# How should I approach a (mathematical) problem?

A strategy proposed by George Polya [Polya1990] generalizes to all sorts of problems:

1. Do you truly understand the problem?
2. After understanding, make a plan on how to apporach the problem.
4. Evaluate your results. How well did you plan play out?

If this approach does not work, then try to find a easier or related problem to solve. Can you play around with a concrete example of the problem? Can you draw it, visualize it?

Herb Simon once made that statement that human insight often arises from a change of problem representation [Simon1996]. I do agree that it is very important to have multiple problem representation methodologies at one's disposal as well as multiple approaches to problems.

## Representation Methodologies

Mathematical problems are usually associated with the following representations (and we follow [Simon1996] with this structure as well)

• Words: Can you reframe the problem in other words?
• Logic: Is there a logical or propositional representation?
• Math: How can the problem be phrased as equation?

At the same time other representation systems are very important and can provide the key for a solution:

• Visual Thinking: Can you draw the problem? How can the problem be represented from different perspectives (e.g., detailed view vs. big picture)? The Back of the Napkin is a great introduction to visual thinking.
• Divergent/Lateral Thinking: Can you gather knowledge and out of the box ideas with mindmaps, random word etc. methods?
• ## Approaches to (Mathematical) Problems

There exist very good works on how to approach mathematical problems.

• [Polya1990] George Polya. How to Solve it: A New Aspect of Mathematical Method. Penguin, ISBN: 978-0140124996, 1990.
• [MaBuSt2010] J. Mason, L. Burton and K. Stacey. Thinking Mathematically. Prentice Hall, 2nd edition, ISBN: 978-0273728917, 2010.
• [Schoenfeld1992] Alan Schoenfeld. Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In Handbook for Research on Mathematics Teaching and Learning, pp 334-370. New Yorn, MacMillan, 1992.

These books provide some introduction to approaching problems in general. I've the books and a short "cheat sheet" for How to solve it in my office.

[Simon1996] Herbert Simon. Machine as Mind. In Machines and Thought, volume 1, pp.81-102, Clarendon Press, Oxford, 1996.