w 9 j ! same place4 ,! use ( well-pr9t$ & well-spac$ pages ( ex]cises relieves ! eye/ra9 ( work+ ) badly made gray figures1 unev5ly & too closely or too widely spac$4 ,i reproduce 9 ,fig4 #be specim5s tak5 at r&om f "o hundr$ r&om samples ( ari?metical "w by pupils 9 grade #h4 ,contra/ ! task ( ! eyes 9 work+ ) ^! & _! task 9 work+ ) pages #baf to #bah4 ,! cu/om>y me?od ( always copy+ ! numb]s to be us$ 9 computation f blackbo>d or book to a %eet ( pap] is an utt]ly unju/ifiable cruelty & wa/e4 #af ,c\rtis f9ds 9 ! case ( addition t ( all ! individuals mak+ mi/akes at any giv5 "t 9 a class1 at lea/ "o ?ird1 & usually two ?irds1 w be mak+ mi/akes 9 c>ry+ or copy+4 ,illu/ration3 ,,fig4 #be;a4--,specim5s tak5 at r&om f ! computation "w ( eigh?-grade pupils4 ,?is computation occurr$ 9 a g5u9e te/4 ,9 ! orig9al gray ( ! p5cil m>ks ! "w is /ill h>d] to make out4 ,illu/ration3 ,,fig4 #be;b4--,specim5s tak5 at r&om f ! computation "w ( eigh?-grade pupils4 ,?is computation occurr$ 9 a g5u9e te/4 ,9 ! orig9al gray ( ! p5cil m>ks ! "w is /ill h>d] to make out4 ,write ! 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 ! miss+ numb]s3 7r /&s = rema9d]47 #be 4444 #cs & 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 & 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 & 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 & 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 ! :ole numb]s or mix$ numb]s :i* ^! 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 ! miss+ figures3-- #f #b #h #a #b - - - - -- - - -- - - #h #d #d #b #aj #e #e #aj #c #f ,write ! miss+ num]ators3-- #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 ,f9d ! products4 ,cancel :5 y c3-- #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 ,,= ,,relat$ ,,activities ,! use ( bodily action1 social games1 & ! l 0 discuss$ 9 ! section on orig9al t5d5cies4 ,significance z a means ( secur+ o!r desir$ ends ?an ari?metical le>n+ itself is !re=e our next topic4 ,su* significance c be giv5 to ari?metical "w by us+ t "w z a means to pres5t & future success 9 problems ( sports1 h\sekeep+1 %opwork1 dressmak+1 self-managem5t1 o!r s*ool /udies ?an ari?metic1 & g5]al s*ool life & affairs4 ,significance z a means to future ends alone c also be m cle>ly & ext5sively atta*$ to x ?an x n[ is4 ,:atev] is done to supply great] /r5g? ( motive 9 /udy+ ari?metic mu/ be c>efully devis$ s z n to get a /rong b wrong motive1 s z n to get abundant int]e/ b 9 some?+ o!r ?an ari?metic1 & s z n to kill ! goose t aft] all lays ! gold5 eggs--! int]e/ 9 intellectual activity & a*ievem5t itself4 ,x is easy to secure an int]e/ 9 lay+ out a baseball diamond1 measur+ +r$i5ts = a cake1 mak+ a balloon ( a c]ta9 capacity1 or decid+ ! add$ co/ ( an extra trimm+ ( ribbon = "o's dress4 ,! problem is to atta* t int]e/ to ari?metical le>n+4 ,nor %\ld a tea*] be satisfi$ ) atta*+ ! int]e/ z a m]e tail t /e]s ! kite1 s long z x /ays on1 or z a sug>-coat+ t deceives ! pupil into swall[+ ! pill1 or z an anodyne ^: dose mu/ be increas$ & increas$ if x is to reta9 its p[]4 ,until ! int]e/ p]meates ! ari?metical activity itself our task is only p>tly done1 & p]haps is made h>d] = ! next "t4 ,"o important means ( really int]fus+ ! ari?metical le>n+ itself ) ^! d]iv$ int]e/s is to lead ! pupil to seek ! help ( ari?metic himself--to lead him1 9 ,dewey's phrase1 to 'feel ! ne$'--to take ! 'problem' attitude--& ?us appreciate ! te*nique :i* he actively hunts = to satisfy ! ne$4 ,9 s f> z ari?metical le>n+ is organiz$ to satisfy ! practical dem&s ( ! pupil's life at ! "t1 he %\ld1 s to speak1 come "p way to get its help4 ,ev5 if we d n make ! mo/ skillful use possible ( ^! int]e/s d]iv$ f ! quantitative problems ( sports1 h\sekeep+1 %opwork1 dressmak+1 self-managem5t1 o!r s*ool /udies1 & s*ool life & affairs1 ! ga9 w /ill be consid]able4 ,to h !m 9 m9d w c]ta9ly pres]ve u f giv+ to *ildr5 ( grades #c & #d problems s devoid ( relation to _! int]e/s z ^? %[n bel[1 all f\nd 79 #aiaj7 9 ?irty successive pages ( a book ( excell5t repute3-- ,a *air has #d legs4 ,h[ _m legs h #h *airs8 #e *airs8 ,a fly has #f legs4 ,h[ _m legs h #c flies8 #i flies8 #g flies8 7,eight m ( ! same sort47 ,9 #ahij ,new ,york _h #a1eac1eja inhabitants1 ,milwaukee _h #bjf1cjh1 ,bo/on _h #ddg1gbj1 ,san ,francisco #big1iij4 ,h[ _m _h ^! cities toge!r8 7,five m ( ! same sort47 ,milton 0 born 9 #afjh & di$ 9 #afgd4 ,h[ _m ye>s did he live8 7,sev]al o!rs ( ! same sort47 ,! population ( a c]ta9 city 0 #ce1fbi 9 #ahhj & #ajf1fgj 9 #ahij4 ,f9d ! increase4 7,sev]al o!rs ( ?is sort47 ,a numb] ( o!rs ab\t ! words 9 v>i\s inaugural addresses & ! ,psalms 9 ! ,bible4 ,x also seems probable t ) en\gh c>e o!r sy/ematic plans ( textbooks c be mu* improv$ 9 ?is respect4 ,f e po9t ( view1 = example1 ! e>ly "w 9 ari?metic %\ld be adapt$ to "s ext5t to ! heal?y *ildi% int]e/s 9 home affairs1 ! behavior ( o!r *ildr51 & ! activities ( mat]ial ?+s1 animals1 & plants4 ,,table #i ,,frequ5cy ,,( ,,appe>ance ,,( ,,c]ta9 ,,words ,,ab\t ,,family ,,life1 ,,play1 ,,& ,,action ,,9 ,,eight ,,elem5t>y ,,textbooks ,,9 ,,ari?metic1 pp4 #a-ej4 ,a ,b ,c ,d ,e ,f ,g ,h ---------------------------------------- ---------------- baby #b #d bro!r #b #f #a #a #a family #b #b #d fa!r #a #c #e #b #a help home #b #d #d #b #b #g #a "m #d #b #i #e #e #a #g si/] #a #b #b #i #a #a =k knife plate #d #b #b #a spoon doll #aj #a #aj #f #aj #i game #a #c #e #e jump #d m>bles #aj #d #aj #aj #a play #a #c run #a #c s+ tag toy #a c> #b #d #b #c #a cut #aj #f #b #h dig #b fl[] #a #d #a #a #b gr[ #a plant #b se$ #c #a /r+ #a #aj #a #a :eel #e #aj ,! words us$ by textbooks give "s indication ( h[ f> ?is aim is be+ realiz$1 or r ( h[ f> %ort we are ( realiz+ x4 ,consid]1 = example1 ! words home1 "m1 fa!r1 bro!r1 si/]1 help1 plate1 knife1 =k1 spoon1 play1 game1 toy1 tag1 m>bles1 doll1 run1 jump1 s+1 plant1 se$1 gr[1 fl[]1 c>1 :eel1 /r+1 cut1 dig4 ,! frequ5cy ( appe>ance 9 ! fir/ fifty pages ( eight beg9n]s' ari?metics 0 z %[n 9 ,table #i4 ,! eight columns ref] to ! eight books 7! fir/ fifty pages ( ea*74 ,! numb]s ref] to ! numb] ( times ! ^w 9 "q appe>$1 ! numb] #aj mean+ #aj or m times 9 ! fifty pages4 ,plurals1 pa/ t5ses1 & ! l 7 c\nt$4 ,help1 =k1 knife1 spoon1 jump1 s+1 & tag did n appe> at all6 ,toy & gr[ appe>$ ea* once 9 ! #djj pages6 ,play1 run1 dig1 plant1 & se$ appe>$ once 9 a hundr$ or m pages4 ,baby did n appe> z (t5 z buggy4 ,family appe>$ no (t5] ?an f5ce or ,friday4 ,fa!r appe>s ab\t a ?ird z (t5 z f>m]4 ,book ,a %[s only #aj ( ^! ?irty words 9 ! fifty pages2 book ,b only #d2 book ,c only #ab2 & books ,d1 ,e1 ,f1 ,g1 & ,h only #ac1 #h1 #ad1 #ac1 #aj1 respectively4 ,! total numb] ( appe>ances 7c\nt+ ! #ajs z only #aj 9 ea* case7 is #dj = ,a1 #i = ,b1 #fj = ,c1 #db = ,d1 #be = ,e1 #fb = ,f1 #cj = ,g1 & #cg = ,h4 ,! five words--apple1 egg1 ,m>y1 milk1 & orange--are us$ (t5] ?an all ^! ?irty toge!r4 ,if x appe>$ t ?is app>5t neglect ( *ildi% affairs & int]e/s 0 delib]ate to provide = a m sy/ematic treatm5t ( pure ari?metic1 a bett] gradation ( problems1 & a bett] prep>ation = lat] g5u9e use ?an c\ld be atta9$ if ! au?or ( ! textbook 7 ti$ to ! *ild's apron /r+s1 ! neglect c\ld be def5d$4 ,x is n at all c]ta9 t *ildr5 9 grade #b get mu* m enjoym5t or ability f add+ ! co/s ( pur*ases = ,*ri/mas or ,f\r? ( ,july1 or multiply+ ! numb] ( cakes ea* *ild is to h at a p>ty by ! numb] ( *ildr5 :o are to be !re1 ?an f add+ grave/ones or multiply+ ! numb] ( hairs ( bald-head$ m54 ,:51 h[ev]1 !re is no?+ ga9$ by sub/itut+ remote facts = ^? ( famili> conc]n to *ildr51 ! safe policy is surely to favor ! latt]4 ,9 g5]al1 ! neglect ( *ildi% data does n seem to be due to provision = "s o!r end1 b to ! same in]tia ( tradition :i* has c>ri$ ov] ! problems ( lay+ walls & digg+ wells into city s*ools ^: *ildr5 nev] saw a /one wall or dug well4 99 99 99 99 99 ,i %all n g into details conc]n+ ! arrangem5t ( c\rses ( /udy1 textbooks1 & lesson-plans to make desirable connections betwe5 ari?metical le>n+ & sports1 h\sework1 %opwork1 & ! re/4 ,x may be wor? :ile1 h[ev]1 to expla9 ! t]m self-managem5t1 s9ce ?is s\rce ( g5u9e problems ( real conc]n to ! pupils has be5 ov]look$ by mo/ writ]s4 ,by self-managem5t is meant ! pupil's use ( 8 "t1 8 abilities1 8 k1 & ! l4 ,by ! "t he rea*es grade #e1 & to "s ext5t be=e !n1 a boy %\ld keep "s acc\nt ( himself1 ( h[ long x takes him to d specifi$ tasks1 ( h[ mu* he gets done 9 a specifi$ "t at a c]ta9 sort ( "w & ) h[ _m errors1 ( h[ mu* improvem5t he makes mon? by mon?1 ( :i* ?+s he c d be/1 & ! l4 ,su* objective1 matt]-(-fact1 quantitative /udy ( "o's behavior is n a /imulus to morbid introspection or egotism2 x is "o ( ! be/ prev5tives ( ^!4 ,to treat oneself imp]sonally is "o ( ! ess5tial elem5ts ( m5tal balance & heal?4 ,x ne$ n1 & %\ld n1 enc\rage priggi%ness4 ,on ! contr>y1 ?is matt]-(-fact /udy ( :at "o is & does may well replace a c]ta9 am\nt ( ! exhortations & admonitions conc]n+ :at "o ought to d & be4 ,all ?is is /ill tru] = a girl4 ,! dem&s :i* su* an acc\nt+ ( "o's own activities make ( ari?metic h ! special value ( connect+ directly ) ! advanc$ "w 9 computation4 ,!y involve ! use ( l>ge numb]s1 decimals1 av]ag+1 p]c5tages1 approximations1 & o!r facts & processes :i* ! pupil has to le>n = lat] life1 b to :i* 8 *ildi% activities z wage-e>n]1 buy] & sell]1 or %opwork] f #aj to #ad d n lead4 ,*ildr5 h little money1 b !y h "t 9 ?\s&s ( units6 ,!y d n get disc\nts or bonuses f comm]cial h\ses1 b !y c disc\nt _! quantity ( examples done = ! errors made1 & cr$it !mselves ) bonuses ( all sorts = extra a*ievem5ts4 ,,intr9sic ,,int]e/ ,,9 ,,ari?metical ,,le>n+ ,!re rema9s ! mo/ important increase ( int]e/ 9 ari?metical le>n+--an increase 9 ! int]e/ directly b\nd to a*ievem5t & success 9 ari?metic itself4 ,ari?metic1 says ,david ,eug5e ,smi?1 is a game & all boys & girls are play]s4 ,x %\ld n be a m]e game = !m & !y %\ld n m]ely play1 b _! unpractical int]e/ 9 do+ x because !y c d x & c see h[ well !y d d x is "o ( ! s*ool's mo/ preci\s assets4 ,any heal?y means to give ?is int]e/ m & bett] /imulus %\ld !re=e be eag]ly s\ght & *]i%$4 ,two su* means h be5 sugge/$ 9 o!r connections4 ,! fir/ is ! ext5sion ( tra9+ 9 *eck+ & v]ify+ "w s t ! pupil may "w to a /&>d ( approximately #ajj success1 & may "k h[ ne>ly he is atta9+ x4 ,! second is ! use ( /&>diz$ practice mat]ial & te/s1 :]eby ! pupil may measure himself aga9/ 8 own pa/1 & h a cle>1 vivid1 & tru/wor?y idea ( j h[ mu* bett] or fa/] he c d ! same tasks ?an he c\ld d a mon? or a ye> ago1 & ( j h[ mu* h>d] ?+s he c d n[ ?an !n4 ,ano!r means ( /imulat+ ! ess5tial int]e/ 9 quantitative ?9k+ itself is ! arrangem5t ( ! "w s t real ari?metical ?9k+ is enc\rag$ m ?an m]e imitation & assiduity4 ,?is means ! avoidance ( long s]ies ( appli$ problems all ( "o type to be solv$ 9 ! same way1 ! avoidance ( miscellane\s s]ies & review s]ies :i* are almo/ v]batim repetitions ( pa/ problems1 & 9 g5]al ! avoidance ( excessive repetition ( any "o problem-situation4 ,/imulation to real ari?metical ?9k+ is weak :5 a :ole day's problem "w requires no *oice ( me?ods1 or :5 a review simply repeats )\t any /ep ( organization or progress1 or :5 a pupil meets a situation 7say ! 'buy x ?+s at y p] ?+1 h[ mu* pay' situation7 = ! five-hundr$? "t4 ,ano!r matt] wor?y ( att5tion 9 ?is connection is ! unwise t5d5cy to omit or pres5t 9 dilut$ =m "s ( ! topics t appeal mo/ to real intellectual int]e/s1 j because !y are h>d4 ,! be/ illu/ration1 p]haps1 is ! problem ( ratio or ,h[ _m times z l>ge 7long1 heavy1 exp5sive1 etc47 z x is y8 ,ma/]y ( ! 'times z' relation is h>d to acquire1 b x is well wor? acquir+1 n only because ( its /rong intellectual appeal1 b also because ( its prime importance 9 ! applications ( ari?metic to sci5ce4 ,9 ! old] ari?metics x 0 confus$ by p$antries & v]bal difficulties & p5aliz$ by unreal problems ab\t fractions ( m5 do+ p>ts ( a job 9 /range & devi\s times4 ,fre$ f ^!1 x %\ld be re9/at$1 beg9n+ z e>ly z grade #e ) su* simple ex]cises z ^? %[n bel[ & progress+ to ! problems ( food values1 nutritive ratios1 ge>s1 spe$s1 & ! l 9 grade #h4 ,john is #d ye>s old4 ,fr$ is #f ye>s old4 ,m>y is #h ye>s old4 ,nell is #aj ye>s old4 ,alice is #ab ye>s old4 ,b]t is #ae ye>s old4 ,:o is twice z old z ,john8 ,:o is half z old z ,alice8 ,:o is ?ree times z old z ,john8 ,:o is "o & "o half times z old z ,nell8 ,:o is two ?irds z old z ,fr$8 etc41 etc41 etc4 ,alice is 4444 times z old z ,john4 ,john is 4444 z old z ,m>y4 ,fr$ is 4444 times z old z ,john4 ,alice is 4444 times z old z ,fr$4 ,fr$ is 4444 z old z ,m>y4 etc41 etc41 etc4 ,f9ally x %\ld be rememb]$ t all improvem5ts 9 mak+ ari?metic wor? le>n+ & help+ ! pupil to le>n x w 9 ! long run add to its int]e/4 ,pupils l to le>n1 to a*ieve1 to ga9 ma/]y4 ,success is int]e/+4 ,if ! measures recomm5d$ 9 ! previ\s *apt]s are c>ri$ out1 !re w be little ne$ to entice pupils to take ari?metic or to sug>-coat x ) illegitimate attractions4 ,,*apt] ,,xiii ,,! ,,conditions ,,( ,,le>n+ ,we %all consid] 9 ?is *apt] ! influ5ce ( "t ( day1 size ( class1 & am\nt ( "t devot$ to ari?metic 9 ! s*ool program1 ! hygi5e ( ! eyes 9 ari?metical "w1 ! use ( concrete objects1 & ! use ( s\nds1 sights1 & ?\ghts z situations & ( spee* & writ+ & ?