Problem Solving

Domain-general knowledge

Domain-specific knowledge

Knowledge transfer from general to specific

The art of problem solving

Constraints and trade-offs

Implicit assumptions and context

Insights

General strategies and thinking

Wishful thinking and making wish lists;
Outside-the-box, or lateral thinking: "putting out fires with sound";
Goal-oriented solving: start at the end, work backward;
Domain analysis, problem breakdown, solution design (blueprinting);
Look at it from real up close, look at it from ten miles away;
search for a solution = proof search in a proof tree;
??? (more)

"Convert, solve, convert back" paradigm

Algebraic topology / number theory / analytic geometry:
- take a problem in topology / numbers / geometry,
- convert to algebra,
- solve the algebraic problem,
- convert back to topology / numbers / geometry.
Laplace transform in differential equations is a perfect example.
Many, many more examples in all areas of math and CS.

Mathematics specific strategies

Explain the mental models and patterns of mathematical ideas:
- structural build-up from axioms by following rules;
- the (cumulative) past determines the present;
- how abstraction works, infinite families of mathematical objects;
- what properties / predicates are and how to detect them;
- ??? (more)

Rules

Discovering the rules

Following the rules

Breaking the rules

Work in progress

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