Lesson 19: Problem Solving Heuristics

Singapore's mathematics syllabus is unique in naming problem-solving heuristics explicitly — twelve strategies that the curriculum teaches, examines, and expects pupils to deploy. "Heuristic" comes from the Greek for "discover": these are not formulas but thinking tools, the moves that experienced problem-solvers make. This lesson teaches the most powerful heuristics with worked examples: draw a diagram, look for patterns, guess and check, work backwards, make suppositions, and simplify. Master these, and no PSLE problem is unfamiliar.

Heuristic 1 — Draw a Diagram or Model

The master heuristic: when stuck, draw. Bar models (Lessons 5–8) are the flagship, but diagrams include tables, timelines, geometric sketches, and graphs. "A farmer has chickens and cows: 20 heads and 56 legs. How many of each?" Draw 20 heads; give each 2 legs (40 legs); the 16 remaining legs go in pairs onto chickens-turned-cows: 8 cows, 12 chickens. The diagram does the algebra.

The principle: externalize thinking. Working memory holds few items; paper holds unlimited. Drawing converts abstract relationships into inspectable objects. Singaporean teachers assess model-drawing explicitly — the diagram earns method marks even when arithmetic slips.

When to draw: any problem with relationships (more than, times as many, before-after), any geometry, any counting problem. The heuristic's only cost is time; its payoff is clarity. In examinations, the diagram is never wasted.

Heuristic 2 — Look for Patterns

Sequences hide rules: 1, 4, 9, 16, ... (squares — next 25); 2, 6, 12, 20, ... (n(n+1) — next 30); 1, 1, 2, 3, 5, 8, ... (Fibonacci — next 13). The method: compute differences between terms; if constant, it is linear; if differences of differences constant, quadratic. Singaporean pupils tabulate until the pattern declares itself.

Patterns appear in geometry: "How many squares in a 4×4 grid?" Count 1×1 (16), 2×2 (9), 3×3 (4), 4×4 (1): total 30. The sequence 1, 5, 14, 30 for grids 1–4 reveals n(n+1)(2n+1)/6 — pattern leading to formula. The heuristic turns counting into conjecture.

The discipline: find the pattern, state it in words, test it on the next case, then generalize. "It looks like squares" is a hypothesis; verifying 5² = 25 as the 5th term is confirmation. Pattern-spotting without verification is numerology; with verification, it is mathematics.

Heuristic 3 — Guess and Check (Systematic Trial)

Guess and check is not random guessing but organized experimentation. Problem: "Ahmad is 3 years older than Bala; the sum of their ages is 27." Guess Bala = 10: Ahmad = 13, sum = 23 (too low). Guess 12: Ahmad = 15, sum = 27 ✓. The guesses bracket the answer; each guess teaches.

The systematic version tabulates: guess, compute, compare, adjust. "Find two numbers with sum 20 and product 96." Guess 10/10: product 100 (close). Guess 12/8: product 96 ✓. Organized trials converge fast — and the table earns method marks.

Singapore's curriculum dignifies guess-and-check as a legitimate heuristic, not a fallback for the weak. It is, in fact, the ancestor of numerical methods: computers solve equations by sophisticated guess-and-check (Newton's method). The heuristic teaches a profound truth: systematic trials, intelligently adjusted, solve what pure deduction cannot.

Heuristic 4 — Work Backwards

When a problem describes a sequence of operations ending in a known result, reverse the operations from the end. "Siti spent half her money plus S$5, and had S$10 left. How much at first?" Backwards: before the final S$10, she had S$10; add back the S$5 → S$15; that was half, so double → S$30. Each forward operation inverts: subtract S$5 becomes add S$5; halve becomes double.

The bar model draws this beautifully (Lesson 8): the final bar is known, and each reversed operation rebuilds the previous bar. Working backwards is the undoing heuristic — applicable to fraction-of-remainder problems, "gave away then had left" problems, and any chained process.

The meta-lesson: some problems are hard forwards and easy backwards — like a maze solved from the exit. Recognizing when to reverse direction is a general thinking skill, from debugging code to planning projects.

More Heuristics — Supposition, Simplification, Restatement

Supposition ("assume all are one type"): the chickens-and-cows problem assumes all 20 are chickens (40 legs), then converts. Also called "excess and shortage": "If each pupil gets 3 sweets, 10 remain; if each gets 4, 2 are short. How many pupils?" Suppose p pupils: 3p + 10 = 4p − 2, so p = 12. Supposition turns two scenarios into one equation.

Simplification: solve a smaller case first. "How many handshakes among 20 people?" Try 2 people (1), 3 (3), 4 (6) — the pattern n(n−1)/2 emerges; for 20: 190. Reducing to tiny cases reveals structure invisible at full scale.

Restatement: rephrase the problem. "A is 20% more than B" restates as "A = 1.2B" or "A:B = 6:5." Changing representation — words to ratio to equation — often dissolves difficulty. Singapore's examinations reward pupils who restate before computing.

Putting Heuristics Together — A Master Problem

Problem: "A box contains red and blue marbles. There are 12 more red marbles than blue marbles. If 8 red marbles are removed, the remaining red marbles are twice the number of blue marbles. How many marbles of each colour were there at first?" This PSLE-style problem yields to combined heuristics.