\ght z responses4#ag #ag ,facts conc]n+ ! conditions ( le>n+ 9 g5]al w be f\nd 9 ! au?or's ,educational ,psy*ology1 ,vol4 #b1 ,*apt] #h1 or 9 ! ,educational ,psy*ology1 ,brief] ,c\rse1 ,*apt] #ae4 ,,ext]nal ,,conditions ,computation ( "o or ano!r sort has be5 us$ by sev]al inve/igators z a te/ ( effici5cy at diff]5t times 9 ! day4 ,:5 fre$ f ! effects ( practice on ! "o h& & lack ( int]e/ due to repetition on ! o!r1 ! results uni=mly %[ an increase 9 spe$ late 9 ! s*ool session ) a fall+ (f 9 accuracy t ab\t balances x4#ah ,!re is no wisdom 9 putt+ ari?metic e>ly 9 ! session because ( its difficulty4 ,lively & sociable ex]cises 9 m5tal ari?metic ) oral answ]s 9 fact seem to be admirably fitt$ = use late 9 ! session4 ,except = ! g5]al pr9ciples 7#a7 ( />t+ ! day ) "w t w set a good /&>d ( *e]ful1 effici5t production & 7#b7 ( gett+ ! lea/ int]e/+ features ( ! day's "w done fairly e>ly 9 ! day1 psy*ology p]mits practical exig5cies to rule ! program1 s f> z pres5t k ext5ds4 ,adequate measurem5ts ( ! effect ( "t ( day on improvem5t h n be5 made1 b !re is no reason to believe t any "o "t betwe5 #i ,a4,m4 & #d ,p4,m4 is appreciably m favorable to ari?metical le>n+ ?an to le>n+ geography1 hi/ory1 spell+1 & ! l4 #ah ,see ,?orndike '#jj1 ,k+ '#jg1 & ,heck '#ac4 ,! influ5ce ( size ( class ^u progress 9 s*ool /udies is v difficult to measure because 7#a7 )9 ! same city sy/em ! av]age ( ! six 7or m7 sizes ( class t a pupil has exp]i5c$ w t5d to approximate closely to ! correspond+ av]age = any o!r *ild2 because fur!r 7#b7 !re may be a t5d5cy ( sup]visory (fic]s to assign m pupils to ! bett] tea*]s2 & because 7#c7 sep>ate sy/ems :i* diff] 9 respect to size ( class probably diff] 9 o!r respects also s t _! diff]5ces 9 a*ievem5t may be ref]able to totally diff]5t diff]5ces4 ,elliott '#ad has made a beg9n+ by not+ size ( class dur+ ! ye> ( te/ 9 connection ) 8 own measures ( ! a*ievem5ts ( sev5te5 hundr$ pupils1 supplem5t$ by records f ov] f\r hundr$ o!r classes4 ,z might be expect$ f ! facts j /at$1 he f9ds no appreciable diff]5ce betwe5 classes ( diff]5t sizes )9 ! same s*ool sy/em1 ! effect ( ! few mon?s 9 a small class be+ swamp$ by ! antec$5ts or concomitants !re(4 ,! effect ( ! am\nt ( "t devot$ to ari?metic 9 ! s*ool program has be5 /udi$ ext5sively by ,rice '#jb & '#jc & ,/one '#jh4 ,dr4 ,rice '#jb measur$ ! ari?metical ability ( "s #fjjj *ildr5 9 #ah diff]5t s*ools 9 #g diff]5t cities4 ,! results ( ^! measurem5ts are summ>iz$ 9 ,table #aj4 ,?is table gives two av]ages = ea* grade z well z = ea* s*ool z a :ole4 ,?us1 ! s*ool at ! top %[s av]ages ( #hj4j & #hc4a1 & ! "o at ! bottom1 #be4c & #ca4e4 ,! fir/ repres5ts ! p]c5tage ( answ]s :i* 7 absolutely correct2 ! second %[s :at p] c5t ( ! problems 7 correct 9 pr9ciple1 i4e4 ! av]age t w\ld h be5 receiv$ if no me*anical errors _h be5 made4 ,! facts ( ,dr4 ,rice's table %[ t !re is a positive relation betwe5 ! g5]al /&+ ( a s*ool sy/em 9 ! te/s & ! am\nt ( "t devot$ to ari?metic by its program4 ,! relation is n close1 h[ev]1 be+ t express$ by a correlation coeffici5t ( 4#cf-a_/b4 ,)9 any "o s*ool sy/em !re is no relation betwe5 ! /&+ ( a p>ticul> s*ool & ! am\nt ( "t devot$ to ari?metic 9 t s*ool's program4 ,x mu/ be kept 9 m9d t ! am\nt ( "t giv5 9 ! s*ool program may be c\nt]balanc$ by emphasiz+ "w at home & dur+ /udy p]iods1 or1 on ! o!r h&1 may be a symptom ( correspond+ly small or great emphasis on ari?metic 9 "w set = ! /udy p]iods at home4 ,a /ill m elaborate inve/igation ( ?is same topic 0 made by ,/one '#jh4 ,i quote some:at fully f x1 s9ce x is an in/ructive sample ( ! sort ( /udies t w d\btless soon be made 9 ! case ( e elem5t>y s*ool subject4 ,he f\nd t s*ool sy/ems diff]$ notably 9 ! a*ievem5ts made by _! six?-grade pupils 9 8 te/s ( computation 7! s-call$ 'fundam5tals'7 & ( ! solution ( v]bally describ$ problems 7! s-call$ 'reason+'74 ,! facts 7 z %[n 9 ,table #aa4 ,,table #aj ,,av]ages ,,= ,,individual ,,s*ools ,,9 ,,ari?metic ,,key ,a3 ,,city ,b3 ,,s*ool ,c3 ,result ,d3 ,pr9ciple ,e3 ,p]c5t ( ,me*anical ,errors ,f3 ,m9utes ,daily #f,,? ,,ye> #g,,? ,,ye> #h,,? ,,ye> ,,s*ool ,,av]age ---------------------------------------- -- ,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 a*ievem5t by a sy/em 9 computation w5t ) high a*ievem5t 9 solv+ ! problems1 ! correlation be+ ab\t 4#ej2 & ! sy/em t scor$ high 9 addition or subtraction or multiplication or division usually %[$ closely simil> excell5ce 9 ! o!r ?ree1 ! correlations be+ ab\t 4#ij4 ,,table #aa ,,scores ,,made ,,by ,,! ,,six?-,,grade ,,pupils ,,( ,,ea* ,,( ,,tw5ty-,,six ,,s*ool ,,sy/ems ,,sy/em ,,score ,,9 ,,te/s ,,) ,,score ,,9 ,,te/s ,,9 ,,problems ,,comput+ ---------------------------------------- ------- #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 ,( ! conditions "u :i* ari?metical le>n+ took place1 ! "o mo/ elaborately /udi$ 0 ! am\nt ( "t devot$ to ari?metic4 ,on ! basis ( replies by pr9cipals ( s*ools to c]ta9 que/ions1 he gave ea* ( ! tw5ty-six s*ool sy/ems a measure = ! probable "t sp5t on ari?metic up ?r\gh grade #f4 ,leav+ home /udy out ( acc\nt1 !re seems to be little or no correlation betwe5 ! am\nt ( "t a sy/em devotes to ari?metic & its score 9 problem-solv+1 & n mu* m betwe5 "t exp5diture & score 9 computation4 ,) home /udy includ$ !re is little relation to ! a*ievem5t ( ! sy/em 9 solv+ problems1 b !re is a cle> effect on a*ievem5t 9 computation4 ,! facts z giv5 by ,/one are3-- ,,table #ab ,,correlation ,,( ,,"t ,,exp5ditures ,,) ,,abilities ,)\t ,home ,/udy ,reason+ & ,"t ,exp5diture -4#ja ,fundam5tals & ,"t ,exp5diture 4#ji ,includ+ ,home ,/udy ,reason+ & ,"t ,exp5diture 4#ac ,fundam5tals & ,"t ,exp5diture 4#di ,^! correlations1 x %\ld be borne 9 m9d1 are = s*ool sy/ems1 n = individual pupils4 ,x might be t1 ?\gh ! sy/em :i* devot$ ! mo/ "t to ari?metic did n %[ correspond+ sup]iority 9 ! product ov] ! sy/em devot+ only half z mu* "t1 ! pupils )9 ! sy/em did a*ieve 9 exact proportion to ! "t !y gave to /udy4 ,nei!r correlation w\ld p]mit inf]5ce conc]n+ ! effect ( diff]5t am\nts ( "t sp5t by ! same pupil4 ,/one consid]$ also ! pr9t$ ann\ncem5ts ( ! c\rses ( /udy 9 ari?metic 9 ^! tw5ty-six sy/ems4 ,n9ete5 judges rat$ ^! ann\nc$ c\rses ( /udy = excell5ce accord+ to ! in/ructions quot$ bel[3-- ,,conc]n+ ,,! ,,rat+ ,,( ,,c\rses ,,( ,,/udy ,judges please read be=e scor+ ,i4 ,"s ,factors ,det]m9+ ,relative ,excell5ce4 7,n4 ,b4 ,! foll[+ enum]ation is meant to be sugge/ive r ?an complete or exclusive4 ,& ea* scor] is urg$ to rely prim>ily on 8 own judgm5t47 #a4 ,helpfulness to ! tea*] 9 tea*+ ! subject matt] outl9$4 #b4 ,social value or concret5ess ( s\rces ( problems4 #c4 ,! arrangem5t ( subject matt]4 #d4 ,! provision made = adequate drill4 #e4 ,a reasonable m9imum requirem5t ) sugge/ions = valuable additional "w4 #f4 ,! relative values ( any pr$om9at+ s-call$ me?ods--su* z ,spe]1 ,grube1 etc4 #g4 ,! place ( oral or s-call$ m5tal ari?metic4 #h4 ,! m]it ( textbook ref]5ces4 ,,ii4 ,cautions & ,directions4 7,judges please foll[ z implicitly z possible47 #a4 ,include ref]5ces to textbooks z p>ts ( ! ,c\rse ( ,/udy4 ,?is necessitates judg+ ! p>ts ( ! texts ref]r$ to4 #b4 ,z f> z possible become equally famili> ) all c\rses be=e scor+ any4 #c4 ,:5 y are ready to beg9 to score1 7#a7 arrange 9 s]ial ord] accord+ to excell5ce1 7#b7 />t+ ) ! middle "o score x #ej1 !n score above & bel[ #ej accord+ z c\rses are bett] or poor]1 indicat+ relative diff]5ces 9 excell5ce by relative diff]5ces 9 scores1 i4e4 9 s f> z y f9d t ! c\rses diff] by ab\t equal /eps1 score ^? bett] ?an ! middle "o #ea1 #eb1 etc41 & ^? poor] #di1 #dh1 etc41 b if y f9d t ! c\rses diff] by unequal /eps %[ ^! inequalities by omitt+ numb]s4 #d4 ,write rat+s on ! slip ( pap] atta*$ to ea* c\rse4 ,! sy/ems ^: c\rses ( /udy 7 ?us rat$ highe/ did n manife/ any great] a*ievem5t 9 ,/one's te/s ?an ! re/4 ,! ?irte5 ) ! mo/ approv$ ann\ncem5ts ( c\rses ( /udy 7 9 fact a little inf]ior 9 a*ievem5t to ! o!r ?irte51 & ! correlation coeffici5ts 7 slightly negative4 ,/one also comp>$ eighte5 sy/ems :]e !re 0 sup]vision ( ! "w by sup]9t5d5ts or sup]visors z well z by pr9cipals ) f\r sy/ems :]e ! pr9cipals & tea*]s _h no su* help4 ,! scores 9 8 te/s 7 v mu* l[] 9 ! f\r latt] cities4 ,,! ,,hygi5e ,,( ,,! ,,eyes ,,9 ,,ari?metic ,we h already not$ t ! task ( read+ & copy+ numb]s is "o ( ! h>de/ t ! eyes h to p]=m 9 ! elem5t>y s*ool1 & t x %\ld be alleviat$ by arrang+ mu* ( ! "w s t only answ]s ne$ be writt5 by ! pupil4 ,! figures to be read & copi$ %\ld obvi\sly be 9 type ( suitable size & /yle1 s arrang$ & spac$ on ! page or blackbo>d z to cause a m9imum ( ef=t & /ra94 ,illu/ration3 ,,fig4 #bf4--,type too l>ge4 ,illu/ration3 ,,fig4 #bg4--ab-po9t1 #aa-po9t1 & #aj-po9t type4 ,size4--,type may be too l>ge z well z too small1 ?\gh ! latt] is ! common] error4 ,if x is too l>ge1 z 9 ,fig4 #bf1 :i* is a duplicate ( type actually us$ 9 a =m ( practice pad1 ! eye has to make too _m fixations to take 9 a giv5 cont5t4 ,all ?+s consid]$1 #ab-po9t type 9 grades #c & #d1 #aa-po9t 9 grades #e & #f1 & #aj-po9t 9 grades #g & #h seem ! mo/ desirable sizes4 ,^! are %[n 9 ,fig4 #bg4 ,too small type occurs (t5e/ 9 fractions & 9 ! dim5sion-numb]s or scale numb]s ( draw+s4 ,figures #bh1 #bi1 & #cj are samples f actual s*ool practice4 ,samples ( ! desirable size are %[n 9 ,figs4 #ca & #cb4 ,! te*nique ( mod]n typesett+ makes x v difficult & exp5sive to make fractions ( ! horizontal type 7#a #c #e - - - #d1 #h1 #f7 l>ge en\gh )\t mak+ ! :ole-numb] figures ) :i* !y are m+l$ too l>ge or giv+ an unc\? appe>ance to ! total4 ,consequ5tly fractions some:at small] ?an are desirable may h to be us$ occasionally 9 textbooks4#ai ,!re is no valid excuse1 h[ev]1 = ! excessively small fractions :i* (t5 are made 9 blackbo>d "w4 #ai ,a special type c\ld be con/ruct$ t w\ld use a l>ge type body1 say #ad po9t1 ) integ]s 9 #aj or #ab po9t & fractions mu* l>g] ?an n[4 ,illu/ration3 ,,fig4 #bh4--,type ( measurem5ts too small4 ,?is is a picture ( ,m>y's g>d54 ,h[ _m feet is x ar\nd ! g>d58 ,illu/ration3 ,,fig4 #bi4--,type too small4 ,illu/ration3 ,,fig4 #cj4--,numb]s too small & badly design$4 ,illu/ration3 ,,fig4 #ca4--;,figure #bh ) suitable numb]s4 ,illu/ration3 ,,fig4 #cb4--;,figure #cj ) suitable numb]s4 ,/yle4--,! ord9>y type =ms (t5 h #c & #h s made z to require /ra9 to di/+ui% !m4 #e is sometimes easily confus$ ) #c & ev5 ) #h4 #a1 #d1 & #g may be less easily di/+ui%able ?an is desirable4 ,figure #cc %[s a specially good type 9 :i* ea* figure is repres5t$ by its ess5tial#bj features )\t any di/ract+ %ad+ or knobs or turns4 ,figure #cd %[s "s ( ! types 9 common use4 ,!re are no demon/rably great diff]5ces among/ ^!4 ,9 fractions !re is a notable ga9 f us+ ! slant =m 7#b_/c1 #c_/d7 = ex]cises 9 addition & subtraction1 & = almo/ all mix$ numb]s4 ,?is appe>s cle>ly to ! eye 9 ! comp>ison ( ,fig4 #ce bel[1 :]e ! same fractions all 9 #aj-po9t type are display$ 9 horizontal & 9 slant =m4 ,! figures 9 ! slant =m are 9 g5]al l>g] & ! space betwe5 !m & ! fraction-l9e is wid]4 ,also ! slant =m makes x easi] = ! eye to exam9e ! d5om9ators to see :e!r r$uctions are necess>y4 ,except = a few cases to %[ t ! op]ations c be done j z truly ) ! horizontal =ms1 ! book & ! blackbo>d %\ld display mix$ numb]s & fractions to be add$ or subtract$ 9 ! slant =m4 ,! slant l9e %\ld be at an angle ( approximately #de degrees4 ,pupils %\ld be taught to use ?is =m 9 _! own "w ( ?is sort4 #bj ,x w be /ill bett] if ! #d is replac$ by an op5-top #d4 ,:5 script figures are pres5t$ !y %\ld be ( simple design1 %[+ cle>ly ! ess5tial features ( ! figure1 ! l9e be+ ev]y:]e ( equal or ne>ly equal wid? 7t is1 )\t %ad+1 & )\t ornam5tation or ecc5tricity ( any sort74 ,! op5+ ( ! #c %\ld be wide to prev5t confusion ) #h2 ! top ( ! #c %\ld be curv$ to aid its diff]5tiation f #e2 ! d[n /roke ( ! #i %\ld be almo/ or q /raight2 ! #a1 #d1 #g1 & #i %\ld be cle>ly di/+ui%able4 ,!re are _m ways ( di/+ui%+ !m cle>ly1 ! be/ probably be+ to use ! /raight l9e = #a1 ! op5 #d ) cle> angul>ity1 a wide top to ! #g1 & a cle>ly clos$ curve = ! top ( ! #i4 ,illu/ration3 ,,fig4 #cc4--;,block type2 a v desirable type except t x is some:at too heavy4 ,illu/ration3 ,,fig4 #cd4--;,common /yles ( pr9t$ numb]s4 ,illu/ration3 ,,fig4 #ce4--;,diagonal & horizontal fractions comp>$4 ,illu/ration3 ,,fig4 #cf4--;,good v]tical spac+4 ,illu/ration3 ,,fig4 #cg4--;,bad v]tical spac+4 ,illu/ration3 ,,figs4 #ch 7above7 & #ci 7bel[74--,good & bad left-"r spac+4 ,! pupil's writ+ ( figures %\ld be cle>4 ,he w !reby be sav$ eye/ra9 & errors 9 8 s*ool "w z well z giv5 a valuable ability = life4 ,h&writ+ ( figures is us$ enorm\sly 9 spite ( ! developm5t ( typewrit]s2 illegible figures are commonly m h>mful ?an illegible lett]s or words1 s9ce ! context f> less (t5 tells :at ! figure is int5d$ to be2 ! habit ( mak+ cle> figures is n s h>d to acquire1 s9ce !y are writt5 unjo9$ & require only ! automatic action ( t5 m9or acts ( skill4 ,! s*ools h miss$ a great opportunity 9 ?is respect4 ,:]eas ! h& writ+ ( words is (t5 bett] ?an x ne$s to be = life's purposes1 ! writ+ ( figures is usually mu* worse4 ,! figures pres5t$ 9 books on p5man%ip are also commonly bad1 %[+ neglect or misund]/&+ ( ! matt] on ! "p ( lead]s 9 p5man%ip4 ,spac+4--,spac+ up & d[n ! column is r>ely too wide1 b v (t5 too n>r[4 ,! specim5s %[n 9 ,figs4 #cf & #cg %[ good practice contra/$ ) ! common fault4 ,spac+ f "r to left is g5]ally fairly satisfactory 9 books1 ?\gh !re is a bad t5d5cy to adopt "s "o r\t9e ?r\gh\t & s to miss *ances to use r$uctions & increases ( spac+ s z to help ! eye & ! m9d 9 special cases4 ,specim5s ( good & bad spac+ are %[n 9 ,figs4 #ch & #ci4 ,9 ! "w ( ! pupils1 ! spac+ f "r to left is (t5 too n>r[4 ,?is cr[d+ ( lett]s1 toge!r ) unev5ness ( spac+1 adds notably to ! task ( eye & m9d4 ,! composition or make-up ( ! page4--,o!r ?