In one paragraph
Pólya splits problem solving into four phases. Understand the problem: what is the unknown, what are the data, what is the condition? Devise a plan: have you seen a related problem? Carry out the plan, checking each step. Look back: can you check the result, derive it differently, use it elsewhere? Most of the book is a dictionary of heuristics to use inside those phases.
My review
[Your verdict: what the book does well, where it falls short.]
What I learned
- Name the unknown before touching the data; a plan for the wrong unknown is wasted work.
- A related, solved problem is the fastest route to a plan.
- Looking back is where the learning happens: checking the result and asking where else the method applies.
- [Your own points.]
Case study: how I used it
I use the four phases as the structure for problem-solving journals in my research. In MegoBin, each design decision is an entry that works through Understand, Plan, Carry out and Look back before any code is written. In my study notebooks for infinite SBMs, the Understand step is literal: the first cell has the headings the unknown, the data and the condition.
One limit I hit: on routine problems the phases feel empty, because every question has an obvious answer. [How you handle that now.]
Going deeper: references worth following
- G. Pólya. Mathematics and Plausible Reasoning, vols. 1–2. Princeton University Press, 1954. The longer treatment of induction and analogy behind the heuristics.
- A. H. Schoenfeld. Mathematical Problem Solving. Academic Press, 1985. Tests Pólya's heuristics with real students, and adds control and beliefs.