Teaching Math Problem-Solving Strategies That Transfer to New Problems

Most students approach an unfamiliar math problem by looking for a formula to plug numbers into. When they can't find one, they stop. Teaching problem-solving strategies gives students a toolkit for when the standard approach fails.

More importantly, it builds the disposition to persist when math gets hard.

Polya's Four-Step Framework

George Polya's How to Solve It (1945) remains the most useful framework for teaching problem-solving:

1. Understand the problem: What are you asked to find? What information is given? What constraints exist?
2. Make a plan: What strategies might work? Have you seen a related problem?
3. Execute the plan: Carry it out. If it doesn't work, try another.
4. Look back: Does the answer make sense? Can you verify it? Could you solve it another way?

Teaching this framework explicitly — and having students verbalize each step — builds metacognitive problem-solving habits.

A Toolkit of Strategies

Specific strategies students should know:

Draw a picture or diagram: visualizes the problem structure. Works for geometry, combinatorics, and many word problems.

Make a table or list: organizes information, often reveals patterns. Works for counting problems, rate problems, and combinatorics.

Look for a pattern: examine specific cases, then generalize. "What happens for 1, 2, 3 objects? Can I find a rule?"

Work backward: start from the solution and work toward the given information. Useful when the endpoint is known.

Solve a simpler version: replace complex numbers with simple ones and solve, then see if the method generalizes.

Guess and check (strategic): make an educated guess, evaluate it, use the result to refine the next guess.

Teaching Strategies, Not Just Using Them

The critical teaching move: name the strategy explicitly, demonstrate it on a problem, then have students identify which strategy to use on new problems (not just how to use one strategy on one problem).

Students who can name "I'm going to draw a picture because this is a spatial problem" are different from students who draw pictures because the teacher said to on this worksheet.

The Persistence Disposition

Problem-solving strategies only help students who are willing to try something. Students who've been trained by math class to stop when they don't immediately know the answer need explicit work on persistence.

Tasks with multiple entry points help: a problem that can be approached visually, numerically, or algebraically lets students enter where they're strongest and work toward other representations.

Normalize: "I'm stuck. Let me try a different strategy." Post the strategy list. Give credit for documented attempts, not just correct answers.

Problem Selection Matters

Good problem-solving instruction uses problems that genuinely require strategy — not standard exercises with one obvious approach. Problems from competitions (AMC 8, Math Olympiad, MATHCOUNTS at appropriate levels), open-ended tasks, and non-routine contexts all work.

If students can solve a problem in 30 seconds using a memorized formula, it isn't developing problem-solving ability. That's fluency practice. Both are valuable; don't conflate them.