+s be+ equal1 t arrangem5t ( ! page is be/ :i* helps a *ild mo/ to keep 8 place on a page & to f9d x aft] hav+ look$ away to "w on ! pap] on :i* he computes1 or = o!r good reasons4 ,a good page & a bad page 9 ?is respect are %[n 9 ,figs4 #dj & #da4 ,illu/ration3 ,,fig4 #dj4--;,a page well made up to suit ! action ( ! eye4 ,illu/ration3 ,,fig4 #da4--,! same matt] z 9 ,fig4 #dj1 mu* less well made up4 ,objective pres5tations4--,pictures1 diagrams1 maps1 & o!r pres5tations %\ld n tax ! eye unduly1 7a7 by requir+ too f9e di/9ctions1 or 7b7 by inconv5i5t arrangem5t ( ! data1 prev5t+ easy c\nt+1 measur+1 comp>ison1 or :atev] ! task is1 or 7c7 by putt+ too _m facts 9 "o picture s t ! eye & m9d1 :5 try+ to make out any "o1 are confus$ by ! o!rs4 ,illu/rations ( bad practices 9 ^! respects are %[n 9 ,figs4 #db to #eb4 ,a few specim5s ( "w well arrang$ = ! eye are %[n 9 ,figs4 #ec to #ef4 ,good rules to rememb] are3-- ,o!r ?+s be+ equal1 make di/9ctions by ! cle>e/ me?od1 fit mat]ial to ! t5d5cy ( ! eye to see an 'eyeful' at a "t 7r\ghly #a-a_/b in* by #a_/b in* 9 a book2 #a-a_/b ft4 by #a_/b ft4 on ! blackbo>d71 & let "o picture tea* only "o fact or relation1 or su* facts & relations z d n int]f]e 9 p]ception4 ,! g5]al conditions ( seat+1 illum9ation1 pap]1 & ! l are ev5 m important :5 ! eyes are us$ ) numb]s ?an :5 !y are us$ ) words4 ,illu/ration3 ,,fig4 #db4--,try to c\nt ! rungs on ! ladd]1 or ! %ocks 9 ! wagon4 ,illu/ration3 ,,fig4 #dc4--;,h[ _m o>s d y see8 ,h[ _m birds8 ,h[ _m fi%8 ,illu/ration3 ,,fig4 #dd4--;,c\nt ! birds 9 ea* ( ! ?ree flocks ( birds4 ,illu/ration3 ,,fig4 #de4--,note ! lack ( cle> division ( ! hundr$s4 ,consid] ! difficulty ( c\nt+ "o ( ^! columns ( dots4 ,illu/ration3 ,,fig4 #df4--,:at d y suppose ^! pictures are int5d$ to %[8 ,illu/ration3 ,,fig4 #dg4--,w\ld a beg9n] "k t aft] ,,?irte5 he 0 to swit* ar\nd & beg9 at ! o!r end8 ,c\ld y read ! ,,six ( ,,tw5ty-,,six if y did n already "k :at x ought to be8 ,:at mean+ w\ld all ! brackets h = a little *ild 9 grade #b8 ,does ?is picture illu/rate or obfuscate8 ,illu/ration3 ,,fig4 #dh4--;,h[ long did x take y to f9d out :at ^! pictures mean8 ,illu/ration3 ,,fig4 #di4--;,c\nt ! figures 9 ! fir/ r[1 us+ y\r eyes alone2 h "s "o make l9es ( #aj1 #aa1 #ab1 #ac1 & m repetitions ( ?is figure spac$ closely z "h4 ,c\nt #bj or #cj su* l9es1 us+ ! eye unaid$ by f+]s1 p5cil1 etc4 ,illu/ration3 ,,fig4 #ej4--;,c y answ] ! "q )\t measur+8 ,c\ld a *ild ( sev5 or eight8 ,illu/ration3 ,,fig4 #ea4--,:at are ^! draw+s int5d$ to %[8 ,:y d !y %[ ! facts only obscurely & dubi\sly8 ,illu/ration3 ,,fig4 #eb4--,:at are ^! draw+s int5d$ to %[8 ,:at simple *ange w\ld make !m %[ ! facts mu* m cle>ly8 ,illu/ration3 ,,fig4 #ec4--;,arrang$ 9 conv5i5t eye-fulls4 ,illu/ration3 ,,fig4 #ed4--;,cle>1 simple1 & easy ( comp>ison4 ,illu/ration3 ,,fig4 #ee4--;,cle>1 simple1 & well spac$4 ,illu/ration3 ,,fig4 #ef4--,well arrang$1 ?\gh a little wid] spac+ betwe5 ! squ>es w\ld make x ev5 bett]4 ,,! ,,use ,,( ,,concrete ,,objects ,,9 ,,ari?metic ,we mean by concrete objects actual ?+s1 ev5ts1 & relations pres5t$ to s5se1 9 contra/ to words & numb]s & symbols :i* mean or /& = ^! objects or = m ab/ract qualities & relations4 ,blocks1 too?-picks1 co9s1 foot rules1 squ>$ pap]1 qu>t measures1 bank books1 & *ecks are su* concrete ?+s4 ,a foot rule put successively along ! ?ree ?irds ( a y>d rule1 a bell rung five times1 & a p\nd weight balanc+ sixte5 ounce weights are su* concrete ev5ts4 ,a p9t beside a qu>t1 an in* beside a foot1 an apple %[n cut 9 halves display su* concrete relations to a pupil :o is att5tive to ! issue4 ,concrete pres5tations are obvi\sly useful 9 ari?metic to tea* mean+s "u ! g5]al law t a ^w or numb] or sign or symbol acquires mean+ by be+ connect$ ) actual ?+s1 ev5ts1 qualities1 & relations4 ,we h also not$ _! usefulness z means to v]ify+ ! results ( ?9k+ & comput+1 z :5 a pupil1 hav+ solv$1 ,h[ _m badges ea* #e in*es long c be made f #c-a_/c yd4 ( ribbon8 by us+ #aj x #ab_/e1 draws a l9e #c-a_/c yd4 long & divides x into #e-;in* l5g?s4 ,concrete exp]i5ces are useful :5ev] ! mean+ ( a numb]1 l #i or #g_/h or 4#jjd1 or ( an op]ation1 l multiply+ or divid+ or cub+1 or ( "s t]m1 l rectangle or hypo!nuse or disc\nt1 or "s proc$ure1 l vot+ or insur+ prop]ty aga9/ fire or borr[+ money f a bank1 is abs5t or incomplete or faulty4 ,concrete "w ?us is by no means conf9$ to ! prim>y grades b may be appropriate at all /ages :5 new facts1 relations1 & proc$ures are to be taught4 ,h[ mu* concrete mat]ial %all be pres5t$ w dep5d ^u ! fact or relation or proc$ure :i* is to be made intelligible1 & ! ability & k ( ! pupil4 ,?us '"o half' w 9 g5]al require less concrete illu/ration ?an 'five six?s'2 & five six?s w require less 9 ! case ( a bright *ild :o already kn[s #b_/c1 #c_/d1 #c_/h1 #e_/h1 #g_/h1 #b_/e1 #c_/e1 & #d_/e ?an 9 ! case ( a dull *ild or "o :o only kn[s #b_/c & #c_/d4 ,z a g5]al rule ! same topic w require less concrete mat]ial ! lat] x appe>s 9 ! s*ool c\rse4 ,if ! mean+s ( ! numb]s are taught 9 grade #b in/ead ( grade #a1 !re w be less ne$ ( blocks1 c\nt]s1 spl9ts1 beans1 & ! l4 ,if #a-a_/b #a_/b #b is taught e>ly 9 grade #c1 !re w be m ga9 f ! use ( #a-a_/b in*es & #a_/b in* on ! foot rule ?an if ! same relations 7 taught 9 connection ) ! g5]al addition ( l fractions late 9 grade #d4 ,sometimes ! und]/&+ c be _h ei!r by connect+ ! idea ) ! reality directly1 or by connect+ ! two indirectly via "s o!r idea4 ,! am\nt ( concrete mat]ial to be us$ w dep5d on its relative advantage p] unit ( "t sp5t4 ,?us x might be m economical to connect #e_/ab1 #g_/ab1 & #aa_/ab ) real mean+s indirectly by call+ up ! resemblance to ! #b_/c1 #c_/d1 #c_/h1 #e_/h1 #g_/h1 #b_/e1 #c_/e1 #d_/e1 & #e_/f already /udi$1 ?an by %[+ #e_/ab ( an apple1 #g_/ab ( a y>d1 #aa_/ab ( a foot1 & ! l4 ,9 g5]al ! economical c\rse is to te/ ! und]/&+ ( ! matt] f "t to "t1 us+ m concrete mat]ial if x is ne$$1 b be+ c>eful to enc\rage pupils to proce$ to ! ab/ract ideas & g5]al pr9ciples z fa/ z !y c4 ,x is we>isome & debau*+ to pupils' intellects = !m to be put ?r\gh elaborate concrete exp]i5ces to get a mean+ :i* !y c\ld h got !mselves by pure ?\ght4 ,we %\ld also rememb] t ! new idea1 say ( ! mean+ ( decimal fractions1 w be improv$ & cl>ifi$ by us+ x 7see page #ahc f471 s t ! atta9m5t ( a p]fect conception ( decimal fractions be=e do+ any?+ ) !m is unnecess>y & probably v wa/eful4 ,a few illu/rations may make ^! pr9ciples m in/ructive4 7a7 ,v l>ge numb]s1 su* z #ajjj1 #aj1jjj1 #ajj1jjj1 & #a1jjj1jjj1 ne$ m concrete aids ?an are commonly giv54 ,guess+ conte/s ab\t ! value 9 doll>s ( ! s*ool build+ & o!r build+s1 ! area ( ! s*oolroom floor & o!r surfaces 9 squ>e in*es1 ! numb] ( m9utes 9 a week1 & ye>1 & ! l1 toge!r ) prop] computations & measurem5ts1 are v useful to re5=ce ! concrete pres5tations & supply g5u9e problems 9 multiplication & subtraction ) l>ge numb]s4 7b7 ,numb]s v mu* small] ?an "o1 su* z #a_/cb1 #a_/fd1 4#jd1 & 4#jjb1 also ne$ "s concrete aids4 ,a diagram l t ( ,fig4 #eg is useful4 7c7 ,majority & plurality %\ld be und]/ood by e citiz54 ,!y c be und]/ood )\t concrete aid1 b an actual vote is well wor? :ile = ! ga9 9 vividness & surety4 ,illu/ration3 ,,fig4 #eg4--;,concrete aid to und]/&+ fractions ) l>ge d5om9ators4 ,a #a_/ajjj sq4 ft42 ,b #a_/ajj sq4 ft42 ,c #a_/ej sq4 ft42 ,d #a_/aj sq4 ft4 7d7 ,insurance aga9/ loss by fire c be taught by explanation & analogy alone1 b x w be economical to h "s actual insur+ & paym5t ( premiums & a g5u9e loss :i* is reimburs$4 7e7 ,f\r play banks 9 ! corn]s ( ! room1 receiv+ deposits1 ca%+ *ecks1 & lat] disc\nt+ notes w give good educational value = ! "t sp5t4 7f7 ,trade disc\nt1 on ! contr>y1 h>dly requires m concrete illu/ration ?an is f\nd 9 ! v problems to :i* x is appli$4 7g7 ,! process ( f9d+ ! numb] ( squ>e units 9 a rectangle by multiply+ ) ! appropriate numb]s repres5t+ l5g? & wid? is probably r h9d]$ ?an help$ by ! ord9>y objective pres5tation z an introduction4 ,! usual =m ( objective introduction is z foll[s3-- ,illu/ration3 ,,fig4 #eh4 ,h[ long is ?is rectangle8 ,h[ l>ge is ea* squ>e8 ,h[ _m squ>e in*es are !re 9 ! top r[8 ,h[ _m r[s are !re8 ,h[ _m squ>e in*es are !re 9 ! :ole rectangle8 ,s9ce !re are ?ree r[s ea* conta9+ #d squ>e in*es1 we h #c x #d squ>e in*es #ab squ>e in*es4 ,draw a rectangle #g in*es long & #b in*es wide4 ,if y divide x into in* squ>es h[ _m r[s w !re be8 ,h[ _m in* squ>es w !re be 9 ea* r[8 ,h[ _m squ>e in*es are !re 9 ! rectangle8 ,illu/ration3 ,,fig4 #ei4 ,x is bett] actually to hide ! individual squ>e units z 9 ,fig4 #ei4 ,!re are f\r reasons3 7#a7 ,! concrete r[s & columns r di/ract att5tion f ! ess5tial ?+ to be le>n$4 ,?is is n t x r[s "o squ>e wide1 y squ>es 9 a r[ w make xy squ>es 9 all1 b t by us+ prop] units & ! prop] op]ation ! area ( any rectangle c be f\nd f its l5g? & wid?4 7#b7 ,*ildr5 h little difficulty 9 le>n+ to multiply r ?an add1 subtract1 or divide :5 comput+ area4 7#c7 ,! habit s =m$ holds good = areas l #a-b_/c by #d-a_/b1 ) fractional dim5sions1 9 :i* any ef=t to c\nt up ! areas ( r[s is v tr\blesome & confus+4 7#d7 ,! notion t a squ>e in* is an area #a' by #a' r ?an #a_/b' by #b' or #a_/c 94 by #c 94 or #a-a_/b 94 by #b_/c 94 is likely to be =m$ too emphatically if mu* "t is sp5t ^u ! sort ( concrete pres5tation %[n above4 ,x is !n bett] to use concrete c\nt+ ( r[s ( small areas z a means ( v]ification aft] ! proc$ure is le>n$1 ?an z a means ( d]iv+ x4 ,!re has be51 especially 9 ,g]many1 mu* argum5t conc]n+ :at sort ( numb]-pictures 7t is1 arrangem5t ( dots1 l9es1 or ! l1 z %[n 9 ,fig4 #fj7 is be/ = use 9 connection ) ! numb] names 9 ! e>ly ye>s ( ! tea*+ ( ari?metic4 ,lay '#ih & '#jg1 ,walsemann '#jg1 ,freeman '#aj1 ,h[ell '#ad1 & o!rs h measur$ ! accuracy ( *ildr5 9 e/imat+ ! numb] ( dots 9 arrangem5ts ( "o or m ( ^! diff]5t types4#ba ,_m writ]s int]pret a diff]5ce 9 favor ( e/imat+1 say1 ! squ>e arrangem5ts ( ,born or ,lay z mean+ t su* is ! be/ arrangem5t to use 9 tea*+4 ,! inf]5ce is1 h[ev]1 unju/ifi$4 ,t c]ta9 numb]-pictures are easi] to e/imate num]ically does n necess>ily mean t !y are m in/ructive 9 le>n+4 ,"o set may be easi] to e/imate j because !y are m famili>1 hav+ be5 (t5] exp]i5c$4 ,ev5 if ! favor$ set 0 s aft] equal exp]i5ce ) all sets1 accuracy ( e/imation w\ld be a sign ( sup]iority = use 9 in/ruction only if all o!r ?+s 7 equal 7or 9 favor ( ! arrangem5t 9 "q74 ,obvi\sly ! way to decide :i* ( ^! is be/ to use 9 tea*+ is by us+ !m 9 tea*+ & measur+ all relevant results1 n by m]ely record+ :i* ( !m are mo/ accurately e/imat$ 9 c]ta9 "t exposures4 #ba ,= an acc\nt 9 ,engli% ( _! ma9 f9d+s see ,h[ell '#ad1 pp4 #adi-bea4 ,x may be not$ t ! ,born1 ,lay1 & ,freeman pictures h claims = special consid]ation on gr\nds ( probable in/ructiv5ess4 ,s9ce !y are also sup]ior 9 ! te/s 9 respect to accuracy ( e/imate1 *oice %\ld probably be made f ^! ?ree by any tea*] :o wi%es to connect "o set ( numb]-pictures sy/ematically ) ! numb] names1 z by drills ) ! blackbo>d or ) c>ds4 ,illu/ration3 ,,fig4 #fj4--,v>i\s propos$ arrangem5ts ( dots = use 9 tea*+ ! mean+s ( ! numb]s #a to #aj4 ,su* drills are probably useful if und]tak5 ) zeal1 & if kept z supplem5t>y to m reali/ic objective "w ) play money1 *ildr5 m>*+1 mat]ial to be di/ribut$1 g>d5-plot l5g?s to be measur$1 & ! l1 & if s adm9i/]$ t ! pupils soon get ! g5]aliz$ ab/ract mean+ ( ! numb]s fre$ f dep5d5ce on an inn] picture ( any sort4 ,?is fre$om is s important t x may make ! use ( _m types ( numb]-pictures advisable r ?an ! use ( ! "o :i* 9 & ( itself is be/4 ,z ,meumann says3 ,p]ceptual reckon+ c be ov]done4 ,x _h its *ief significance = ! surety & cle>ness ( ! fir/ f\ndation ( ari?metical in/ruction4 ,if1 h[ev]1 x is cont9u$ aft] ! fir/ op]ations become famili> to ! *ild1 & ext5d$ to op]ations :i* develop f ^! elem5t>y ones1 x necess>ily works z a ret>d+ =ce & holds back ! natural developm5t ( ari?metic4 ,?is moves on to "w ) ab/ract numb] & ) me*anical association & reproduction4 '#jg1 ,vol4 #b1 p4 #ceg4 ,su* drills are commonly ov]done by ^? :o make use ( !m1 be+ giv5 too (t51 & cont9u$ aft] _! in/ructiv5ess has wan$1 & us$ in/ead ( m significant1 int]e/+1 & v>i$ "w 9 c\nt+ & e/imat+ & measur+ real ?+s4 ,consequ5tly1 !re is n[ r a prejudice aga9/ !m 9 our bett] s*ools4 ,!y %\ld probably be re9/at$ b to a mod]ate & judici\s use4 ,,oral1 ,,m5tal1 ,,& ,,writt5 ,,ari?metic ,!re has be5 mu* dispute ov] ! relative m]its ( oral & writt5 "w 9 ari?metic--a "q :i* is mu* confus$ by ! diff]5t mean+s ( 'oral' & 'writt54' ,oral has meant 7#a7 "w :]e ! situations are pres5t$ orally & ! pupil's f9al responses are giv5 orally1 or 7#b7 "w :]e ! situations are pres5t$ orally & ! pupils' f9al responses are writt5 or p>tly writt5 & p>tly oral1 or 7#c7 "w :]e ! situations are pres5t$ 9 writ+ or pr9t & ! f9al responses are oral4 ,writt5 has meant 7#a7 "w :]e ! situations are pres5t$ 9 writ+ or pr9t & ! f9al responses are made 9 writ+1 or 7#b7 "w :]e also _m ( ! int]m$iate responses are writt51 or 7#c7 "w :]e ! situations are pres5t$ orally b ! f9al responses & a l>ge p]c5tage ( ! int]m$iate computational responses are writt54 ,!re are o!r mean+s ?an ^!4 ,x is bett] to drop ^! v ambigu\s t]ms & ask cle>ly :at are ! m]its & dem]its1 9 ! case ( any specifi$ ari?metical "w1 ( auditory & ( visual pres5tation ( ! situations1 & ( say+ & ( writ+ ea* specifi$ /ep 9 ! response4 ,! disputes ov] m5tal v]sus writt5 ari?metic are also confus$ by ambiguities 9 ! use ( 'm5tal4' ,m5tal has be5 us$ to mean done )\t p5cil & pap] & also done ) few ov]t responses1 ei!r writt5 or spok51 betwe5 ! sett+ ( ! task & ! ann\ncem5t ( ! answ]4 ,9 nei!r case is ! ^w m5tal specially appropriate z a description ( ! total fact4 ,z be=e1 we %\ld ask cle>ly1 ,:at are ! m]its & dem]its ( mak+ c]ta9 specifi$ int]m$iate responses 9 inn] spee* or imag$ s\nds or visual images or imageless ?