Thursday, 20 April 2017

There's so much maths to learn...or is that so little?

The problem of remembering

Observation has led me to the conclusion that there are two very different kinds of mathematics student.
  1. One kind thinks one of the disadvantages of mathematics is that there is so much to learn.
  2. The other thinks one of the advantages of mathematics is that there is so little to learn.
The strange thing is that both are correct: where they differ is they way that they approach the process of learning the mathematics.

The student of the first kind is the one who simply tries to learn everything by heart, whether it is a brute fact, a concept, or a standard proof. It has to be memorized exactly, in the way one learns a poem in a language one doesn't speak.

The student of the other kind has different strategies for remembering different types of information.

I see it as part of my job to try to help students of the first kind become students of the second. So, how to go about that?

The types of things that need remembering

I'll break the types of information up into four categories, each with its associated method or learning. To a first approximation, the first two categories are school level, and the second two undergraduate level. But that's only a first approximation, and some particular items will straddle categories, in some cases depending on the level of the learner.
  1. Arbitrary Stuff, eg order of precedence for arithmetic operators, names of functions...
  2. Formulae and Statements of Theorems, eg binomial expansion, trig identities, integrals and derivatives, product rule for differentiation...
  3. Conceptual Definitions, eg limit, continuity, differentiability ...
  4. Proofs, eg sum of limits is limit of sums, a bounded monotone sequence has a convergent subsequence ...

How to remember them

One approach to the problem of having stuff to learn is simply to deny the necessity. After all, you can just look things up whenever you need to know them, so you don't need to remember them. Alas, that just isn't so. You need stuff to be part of your mental toolkit in a way that makes using it effortless so that you can devote the effort to solving the problem that you need to know the stuff for. If you're spending significant effort on trying to hold in short term memory things which aren't bedded into your long term memory, then you're robbing the real job of that effort.

So we accept that some things just have to live in our heads, and need strategies for getting them there. If possible, the strategies ought to be efficient.

Type 1

The things of the first category are what I think of as 'brute facts'; there's no underlying pattern or reason to aid you in committing them to memory, you just have to do it.

I'll be blunt: I am not an enemy of memorization or rote learning. This is not a guilty confession, it is a simple statement. In some contexts memorized material can be useful: it's a Good Thing to have a mind furnished with interesting bits of literature, or know the general timeline of an aspect of history, or to know that Bach precedes Beethoven chronologically, for example. And there isn't really any way out of it for some things. If you want to remember the order of precedence of arithmetic operations, or which of \(\sin(\theta)\) and \(\cos(\theta)\) is the \(x\) and which is the \(y\) coordinate of a point on the unit circle an angle \(\theta\) round from the \(x\) axis, or the definition of \(\sinh(x)\) in terms of \(e^x\) and \(e^{-x}\) then you pretty much just have to commit them to memory.

Mnemonics help, and flash cards with spaced rehearsal give a very effective method for fixing these things in memory. If there's a meaningful picture, draw it. (This goes for all categories!) Constant usage will also help: and why would you memorize something you don't need to use a lot?

Type 2

In the second category I place standard formulae and statements of theorems: I'll just refer to both as results here. Some are easily derived, others not so much. I am not in favour of using mnemonic devices to remember results, as I think these can distract from the meaning. For these, I recommend keeping a list (or lists), and referring to it/them at need.

But it is important to rehearse mentally the meaning of the result every time it is looked up. It is the meaning that needs to be remembered, not a literal pattern of words, letters and function names. (As an aside, I can say I find it distressing to see a student confused by being asked to differentiate \(v/u\) when they remember the quotient rule as something that tells them how to differentiate \(u/v\).)

Its also important to look for patterns, or special cases that make the general case easier to remember. I never learned the addition rules for \(\sin\) and \(\cos\), but I did learn the double angle formulae and then worked back to the 'obvious' thing that reduced to the double angle formulae.

