Math Problem Solving Tools: An Overview

Following the posts on note-making in math and on basic thinking tools, here come first ideas on thinking tools for math problem solving.

Finding a solution to a math problem can be broken down into parts in several ways – for example by asking questions.
A first useful catalogue of questions could look like this, where each main question is followed by more specific ones.
(The catalogue is massively inspired by the famous list of questions from George Polya’s classic book “How to Solve It”. – Polya’s list is reproduced here.)
The next posts shall give some answers to these questions.

How can I get started?

  • How can I approach a problem? Are there established methods that work for a large variety of cases? How can I use them?
  • How can I represent a given problem, using equations, diagrams and other means that allow a math treatment?

How can I construct a solution?

  • How can I generate seminal ideas? What methods can I use to generate ideas?
  • How can I exploit these ideas?
  • How can I start from what is given and work forward?
    How can I start from the end and work backward?
  • How can I decide whether to proceed with an idea or whether to try a new one?
  • How can I deal with multiple ideas?
    Should I produce several of them and pick the most promising one, or should I start with one and remodel it until it works?

How to deal with obstacles?
(This part is closely connected to the previous and the next one.)

  • How can I make sure I realize there are obstacles that need attention?
  • How can I use established ways of dealing with obstacles?
  • How can I analyze an obstacle?
  • How can I generate ideas to overcome an obstacle?

How to deal with other troubles?

  • How can I deal with frustration, with the impression that I’m no good at math, with a lack of interest?

How to look back?
(This part is not exactly necessary, but a great opportunity to improve problem solving skills.)
Remember: Looking back should happen not only at the end, but also on the way!

  • What’s bothering me? What doesn’t feel right?
  • How can I clarify these intuitions?
  • How can I check if my solution is correct?
  • What can I learn from what I’ve done so far?

Finding answers to these questions may lead to a collection of math problem solving tools. Read more in the next post.


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