Word Problems & Problem Solving
All ages Problem solving
Word problems scare people not because the math is hard but because the translation is. The math inside is usually just adding, multiplying, or dividing — the work is figuring out which. There's a reliable method, and it's nearly 80 years old.
Pólya's four steps
- Understand it. What's actually being asked? Say it in your own words before touching numbers.
- Make a plan. Draw it, guess at it, work backward, or solve a smaller version first.
- Carry out the plan. Do the arithmetic — slowly enough to not lose the thread.
- Look back. Does the answer make sense? Is it the right size? Does it answer the actual question?
Worked example
The problem
"A farmer has chickens and cows — 30 heads and 74 legs. How many of each?"
- Understand: two kinds of animals, 30 animals total, 74 legs total.
- Plan: if all 30 were chickens there'd be 60 legs. We're 14 legs short. Each cow adds 2 extra legs.
- Solve: 14 ÷ 2 = 7 cows → 30 − 7 = 23 chickens.
- Check: 23×2 + 7×4 = 46 + 28 = 74 legs. ✓
Unpacking tricks that always help
- Draw it. A sketch exposes relationships text hides.
- Name the unknown. Write "let c = chickens" — naming it is halfway to finding it.
- Try a tiny version. 30 heads is scary; 3 heads isn't. Solve the small case, then scale the method.
- Work backward. Some problems only unwind from the end.
- Track units. "74 legs" not "74" — units catch wrong operations automatically.
The classics & the research
- Pólya, G. (1945/2004). How to Solve It. Princeton University Press. — the founding text of problem-solving instruction; the four steps above are his.
- Schoenfeld, A. H. (1985). Mathematical Problem Solving. Academic Press. — the research foundation: what expert solvers actually do differently (they manage strategy, not just computation).
- Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177. arXiv:math/9404236 · doi:10.1090/S0273-0979-1994-00502-6 — a Fields medalist on how mathematicians really think: understanding first, formalism second.
Try it yourself:
- Movie tickets: $8 child, $12 adult; 10 tickets sold for $96. How many of each? (5 and 5)
- A number doubled then plus 6 gives 30 — what's the number? (12)
- Two painters paint a fence in 6 and 3 hours alone; together? (2 hours)
Related: estimation for checking answers · order of operations.