\ght--t is1 )\t actual writ+ or ov]t spee*8 ,x may be said at ! outset t oral1 writt51 & inn] pres5tations ( initial situations1 oral1 writt51 & inn] ann\ncem5ts ( f9al responses1 & oral1 writt51 & inn] managem5t ( int]m$iate processes h v>y+ degrees ( m]it accord+ to ! p>ticul> ari?metical ex]cise1 pupil1 & context4 ,devotion to oralness or m5talness z su* is simply fanatical4 ,v>i\s comb9ations1 su* z ! writt5 pres5tation ( ! situation ) inn] managem5t ( ! int]m$iate responses & oral ann\ncem5t ( ! f9al response h _! special m]its = p>ticul> cases4 ,^! m]its ! read] c evaluate = himself = any giv5 sort ( "w = a giv5 class by consid]+3 7#a7 ,! am\nt ( practice receiv$ by ! class p] h\r sp5t2 7#b7 ! ease ( correction ( ! "w2 7#c7 ! ease ( und]/&+ ! tasks2 7#d7 ! prev5tion ( *eat+2 7#e7 ! *e]fulness & sociability ( ! "w2 7#f7 ! fre$om f eye/ra91 & o!r less important desid]ata4 ,x %\ld be not$ t ! /ock s*emes ,a1 ,b1 ,c1 & ,d bel[ are only a few ( ! _m t are possible & t s*emes ,e1 ,f1 ,g1 & ,h h special m]its4 ,,pres5tation ,,( ,,managem5t ,,( ,,ann\ncem5t ,,( ,,initial ,,situation ,,int]m$iate ,,processes ,,f9al ,,response ,a4 ,pr9t$ or writt5 ,writt5 ,writt5 ,b4 ,inn] ,oral by "o pupil1 inn] by ! re/ ,c4 ,oral 7by tea*]7 ,writt5 ,writt5 ,d4 ,inn] ,oral by "o pupil1 inn] by ! re/ ,e4 ,z 9 ,a or ,c ,a mixture1 ! pupil ,z 9 ,a or ,b or ,h writ+ :at he ne$s ,f4 ,! real situation ,z 9 ,e ,z 9 ,a or ,b or ,h itself1 9 "p at lea/ ,g4 ,bo? read by ! pupil ,z 9 ,e ,z 9 ,a or ,b or ,h & put orally by ! tea*] ,h4 ,z 9 ,a or ,c or ,g ,z 9 ,e ,writt5 by all pupils1 ann\nc$ orally by "o pupil ,! common practice ( ei!r hav+ no use made ( p5cil & pap] or hav+ all computations & ev5 mu* v]bal analysis writt5 out elaborately = exam9ation is unfavorable = le>n+4 ,! dem&s :i* life itself w make ( ari?metical k & skill w range f tasks done ) e p]c5tage ( writt5 "w f z]o up to ! case :]e e ma9 result obta9$ by ?\ght is record$ = lat] use by fur!r ?\ght4 ,9 s*ool ! be/ way is t :i*1 = ! pupils 9 "q1 has ! be/ total effect ^u quality ( product1 spe$1 & ease ( production1 re5=cem5t ( tra9+ already giv51 & prep>ation = tra9+ to be giv54 ,!re is no?+ intellectually crim9al ab\t us+ a p5cil z well z inn] ?\ght2 on ! o!r h& !re is no magical value 9 writ+ out = ! tea*]'s inspection figures t ! pupil does n ne$ 9 ord] to atta91 pres]ve1 v]ify1 or correct 8 result4 ,! common practice ( hav+ ! f9al responses ( all easy tasks giv5 orally has no sure ju/ification4 ,on ! contr>y1 ! great advantage ( hav+ all pupils really d ! "w %\ld be secur$ 9 ! easy "w m ?an any:]e else4 ,if ! "t co/ ( copy+ ! figures is elim9at$ by ! simple plan ( hav+ !m pr9t$1 & if ! sup]vision co/ ( exam9+ ! pap]s is elim9at$ by hav+ ! pupils correct ea* o!r's "w 9 ^! easy tasks1 writt5 answ]s are (t5 sup]ior to oral except = ! elem5ts ( sociability & 'g' & fre$om f eye/ra9 ( ! oral ex]cise4 ,su* writt5 "w provides ! gift$ pupils ) f two to t5 times z mu* practice z !y w\ld get 9 an oral drill on ! same mat]ial1 suppos+ !m to give inn] answ]s to e ex]cise done by ! class z a :ole2 x makes sure t ! dull pupils :o w\ld r>ely get an inn] answ] at ! rate dem&$ by ! oral ex]cise1 d z mu* z !y are able to d4 ,two argum5ts (t5 made = ! oral /atem5t ( problems by ! tea*] are t problems s put are bett] und]/ood1 especially 9 ! grades up ?r\gh ! fif?1 & t ! problems are m likely to be g5u9e & relat$ to ! life ! pupils "k4 ,:5 ^! /atem5ts are true1 ! fir/ is a /ill bett] argum5t = hav+ ! pupils read ! problems aid$ by ! tea*]'s oral /atem5t ( !m4 ,= ! difficulty is l>gely t ! pupils _c read well en\gh2 & x is bett] to help !m to surm\nt ! difficulty r ?an simply evade x4 ,! second is n an argum5t = oralness v]sus writt5ness1 b = good problems v]sus bad2 ! tea*] :o makes up su* good problems %\ld1 9 fact1 take special c>e to write !m d[n = lat] use1 :i* may be by voice or by ! blackbo>d or by pr9t$ %eet1 z is be/4 ,,*apt] ,,xiv ,,! ,,conditions ,,( ,,le>n+3 ,,! ,,problem ,,attitude ,dewey1 & o!rs foll[+ him1 h emphasiz$ ! desirability ( hav+ pupils d _! "w z active seek]s1 consci\s ( problems ^: solution satisfies "s real ne$ ( _! own natures4 ,o!r ?+s be+ equal1 x is unwise1 !y argue1 = pupils to be l$ along bl9dfold z x 7 by ! tea*] & textbook1 n kn[+ :]e !y are go+ or :y !y are go+ !re4 ,!y ought r to h "s liv+ purpose1 & be zeal\s = its atta9m5t4 ,?is doctr9e is 9 g5]al s\nd1 z we %all see1 b x is (t5 misus$ z a def5se ( practices :i* neglect ! =mation ( fundam5tal habits1 or z a recomm5dation to practices :i* are q unworkable "u ord9>y classroom conditions4 ,s x seems probable t its nature & limitations are n ?or\ghly kn[n1 ev5 to its foll[]s1 & t a r detail$ treatm5t ( x %\ld be giv5 "h4 ,,illu/rative ,,cases ,consid] fir/ "s cases :]e "t sp5t 9 mak+ pupils und]/& ! end to be atta9$ be=e attack+ ! task by :i* x is atta9$1 or c>e ab\t atta9+ ! end 7well or ill und]/ood7 is well sp5t4 ,x is well = a pupil :o has le>n$ 7#a7 ! mean+s ( ! numb]s "o to t51 7#b7 h[ to c\nt a collection ( t5 or less1 & 7#c7 h[ to measure 9 in*es a magnitude ( t51 n9e1 eight in*es1 etc41 to be confront$ ) ! problem ( true add+ )\t c\nt+ or measur+1 z 9 'hidd5' addition & measurem5t by inf]5ce4 ,= example1 ! tea*] has ?ree p5cils c\nt$ & put "u a book2 has two m c\nt$ & put "u ! book2 & asks1 ,h[ _m p5cils are !re "u ! book8 ,answ]s1 :5 obta9$1 are v]ifi$ or refut$ by actual c\nt+ & measur+4 ,! "t "h is well sp5t because ! *ildr5 c d ! necess>y ?9k+ if ! tasks are well *os52 because !y are !reby prev5t$ f beg9n+ _! /udy ( addition by ! bad habit ( pseudo-add+ by look+ at ! two gr\ps ( objects & c\nt+ _! numb] in/ead ( real add+1 t is1 ?9k+ ( ! two numb]s & inf]r+ _! sum2 & fur!r1 because fac+ ! problem ( add+ z a real problem is 9 ! end m economical = le>n+ ari?metic & = intellectual tra9+ 9 g5]al ?an be+ entic$ into add+ by objective or o!r processes :i* conceal ! difficulty :ile help+ ! pupil to ma/] x4 ,! manipulation ( %ort multiplication may be introduc$ by confront+ ! pupils ) su* problems z1 ,h[ to tell h[ _m ,une$a biscuit !re are 9 f\r boxes1 by op5+ only "o box4 ,correct solutions by addition %\ld be accept$4 ,correct solutions by multiplication1 if any gift$ *ildr5 ?9k ( ?is way1 %\ld be accept$1 ev5 if ! *ildr5 _c ju/ify _! proc$ure4 7,inf]r+ ! manipulation f ! place-values ( numb]s is beyond all save ! mo/ gift$ & probably beyond !m47 ,correct solution by multiplication by "s *ild :o happ5s to h le>n$ x else:]e %\ld be accept$4 ,let ! ma9 pro( ( ! tru/wor?9ess ( ! manipulation be by measurem5t & by addition4 ,pro( by ! /ock argum5ts f ! place-values ( numb]s may also be us$4 ,if no *ild hits on ! manipulation 9 "q1 ! problem ( f9d+ ! l5g? )\t add+ may be set4 ,if !y /ill fail1 ! problem may be made easi] by be+ put z #d times #bb gives ! answ]4 ,write d[n :at y ?9k #d times #bb w be4 ,o!r r$uctions ( ! difficulty ( ! problem may be made1 or ! tea*] may give ! answ] )\t v great h>m be+ done4 ,! important requirem5t is t ! pupils %\ld be aw>e ( ! problem & treat ! manipulation z a solution ( x1 n z a =m ( educational c]emonial :i* !y le>n to satisfy ! :ims ( p>5ts & tea*]s4 ,9 ! case ( any p>ticul> class a situation t is m appeal+ to ! pupils' practical int]e/s ?an ! situation us$ "h c probably be devis$4 ,! "t sp5t 9 ?is way is well sp5t 7#a7 because all b ! v dull pupils c solve ! problem 9 "s way1 7#b7 because ! significance ( ! manipulation z an economy ov] addition is wor? br++ out1 & 7#c7 because !re is no way ( beg9n+ tra9+ 9 %ort multiplication t is mu* bett]4 ,9 ! same fa%ion multiplication by two-place numb]s may be introduc$ by confront+ pupils ) ! problem ( ! numb] ( %eets ( pap] 9 #gb pads1 or pieces ( *alk 9 #bd boxes1 or squ>e in*es 9 #ce squ>e feet1 or ! numb] ( days 9 #cb ye>s1 or :atev] simil> problem c be br\ght up s z to be felt z a problem4 ,suppose t x is ! #ce squ>e feet4 ,solutions by 7#e x #add7 7#cj x #add71 h[ev] arrang$1 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 h[ev] arrang$1 %\ld all be li/$ = v]ification or rejection4 ,! pupils ne$ n be requir$ to ju/ify _! proc$ures by a v]bal /atem5t4 ,answ]s l #dcb1gbj1 or #gbj1dcb1 or #aaeb1 or #dbbj1 or #cbbj %\ld be li/$ = v]ification or rejection4 ,v]ification may be by a mixture ( %ort multiplication & objective "w1 or by a mixture ( %ort multiplication & addition1 or by addition abbreviat$ by tak+ t5 #adds z #addj1 or ev5 7= v /upid pupils7 by ! au?ority ( ! tea*]4 ,or ! manipulation 9 cases l #ec x #i or #hd x #g may be v]ifi$ by ! rev]se %ort multiplication4 ,! d$uctive pro( ( ! correctness ( ! manipulation may be giv5 9 :ole or 9 "p 9 connection ) ex]cises l #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 add$ to #e x #add #c x #ad ,c]ta9 wrong answ]s may be %[n to be wrong 9 _m ways2 e4g41 #dcb1gbj is too big1 = #ce times a ?\s& squ>e in*es is only #ce1jjj2 #aaeb is too small1 = #ce times a hundr$ squ>e in*es w\ld be #cejj1 or m ?an #aaeb4 ,! "t sp5t 9 realiz+ ! problem "h is fairly well sp5t because 7#a7 any successful orig9al manipulation 9 ?is case repres5ts an excell5t ex]cise ( ?\ght1 because 7#b7 failures %[ t x is useless to juggle ! figures at r&om1 & because 7#c7 ! previ\s exp]i5ce ) %ort multiplication makes x possible = ! pupils to realize ! problem 9 a v few m9utes4 ,x may1 h[ev]1 be /ill bett] to give ! pupils ! "r me?od j z soon z ! problem is realiz$1 )\t hav+ !m sp5d m "t 9 try+ to solve x4 ,?us3-- #a squ>e foot has #add squ>e in*es4 ,h[ _m squ>e in*es are !re 9 #ce squ>e feet 7m>k$ out 9 *alk on ! floor z a piece #aj ft4 x #c ft4 plus a piece #e ft4 x #a ft478 #a y>d #cf in*es4 ,h[ _m in*es long is ?is wall 7f\nd by measure to be #ac y>ds78 ,"h is a quick way to f9d ! answ]s3-- #add #ce ---- #gbj #dcb ---- #ejdj sq4 in*es 9 #ce sq4 ft4 #cf #ac --- #ajh #cf --- #dfh in*es 9 #ac yd4 ,consid] n[ ! foll[+ introduction to divid+ by a decimal3-- ,divid+ by a ,decimal #a4 ,h[ _m m9utes w x take a motorcycle1 to g #ab4fge miles at ! rate ( 4#ge mi4 p] m9ute8 #af4i 4#geab4fge #g #e --- #e #ag #d #ej ---- #fge #fge --- #b4 ,*eck by multiply+ #af4i by 4#ge4 #c4 ,h[ d y "k t ! quoti5t _c be z little z #a4fi8 #d4 ,h[ d y "k t ! quoti5t _c be z l>ge z #afi8 #e4 ,f9d ! quoti5t = #c4ge _/ #a4e4 #f4 ,*eck y\r result by multiply+ ! quoti5t by ! divisor4 #g4 ,h[ d y "k t ! quoti5t _c be 4#be or #be8 #h4 ,look at ?is problem4 4#beg4e ,h[ d y "k t #c4j is wrong = ! quoti5t8 ,h[ d y "k t #cjj is wrong = ! quoti5t8 ,/ate :i* quoti5t is "r = ea* ( ^!3-- 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 ?is rule true8 ,if x is true1 le>n x4 ,9 a correct result1 ! numb] ( decimal places 9 ! divisor & quoti5t toge!r equals ! numb] ( decimal places 9 ! divid5d4 ,^! & simil> ex]cises excite ! problem attitude 9 *ildr5 :o h a g5]al int]e/ 9 gett+ "r answ]s4 ,su* a s]ies c>efully arrang$ is a desirable introduction to a /atem5t ( ! rule = plac+ ! decimal po9t 9 division ) decimals4 ,= x attracts att5tion to ! g5]al pr9ciple 7divisor x quoti5t %\ld equal divid5d71 :i* is m important ?an ! rule = conv5i5t location ( ! decimal po9t1 & x gives tra9+ 9 plac+ ! po9t by inspection ( ! divisor1 quoti5t1 & divid5d1 :i* suffices = n9ete5 out ( tw5ty cases t ! pupil w ev] enc\nt] outside ( s*ool4 ,he is likely to rememb] ?is me?od by inspection long aft] he has =gott5 ! fix$ rule4 ,x is well = ! pupil to be introduc$ to _m ari?metical facts by way ( problems ab\t _! common uses4 ,! clockface1 ! railroad di/ance table 9 hundr$?s ( a mile1 ! cyclomet] & spe$omet]1 ! recipe1 & ! l (f] problems :i* enli/ 8 int]e/ & en]gy & also connect ! result+ ari?metical le>n+ ) ! activities :]e x is ne$$4 ,!re is no "t co/1 b a "t-sav+1 = ! le>n+ z a means to ! solution ( ! problems is quick] ?an ! m]e le>n+ ( ! ari?metical facts by !mselves alone4 ,a few samples ( su* proc$ure are %[n bel[3-- ,,grade #c ,to be ,done at ,home ,look at a wat*4 ,has x any h&s besides ! h\r h& & ! m9ute h&8 ,f9d out all t y c ab\t h[ a wat* tells seconds1 h[ long a second is1 & h[ _m seconds make a m9ute4 ,,grade #e ,measur+ ,ra9fall ,ra9fall p] ,week 7cu4 94 p] sq4 94 ( 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 ,9 :i* weeks 0 ! ra9fall #a or m8 #b4 ,:i* week ( ,augu/ _h ! l>ge/ ra9fall = t mon?8 #c4 ,:i* 0 ! drie/ week ( ! summ]8 7,drie/ means ) ! lea/ ra9fall47 #d4 ,:i* week 0 ! next to ! drie/8 #e4 ,9 :i* weeks 0 ! ra9fall betwe5 4#hjj & #a4jjj8 #f4 ,look d[n ! table & e/imate :e!r ! av]age ra9fall = "o week 0 ab\t 4#e1 or ab\t 4#f1 or ab\t 4#g1 or ab\t 4#h1 or ab\t 4#i4 ,dairy ,records ,record ( ,/> ,elsie ,p\nds ( ,milk ,butt]-,fat p] ,p\nd ( ,milk ,jan4 #agdb 4#jdfa ,feb4 #afij 4#jdhe ,m>4 #aegd 4#jejd ,apr4 #abbf 4#jdij ,may #abjb 4#jdff ,june #abea 4#jdha ,read ?is record ( ! milk giv5 by ! c[ ,/> ,elsie4 ,! fir/ column tells ! numb] ( p\nds ( milk giv5 by ,/> ,elsie ea* mon?4 ,! second column tells :at fraction ( a p\nd ( butt]-fat ea* p\nd ( milk conta9$4 #a4 ,read ! fir/ l9e1 say+1 ,9 ,janu>y ?is c[ gave #agdb p\nds ( milk4 ,!re 7 #dfa t5 ?