Survey: Polya's questioning methodology from How to Solve It #8
Description
Survey: Polya's Way of Asking Questions in "How to Solve It"
George Polya's How to Solve It (1945) presents a systematic questioning methodology for problem solving. This issue documents the core framework for potential integration into the brainstorming/ideation workflow.
The Four Phases and Their Questions
Phase 1: Understanding the Problem
- "What is the unknown?"
- "What are the data?"
- "What is the condition?"
- "Is it possible to satisfy the condition?"
- "Is the condition sufficient to determine the unknown? Or insufficient? Or redundant? Or contradictory?"
- "Draw a figure." / "Introduce suitable notation."
- "Separate the various parts of the condition. Can you write them down?"
Phase 2: Devising a Plan
- "Have you seen it before? Or have you seen the same problem in a slightly different form?"
- "Do you know a related problem? Do you know a theorem that could be useful?"
- "Look at the unknown! Try to think of a familiar problem having the same or a similar unknown."
- "Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method?"
- "Could you restate the problem? Could you restate it still differently?"
- "Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem?"
- "Could you solve a part of the problem?" / "Keep only a part of the condition, drop the other part; how far is the unknown then determined?"
- "Could you derive something useful from the data?"
- "Did you use all the data? Did you use the whole condition?"
Phase 3: Carrying Out the Plan
- "Can you see clearly that the step is correct?"
- "Can you prove that it is correct?"
Phase 4: Looking Back
- "Can you check the result?"
- "Can you check the argument?"
- "Can you derive the result differently?"
- "Can you see it at a glance?"
- "Can you use the result, or the method, for some other problem?"
The Socratic Architecture
Funnel of generality — Questions move from maximally general ("What is the unknown?") to progressively specific ("Keep only a part of the condition, drop the other part"). The final leap of insight belongs to the student.
Questions = Suggestions — "Question and suggestion aim at the same effect; they tend to provoke the same mental operation." Polya alternates between interrogative and imperative forms strategically.
Internalization — The ultimate goal is to install these questions as the solver's own internal monologue. The Socratic interlocutor eventually lives inside the solver's head.
Key Heuristic Concepts (from the Short Dictionary, 67 entries)
| Entry | Core Idea |
|---|---|
| Analogy | Find relational correspondence to a different system |
| Working backwards | Assume the solution exists, trace back to knowns (from Pappus) |
| Inventor's paradox | "The more ambitious plan may have more chances of success" — sometimes a general problem is easier than a specific one |
| Decomposing and recombining | Break the problem into parts, recombine in new ways |
| Specialization | Test a concrete case to gain insight about the general |
| Generalization | Expand to a broader class for structural leverage |
| Subconscious work | Step away; let the mind process in the background |
| Test by dimension | Quick-check: do the units/dimensions of the result make sense? |
| Signs of progress | Increasing connection between data and unknown signals you're on the right track |
Question Taxonomy (Functional Categories)
| Category | Purpose | Examples |
|---|---|---|
| Comprehension | Establish what the problem is | "What is the unknown?", "What are the data?" |
| Connection | Bridge to prior knowledge | "Have you seen it before?", "Do you know a related problem?" |
| Transformation | Modify the problem to make it tractable | "Could you restate the problem?", "Could you imagine a more general problem?" |
| Decomposition | Break into parts | "Could you solve a part of the problem?" |
| Completeness-check | Verify nothing was overlooked | "Did you use all the data?" |
| Verification | Check the solution | "Can you check the result?", "Can you derive it differently?" |
| Transfer | Extend beyond the immediate problem | "Can you use the result for some other problem?" |
Research Landscape Questions (Polya-Inspired, for Scientific Brainstorming)
These questions adapt Polya's spirit to the task of evaluating a research direction:
Understanding the Field
- What is the basic landscape of this field? — What are the key sub-areas, major results, active groups, and temporal trends?
- What are the key problems in this field? — What are the central open questions that the community considers most important?
- What are the key bottlenecks of these key problems? — What specific obstacles are preventing progress on these open questions?
Devising Your Approach
- Why can you solve this key bottleneck? — What unique insight, method, or resource do you bring that others lack?
- Why can only you solve this bottleneck now, and why couldn't others solve it 10 years ago? — What has changed (new data, new tools, new theory) that makes this the right moment for this approach?
Looking Back / Kill Criteria
- What would it look like if the problem were solved? — What observable outcome or result would constitute success?
- What would it look like if the problem isn't solved yet, but you still have hope? — What partial progress or intermediate signals would justify continuing?
- What would it look like if your approach is fundamentally wrong — time to abandon or pivot? — What failure modes or negative signals should trigger a change of direction?
Influence and Legacy
- Mathematics education: NCTM adopted Polya's four-step model; Schoenfeld extended it with metacognition
- AI: Newell & Simon named "heuristics" after Polya; General Problem Solver operationalized means-ends analysis
- Modern LLM reasoning: Chain-of-Thought, Tree of Thoughts, and self-verification echo Polya's phases
- Software engineering: The four phases map to requirements → design → implementation → testing
- Over 1 million copies sold, 17+ languages, continuously in print for 80+ years
References
- Polya, G. (1945). How to Solve It. Princeton University Press.
- Schoenfeld, A. H. (1985). Mathematical Problem Solving. Academic Press.
- Newell, A. (1981). "The Heuristic of George Polya and Its Relation to Artificial Intelligence." Carnegie Mellon University.
- Rhee, R. J. "The Socratic Method and the Mathematical Heuristic of George Polya." Florida Law Review.