## Personal Notes Are For One's Self

I recently stumbled across *Thinking Mathematically* by J. Mason, L. Burton, K. Stacey (Amazon) which has an interesting approach to writing personal mathematical notes.

I say "personal" as opposed to "expository" because they are really personal scratchwork rather than explanations.

What's really cute is Mason, et al., espouse a sort of "markup language" approach (they call it a "rubric"). Let me review this a little.

When "entering" a mathematics problem, there are three things to ask ourselves, which constitute the entry phase:

- What am I know? (What is given?)
- What do I want?
- What can I introduce?

With introducing stuff. . . we can do several types of introduction.

- Notation:
- Assigning values and meanings to variables.
- Organization:
- Recording and arranging what you know.
- Representation:
- Choose particular representatives which are easier to manipulate.

Now, reviewing your work is also critical. There are several different ways of doing this:

- Check:
- the resolution;
- Reflect:
- on the key ideas and key moments;
- Extend:
- to a wider context.

We can check several things:

- Check calculations;
- Check arguments to ensure computations are appropriate;
- Consequences of conclusions to see if they are reasonable;
- Check that the resolution fits the question.

We also reflect in finitely many ways:

- What are the key ideas and key moments?
- What are the implications of conjectures and arguments?
- Can the resolution be made clearer?

As far as **extending**, one should really be *generalizing* the problem. For example: how many squares are on a 3 × 3 chess board? There are 9 instances of 1 × 1 squares, 4 instances of 2 × 2 squares, and a single 3 × 3 square. Thus there are 14 squares altogether. Now, to extend:

- How many squares are on an n × n board?
- How many rectangles are on a 3 × 3 board? Extend this to n × n boards.
- What if we start with an m × n board? How many squares are there in it?
- Why work only in two dimensions?
- Why count squares with edges parallel to the original?

When **stuck**, try re-entering the entry phase. This can be done through:

- Summarize everything known and wanted;
- Rephrase the question in a more appealing way;
- Re-read or re-digest the problem.

Conjecturing is a cyclic four-step procedure:

- Articulate a conjecture (and while making it, believe it);
- Check the conjecture covers all known cases and examples;
- Distrust the conjecture. Try to refute it by finding a nasty case or example; use it to make predictions which can be checked;
- Get a sense of why the conjecture is right, or how to modify it, on new examples (go back to step 1).

It's not too long until one gets to a state where one says "I don't believe it's possible" which leads to the questions

- Why can it not be done?
- All right, what can be done?

Now, critical mathematical thinking should be nurtured by thinking three things while doing or reading a proof:

- Every statement made should be treated as a conjecture.
- Try to defeat and prove conjectures simultaneously.
- Look critically at other people's proofs.

- I Know:
- What is given? What is known?
- I Want:
- What do we want to prove?
- Introduce:
- Try contributing some:
- Notation:
- Assigning values and meanings to variables.
- Organization:
- Recording and arranging what you know.
- Representation:
- Choose particular representatives which are easier to manipulate.

- Stuck!:
- "I do not understand...", "I do not know what to do about...", "I cannot see how to...", "I cannot see why..." — try going back to the entry portion "want/know/introduce", or make a conjecture.
- AHA!:
- Whenever you have a good idea, write it down. Usually, they are of the form: "AHA! Try...", "AHA! Maybe...", or "AHA! But why...".
- Check:
- the mathematics. This means:
- Check calculations;
- Check arguments to ensure computations are appropriate;
- Consequences of conclusions to see if they are reasonable;
- Check that the resolution fits the question, i.e., our answer is the answer to the question asked.

- Reflect:
- meditate on:
- What are the key ideas and key moments?
- What are the implications of conjectures and arguments?
- Can the resolution be made clearer?

- Extend:
- generalize to other settings.

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