**Problem solving**

A problem of the form *What can I figure out about this new idea?*
focuses on a *mathematical object,* abstracted from the trappings of
the real world. For example, the problem may be about understanding the
notion of graph, tree, geometric shape, permutation, or function.

**Getting to know an idea**

You begin work on this kind of problem by *brainstorming.* To
brainstorm, you list as many observations and questions as you can without
evaluating them. While still brainstorming, you look at lots of examples,
examining them for patterns, looking for properties (characteristics) of
the mathematical object being examined.

Solving a problem of the form *How can I solve all problems of this
type?* requires finding a general method, called an *algorithm.*
As you might expect, working on several problems of the same type can give
you ideas about how to work such problems in general.

Sometimes algorithms are expressed in a formal structured language so that their correctness can be checked easily. Then they can be used in proofs. Often algorithms are expressed less formally, however.

However you express the algorithm, you need to test it on as many examples of problems as you can.

**Monitoring your work**

In solving any kind of problem, you might want to monitor your work as you
go. Here is an idea from John Mason, Leone Burton, and Kaye Stacey, in
their book
*Thinking Mathematically.*

While working on a problem, make these marks in your notes to keep tabs on how you're doing:

- STUCK
- AHA!
- TRY
- CHECK
- REFLECT
- KNOW
- WANT
- QUESTION
- THINKING
- WARMING UP
- ASSUME

All mathematicians get stuck while trying to solve problems. Galovich, in
the book *Doing Mathematics: An Introduction to Proofs and Problem
Solving*, compiles ideas about being stuck from Polya and Schoenfeld:

- Write down what you know
- Write down what you want
- Draw a picture
- Label some variables
- Try one or more special cases (specialize) and look for patterns
- Consider extreme cases
- Consider using symmetry properties
- Restate conditions in equivalent form
- Recombine elements of the problem
- Introduce auxiliary elements
- Change perspective or notation
- Assume you have a solution and determine its properties
- Try a simpler problem
- Try an analogous problem
- Generalize the problem
- Isolate a variable

We would add:

- Look up stuff
- Make sure you're aware of relevant results by others.
- Think backward
- Imagine that you have the information that is asked for in the problem but donŐt have the information in the question. Does this give you any ideas about the relationship between the question and the solution?

When you're not stuck and are going fine, keep asking

- Is this taking too long?
- Should I keep going on this track or try something else?

When you've been working on one track but haven't made progress for awhile, consider yourself stuck and ask those questions again.

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