Mathematical Problem Solving

Mathematics is the original proving ground for problem-solving theory. Polya's How to Solve It introduced the four phases (understand, plan, execute, review) that every subsequent framework builds on. Schoenfeld showed that phases alone are insufficient: without active monitoring ("control"), novices spend 20 minutes on a dead end without noticing. This skill combines the two: Polya's phases as the scaffold, Schoenfeld's control as the supervisor.

Agent affinity: polya-ps (overall framing), schoenfeld (control and monitoring), simon (search structure)

Concept IDs: prob-problem-representation, prob-goal-decomposition, prob-pattern-recognition, prob-simplification, prob-systematic-listing

The Four Phases at a Glance

Phase	Polya question	Schoenfeld control check
1. Understand	What is the unknown? What is given? What is the condition?	Do I actually understand this, or am I about to solve the wrong problem?
2. Plan	Do I know a related problem? Can I solve part of it?	Is this plan likely to work, and how much budget do I give it?
3. Execute	Can I check each step?	Is this step still making progress, or have I wandered?
4. Look back	Can I verify the result? Can I use it for another problem?	Does the answer actually answer the original question?

Phase 1 — Understand the Problem

Goal: Produce a clean problem representation. Most of this is already covered by problem-comprehension, but math adds specific operations.

Math-specific operations:

• Identify the unknown. What is being asked for? Is it a number, a set, a function, a proof?
• Identify the data. What is given? Numerical values, geometric conditions, functional relationships.
• Identify the condition. What links the data to the unknown?
• Introduce notation. Assign variables, name points, label sides.
• Draw a figure. Geometry problems often solve themselves once the figure is accurate.
• Restate in your own symbols.

Control check: "Can I solve this problem without the original statement by looking only at my notation and figure?" If not, return to understanding.

Phase 2 — Devise a Plan

Goal: Choose a method to connect the data to the unknown. Polya's heuristics are central here.

Polya heuristics:

• Have you seen this problem before, perhaps in a slightly different form?
• Do you know a related problem?
• Look at the unknown. Can you think of a familiar problem with the same unknown?
• Can you restate the problem?
• Can you solve part of the problem?
• If you cannot solve the proposed problem, first try to solve a related one.
• Specialize: solve a simpler or more specific case first.
• Generalize: can the problem be viewed as a special case of something larger?
• Vary the problem: weaken an assumption, strengthen a conclusion.
• Introduce auxiliary elements: a point, a line, an equation that did not appear in the original.

Control check: "Does this plan connect the data to the unknown? What is my time budget for this plan? What is my fallback if it fails?"

Phase 3 — Carry Out the Plan

Goal: Execute the plan carefully, checking each step.

Rules:

• Work in small, verifiable steps.
• Write enough that a reader (including future you) can follow each step.
• When a step introduces a result, note whether it is exact or approximate.
• When a step is ad hoc, flag it — ad hoc steps are common error sites.

Schoenfeld's observation: Novices spend 90% of their time in this phase; experts allocate more to Phase 2 and return to Phase 2 when execution stalls. The boundary between phases is porous.

Control check (every few steps): "Is this still on the plan? Am I making progress? Should I reconsider the plan?"

Phase 4 — Look Back

Goal: Verify the answer, extract the lesson, build transfer.

Operations:

• Check the result. Does the answer satisfy the original conditions? Is it dimensionally consistent? Does it have the right sign or magnitude?
• Check the argument. Is every step justified? Any unstated assumptions?
• Is there a different way? Alternative solutions often reveal more about the problem.
• Can the method be used for another problem? Transfer is the long-term payoff of problem solving.
• Is the answer the answer to the original question? Easy to lose track during execution.

Control check: "Am I confident in this answer? On what basis?"

Schoenfeld's Control Layer

Schoenfeld's contribution is that the four phases are not enough. Novices skip Phase 1, short-circuit Phase 2, grind through Phase 3, and neglect Phase 4. The fix is control: a supervisor process that interrupts execution at regular intervals to ask whether the current activity is the right activity.

Control operations:

1. Plan monitoring. Is the current plan still the best available?
2. Progress monitoring. Am I moving toward the goal or wandering?
3. Time budgeting. How much time have I spent vs. budgeted?
4. Strategy switching. When to abandon a plan and try another.
5. Resource assessment. What do I know that I haven't used?

Without control, problem solving is a random walk through the strategy space. With control, it is a bounded search.

Worked Example — A Geometry Problem

Prove: In any triangle, the sum of the interior angles is 180 degrees.

Phase 1 — Understand. Unknown: a proof. Data: any triangle (three vertices, three sides, three interior angles). Condition: the angles must sum to 180. Draw a figure: triangle ABC with angles alpha, beta, gamma.

Phase 2 — Plan. Related problem: parallel lines cut by a transversal produce equal alternate angles. Can we introduce a parallel line through one vertex? Yes — a line through A parallel to BC creates two transversals (AB and AC) producing angles equal to beta and gamma on the other side of A. Then alpha + beta + gamma = straight angle at A = 180.

Phase 3 — Execute. Draw line l through A parallel to BC. By alternate interior angles on transversal AB: the angle on the other side of A equal to beta. By alternate interior angles on transversal AC: the angle on the other side equal to gamma. The three angles at A (gamma, alpha, beta) form a straight line, so gamma + alpha + beta = 180.

Phase 4 — Look back. Check: does this use any assumption beyond Euclidean geometry? Yes — the parallel postulate. On a sphere, triangles sum to more than 180. The proof is correct for Euclidean geometry only. The auxiliary line (introduce a parallel through A) was the key move; this generalizes to many problems where a single auxiliary element unlocks the solution.

When Mathematical Problem Solving Fails

• Racing to Phase 3. Execution without understanding produces symbol manipulation that does not answer the question.
• No plan. Writing equations with no idea where they lead wastes time and produces no insight.
• No control. Ten minutes in, the solver has no idea whether they are making progress. Control checks every 2-3 minutes catch this early.
• Skipping Phase 4. The answer is written, the problem is "done," but the lesson is not extracted. Without Phase 4, problem-solving skill does not accumulate.

Cross-References

• problem-comprehension provides the base comprehension operations that Phase 1 builds on
• strategy-selection is the broader context for Polya's Phase 2 heuristics
• metacognitive-monitoring implements Schoenfeld's control layer explicitly
• design-thinking-ps replaces proof-style execution with iterative prototyping for ill-defined problems
• collaborative-problem-solving distributes phases across team members