\s&?s ( a p\nd ( butt]-fat p] p\nd ( milk4 ,read ! o!r l9es 9 ! same way4 #b4 ,h[ _m p\nds ( butt]-fat did ! c[ produce 9 ,jan48 #c4 ,9 ,feb48 #d4 ,9 ,m>48 #e4 ,9 ,apr48 #f4 ,9 ,may8 #g4 ,9 ,june8 ,,grade #e ,,or ,,lat] ,us+ ,recipes to ,make ,l>g] or ,small] ,quantities ,i4 ,/ate h[ mu* y w\ld use ( ea* mat]ial 9 ! foll[+ recipes3 7a7 ,to make d\ble ! quantity4 7b7 ,to make half ! quantity4 7c7 ,to make #a-a_/b times ! quantity4 ,y may use p5cil & pap] :5 y _c f9d ! "r am\nt m5tally4 #a4 ,,peanut ,,p5u*e #a tablespoon butt] #b cups br[n sug> #a_/c cup milk or cream #c_/d cup *opp$ peanuts #a_/c teaspoon salt #b4 ,,molasses ,,c&y #a_/b cup butt] #b cups sug> #a cup molasses #a-a_/b cups boil+ wat] #c4 ,,rais9 ,,op]a ,,c>amels #b cups light br[n sug> #g_/h cup ?9 cream #a_/b cup rais9s #d4 ,,walnut ,,molasses ,,squ>es #b tablespoons butt] #a cup molasses #a_/c cup sug> #a_/b cup walnut meats #e4 ,,reception ,,rolls #a cup scald$ milk #a-a_/b tablespoons sug> #a teaspoon salt #a_/d cup l>d #a yea/ cake #a_/d cup lukew>m wat] ,:ite ( #a egg #c-a_/b cups fl\r #f4 ,,graham ,,rais$ ,,loaf #b cups milk #f tablespoons molasses #a-a_/b teaspoons salt #a_/c yea/ cake #a_/d cup lukew>m wat] #b cups sift$ ,graham fl\r #a_/b cup ,graham bran #c_/d cup fl\r 7to knead7 ,,ii4 ,h[ mu* w\ld y use ( ea* mat]ial 9 ! foll[+ recipes3 7a7 ,to make #b_/c z l>ge a quantity8 7b7 ,to make #a-a_/c times z mu*8 7c7 ,to make #b-a_/b times z mu*8 #a4 ,,engli% ,,dumpl+s #a_/b p\nd beef suet #a-a_/d cups fl\r #c teaspoons bak+ p[d] #a teaspoon salt #a_/b teaspoon pepp] #a teaspoon m9c$ p>sley #a_/b cup cold wat] #b4 ,,:ite ,,m\nta9 ,,angel ,,cake #a-a_/b cups egg :ites #a-a_/b cups sug> #a teaspoon cream ( t>t> #a cup bread fl\r #a_/d teaspoon salt #a teaspoon vanilla ,9 _m cases ari?metical facts & pr9ciples c be well taught 9 connection ) "s problem or project :i* is n ari?metical1 b :i* has special pot5cy to ar\se an intellectual activity 9 ! pupil :i* by "s +5uity c be direct$ to ari?metical le>n+4 ,play+ /ore is ! mo/ fundam5tal case4 ,plann+ = a p>ty1 see+ :o w9s a game ( bean bag1 und]/&+ ! cal5d> = a mon?1 select+ ,*ri/mas pres5ts1 plann+ a picnic1 arrang+ a g>d51 ! clock1 ! wat* ) second h&1 & draw+ v simple maps are situations sugge/+ problems :i* may br+ a liv+ purpose into ari?metical le>n+ 9 grade #b4 ,^! are all available "u ord9>y conditions ( class in/ruction4 ,a sample ( su* problems = a high] grade 7#f7 is %[n bel[4 ,e/imat+ ,areas ,! *ildr5 9 ! geography class _h a conte/ 9 e/imat+ ! areas ( diff]5t surfaces4 ,ea* *ild wrote 8 e/imates = ea* ( ^! maps1 ,a1 ,b1 ,c1 ,d1 & ,e4 7,only ,c & ,d are %[n "h47 ,9 ! ari?metic class !y le>n$ h[ to f9d ! exact areas4 ,!n !y comp>$ _! e/imates ) ! exact areas to f9d :o came ne>e/4 ,illu/ration ,write y\r e/imates = ,a1 ,b1 ,c1 ,d1 & ,e4 ,!n /udy ! next #f pages & le>n h[ to f9d ! exact areas4 7,! next #f pages comprise tra9+ 9 ! m5suration ( p>allelograms & triangles47 ,9 "s cases ! affairs ( individual pupils include problems :i* may be us$ to guide ! individual 9 "q to a zeal\s /udy ( ari?metic z a means ( a*iev+ 8 purpose--( mak+ a canoe1 survey+ an isl&1 keep+ ! acc\nts ( a ,girls' ,cann+ ,club1 or ! l4 ,x requires mu* "t & v great skill to direct ! "w ( ?irty or m pupils ea* busy ) a special type ( 8 own1 s z to make ! "w in/ructive = ea*1 b 9 "s cases ! exp5se ( "t & skill is ju/ifi$4 ,,g5]al ,,pr9ciples ,9 g5]al :at %\ld be meant :5 "o says t x is desirable to h pupils 9 ! problem-attitude :5 !y are /udy+ ari?metic is sub/antially z foll[s3-- ,fir/4--,in=mation t comes z an answ] to que/ions is bett] att5d$ to1 und]/ood1 & rememb]$ ?an in=mation t j comes4 ,second4--,simil>ly1 movem5ts t come z a /ep t[>d a*iev+ an end t ! pupil has 9 view are bett] connect$ ) _! appropriate situations1 & su* connections are long] reta9$1 ?an is ! case ) movem5ts t j happ54 ,?ird4--,! m ! pupil is set t[>d gett+ ! "q answ]$ or gett+ ! end a*iev$1 ! great] is ! satisfy+ness atta*$ to ! bonds ( k or skill :i* mean progress !reto4 ,f\r?4--,x is bad policy to rely exclusively on ! purely intellectuali/ic problems ( ,h[ c ,i d ?is8 ,h[ c ,i get ! "r answ]8 ,:at is ! reason = ?is8 ,is !re a bett] way to d t8 & ! l4 ,x is bad policy to supplem5t ^! intellectuali/ic problems by only ! remote problems ( ,h[ c ,i be fitt$ to e>n a high] wage8 ,h[ c ,i make sure ( graduat+8 ,h[ c ,i please my p>5ts8 & ! l4 ,! purely intellectuali/ic problems h too weak an appeal = _m pupils2 ! remote problems are weak s long z !y are remote &1 :at is worse1 may be depriv$ ( ! /r5g? t !y w\ld h 9 due "t if we attempt to use !m too soon4 ,x is ! extreme ( bad policy to neglect ^? p]sonal & practical problems furni%$ by life outside ! class 9 ari?metic ! solution ( :i* c really be fur!r$ by ari?metic !n & !re4 ,x is good policy to sp5d "t 9 e/abli%+ c]ta9 m5tal sets--/imulat+1 or ev5 creat+1 c]ta9 ne$s--sett+ up problems !mselves--:5 ! "t s sp5t br+s a suffici5t improvem5t 9 ! quality & quantity ( ! pupils' int]e/ 9 ari?metical le>n+4 ,fif?4--,x w\ld be /ill worse policy to rely exclusively on problems aris+ outside ari?metic4 ,to le>n ari?metic is itself a s]ies ( problems ( intr9sic int]e/ & wor? to heal?y-m9d$ *ildr54 ,! ne$ = ability to multiply ) ,unit$ ,/ates money or to add fractions or to compute p]c5ts may be z truly vital & engag+ z ! ne$ = skill to make a p>ty dress or = money to buy x or = "t to play baseball4 ,! intellectuali/ic ne$s & problems %\ld be consid]$ along ) all o!rs1 & giv5 :atev] weight _! educational value des]ves4 ,,difficulty ,,& ,,success ,,z ,,/imuli ,!re are c]ta9 misconceptions ( ! doctr9e ( ! problem-attitude4 ,! mo/ notewor?y is t difficulty--tempor>y failure--an inadequacy ( already exi/+ bonds--is ! ess5tial & necess>y /imulus to ?9k+ & le>n+4 ,dewey himself does n1 z ,i und]/& him1 mean ?is1 b he has be5 int]pret$ z mean+ x by "s ( 8 foll[]s4#bb #bb ,9 8 ,h[ ,we ,?9k4 ,difficulty--tempor>y failure1 inadequacy ( exi/+ bonds--on ! contr>y does no?+ :atsoev] 9 & ( itself2 & :at is done by ! annoy+ lack ( success :i* sometimes accompanies difficulty sometimes h9d]s ?9k+ & le>n+4 ,m]e difficulty1 m]e failure1 m]e inadequacy ( exi/+ bonds1 does no?+4 ,x is h>d = me to add ?ree eight-place numb]s at a glance2 ,i h fail$ to f9d z effective illu/rations = pages #bgf to #bgg z ,i wi%$2 my exi/+ s5sori-motor connections are inadequate to play+ a golf c\rse 9 #fe4 ,b ^! ev5ts & conditions h done no?+ to /imulate me 9 respect to ! behavior 9 "q4 ,9 ! fir/ ( ! ?ree !re is no annoy+ lack & no dynamic influ5ce at all2 9 ! second !re 0 to "s degree an annoy+ lack--a slight irritation at n gett+ j :at ,i want$1--& ?is might h impell$ me to fur!r ?9k+ 7?\gh x did n1 & gett+ "o tiptop illu/ration w\ld z a rule /imulate me to hunt = o!rs m ?an fail+ to get su*74 ,9 ! ?ird case ! lack ( ! #fe does n annoy me or h any notewor?y dynamic effect4 ,! lack ( #ij in/ead ( #ie-ajj is annoy+ & is at times a /imulus to fur!r le>n+1 ?\gh n ne>ly s /rong a /imulus z ! atta9m5t ( ! #ij w\ld be6 ,at o!r times ?is annoy+ lack is di/9ctly inhibitory--a /imulus to ceas+ to le>n4 ,9 ! intellectual life ! inhibitory effect seems f> ! common] ( ! two4 ,n gett+ answ]s seems z a rule to make u /op try+ to get !m4 ,! annoy+ lack ( success ) a !oretical problem mo/ (t5 makes u des]t x = problems to ^: solution ! exi/+ bonds promise to be m adequate4 ,! real issue 9 all ?is conc]ns ! relative /r5g?1 9 ! pupil's intellectual life1 ( ! negative reaction ( behavior 9 g5]al4 ,an animal ^: life processes are int]f]$ ) s t an annoy+ /ate ( affairs is set up1 *anges 8 behavior1 mak+ "o aft] ano!r responses z 8 in/9cts & le>n$ t5d5cies prescribe1 until ! annoy+ /ate ( affairs is t]m9at$1 or ! animal dies1 or suff]s ! annoyance z less ?an ! alt]natives :i* 8 responses h produc$4 ,:5 ! annoy+ /ate ( affairs is *>act]iz$ by ! failure ( ?+s z !y are to m9i/] to a crav+--z 9 cases ( hung]1 lonel9ess1 sex-pursuit1 & ! l1--we h /imulus to action by an annoy+ lack or ne$1 ) relief f action by ! satisfaction ( ! ne$4 ,su* is 9 "s measure true ( man's intellectual life4 ,9 recall+ a =gott5 "n1 9 solv+ c]ta9 puzzles1 or 9 simplify+ an algebraic complex1 !re is an annoy+ lack ( ! "n1 solution1 or factor1 a trial ( "o aft] ano!r response1 until ! annoyance is reliev$ by success or made less pot5t by fatigue or di/raction4 ,ev5 "h ! difficulty does n d any?+--b only ! annoy+ int]f]5ce ) our intellectual peace by ! problem4 ,fur!r1 al?\gh = ! p>ticul> problem1 ! annoy+ lack /imulates1 & ! successful atta9m5t /ops ?9k+1 ! lat] & m important g5]al effect on ?9k+ is ! rev]se4 ,successful atta9m5t /ops our ?9k+ on t problem b makes u m pr$ispos$ lat] to ?9k+ 9 g5]al4 ,ov]t negative reaction1 h[ev]1 plays a relatively small "p 9 man's intellectual life4 ,fill+ intellectual voids or reliev+ intellectual /ra9s 9 ?is way is mu* less frequ5t ?an be+ /imulat$ positively by ?+s se51 words read1 & pa/ connections act+ "u modifi$ circum/ances4 ,! notion ( ?9k+ z com+ to a lack1 fill+ x1 meet+ an ob/acle1 dodg+ x1 be+ held up by a difficulty & ov]com+ x1 is s "o-sid$ z to v]ge on phantasy4 ,! ov]t lacks1 /ra9s1 & difficulties come p]haps once 9 five h\rs ( smoo? /raight=w>d use & adaptation ( exi/+ connections1 & !y might z truly be call$ h9drances to ?\ght--b>ri]s :i* pa/ successes help ! ?9k] to surm\nt4 ,problems !mselves come m (t5 z *]i%$ issues :i* new facts reveal1 & ^: contemplation ! ?9k] enjoys1 ?an z /ra9s or lacks or 'problems :i* ,i ne$ to solve4' ,x is j z true t ! ?9k] gets _m ( 8 problems z results f1 or bonuses along )1 8 in=mation1 z t he gets mu* ( 8 in=mation z results ( 8 ef=ts to solve problems4 ,z betwe5 difficulty & success1 success is 9 ! long run m productive ( ?9k+4 ,necessity is n ! "m ( inv5tion4 ,k ( previ\s inv5tions is ! "m2 orig9al ability is ! fa!r4 ,! solutions ( previ\s problems are m pot5t 9 produc+ bo? new problems & _! solutions ?an is ! m]e aw>5ess ( problems & desire to h !m solv$4 ,9 ! case ( ari?metic1 le>n+ to cancel in/ead ( gett+ ! product ( ! divid5ds & ! product ( ! divisors & divid+ ! =m] by ! latt]1 is a cle> case ( v valuable le>n+1 ) ease emphasiz$ r ?an difficulty1 ) ! adequacy ( exi/+ bonds 7:5 slightly r$irect$7 z ! prime feature ( ! process r ?an _! inadequacy1 & ) no s5se ( failure or lack or conflict4 ,x w\ld be absurd to sp5d "t 9 ar\s+ 9 ! pupil1 be=e beg9n+ cancellation1 a s5se ( a difficulty--viz41 t ! full multiply+ & divid+ takes long] ?an "o w\ld l4 ,a pupil 9 grade #d or #e might well contemplate t difficulty = ye>s to no advantage4 ,he %\ld at once beg9 to cancel & prove by *eck+ t errorless cancellation always gives ! "r answ]4 ,to emphasize be=e tea*+ cancellation ! inadequacy ( ! old full multiply+ & divid+ w\ld1 moreov]1 n only be uneconomical z a means to tea*+ cancellation2 x w\ld am\nt to ca/+ ne$less slurs on valuable pa/ acquisitions1 & x w\ld1 sci5tifically1 be false4 ,=1 until a pupil has le>n$ to cancel1 ! old full multiply+ is n inadequate2 x is admirable 9 e respect4 ,! issue ( its inadequacy does n truly appe> until ! new me?od is f\nd4 ,x is ! be/ way until ! bett] way is ma/]$4 ,9 ! same way x is unwise to sp5d "t 9 mak+ pupils aw>e ( ! annoy+ lacks to be suppli$ by ! multiplication tables1 ! division tables1 ! ca/+ out ( n9es1 or ! use ( ! product ( ! l5g? & bread? ( a rectangle z its area1 ! unit be+ *ang$ to ! squ>e erect$ on ! l9e> unit z base4 ,! annoy+ lack w be unproductive1 :ile ! le>n+ takes place readily z a modification ( exi/+ habits1 & is suffici5tly appreciat$ z soon z x does take place4 ,! multiplication tables come :5 in/ead ( m]ely c\nt+ by #gs f #j up say+ #g1 #ad1 #ba1 etc41 ! pupil c\nts by #gs f #j up say+ ,two sev5s make #ad1 ?ree sev5s make #ba1 f\r sev5s make #bh1 etc4 ,! division tables come z easy selections f ! kn[n multiplications2 ! ca/+ out ( n9es comes z an easy device4 ,! computation ( ! area ( a rectangle is be/ facilitat$1 n by aw>5ess ( ! lack ( a process = do+ x1 b by aw>5ess ( ! success ( ! process z v]ifi$ objectively4 ,9 all ^! cases1 too1 ! pupil w\ld be misl$ if we ar\s$ fir/ a s5se ( ! inadequacy ( c\nt+1 add+1 & objective division1 an aw>5ess ( ! difficulties :i* ! multiplication & division tables & n9es device & area !orem relieve4 ,! displac$ processes are admirable & no unnecess>y fault %\ld be f\nd ) !m1 & !y are n inadequate until ! %ort] ways h be5 le>n$4 ,,false ,,inf]5ces ,"o false inf]5ce ab\t ! problem-attitude is t ! pupil %\ld always und]/& ! aim or issue be=e beg9n+ to =m ! bonds :i* give ! me?od or process t provides ! solution4 ,on ! contr>y1 he w (t5 get ! process m easily & value x m highly if he is taught :at x is = gradually :ile he is le>n+ x4 ,! sy/em ( decimal notation1 = example1 may bett] be tak5 fir/ z a m]e fact1 j z we tea* a *ild to talk )\t try+ fir/ to h him und]/& ! value ( v]bal int]c\rse1 or to keep clean )\t try+ fir/ to h him und]/& ! bact]iological consequ5ces ( fil?4 ,a second inf]5ce--t ! pupil %\ld always be taught to c>e ab\t an issue & crave a process = manag+ x be=e beg9n+ to le>n ! process--is equally false4 ,on ! contr>y1 ! be/ way to become int]e/$ 9 c]ta9 issues & ! ways ( h&l+ !m is to le>n ! process--ev5 to le>n x by %e] habituation--& !n note :at x does = u4 ,su* is ! case ) 4#afff-b_/c x divide by #f1 4#ccc-a_/c x divide by #c1 multiply by 4#hge divide ! numb] by #h & subtract ! quoti5t f ! numb]4 ,a ?ird unwise t5d5cy is to degrade ! m]e giv+ ( in=mation--to belittle ! value ( facts acquir$ 9 any o!r way ?an 9 ! c\rse ( delib]ate ef=t by ! pupil to relieve a problem or conflict or difficulty4 ,z a prote/ aga9/ m]ely v]bal k1 & m]ely memorit] k1 & neglect ( ! active1 que/ion+ se>* = k1 ?is t5d5cy to belittle m]e facts has be5 heal?y1 b z a g5]al doctr9e x is itself equally "o-sid$4 ,m]e facts n got by ! pupil's ?9k+ are (t5 ( enorm\s value4 ,!y may /imulate to active ?9k+ j z truly z t may /imulate to ! reception ( facts4 ,9 ari?metic1 = example1 ! names ( ! numb]s1 ! use ( ! fractional =m to signify t ! upp] numb] is divid$ by ! l[] numb]1 ! e>ly use ( ! decimal po9t 9 ,u4 ,s4 money to di/+ui% doll>s f c5ts1 & ! mean+s ( ea*1 :ole1 "p1 toge!r1 9 all1 sum1 diff]5ce1 product1 quoti5t1 & ! l are self-ju/ify+ facts4 ,a f\r? false inf]5ce is t :atev] tea*+ makes ! pupil face a "q & ?9k out its answ] is !reby ju/ifi$4 ,?is is n necess>ily s unless ! "q is a wor?y "o & ! answ] t is ?