FAQ - How to approach problems?

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.
  3. Pursue your plan.
  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.