| name | mathematical-problem-solving |
| description | Polya's four-phase method applied to mathematical problems, extended with Schoenfeld's control/monitoring layer. Covers understand, devise plan, carry out, look back. Includes heuristics specific to math (specialize, generalize, vary the problem, introduce auxiliary elements) and the metacognitive discipline that separates routine computation from genuine problem solving. |
| type | skill |
| category | problem-solving |
| status | stable |
| origin | tibsfox |
| modified | false |
| first_seen | "2026-04-12T00:00:00.000Z" |
| first_path | examples/skills/problem-solving/mathematical-problem-solving/SKILL.md |
| superseded_by | null |
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.