\ght out an intr9sically valuable "o & ! process ( ?9k+ us$ "o t is appropriate = t pupil = t "q4 ,m]ely to ?9k may be ( little value4 ,to rely mu* on =mal discipl9e is j z p]nici\s "h z else:]e4 ,! t5d5cy to emphasize ! me?ods ( le>n+ ari?metic at ! exp5se ( :at is le>n$ is likely to lead to abuses diff]5t 9 nature b z bad 9 effect z t to :i* ! emphasis on discipl9>y r ?an cont5t value has l$ 9 ! /udy ( languages & gramm>1 or 9 ! old puzzle problems ( ari?metic4 ,! la/ false inf]5ce t ,i %all discuss "h is ! inf]5ce t mo/ ( ! problems by :i* ari?metical le>n+ is /imulat$ _h bett] be ext]nal to ari?metic itself--problems ab\t ,noah's ,ark or ,ea/] ,fl[]s or ! ,m]ry ,g ,r\nd or ,a ,trip d[n ! ,rh9e4 ,outside int]e/s %\ld be kept 9 m9d1 z has be5 abundantly illu/rat$ 9 ?is volume1 b x is folly to neglect ! p[]1 ev5 = v "y or = v /upid *ildr51 ( ! problem ,h[ c ,i get ! "r answ]8 ,*ildr5 d h intellectual int]e/s4 ,!y d l dom9oes1 *eck]s1 anagrams1 & riddles z truly z play+ tag1 pick+ fl[]s1 & bak+ cake4 ,) c>efully grad$ "w t is )9 _! p[]s !y l to le>n to add1 subtract1 multiply1 & divide ) integ]s1 fractions1 & decimals1 & to "w out quantitative relations4 ,9 "s measure1 le>n+ ari?metic is l le>n+ to typewrite4 ,! le>n] ( ! latt] has little desire to pres5t attractive-look+ excuses = be+ late1 or to save exp5se = pap]4 ,he has no desire to ho>d copies ( su* & su* lit]>y gems4 ,he may ga9 zeal f ! fact t a s*ool p>ty is to be giv5 & invitations are to be s5t out1 b ! problem ,to typewrite bett] is aft] all 8 ma9 problem4 ,le>n+ ari?metic is 9 "s measure a game ^: moves are motivat$ by ! g5]al set ( ! m9d t[>d victory--w9n+ "r answ]s4 ,z a ball-play] le>ns to ?r[ ! ball accurately to fir/-base1 n prim>ily because ( any p>ticul> problem conc]n+ gett+ rid ( ! ball1 or hav+ ! man at fir/-base possess x1 or putt+ out an oppon5t aga9/ :om he has a grudge1 b because t skill is requir$ by ! game z a :ole1 s ! pupil1 9 "s measure1 le>ns ! te*nique ( ari?metic1 n because ( p>ticul> concrete problems ^: solutions x furni%es1 b because t te*nique is requir$ by ! game ( ari?metic--a game t has intr9sic wor? & _m g5]al recomm5dations4 ,,*apt] ,,xv ,,individual ,,diff]5ces ,! g5]al facts conc]n+ individual v>iations 9 abilities--t ! v>iations are l>ge1 t !y are cont9u\s1 & t = *ildr5 ( ! same age !y usually clu/] ar\nd "o typical or modal ability1 becom+ less & less frequ5t z we pass to v high or v l[ degrees ( ! ability--are all well illu/rat$ by ari?metical abilities4 ,,nature ,,& ,,am\nt ,! surfaces ( frequ5cy %[n 9 ,figs4 #fa1 #fb1 & #fc are samples4 ,9 ^! diagrams ea* space along ! basel9e repres5ts a c]ta9 score or degree ( ability1 & ! height ( ! surface above x repres5ts ! numb] ( individuals obta9+ t score4 ,?us 9 ,fig4 #fa1 #fc out ( #ajjj soldi]s _h no correct answ]1 #cf out ( #ajjj _h "o correct answ]1 #di _h two1 #ee _h ?ree1 #fg _h f\r1 & s on1 9 a te/ ) problems 7/at$ 9 words74 ,figure #fa %[s t ^! adults v>i$ f no problems solv$ correctly to eighte51 ar\nd eight z a c5tral t5d5cy4 ,figure #fb %[s t *ildr5 ( ! same ye>-age 7!y 7 also f ! same neighborhood & 9 ! same s*ool7 v>i$ f "u #dj to ov] #bjj figures correct4 ,figure #fc %[s t ev5 among *ildr5 :o h all rea*$ ! same s*ool grade & s _h r simil> educational opportunities 9 ari?metic1 ! v>iation is /ill v great4 ,x requires a range f #ae to ov] #cj examples "r to include ev5 n9e t5?s ( !m4 ,illu/ration3 ,,fig4 #fa4--,! scores ( #ajjj soldi]s 9 ! ,national ,army born 9 ,engli%-speak+ c\ntries1 9 ,te/ #b ( ! ,army ,alpha4 ,! score is ! numb] ( correct answ]s obta9$ 9 five m9utes4 ,probably #aj to #ae p]c5t ( ^! m5 7 unable to read or able to read only v easy s5t5ces at a v sl[ rate4 ,data furni%$ by ! ,division ( ,psy*ology 9 ! (fice ( ! ,surgeon ,g5]al4 ,x %\ld1 h[ev]1 be not$ t if ea* individual _h be5 scor$ by ! av]age ( 8 "w on eight or t5 diff]5t days in/ead ( by 8 "w 9 j "o te/1 ! v>iability w\ld h be5 some:at less ?an appe>s 9 ,figs4 #fa1 #fb1 & #fc4 ,illu/ration3 ,,fig4 #fb4--,! scores ( #ajj #aa-ye>-old pupils 9 a te/ ( computation4 ,e/imat$ f ! data giv5 by ,burt '#ag1 p4 #fh = #aj-1 #aa-1 & #ab-ye>-olds4 ,! score equals ! numb] ( correct figures4 ,x is also ! case t if ea* individual _h be5 scor$1 n 9 problem-solv+ alone or division alone1 b 9 an elaborate exam9ation on ! :ole field ( ari?metic1 ! v>iability w\ld h be5 some:at less ?an appe>s 9 ,figs4 #fa1 #fb1 & #fc4 ,on ! o!r h&1 if ! (fic]s & ! soldi]s reject$ = feeblem9d$ness _h be5 includ$ 9 ,fig4 #fa1 if ! #aa-ye>-olds 9 special classes = ! v dull _h be5 includ$ 9 ,fig4 #fb1 & if all *ildr5 :o _h be5 to s*ool six ye>s _h be5 includ$ 9 ,fig4 #fc1 no matt] :at grade !y _h rea*$1 ! effect w\ld h be5 to increase ! v>iability4 ,illu/ration3 ,,fig4 #fc4--,! scores ( pupils 9 grade #f 9 city s*ools 9 ! ,woody ,division ,te/ ,a4 ,! score is ! numb] ( correct answ]s obta9$ 9 #bj m9utes4 ,f ,woody '#af1 p4 #fa4 ,9 spite ( ! ef=t by s*ool (fic]s to collect 9 any "o s*ool grade ^? some:at equal 9 ability or 9 a*ievem5t or 9 a mixture ( ! two1 ! population ( ! same grades 9 ! same s*ool sy/em %[s a v wide range 9 any ari?metical ability4 ,?is is p>tly because promotion is on a m g5]al basis ?an ari?metical ability s t "s v able ari?meticians are delib]ately held back on acc\nt ( o!r defici5cies1 & "s v incompet5t ari?meticians are advanc$ on acc\nt ( o!r excell5cies4 ,x is p>tly because ( g5]al inaccuracy 9 classify+ & promot+ pupils4 ,9 a composite score made up ( ! sum ( ! scores 9 ,woody te/s1--,add4 ,a1 ,subt4 ,a1 ,mult4 ,a1 & ,div4 ,a1 & two te/s 9 problem-solv+ 7t5 & six grad$ problems1 ) maximum atta9able cr$its ( #cj & #ah71 ,kruse '#ah f\nd facts f :i* ,i compute ^? ( ,table #ac1 & ,figs4 #fd to #ff1 = pupils all hav+ ! tra9+ ( ! same city sy/em1 "o :i* s\ght to grade its pupils v c>efully4 ,illu/ration3 ,,figs4 #fd1 #fe1 & #ff4--,! scores ( pupils 9 grade #f 7,fig4 #fd71 grade #g 7,fig4 #fe71 & grade #h 7,fig4 #ff7 9 a composite ( te/s 9 computation & problem-solv+4 ,! "t 0 ab\t #abj m9utes4 ,! maximum score atta9able 0 #aif4 ,! ov]lapp+ ( grade ^u grade %\ld be not$4 ,( ! pupils 9 grade #f ab\t #ah p]c5t d bett] ?an ! av]age pupil 9 grade #g1 & ab\t #g p]c5t d bett] ?an ! av]age pupil 9 grade #h4 ,( ! pupils 9 grade #h ab\t #cc p]c5t d worse ?an ! av]age pupil 9 grade #g & ab\t #ab p]c5t d worse ?an ! av]age pupil 9 grade #f4 ,,table #ac ,,relative ,,frequ5cies ,,( ,,scores ,,9 ,,an ,,ext5sive ,,team ,,( ,,ari?metical ,,te/s4#bc ,,9 ,,p]c5ts ,,score ,,grade #f ,,grade #g ,,grade #h ---------------------------------------- --- #gj to #gi #a4c 4#i 4#d #hj #hi #e4e #b4c 4#d #ij #ii #aj4f #d4c #b4i #ajj #aji #ai4d #e4b #d4d #aaj #aai #ai4h #ah4e #e4h #abj #abi #bc4e #af4b #af4h #acj #aci #ab4f #ag4e #af4h #adj #adi #d4f #ac4i #bb4i #aej #aei #a4g #ac4f #ag4a #afj #afi #a4b #d4h #i4d #agj #agi #b4e #c4c #bc ,compil$ f data on p4 #hi ( ,kruse '#ah4 ,,diff]5ces ,,)9 ,,"o ,,class ,! v>iation )9 a s+le class = :i* a s+le tea*] has to provide is great4 ,ev5 :5 tea*+ is dep>tm5tal & promotion is by subjects1 & :5 also ! s*ool is a l>ge "o & classification )9 a grade is by ability--!re may be a wide range = any giv5 special compon5t ability4 ,"u ord9>y circum/ances ! range is s great z to be "o ( ! *ief limit+ conditions = ! tea*+ ( ari?metic4 ,_m me?ods appropriate to ! top qu>t] ( ! class w be almo/ useless = ! bottom qu>t]1 & vice v]sa4 ,illu/ration3 ,,figs4 #fg & #fh4--,! scores ( t5 #f ,b classes 9 a #ab-m9ute te/ 9 computation ) integ]s 7! ,c\rtis ,te/ #g74 ,! score is ! numb] ( units done4 ,c]ta9 long tasks are c\nt$ z two units4 ,figures #fg & #fh %[ ! scores ( t5 classes tak5 at r&om f n9ety #f ,b classes 9 "o city by ,c\rtis '#ac1 p4 #fd 9 am\nt ( computation done 9 #ab m9utes4 ,obs]ve ! v wide v>iation pres5t 9 ! case ( e class4 ,! v>iation )9 a class w\ld be some:at r$uc$ if ea* pupil 7 measur$ by 8 av]age 9 eight or t5 su* te/s giv5 on diff]5t days4 ,if a r g5]\s all[ance is made = ?is we /ill h a v>iation 9 spe$ z great z t %[n 9 ,fig4 #fi1 z ! fact to be expect$ = a class ( ?irty-two #f ,b pupils4 ,illu/ration3 ,,fig4 #fi4--;,a cons]vative e/imate ( ! am\nt ( v>iation to be expect$ )9 a s+le class ( #cb pupils 9 grade #f1 9 ! numb] ( units done 9 ,c\rtis ,te/ #g :5 all *ance v>iations are elim9at$4 ,! v>iations )9 a class 9 respect to :at processes are und]/ood s z to be done ) only occasional errors may be illu/rat$ fur!r z foll[s3--,a tea*] 9 grade #d at or ne> ! middle ( ! ye> 9 a city do+ ! cu/om>y "w 9 ari?metic w probably f9d "s pupil 9 h] class :o _c d column addition ev5 )\t c>ry+1 or ! easie/ writt5 subtraction 7#h #i #gh7 7#e #c or #cg7 7- - --71 :o does n "k 8 multiplication tables or h[ to d]ive !m1 or und]/& ! mean+s ( - x & _/1 or h any useful ideas :atev] ab\t division4 ,!re w probably be "s *ild 9 ! class :o c d su* "w z t %[n bel[1 & ) v 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 94 #c4di #b yd4 #b ft4 #c 94 ----- ---------------------- ,multiply #a-a_/d x #h #af #ade #b-e_/h #bjf ------ --- ,divide #b7ac4ej #be7igej ,! inv5tion ( means ( tea*+ ?irty s diff]5t *ildr5 at once ) ! maximum help & m9imum h9drance f _! diff]5t capacities & acquisitions is "o ( ! great opportunities = appli$ sci5ce4 ,c\rtis1 emphasiz+ ! social dem& = a c]ta9 mod]ate ari?metical atta9m5t 9 ! case ( ne>ly all elem5t>y s*ool *ildr5 (1 say1 grade #f1 has urg$ t def9ite special means be tak5 to br+ ! defici5t *ildr5 up to c]ta9 /&>ds1 )\t caus+ undesirable 'ov]le>n+' by ! m gift$ *ildr54 ,c]ta9 exp]im5tal "w to ?is end has be5 c>ri$ out by him & o!rs1 b probably mu* m mu/ be done be=e an au?oritative program = secur+ c]ta9 m9imum /&>ds = all or ne>ly all pupils c be arrang$4 ,,! ,,causes ,,( ,,individual ,,diff]5ces ,! diff]5ces f\nd among *ildr5 ( ! same grade 9 ! same city are due 9 l>ge measure to inborn diff]5ces 9 _! orig9al natures4 ,if1 by a miracle1 ! *ildr5 /udi$ by ,c\rtis1 or by ,woody1 or by ,kruse _h all receiv$ exactly ! same nurture f bir? to date1 !y w\ld /ill h v>i$ greatly 9 ari?metical ability1 p]haps almo/ z mu* z !y n[ d v>y4 ,! evid5ce = ?is is ! g5]al evid5ce t v>iation 9 orig9al nature is responsible = mu* ( ! ev5tual v>iation f\nd 9 intellectual & moral traits1 plus c]ta9 special evid5ce 9 ! case ( ari?metical abilities !mselves4 ,?orndike f\nd '#je t 9 te/s ) addition & multiplication tw9s 7 v mu* m alike ?an sibl+s#bd two or ?ree ye>s ap>t 9 age1 ?\gh ! resemblance 9 home & s*ool tra9+ 9 ari?metic %\ld be ne>ly z great = ! latt] z = ! =m]4 ,also ! "y tw9s 7#i-aa7 %[$ z close a resemblance 9 addition & multiplication z ! old] tw9s 7#ab-ae71 al?\gh ! simil>ities ( tra9+ 9 ari?metic h _h twice z long to op]ate 9 ! latt] case4 #bd ,sibl+s is us$ = *ildr5 ( ! same p>5ts4 ,if ! diff]5ces f\nd1 say among *ildr5 9 grade #f 9 addition1 7 due to diff]5ces 9 ! quantity & quality ( tra9+ 9 addition :i* !y h _h1 !n by giv+ ea* ( !m #bjj m9utes ( additional id5tical tra9+ ! diff]5ces %\ld be r$uc$4 ,= ! #bjj m9utes ( id5tical tra9+ is a /ep t[>d equaliz+ tra9+4 ,x has be5 f\nd 9 _m inve/igations ( ! matt] t :5 we make tra9+ 9 ari?metic m ne>ly equal = any gr\p ! v>iation )9 ! gr\p is n r$uc$4 ,on ! contr>y1 equaliz+ tra9+ seems r to increase diff]5ces4 ,! sup]ior individual seems to h atta9$ 8 sup]iority by 8 own sup]iority ( nature r ?an by sup]ior pa/ tra9+1 =1 dur+ a p]iod ( equal tra9+ = all1 he increases 8 lead4 ,= example1 comp>e ! ga9s ( diff]5t individuals due to ab\t #cjj m9utes ( practice 9 m5tal multiplication ( a ?ree-place numb] by a ?ree-place numb] %[n 9 ,table #ad bel[1 f data obta9$ by ! au?or '#jh4be #be ,simil> results h be5 obta9$ 9 ! case ( ari?metical & o!r abilities by ,?orndike '#jh1 '#aj1 '#ae1 '#af1 ,:itley '#aa1 ,/>* '#aa1 ,wells '#ab1 ,kirby '#ac1 ,donovan & ,?orndike '#ac1 ,hahn & ,?orndike '#ad1 & on a v l>ge scale by ,race 9 a /udy z yet unpubli%$4 ,,table #ad ,,! ,,effect ,,( ,,equal ,,am\nts ,,( ,,practice ,,^u ,,individual ,,diff]5ce ,,9 ,,! ,,multiplication ,,( ,,?ree-,,place ,,numb]s ,,am\nt ,,p]c5tage ,,( ,,correct ,,figures ------------------------------- ,initial ,initial ,score ,ga9 ,score ,ga9 ---------------------------------------- ------------------------ ,initially highe/ five individuals #he #fa #gj #ah next five #ef #ea #fh #aj next six #df #bb #gd #h next six #ch #h #eh #ab next six #bi #bd #ef #ad ,,! ,,int]relations ,,( ,,individual ,,diff]5ces ,a*ievem5t 9 ari?metic dep5ds ^u a numb] ( diff]5t abilities4 ,= example1 accuracy 9 copy+ numb]s dep5ds ^u eyesight1 ability to p]ceive visual details1 & %ort-t]m memory = ^!4 ,long column addition dep5ds *iefly ^u great /r5g? ( ! addition comb9ations especially 9 high] decades1 'c>ry+1' & keep+ "o's place 9 ! column4 ,! solution ( problems fram$ 9 words requires und]/&+ ( language1 ! analysis ( ! situation describ$ into its elem5ts1 ! selection ( ! "r elem5ts = use at ea* /ep & _! use 9 ! "r relations4 ,s9ce ! abilities :i* toge!r con/itute ari?metical ability are ?us specializ$1 ! individual :o is ! be/ ( a ?\s& ( 8 age or grade 9 respect to1 say1 add+ integ]s1 may occupy diff]5t /ations1 p]haps f #a/ to #fjj?1 9 multiply+ ) integ]s1 plac+ ! decimal po9t 9 division ) decimals1 solv+ novel problems1 copy+ figures1 etc41 etc4 ,su* specialization is 9 "p due to 8 hav+ _h1 relatively to ! o!rs 9 ! ?\s&1 m or bett] tra9+ 9 c]ta9 ( ^! abilities ?an 9 o!rs1 & to v>i\s circum/ances ( life :i* h caus$ him to h1 relatively to ! o!rs 9 ! ?