How to Solve It¶
Pólya, G. (1945). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press.
Cited by¶
8 citations across 8 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Analogy
- Mathematics uses analogy to transfer proof techniques across structurally similar problem classes, and category theory formalizes structural correspondence as a universal construct.
This sourceFoundational pedagogical pattern catalog: codifies recurring problem-solving heuristics (understand the problem, devise a plan, carry out the plan, look back) into teachable vocabulary; prototype of pattern-language thinking in pedagogy.
- Mathematics uses analogy to transfer proof techniques across structurally similar problem classes, and category theory formalizes structural correspondence as a universal construct.
- Decomposition
- Each piece becomes a bounded focus of attention; patterns within pieces become visible; relationships between pieces can be designed deliberately—a heuristic Pólya (1945) codified as "if you cannot solve the proposed problem, try to solve first some related problem" via sub-problem reduction.
This sourceCodifies problem-solving heuristics including 'if you cannot solve the proposed problem, try to solve first some related problem' (sub-problem reduction).
- Each piece becomes a bounded focus of attention; patterns within pieces become visible; relationships between pieces can be designed deliberately—a heuristic Pólya (1945) codified as "if you cannot solve the proposed problem, try to solve first some related problem" via sub-problem reduction.
- Design Patterns
This sourceCodifies recurring problem-solving heuristics (understand the problem, devise a plan, carry it out, look back) into a teachable vocabulary, including 'look at the unknown… think of a familiar problem with the same/similar unknown.'
- Heuristic
This sourceFoundational pedagogical pattern catalog: codifies recurring problem-solving heuristics (understand the problem, devise a plan, carry out the plan, look back) into teachable vocabulary; prototype of pattern-language thinking in pedagogy.
- Inversion
- Mathematics and formal systems: Inverse functions (f and f^(-1)), matrix inversion and linear algebra, group inverses and multiplicative structure, duality in category theory and functional analysis, inversion in permutation theory (cycle structure and sign).
This sourceFoundational pedagogical pattern catalog: codifies recurring problem-solving heuristics (understand the problem, devise a plan, carry out the plan, look back) into teachable vocabulary; prototype of pattern-language thinking in pedagogy.
- Mathematics and formal systems: Inverse functions (f and f^(-1)), matrix inversion and linear algebra, group inverses and multiplicative structure, duality in category theory and functional analysis, inversion in permutation theory (cycle structure and sign).
- Problem Representation
- Mathematicians solve problems by changing representation: polar to Cartesian, time to frequency, primal to dual.
This sourceReformulation heuristics, including "consider a related problem you can solve," as moves that change the representation to make a problem tractable.
- Mathematicians solve problems by changing representation: polar to Cartesian, time to frequency, primal to dual.
- Problem Space
- Problem space trains reasoners to ask a core set of diagnostic questions, paralleling the heuristic decomposition Pólya (1945) sets out in How to Solve It (understand the problem, devise a plan, carry out the plan, look back).
This sourceFoundational pedagogical pattern catalog: codifies recurring problem-solving heuristics (understand the problem, devise a plan, carry out the plan, look back) into teachable vocabulary; prototype of pattern-language thinking in pedagogy.
- Problem space trains reasoners to ask a core set of diagnostic questions, paralleling the heuristic decomposition Pólya (1945) sets out in How to Solve It (understand the problem, devise a plan, carry out the plan, look back).
- Progressive Refinement from Core Model
This sourceFoundational pedagogical pattern catalog: codifies recurring problem-solving heuristics (understand the problem, devise a plan, carry out the plan, look back) into teachable vocabulary; prototype of pattern-language thinking in pedagogy.
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