Tuesday, December 13, 2011

Personal v. Expository notes

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:

  1. What am I know? (What is given?)
  2. What do I want?
  3. What can I introduce?
Know, want, introduce. These three things we should always, always, always ask ourselves! Even in long proofs, we should ask ourselves these questions! We should ask ourselves when we get stuck!

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.
Bear in mind that rephrasing the question is particularly useful.

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.
The best way to get the most out of reviewing is to write up your resolution for someone else, in such a way that one can follow what you have done and why.

We can check several things:

  1. Check calculations;
  2. Check arguments to ensure computations are appropriate;
  3. Consequences of conclusions to see if they are reasonable;
  4. Check that the resolution fits the question.
This is sort of subconsciously done.

We also reflect in finitely many ways:

  1. What are the key ideas and key moments?
  2. What are the implications of conjectures and arguments?
  3. Can the resolution be made clearer?
It helps one see things that otherwise would have been missed.

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:

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

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

  1. Summarize everything known and wanted;
  2. Rephrase the question in a more appealing way;
  3. Re-read or re-digest the problem.
This is useful sometimes.

Conjecturing is a cyclic four-step procedure:

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

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

  1. Why can it not be done?
  2. All right, what can be done?
Asking "What can be done" is a critical step in conjecturing.

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

  1. Every statement made should be treated as a conjecture.
  2. Try to defeat and prove conjectures simultaneously.
  3. Look critically at other people's proofs.
There are some mathematical registers that some authors suggest using while writing notes. The collection is sometimes called a rubric:
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:
  1. Check calculations;
  2. Check arguments to ensure computations are appropriate;
  3. Consequences of conclusions to see if they are reasonable;
  4. Check that the resolution fits the question, i.e., our answer is the answer to the question asked.
Reflect:
meditate on:
  1. What are the key ideas and key moments?
  2. What are the implications of conjectures and arguments?
  3. Can the resolution be made clearer?
Extend:
generalize to other settings.
The real trick is to change "I'm stuck, panic!" to "I'm stuck, okay, so what can be done about it?"

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