\s&1 great] int]e/ 9 c]ta9 ( ^! a*ievem5ts ?an 9 o!rs4 ,! specialization is n :olly due !reto1 h[ev]4 ,c]ta9 inborn *>act]i/ics ( an individual pr$ispose him to diff]5t degrees ( sup]iority or inf]iority to o!r m5 9 diff]5t features ( ari?metic4 ,we measure ! ext5t to :i* ability ( "o sort goes ) or fails to g ) ability ( "s o!r sort by ! coeffici5t ( correlation betwe5 ! two4 ,if e individual keeps ! same rank 9 ! second ability--if ! individual :o is ! be/ ( ! ?\s& 9 "o is ! be/ ( ! gr\p 9 ! o!r1 & s on d[n ! li/--! correlation is #a4jj4 ,9 proportion z ! ranks ( individuals v>y 9 ! two abilities ! coeffici5t drops f #a4jj1 a coeffici5t ( #j mean+ t ! be/ individual 9 ability ,a is no m likely to be 9 fir/ place 9 ability ,b ?an to be 9 any o!r rank4 ,! mean+s ( coeffici5ts ( correlation ( 4#ij1 4#gj1 4#ej1 & #j are %[n by ,tables #ae1 #af1 #ag & #ah4bf #bf ,unless he has a ?or\gh und]/&+ ( ! und]ly+ !ory1 ! /ud5t %\ld be v cauti\s 9 mak+ inf]5ces f coeffici5ts ( correlation4 ,,table #ae ,,di/ribution ,,( ,,arrays ,,9 ,,successive ,,t5?s ,,( ,,! ,,gr\p ,,:5 r 4#ij #aj,,? #i,,? #h,,? #g,,? #f,,? #e,,? #d,,? #c;,d #b;,d #a,,/ ---------------------------------------- -------------------- #a/ t5? 4#a 4#d #a4h #f4f #bb4d #fh4g #b;d t5? 4#a 4#d #a4d #d4g #aa4e #bc4e #cf4j #bb4d #c;d t5? 4#a 4#e #b4a #e4h #ab4h #ba4a #bg4d #bc4e #f4f #d? t5? 4#d #b4a #f4d #ab4h #bj4a #bc4h #ba4b #aa4e #a4h #e? t5? 4#a #a4d #e4h #ab4h #ai4c #bb4f #bj4a #ab4h #d4g 4#d #f? t5? 4#d #d4g #ab4h #bj4a #bb4f #ai4c #ab4h #e4h #a4d 4#a #g? t5? #a4h #aa4e #ba4b #bc4h #bj4a #ab4h #f4d #b4a 4#d #h? t5? #f4f #bc4e #bg4d #ba4a #ab4h #e4h #b4a 4#e 4#a #i? t5? #bb4d #cf4j #bc4e #aa4e #d4g #a4d 4#d 4#a #aj? t5?#fh4g #bb4d #f4f #a4h 4#d 4#a ,,table #af ,,di/ribution ,,( ,,arrays ,,9 ,,successive ,,t5?s ,,( ,,! ,,gr\p ,,:5 r 4#gj #aj,,? #i,,? #h,,? #g,,? #f,,? #e,,? #d,,? #c;,d #b;,d #a,,/ ---------------------------------------- -------------------- #a/ t5? 4#b 4#g #a4e #b4h #d4h #h4j #ac4j #bb4c #df4g #b;d t5? 4#b #a4b #b4f #d4e #g4j #i4h #ac4d #ag4c #ba4g #bb4c #c;d t5? 4#g #b4f #e4j #g4c #aj4j #ab4e #ad4i #af4g #ag4c #ac4j #d? t5? #a4e #d4e #g4c #i4h #ab4j #ac4g #ad4h #ad4i #ac4d #h4j #e? t5? #b4h #g4j #aj4j #ab4j #ac4d #ad4j #ac4g #ab4e #i4h #d4h #f? t5? #d4h #i4h #ab4e #ac4g #ad4j #ac4d #ab4j #aj4j #g4j #b4h #g? t5? #h4j #ac4d #ad4i #ad4h #ac4g #ab4j #i4h #g4c #d4e #a4e #h? t5? #ac4j #ag4c #af4g #ad4i #ab4e #aj4j #g4c #e4j #b4f 4#g #i? t5? #bb4c #ba4g #ag4c #ac4d #i4h #g4j #d4e #b4f #a4b 4#b #aj? t5?#df4g #bb4c #ac4j #h4j #d4h #b4h #a4e 4#g 4#b ,,table #ag ,,di/ribution ,,( ,,arrays ,,( ,,successive ,,t5?s ,,( ,,! ,,gr\p ,,:5 r 4#ej #aj,,? #i,,? #h,,? #g,,? #f,,? #e,,? #d,,? #c;,d #b;,d #a,,/ ---------------------------------------- -------------------- #a/ t5? 4#h #b4j #c4b #d4f #f4b #h4a #aj4e #ac4i #ah4j #ca4h #b;d t5? #b4j #d4a #e4g #g4c #h4h #aj4e #ab4b #ad4a #af4d #ah4i #c;d t5? #c4b #e4g #g4d #h4i #aj4j #aa4b #ab4c #ac4c #ad4a #ac4i #d? t5? #d4f #g4c #h4h #i4i #aj4h #aa4f #ab4j #ab4c #ab4b #aj4e #e? t5? #f4b #h4h #aj4j #aj4h #aa4c #aa4e #aa4f #aa4b #aj4e #h4a #f? t5? #h4a #aj4e #aa4b #aa4f #aa4e #aa4c #aj4h #aj4j #h4h #f4b #g? t5? #aj4e #ab4b #ab4c #ab4j #aa4f #aj4h #i4i #h4h #g4e #d4f #h? t5? #ac4i #ad4a #ac4c #ab4c #aa4b #aj4j #h4h #g4d #e4g #c4b #i? t5? #ah4i #af4d #ad4a #ab4b #aj4e #h4h #g4c #e4g #d4a #b4j #aj? t5?#ca4h #ah4i #ac4i #aj4e #h4a #f4b #d4f #c4b #b4j 4#h ,,table #ah ,,di/ribution ,,( ,,arrays1 ,,9 ,,successive ,,t5?s ,,( ,,! ,,gr\p ,,:5 r 4#j #aj,,? #i,,? #h,,? #g,,? #f,,? #e,,? #d,,? #c;,d #b;,d #a,,/ ---------------------------------------- -------------------- #a/ t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #b;d t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #c;d t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #d? t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #e? t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #f? t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #g? t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #h? t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #i? t5? #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj #aj? t5?#aj #aj #aj #aj #aj #aj #aj #aj #aj #aj ,! significance ( any coeffici5t ( correlation dep5ds ^u ! gr\p ( individuals = :i* x is det]m9$4 ,a correlation ( 4#dj betwe5 computation & problem-solv+ 9 eigh?-grade pupils ( #ad ye>s w\ld mean a mu* clos] real relation ?an a correlation ( 4#dj 9 all #ad-ye>-olds1 & a v1 v mu* clos] relation ?an a correlation ( 4#dj = all *ildr5 #h to #ae4 ,unless ! individuals conc]n$ are v elaborately te/$ on sev]al days1 ! correlations obta9$ are att5uat$ t[>d #j by ! accid5tal errors 9 ! orig9al measurem5ts4 ,?is effect 0 n kn[n until #aijd2 consequ5tly ! correlations 9 ! e>li] /udies ( ari?metic are all too l[4 ,9 g5]al1 ! correlation betwe5 ability 9 any "o important feature ( computation & ability 9 any o!r important feature ( computation is high4 ,if we make en\gh te/s to measure ea* individual exactly 93-- 7,a7 ,subtraction ) integ]s & decimals1 7,b7 ,multiplication ) integ]s & decimals1 7,c7 ,division ) integ]s & decimals1 7,d7 ,multiplication & division ) common fractions1 & 7,e7 ,comput+ ) p]c5ts1 we %all probably f9d ! int]correlations = a ?\s& #ad-ye>-olds to be ne> 4#ij4 ,addition ( integ]s 7,f7 w1 h[ev]1 correlate less closely ) any ( ! above1 be+ app>5tly dep5d5t on simpl] & m isolat$ abilities4 ,! correlation betwe5 problem-solv+ 7,g7 & computation w be v mu* less1 probably n ov] 4#fj4 ,x %\ld be not$ t ev5 :5 ! correlation is z high z 4#ij1 !re w be "s individuals v high 9 "o ability & v l[ 9 ! o!r4 ,su* disp>ities are to "s ext5t1 z ,c\rtis '#ac1 pp4 #fg-ge & ,cobb '#ag h argu$1 due to inborn *>act]i/ics ( ! individual 9 "q :i* pr$ispose him to v special sorts ( /r5g? & weakness4 ,!y are (t5 due1 h[ev]1 to defects 9 8 le>n+ :]eby he has acquir$ m ability ?an he ne$s 9 "o l9e ( "w or has fail$ to acquire "s ne$$ ability :i* 0 well )9 8 capacity4 ,9 g5]al1 all correlations betwe5 an individual's div]g5ce f ! common type or av]age ( 8 age = "o ari?metical function1 & 8 div]g5ces f ! av]age = any o!r ari?metical function1 are positive4 ,! correlation due to orig9al capacity m ?an c\nt]balances ! effects t robb+ ,pet] to pay ,paul may h4 ,spe$ & accuracy are ?us positively correlat$4 ,! individuals :o d ! mo/ "w 9 t5 m9utes w be above ! av]age 9 a te/ ( accuracy4 ,! common notion t spe$ is oppos$ to accuracy is correct :5 x means t ! same p]son w t5d to make m errors if he works at too rapid a rate2 b x is entirely wrong :5 x means t ! k9d ( p]son :o works m rapidly ?an ! av]age p]son is likely to be less accurate ?an ! av]age p]son4 ,int]e/ 9 ari?metic & ability at ari?metic are probably correlat$ positively 9 ! s5se t ! pupil :o has m int]e/ ?an o!r pupils ( 8 age t5ds 9 ! long run to h m ability ?an !y4 ,!y are c]ta9ly correlat$ 9 ! s5se t ! pupil :o 'likes' ari?metic bett] ?an geography or hi/ory t5ds to h relatively m ability 9 ari?metic1 or1 9 o!r words1 t ! pupil :o is m gift$ at ari?metic ?an at draw+ or ,engli% t5ds also to l x bett] ?an he likes ^!4 ,^! correlations are high4 ,x is correct !n to ?9k ( ma!matical ability z1 9 a s5se1 a unit>y ability ( :i* any "o individual may h mu* or little1 mo/ individuals possess+ a mod]ate am\nt ( x4 ,?is is consi/5t1 h[ev]1 ) ! occasional appe>ance ( individuals possess$ ( v great tal5ts = ?is or t p>ticul> feature ( ma!matical ability & equally notable defici5cies 9 o!r features4 ,f9ally x may be not$ t ability 9 ari?metic1 ?\gh occasionally f\nd 9 m5 o!rwise v /upid1 is usually associat$ ) sup]ior intellig5ce 9 deal+ ) ideas & symbols ( all sorts1 & is "o ( ! be/ e>ly indications !re(4 ,,bibliography ,,( ,,ref]5ces ,,made ,,9 ,,! ,,text ,ames1 ,a4 ,f41 & ,mc,lellan1 ,j4 ,f42 '#jj ,public ,s*ool ,ari?metic4 ,ball\1 ,f4 ,w42 '#af ,det]m9+ ! ,a*ievem5ts ( ,pupils 9 ! ,addition ( ,fractions4 ,s*ool ,docum5t ,no4 #c1 #aiaf1 ,bo/on ,public ,s*ools4 ,br&ell1 ,g42 '#ac ,skolb>ns intress54 ,translat$ '#ae by ,w4 ,/]n z1 ,das ,int]esse d] ,s*ulk9d] an d5 ,unt]ri*tsfae*]n4 ,br&=d1 ,b42 '#jh ,a ,/udy ( ,ma!matical ,education4 ,br[n1 ,j4 ,c42 '#aa1 '#ab ,an ,inve/igation on ! ,value ( ,drill ,"w 9 ! ,fundam5tal ,op]ations 9 ,ari?metic4 ,j\rnal ( ,educational ,psy*ology1 vol4 #b1 pp4 #ha-hh1 vol4 #c1 pp4 #dhe-dib & #efa-egj4 ,br[n1 ,j4 ,c4 & ,c(fman1 ,l4 ,d42 '#ad ,h[ to ,tea* ,ari?metic4 ,burg]/e91 ,l42 '#ia ,die ,arbeitscurve e9] ,s*ul/unde4 ,zeits*rift fu] ,s*ulgesundheitspflege1 vol4 #d1 pp4 #edc-efb & #fjg-fbg4 ,burnett1 ,c4 ,j42 '#jf ,! ,e/imation ( ,numb]4 ,h>v>d ,psy*ological ,/udies1 vol4 #b1 pp4 #cdi-djd4 ,burt1 ,c42 '#ag ,! ,di/ribution & ,relations ( ,educational ,abilities4 ,report ( ,! ,london ,c\nty ,c\ncil1 ,no4 #ahfh4 ,*apman1 ,j4 ,c42 '#ad ,individual ,diff]5ces 9 ,ability & ,improvem5t & ,_! ,correlations4 ,tea*]s ,college ,contributions to ,education1 ,no4 #fc4 ,*apman1 ,j4 ,c42 '#ag ,! ,sci5tific ,measurem5t ( ,classroom ,products4 7,) ,g4 ,p4 ,ru%47 ,cobb1 ,m4 ,v42 '#ag ,a ,prelim9>y ,/udy ( ! ,inh]itance ( ,ari?metical ,abilities4 ,j\r4 ( ,educational ,psy*ology1 vol4 #h1 pp4 #a-bj4 ,jan41 #aiag4 ,c(fman1 ,l4 ,d41 & ,br[n1 ,j4 ,c42 '#ad ,h[ to ,tea* ,ari?metic4 ,c(fman1 ,l4 ,d41 & ,jessup1 ,w4 ,a42 '#af ,! ,sup]vision ( ,ari?metic4 ,c\rtis1 ,s4 ,a42 '#ji1 '#aj1 '#aa ,measurem5t ( ,gr[? & ,effici5cy 9 ,ari?metic4 ,elem5t>y ,s*ool ,tea*]1 vol4 #aj1 pp4 #eh-gd & #agg-aii1 vol4 #aa1 pp4 #aga-ahe1 #cfj-cgj1 & #ebh-eci4 ,c\rtis1 ,s4 ,a42 '#aa1 '#ab ,report on ,educational ,aspects ( ! ,public ,s*ool ,sy/em ( ! ,city ( ,new ,york4 ,"p ,,ii1 ,subdivision #a1 ,section ,d4 ,report on ! ,c\rtis ,te/s 9 ,ari?metic4 ,c\rtis1 ,s4 ,a42 '#ac ,c\rtis ,/&>d ,te/s4 ,second ,annual ,acc\nt+4 ,c\rtis1 ,s4 ,a42 '#ad ,manual ( ,in/ructions = ,giv+ & ,scor+ ! ,c\rtis ,/&>d ,te/s 9 ! ,?ree ,r's4 ,dep>tm5t ( ,comp>ative ,rese>*1 #hb ,elliot ,/41 ,detroit1 ,mi*41 #aiad4 ,decroly1 ,m41 & ,deg&1 ,j42 '#ab ,l'evolution des notions de quantites cont9ues et discont9ues *ez l'enfant4 ,ar*ives de psy*ologie1 vol4 #ab1 pp4 #ha-aba4 ,deg&1 ,j4 ,see ,decroly4 ,de ,voss1 ,j4 ,c4 ,see ,monroe1 ,de ,voss1 & ,kelly4 ,dewey1 ,j42 '#aj ,h[ ,we ,?9k4 ,dewey1 ,j41 & ,mc,lellan1 ,j4 ,a42 '#ie ,psy*ology ( ,numb] & ,its ,applications to ,me?ods ( ,tea*+ ,ari?metic4 ,donovan1 ,m4 ,e41 & ,?orndike1 ,e4 ,l42 '#ac ,improvem5t 9 a ,practice ,exp]im5t "u ,s*ool ,conditions4 ,am]ican ,j\rnal ( ,psy*ology1 vol4 #bd1 pp4 #dbf-dbh4 ,elliott1 ,c4 ,h42 '#ad ,v>iation 9 ! ,a*ievem5ts ( ,pupils4 ,tea*]s ,college1 ,columbia ,univ]sity1 ,contributions to ,education1 ,no4 #gb4 ,flynn1 ,f4 ,j42 '#ab ,ma!matical ,games--,adaptations f ,games ,old & ,new4 ,tea*]s ,college ,record1 vol4 #ac1 pp4 #cii-dab4 ,freeman1 ,f4 ,n42 '#aj ,unt]su*ung5 ueb] d5 ,aufm]ksamkeitsumfang und die ,zahlauffassung4 ,pa$agogis*e-,psy*ologis*e ,arbeit51 ,i1 #hh-afh4 ,fri$ri*1 ,j42 '#ig ,unt]su*ung5 ueb] die ,e9fluesse d] ,arbeitsdau] und die ,arbeitspaus5 auf die gei/ige ,lei/ungsfaehigkeit d] ,s*ulk9d]4 ,zeits*rift fu] ,psy*ologie1 vol4 #ac1 pp4 #a-ec4 ,gilb]t1 ,j4 ,a42 '#id ,rese>*es on ! ,m5tal & ,physical ,developm5t ( ,s*ool ,*ildr54 ,/udies f ! ,yale ,psy*ological ,laboratory1 vol4 #b1 pp4 #dj-ajj4 ,gre5leaf1 ,b42 '#gc ,practical ,ari?metic4 ,hahn1 ,h4 ,h41 & ,?orndike1 ,e4 ,l42 '#ad ,"s ,results ( ,practice 9 ,addition "u ,s*ool ,conditions4 ,j\rnal ( ,educational ,psy*ology1 vol4 #e1 ,no4 #b1 pp4 #fe-hd4 ,hall1 ,g4 ,s42 '#hc ,! ,cont5ts ( ,*ildr5's ,m9ds on ,ent]+ ,s*ool4 ,pr9ceton ,review1 vol4 ,,ii1 pp4 #bdi-bgb1 ,may1 #ahhc4 ,repr9t$ 9 ,aspects ( ,*ild ,life & ,education1 #aijg4 ,h>tmann1 ,b42 '#ij ,die ,analyze des ,k9dli*5 ,g$ank5-,kreises als die ,naturgemaess$es ,er/5 ,s*ulunt]ri*ts1 #ahij4 ,heck1 ,w4 ,h42 '#ac ,a ,/udy ( ,m5tal ,fatigue4 ,heck1 ,w4 ,h42 '#ac ,a ,second ,/udy 9 ,m5tal ,fatigue 9 ! ,daily ,s*ool ,program4 ,psy*ological ,cl9ic1 vol4 #g1 pp4 #bi-cd4 ,h(fmann1 ,p42 '#aa ,das ,int]esse d] ,s*uel] an d5 ,unt]ri*tsfae*]n4 ,zeits*rift fu] pa$agogis*e ,psy*ologie1 ,,xii1 #deh-dgj4 ,hoke1 ,k4 ,j41 & ,wilson1 ,g4 ,m42 '#bj ,h[ to ,measure4 ,holmes1 ,m4 ,e42 '#ie ,! ,fatigue ( a ,s*ool ,h\r4 ,p$agogical ,sem9>y1 vol4 #c1 pp4 #bac-bcd4 ,h[ell1 ,h4 ,b42 '#ad ,a ,f\ndation ,/udy 9 ! ,p$agogy ( ,ari?metic4 ,hunt1 ,c4 ,w42 '#ab ,play & ,recreation 9 ,ari?metic4 ,tea*]s ,college ,record1 vol4 #ac1 pp4 #chh-cih4 ,jessup1 ,w4 ,a41 & ,c(fman1 ,l4 ,d42 '#af ,! ,sup]vision ( ,ari?metic4 ,kelly1 ,f4 ,j4 ,see ,monroe1 ,de ,voss & ,kelly4 ,k+1 ,a4 ,c42 '#jg ,! ,daily ,program 9 ,elem5t>y ,s*ools4 ,,mss4 ,kirby1 ,t4 ,j42 '#ac ,practice 9 ! ,case ( ,s*ool ,*ildr54 ,tea*]s ,college ,contributions to ,education1 ,no4 #eh4 ,klapp]1 ,p42 '#af ,! ,tea*+ ( ,ari?metic4 ,kruse1 ,p4 ,j42 '#ah ,! ,ov]lapp+ ( ,atta9m5ts 9 ,c]ta9 ,six?1 ,sev5?1 & ,eigh? ,grades4 ,tea*]s ,college1 ,columbia ,univ]sity1 ,contributions to ,education1 ,no4 #ib4 ,las]1 ,h42 '#id ,ueb] gei/ige ,ermu$ung beim ,s*ulunt]ri*t4 ,zeits*rift fu] ,s*ulgesundheitspflege1 vol4 #g1 pp4 #b-bb4 ,lay1 ,w4 ,a42 '#ih ,fuehr] dur* d5 er/5 ,re*5unt]ri*t4 ,lay1 ,w4 ,a42 '#jg ,fuehr] dur* d5 ,re*5unt]ri*t d] ,unt]/ufe4 ,lewis1 ,e4 ,o42 '#ac ,popul> & ,unpopul> ,s*ool-,subjects4 ,! ,j\rnal ( ,exp]im5tal ,p$agogy1 vol4 #b1 pp4 #hi-ih4 ,lobsi51 ,m42 '#jc ,k9d]ideale4 ,zeits*rift fu] pa$agogis*e ,psy*ologie1 ,v1 #cbc-cdd & #deg-did4 ,lobsi51 ,m42 '#ji ,belieb!it und ,unbelieb!it d] ,unt]ri*tsfae*]4 ,pa$agogis*es ,magaz91 ,heft #cfa4 ,mc,call1 ,w4 ,a42 '#ba ,h[ to ,measure 9 ,education4 ,mc,dougle1 ,e4 ,c42 '#ad ,a ,contribution to ! ,p$agogy ( ,ari?metic4 ,p$agogical ,sem9>y1 vol4 #ba1 pp4 #afa-bah4 ,mc,knight1 ,j4,a42 '#jg ,diff]5tiation ( ! ,curriculum 9 ! ,upp] ,gramm> ,grades4 ,,mss4 9 ! libr>y ( ,tea*]s ,college1 ,columbia ,univ]sity4 ,mc,lellan1 ,j4,a41 & ,dewey1 ,j42 '#ie ,psy*ology ( ,numb] & ,its ,applications to ,me?ods ( ,tea*+4 ,mc,lellan1 ,j4,a41 & ,ames1 ,a4,f42 '#jj ,public ,s*ool ,ari?metic4 ,mess5g]1 ,j4,f42 '#jc ,! ,p]ception ( ,numb]4 ,psy*ological ,review1 ,monograph ,supplem5t ,no4 #bb4 ,meumann1 ,e42 '#jg ,vorlesung5 zur ,e9fuehrung 9 die exp]im5telle ,pa$agogik4 ,mit*ell1 ,h4,e42 '#bj ,unpubli%$ /udies ( ! uses ( ari?metic 9 factories1 %ops1 f>ms1 & ! l4 ,monroe1 ,w4,s41 ,de ,voss1 ,j4,c41 & ,kelly1 ,f4,j42 '#ag ,educational ,te/s & ,measurem5ts4 ,nanu1 ,h4,a42 '#jd ,zur ,psy*ologie d] ,zahl ,auffassung4 ,national ,intellig5ce ,te/s2 '#bj ,scale ,a1 ,=m #a1 ,edition #a4 ,phillips1 ,d4,e42 '#ig ,numb] & ,its ,application ,psy*ologically ,consid]$4 ,p$agogical ,sem9>y1 vol4 #e1 pp4 #bba-bha4 ,pomm]1 ,o42 '#ad ,die ,er=s*ung d] ,belieb!it d] ,unt]ri*tsfae*]4 ,ihre psy*ologis*5 ,grundlag5 und ihre pa$agog4 ,b$eutung4 ,,vii4 ,jahresb]4 des k4k4 ,ssaatsgymn4 im ,,xviii ,bez4 v4 ,wi54 ,rice1 ,j4,m42 '#jb ,te/ 9 ,ari?metic4 ,=um1 vol4 #cd1 pp4 #bha-big4 ,rice1 ,j4,m42 '#jc ,causes ( ,success & ,failure 9 ,ari?metic4 ,=um1 vol4 #cd1 pp4 #dcg-deb4 ,ru%1 ,g4,p42 '#ag ,! ,sci5tific ,measurem5t ( ,classroom ,products4 7,) ,j4 ,c4 ,*apman47 ,seekel1 ,e42 '#ad ,ueb] die ,beziehung zwis*5 d] ,belieb!it und d] ,s*wi]igkeit d] ,s*ulfae*]4 ,ergebnisse e9] ,erhebung4 ,zeits*rift fu] ,angew&te ,psy*ologie #i4 ,s4 #bfh-bgg4 ,selk91 ,f4 ,b42 '#ab ,numb] ,games ,bord]+ on ,ari?metic & ,algebra4 ,tea*]s ,college ,record1 vol4 #ac1 pp4 #deb-dic4 ,smi?1 ,d4 ,e42 '#ja ,! ,tea*+ ( ,elem5t>y ,ma!matics4 ,smi?1 ,d4 ,e42 '#aa ,! ,tea*+ ( ,ari?metic4 ,spe]1 ,w4 ,w42 '#ig ,ari?metic3 ,elem5t>y = ,pupils4 ,/>*1 ,d42 '#aa ,transf] ( ,tra9+ 9 ,ari?metical ,op]ations4 ,j\rnal ( ,educational ,psy*ology1 vol4 #b1 pp4 #cjf-caj4 ,/>*1 ,d42 '#af ,educational ,measurem5ts4 ,/]n1 ,w42 '#je ,ueb] ,belieb!it und ,unbelieb!it d] ,s*ulfae*]4 ,zeits*rift fu] pa$agogis*e ,psy*ologie1 ,,vii1 #bfg-bif4 ,/]n1 ,c41 & ,/]n1 ,w42 '#ac ,belieb!it und ,s*wi]igkeit d] ,s*ulfae*]4 7,freie ,s*ulgeme9de ,wick]sdorf47 ,auf ,grund d] von ,h]rn ,lus]ke bes*afft5 ,mat]iali5 be>beitet4 ,93 ,die ,aus/ellung zur v]glei*5d5 ,jung5dkunde d] ,ges*le*t] 9 ,breslau4 ,arbeit #g des ,bundes fu] ,s*ulre=m4 ,s4 #bd-bf4 ,/]n1 ,w42 '#ad ,zur v]glei*5d5 ,jug5dkunde d] ,ges*le*t]4 ,vortrag4 ,,iii4 ,deuts*4 ,kongr4 f4 ,jug5dkunde usw4 ,arbeit5 #h des ,bundes fu] ,s*ulre=m4 ,s4 #ag-ch4 ,/one1 ,c4,w42 '#jh ,ari?metical ,abilities & ,"s ,factors ,det]m9+ ,!m4 ,tea*]s ,college ,contributions to ,education1 ,no4 #ai4 ,suzzallo1 ,h42 '#aa ,! ,tea*+ ( ,prim>y ,ari?metic4 ,?orndike1 ,e4,l42 '#jj ,m5tal ,fatigue4 ,psy*ological ,review1 vol4 #g1 pp4 #dff-dhb & #edg-egi4 ,?orndike1 ,e4,l42 '#jh ,! ,effect ( ,practice 9 ! ,case ( a ,purely ,intellectual ,function4 ,am]ican ,j\rnal ( ,psy*ology1 vol4 #aj1 pp4 #cgd-chd4 ,?orndike1 ,e4,l42 '#aj ,practice 9 ! case ( ,addition4 ,am]ican ,j\rnal ( ,psy*ology1 vol4 #ba1 pp4 #dhc-dhf4 ,?orndike1 ,e4,l41 & ,donovan1 ,m4,e42 '#ac ,improvem5t 9 a ,practice ,exp]im5t "u ,s*ool ,conditions4 ,am]ican ,j\rnal ( ,psy*ology1 vol4 #bd1 pp4 #dbf-dbh4 ,?orndike1 ,e4,l41 & ,donovan1 ,m4,e41 & ,hahn1 ,h4,h42 '#ad ,"s ,results ( ,practice 9 ,addition "u ,s*ool ,conditions4 ,j\rnal ( ,educational ,psy*ology1 vol4 #e1 ,no4 #b1 pp4 #fe-hd4 ,?orndike1 ,e4,l42 '#ae ,! ,relation betwe5 ,initial ,ability & ,improvem5t 9 a ,sub/itution ,te/4 ,s*ool & ,society1 vol4 #ab1 p4 #dbi4 ,?orndike1 ,e4,l42 '#af ,notes on ,practice1 ,improvability & ! ,curve ( ,"w4 ,am]ican ,j\rnal ( ,psy*ology1 vol #bg1 pp4 #eej-efe4 ,wal%1 ,j4,h42 '#jf ,gramm> ,s*ool ,ari?metic4 ,wells1 ,f4,l42 '#ab ,! ,relation ( ,practice to ,individual ,diff]5ces4 ,am]ican ,j\rnal ( ,psy*ology1 vol4 #bc1 pp4 #ge-hh4 ,:ite1 ,e4 ,e42 '#hc ,a ,new ,elem5t>y ,ari?metic4 ,:itley1 ,m4 ,t42 '#aa ,an ,empirical ,/udy ( ,c]ta9 ,te/s = ,individual ,diff]5ces4 ,ar*ives ( ,psy*ology1 ,no4 #ai4 ,wi$]kehr1 ,g42 '#jg ,/ati/i*e ,unt]su*ung5 ueb] die ,art und d5 ,grad des ,int]esses bei ,k9d]n d] ,volkss*ule4 ,neue ,bahn51 vol4 #ai1 pp4 #bda-bea1 #bhi-bii4 ,wilson1 ,g4 ,m42 '#ai ,a ,survey ( ! ,social & ,bus9ess ,usage ( ,ari?metic4 ,tea*]s ,college ,contributions to ,education1 ,no4 #ajj4 ,wilson1 ,g4 ,m41 & ,hoke1 ,k4 ,j42 '#bj ,h[ to ,measure4 ,woody1 ,c42 '#aj ,measurem5ts ( ,"s ,a*ievem5ts 9 ,ari?metic4 ,tea*]s ,college ,contributions to ,education1 ,no4 #hj4 ,,index ,abilities1 ari?metical1 nature (1 #a ff42 measurem5t (1 #bg ff42 con/itution (1 #ea ff42 organization (1 #acg ff4 ,ab/ract numb]s1 #he ff4 ,ab/raction1 #afi ff4 ,accuracy1 9 relation to spe$1 #ca2 9 fundam5tal op]ations1 #ajb ff4 ,addition1 measurem5t (1 #bg ff41 #cd2 con/itution (1 #eb f42 habit 9 relation to1 #ga f42 9 ! high] decades1 #ge f42 accuracy 91 #ajh f42 am\nt ( practice 91 #abb ff42 int]e/ 9 #aif f4 ,aims ( ! tea*+ ( ari?metic1 #bc f4 ,,ames1 ,a4 ,f41 #hi ,analysis1 le>n+ by1 #afi ff42 sy/ematic & opportuni/ic /imuli to1 #agh f42 gradual progress 91 #ahj ff4 ,area1 #beg f41 #bge ,ari?metic1 sociology (1 #bd ff4 ,ari?metical abilities4 ,see ,abilities4 ,ari?metical language1 #h f41 #ai1 #hi ff41 #id ff4 ,ari?metical le>n+1 be=e s*ool1 #aii ff42 conditions (1 #bbg ff42 9 relation to "t ( day1 #bbg ff42 9 relation to "t devot$ to ari?metic1 #bbh ff4 ,ari?metical reason+4 ,see ,reason+4 ,ari?metical t]ms1 #h1 #ai ,av]ages1 #dj f42 #ace f4 ,,ball\1 ,f4 ,w41 #cd1 #ch ,bank+1 #bef f4 ,,b9et1 ,a41 #bja ,bonds1 selection (1 #gj ff42 /r5g? (1 #ajb ff42 = tempor>y s]vice1 #aaa ff42 ord] ( =mation (1 #ada ff4 ,see also ,habits4 ,,br&ell1 ,g41 #baa ,,br&=d1 ,b41 #aih f4 ,,br[n1 ,j4 ,c41 xvi1 #ajc ,,burg]/e91 ,l41 #ajc ,,burnett1 ,c4 ,j41 #bjb ,,burt1 ,c41 #bhf ,c>d9al & ord9al numb]s confus$1 #bjf ,cat* problems1 #ba ff4 ,,*apman1 ,j4 ,c41 #di ,class1 size (1 9 relation to ari?metical le>n+1 #bbh2 v>iation )9 a1 #bhi ff4 ,,cobb1 ,m4 ,v41 #bii ,,c(fman1 ,l4 ,d41 xvi ,collection mean+ ( numb]s1 #c ff4 ,computation1 measurem5ts (1 #cc ff42 explanations ( ! processes 91 #fj ff42 accuracy 91 #ajb ff4 ,see also ,addition1 ,subtraction1 ,multiplication1 ,division1 ,fractions1 ,decimal numb]s1 ,p]c5ts4 ,concomitants1 law ( v>y+1 #agb ff42 law ( contra/+1 #agc ff4 ,concrete numb]s1 #he ff4 ,concrete objects1 use (1 #bec ff4 ,conditions ( ari?metical le>n+1 #bbg ff4 ,con/itution ( ari?metical abilities1 #ea ff4 ,copy+ ( numb]s1 eye/ra9 due to1 #bab f4 ,correlations ( ari?metical abilities1 #bie ff4 ,c\rses ( /udy1 #bcb f4 ,,c\rtis1 ,s4 ,a41 #bh ff41 #dc ff41 #di1 #ajc1 #bia1 #bic1 #bii ,crut*es1 #aab f4 ,culture-epo* !ory1 #aih f4 ,dairy records1 #bgc ,decimal numb]s1 uses (1 #bd f42 measurem5t ( ability )1 #cf ff42 le>n+1 #aha ff42 division by1 #bgj f4 ,,de ,,croly1 ,m41 #bje ,d$uctive reason+1 #fj ff41 #ahe ff4 ,,deg&1 ,j41 #bje ,d5om9ate numb]s1 #ada f41 #adg f4 ,describ$ problems1 #aj ff4 ,developm5t ( k ( numb]1 #bje ff4 ,,de ,,voss1 ,j4 ,c41 #di ,,dewey1 ,j41 #c1 #hc1 #aej1 #bje1 #bjg1 #bjh1 #bai1 #bff1 #bgg ,diff]5ces 9 ari?metical ability1 #bhe ff42 )9 a class1 #bhi ff4 ,difficulty z a /imulus1 #bgg ff4 ,drill1 #ajb ff4 ,discipl9e1 m5tal1 #bj ,di/ribution ( practice1 #aef ff4 ,division1 measurem5t (1 #ce f41 #cg2 con/itution (1 #eg ff42 d$uctive explanations (1 #fc1 #fd f42 inductive explanations (1 #fc f41 #fe f42 habit 9 relation to1 #gb2 ) rema9d]s1 #gf2 ) fractions1 #gh ff42 am\nt ( practice 91 #abb ff42 di/ribution ( practice 91 #afg2 use ( ! problem attitude 9 tea*+1 #bgj f4 ,,donovan1 ,m4 ,e41 #bie ,elem5ts1 responses to1 #afi ff4 ,elev51 multiples (1 #he ,,elliott1 ,c4 ,h41 #bbh ,equation =m1 importance (1 #gg f4 ,explanations ( ! processes ( computation1 #fj ff42 memory (1 #aae f42 "t = giv+1 #aed ff4 ,eye/ra9 9 ari?metical "w1 #bab ff4 ,facilitation1 #adc ff4 ,figures1 pr9t+ (1 #bce ff42 writ+ (1 #bad f41 #bda ,,flynn1 ,f4 ,j41 #aif ,fractions1 uses (1 #bd f42 measurem5t ( ability )1 #cf ff42 k ( ! mean+ (1 #ed ff4 ,,freeman1 ,f4 ,n41 #bei1 #bfa ,,fri$ri*1 ,j41 #ajc ,g5]alization1 #afi ff4 ,,gilb]t1 ,j4 ,a41 #bjc ,grad$ te/s1 #bh ff41 #cf ff4 ,greate/ common divisor1 #hh f4 ,habits1 importance (1 9 ari?metical le>n+1 #gj ff42 n[ neglect$1 #ge ff42 h>mful or wa/eful1 #hc ff42 #ia ff42 propa$eutic1 #aag ff42 organization (1 #acg ff42 arrangem5t (1 #ada ff4 ,,hahn1 ,h4 ,h41 #bie ,,hall1 ,g4 ,s41 #bjj f4 ,,h>tmann1 ,b41 #bjj f4 ,,heck1 ,w4 ,h41 #bbg ,h]$ity 9 ari?metical abilities1 #bic ff4 ,highe/ common factor1 #hh f4 ,,hoke1 ,k4 ,j41 #di ,,holmes1 ,m4 ,e41 #ajc ,,h[ell1 ,h4 ,b41 #bei ,,hunt1 ,c4 ,w41 #aif ,hygi5e ( ari?metic1 #bab ff41 #bcd ff4 ,individual diff]5ces1 #bhe ff4 ,inductive reason+1 #fj ff41 #afi ff4 ,insurance1 #bef ,int]e/ z a pr9ciple det]m9+ ! ord] ( topics1 #aej ff4 ,int]e/s1 in/9ctive #aie ff42 c5suses (1 #bji ff42 neglect ( *ildi%1 #bbf ff42 9 self-managem5t1 #bbc f42 intr9sic1 #bbd ff4 ,int]f]5ce1 #adc ff4 ,inv5tories ( ari?metical k & skill1 #aii ff4 ,,jessup1 ,w4 ,a41 xvi ,,kelly1 ,f4 ,j41 #di ,,k+1 ,a4 ,c41 #ajc1 #bbg ,,kirby1 ,t4 ,j41 #gf f41 #ajd1 #bie ,,klapp]1 ,p41 xvi ,,kruse1 ,p4 ,j41 #bhi1 #bic ,ladd] te/s1 #bh ff41 #cf ff4 ,language 9 ari?metic1 #h f41 #ai1 #hi ff41 #id ff4 ,,las]1 ,h41 #ajc ,,lay1 ,w4 ,a41 #bei1 #bfa ,le>n+1 nature ( ari?metical1 #a ff4 ,lea/ common multiple1 #hh f4 ,,lewis1 ,e4 ,o41 #baj f4 ,,lobsi51 ,m41 #bji f4 ,,mccall1 ,w4 ,a41 #di ,,mcd\gle1 ,e4 ,c41 #he ff4 ,,mcknight1 ,j4 ,a41 #baj ,,mclellan1 ,j4 ,a41 #c1 #hc1 #hi1 #bje1 #bjg ,manipulation ( numb]s1 #fj ff4 ,mean+1 ( numb]s1 #b ff41 #aga2 ( a fraction1 #ed ff42 ( decimals1 #aha f4 ,measurem5t ( ari?metical abilities1 #bg ff4 ,m5tal ari?metic1 #bfb ff4 ,,mess5g]1 ,j4 ,f41 #bjb ,metric sy/em1 #adg ,,meumann1 ,e41 #bfa ,,mit*ell1 ,h4 ,e41 #bd ,,monroe1 ,w4 ,s41 #di ,multiplication1 measurem5t (1 #ce1 #cf2 con/itution (1 #ea2 d$uctive explanations (1 #fa2 inductive explanations (1 #fa f42 ) fractions1 #gh ff42 by elev51 #he2 am\nt ( practice 91 #abb ff42 ord] ( le>n+ ! elem5t>y facts (1 #add f42 di/ribution ( practice 91 #aeh ff42 use ( ! problem attitude 9 tea*+1 #bfg ff4 ,,nanu1 ,h4 ,a41 #bjb ,national ,intellig5ce ,te/s1 #di f4 ,negative reaction 9 intellectual life1 #bgh f4 ,numb] pictures1 #bei ff4 ,numb]s1 mean+ (1 #b2 z measures ( cont9u\s quantities1 #ge2 ab/ract & concrete1 #he ff42 d5om9ate1 #ada f41 #adg f42 use ( l>ge1 #ade f42 p]ception (1 #bje ff42 e>ly aw>5ess (1 #bje ff42 confusion ( c>d9al & ord9al1 #bjf4 ,see also ,decimal numb]s & ,fractions4 ,objective aids1 us$ = v]ification1 #aed2 9 g5]al1 #bdc ff4 ,oral ari?metic1 #bfb ff4 ,ord] ( topics1 #ada ff4 ,ord9al numb]s1 confus$ ) c>d9al1 #bjf ,orig9al t5d5cies & ari?metic1 #aie ff4 ,ov]le>n+1 #acd ff4 ,p]c5ts1 #hj f4 ,p]ception ( numb]1 #bjb ff4 ,,phillips1 ,d4 ,e41 #c1 #d1 #bje1 #bjg ,pictures1 hygi5e (1 #bdf ff42 numb]1 #bei ff4 ,,pomm]1 ,o41 #bab ,practice1 am\nt (1 #abb ff42 di/ribution (1 #aef ff4 ,precision 9 fundam5tal op]ations1 #ajb ff4 ,problem attitude1 #bff ff4 ,problems1 #i ff42 cat*1 #ba ff42 measurem5t ( ability )1 #db ff42 ^: answ] mu/ be kn[n 9 ord] to frame !m1 #ic f42 v]bal =m (1 #aaa f42 int]e/ 91 #bbj ff42 z introductions to ari?metical le>n+1 #bff ff4 ,propa$eutic bonds1 #aag ff4 ,purposive ?9k+1 #aic ff4 ,quantity1 numb] &1 #he ff42 p]ception (1 #bjb ff4 ,,race1 ,h41 #bie ,ra9fall1 #bgb ,ratio1 #bbe f42 mean+ ( numb]s1 #c ff4 ,reaction1 negative1 #bgh f4 ,reality1 9 problems1 #i ff4 ,reason+1 ari?metical1 nature (1 #ai ff42 measurem5t ( ability 91 #db ff42 d]ivation ( tables by1 #eh f42 ab\t ! rationale ( computations1 #fj ff42 habit 9 relation to1 #gc f41 #aij ff42 problems :i* provoke false1 #ajj f42 ! ess5tials ( ari?metical1 #ahe ff42 selection 91 #ahg ff42 z ! cooep]ation ( organiz$ habits1 #aij ff4 ,recapitulation !ory1 #aih f4 ,recipes1 #bgc f4 ,rectangle1 area (1 #beg f4 ,,rice1 ,j4 ,m41 #bbh ff4 ,,ru%1 ,g4 ,p41 #di ,,seekel1 ,e41 #bab ,,selk91 ,f4 ,b41 #aif f4 ,sequ5ce ( topics1 #ada ff4 ,s]ies mean+ ( numb]s1 #b ff4 ,size ( class 9 relation to ari?metical le>n+1 #bbh ,,smi?1 ,d4 ,e41 xvi1 #bbd ,social in/9cts1 use (1 #aie f4 ,sociology ( ari?metic1 #bd ff4 ,spe$ 9 relation to accuracy1 #ca1 #ajh ,,spe]1 ,w4 ,w41 #c1 #e1 #hc ,spiral ord]1 #ada1 #ade ,,/>*1 ,d41 #di1 #bie ,,/]n1 ,w41 #baj1 #bab ,,/one1 ,c4 ,w41 #bg ff41 #db ff41 #bbh ff4 ,subtraction1 measurem5t (1 #cd f42 con/itution (1 #eg f42 am\nt ( practice 91 #abb ff4 ,sup]vision1 #bcc f4 ,,suzzallo1 ,h41 xvi ,tempor>y bonds1 #aaa ff4 ,t]ms1 #aac f4 ,te/s ( ari?metical abilities1 #bg ff4 ,,?orndike1 ,e4 ,l41 #cd1 #ch ff41 #bbg1 #bid ,"t1 devot$ to ari?metic1 #bbh ff42 ( day1 9 relation to ari?metical le>n+1 #bbg f4 ,type1 hygi5e (1 #bce ff4 ,und]le>n+1 #acd ff4 ,unit$ ,/ates money1 #adh ff4 ,units ( measure1 arbitr>y1 #e1 #hc f4 ,v>iation1 among individuals1 #bhe ff4 ,v>iety1 9 tea*+1 #aec ,v]ification1 #ha f42 aid$ by great] /r5g? ( ! fundam5tal bonds1 #ajg ff4 ,,wal%1 ,j4 ,h41 #aa ,,wells1 ,f4 ,l41 #bie ,,:ite1 ,e4 ,e41 #e ,,:itley1 ,m4 ,t41 #bie ,,wi$]kehr1 ,g41 #bab ,,wilson1 ,g4 ,m41 #bd1 #di ,,woody1 ,c41 #bi ff41 #eb1 #bhg1 #bic ,words4 ,see ,language & ,t]ms4 ,writt5 ari?metic1 #bfb ff4 ,z]o 9 multiplication1 #agi f4 ,,transcrib]',s ,,notes3 #a4 ,passages 9 italics are surr\nd$ by und]scores4 #b4 ,passages 9 bold are indicat$ by bold4 #c4 ,mix$ fractions are repres5t$ us+ =w>d sla% & hyph5 9 ?is text v]sion4 ,= example1 #e-a_/b repres5ts five & a half4 #d4 ,images & footnotes h be5 mov$ f ! middle ( a p>agraph to ! close/ p>agraph break4 #e4 ,obvi\s errors 9 spell+ & punctuation h be5 sil5tly clos$4