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Collecting Math Problem Solving Tools

In the post Math Problem Solving Tools: An Overview I presented a list of questions like the following ones:

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?

It seems a reasonable next step to collect answers to these questions, thus composing a tool collection. Some of the following tools are taken from the list in George Polya’s book “How to Solve It”.

• Introduce mathematical notation.
• Write down what is given and what is unknown.
• Draw a figure.
• Arrange data in tables.
• Check diverse representations:
  – Use numbers – decimal, binary, complex, factorized in primes…
  – Use coordinates – cartesian, spheric… Use a suitable origin.
  – Consider features of the problem (like symmetry) to make the representation simpler.

Starting with these simple and fairly heterogeneous examples, here are some ideas on tool collections.

 

Tool collections should help individuals

I should aim for a tool collection that is of help to me – adapted to my knowledge, my experience and my fields of interest. Such a tool collection is evidently not static, but changes with time.

 

How can a tool collection be recorded?

From a tool collection that exists only in my mind to handwritten lists and vast computer-based tool libraries, there are all sorts of possible representations.
Having it in a written form however has several advantages:

• I know explicitly some of the (most important) tools I am using,
• I can discuss such a list with others and
• I can try to improve such a list systematically.

 

What are the main challenges in using a tool collection?

There are many ways of using a tool collection:

• I can use it before the actual problem solving process, to remind me of important processes, tools and practices.
• I can use it during problem solving to get cues.
• I can use it after problem solving to improve it for further use.

The crucial questions for me are the following:

• How do I find – in each problem solving situation where I look for help – the tools that are useful just now?
• How can I apply these tools?
• In most cases applying one tool will not lead to an entire solution – so how can I deal with a sequence or with a combination of several tools?

(Given my obsession with note-making in math, the reader may guess what I will offer as an answer.)

 

What should be in it, and what should be left out?

This question has no general answer. In the following posts on tools, I will focus on general, domain-independent tools that seem to me very useful.

For the time being, I ask my readers to compile their own tool collections. There are some starting points I have worked out in the past, see here and here.

 

Some resources

• An example of an online repository is the Tricki (started by Timothy Gowers). Regrettably, it seems no longer under active development.
• Besides, there are all sorts of math cheat sheets on the internet – here is a first impression.
  From my experience, they are of little help for solving most non-routine math problems.