Restate with units: let blue = 1 unit. Then red at first = 1 unit + 12. After removing 8 red: red = 1 unit + 12 − 8 = 1 unit + 4. The condition says this equals twice the blue: 1 unit + 4 = 2 units. So 1 unit = 4. Blue = 4; red at first = 4 + 12 = 16. Check: remove 8 red → 8 red remain; twice the blue (2 × 4) = 8 ✓.

The heuristics combined: restate (words → units), draw the before-after bars (before: red longer by 12; after: red = 2 × blue), suppose the unit, solve, and verify against both conditions. No single heuristic suffices; together they crack it. This is the master pattern: translate, diagram, unitize, solve, check.

The deeper lesson: hard problems are combinations of familiar structures — a comparison (12 more) plus a before-after (8 removed) plus a multiplicative comparison (twice as many). The experienced solver decomposes the monster into its parts, applies the right heuristic to each, and reassembles. That decomposition skill is what the PSLE's hardest questions truly test.

Conclusion — Why This Matters

Heuristics are the explicit curriculum of mathematical thinking — Singapore's distinctive contribution to pedagogy. Where other systems hope pupils absorb problem-solving by osmosis, Singapore teaches it directly: name the strategy, practise it, examine it.

The twelve heuristics (act it out, draw a diagram, look for patterns, guess and check, work backwards, make suppositions, simplify, restate, and more) are a complete thinking toolkit. Internalize them, and unfamiliar problems become familiar — because you have moves for every situation.

Final lesson next: examination preparation — turning all this knowledge into PSLE and O-level results.

Polya's Four Steps — The Classic Method

George Polya's "How to Solve It" (1945) gave problem-solving its canonical four steps: (1) Understand the problem — what is unknown? What are the data? Draw a figure. (2) Devise a plan — have you seen a similar problem? Which heuristic fits? (3) Carry out the plan — execute carefully, checking each step. (4) Look back — check the result, find a neater solution, generalize. Singapore's heuristics curriculum is Polya, made explicit and examinable.

Step 1 is where most failures occur: pupils compute before understanding. Polya's insistence — restate the problem in your own words, identify the unknown, draw — is the antidote. The bar model is Step 1 and 2 combined: drawing the figure is understanding it, and the figure suggests the plan.

Step 4 (look back) is the most neglected and most valuable: checking by substitution, seeking alternative solutions, asking "what if the numbers changed?" The pupil who looks back learns from every problem; the pupil who doesn't, merely finishes it. Singapore's top performers all have look-back routines.

Polya's wisdom, compressed: slow down at the start (understand), be systematic in the middle (plan and execute), and reflect at the end (look back). Four steps, seventy years old, still the best advice in problem-solving.

Worked Example — Which Heuristic Wins?

Problem: "Eight teams enter a round-robin tournament in which every team plays every other team once. How many matches are played?" Heuristic A — simplify and list: Team 1 needs 7 matches, Team 2 needs 6 *new* ones (its match with Team 1 is counted), then 5, 4, 3, 2, 1. Total = 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28 matches. Heuristic B — generalize into a formula: each of 8 teams plays 7 others, giving 8 × 7 = 56, but every match is counted twice, so 56 ÷ 2 = 28. For n teams the count is n(n − 1)/2.

Listing is unbeatable for small numbers and for *understanding*; the formula wins when the numbers grow (100 teams: 4,950 matches, instantly). The examination skill is choosing — and being able to show the small case that justifies the big one.

Common Mistakes

• Unsystematic guessing. Guess-and-check without a table degenerates into revisiting the same guesses. Record every trial in order — a pattern in the failures is information.
• Generalising from one case. A pattern seen once (n = 1) is a coincidence; check n = 2 and n = 3 before believing n(n − 1)/2.
• Skipping the look-back. Polya's final step — does the answer fit the story? — catches the answer "13.5 rabbits". Answers about people, animals and matches must be whole numbers.

Exam Experience

Question 1 [2 marks]. The first four terms of a sequence are 3, 6, 11, 18, … (a) Find the next two terms. (b) Write down the general term of the sequence.

Question 2 [3 marks]. A number is multiplied by 3, then 6 is added, then the result is divided by 3. The final answer is 9. Use the working-backwards heuristic to find the number.

Question 3 [6 marks]. A farmer keeps chickens and rabbits. Altogether the animals have 30 heads and 86 legs. Use the method of supposition: (a) Suppose all 30 animals are chickens. How many legs would there be? (b) How many extra legs are there in reality, and how many extra legs does each rabbit account for? (c) Hence find the number of rabbits and the number of chickens.

Marking scheme. Q1: (a) B1 for 27 and 38 (differences 3, 5, 7, 9, 11). (b) B1 for n² + 2. Q2: M1 for reversing the last step (9 × 3 = 27); M1 for reversing the addition (27 − 6 = 21); A1 for 21 ÷ 3 = 7. Q3: (a) B1 for 60 legs. (b) M1 for 86 − 60 = 26 extra legs; A1 for 2 extra legs per rabbit. (c) M1 for 26 ÷ 2 = 13 rabbits; A1 for 13 rabbits; A1 for 17 chickens. (Check: 13 × 4 + 17 × 2 = 86.)