Patterns are important. (I know I just said that.) When a pattern is spotted, the list should be rearranged to take advantage of that: put related results together with a comment on the relationship.

And sometimes it is easy to derive a result; whenever it is, the derivation should be worked through every time the result has to be looked up.

As before, if there is a good picture which illustrates the result, draw it and think about the relationship.

All this sounds like effort, and it is. But the effort of processing the formulae over and over makes it much easier to remember them. In the long run, it is more effective and gives more meaningful knowledge of the formulae than rote memorization of the strings of symbols does. It also a lot less drudgery (and more interesting!) than writing formulae out five hundred times and hoping that that will make tjem stick.

Type 3

The third category is less commonly encountered in secondary school. It contains definitions of mathematical ideas: concepts like limit, continuity, differentiability, for example, or objects like metric or topological spaces where there is a significant amount of structure to work with.

I suspect that part of the problem faced by many undergraduates is that their strategies for remembering mathematics have been developed to deal with what is mostly information of the first two kinds. However, as they progress, the type of material that has to be remembered increases considerably in complexity, and they may not receive much explicit guidance in how to cope with this new problem. Certainly it took me a long time to realize that it was an issue, and so to start to provide such guidance.

Again, I strongly recommend to my students that they don't try to learn off by heart the definition I give in class. I tell them to think about what the definition means, re-express it in their own words, in different ways, draw pictures, think of examples of things which satisfy the definition (and which don't). To try to distill out a brief expression of the idea in their own words, and use that as a skeleton for a formal statement.

This advice is heavily biased by the fact that the only way I can retain this type of material is by working with the idea until it is a natural tool, and then I can construct a formal definition whenever it is needed. And it is rarely expressed exactly the same way as the previous time.

Underneath it all, it isn't really a matter of learning the definition. It's a matter of absorbing the definition by working with it, and then constructing a statement of the definition when it is required.

Type 4

Learning proofs is something of a bugbear for many students. They often seem to see being asked to produce proofs as part of an exam as analogous to having some jack-booted fascist insist that they jump through a flaming hoop just because they can. So the first task is to explain that there is a rationale to why knowledge of some proofs is required: it is because they use techniques, or ideas, which can also be used in other situations. It helps to make this plausible if the proof techniques are in fact used in other situations.

So, let's assume that the students have been persuaded that knowing how to prove some standard results is a Good Thing for reasons other than to obtain an extra 5 marks in the exam. (In my experience, if the promise of 5 marks is the only motivation, it is a pretty weak one,, judging by the level of performance it elicits.)

Again, I strongly advise the students against trying to learn the proof off by heart.

The first thing I advise is to work through the proof to understand it. This seems obvious, but to at least some students it seems like additional work, not a labour-saving device. It's hard to remember something you don't understand.

But there is the question of what it means to understand it. A necessary, but not sufficient condition is that the logic of the step from each line to the next be clearly understood. But that certainly isn't enough. In addition to that there has to be an understanding of how the steps accumulate to provide the proof. There is generally a core idea that the structure of the proof is built around, and a firm grip of that central idea (or possibly ideas) together with how the individual steps realize the idea combine to give a solid feel for the truth of the result. Ideally, it shouldn't just be an intellectual acceptance that the steps go from premises to conclusion, so the conclusion must be right, but a more visceral feeling for the truth of the conclusion.

And once the proof has been thought through, wrestled with, and boiled down to a core idea or two, there should be no necessity to 'learn the proof'. The statement of the theorem automatically throws up the core ideas of the proof, and fleshing those ideas out to a formal proof is then just (!) a matter of putting in the detail.

The philosophy here is similar to that for Type 3; you don't learn the proof, you wrestle with it and work with it until it is absorbed into your mathematical toolkit. Eventually you remember it. At no point do you memorize it.


It may have been patently obvious, but the above is not the result of me carrying out carefully designed scientific experiments into just which strategies for learning different types of material are most effective. Some of it is based on reading about how memory works, some on what works for me, and some on the responses I've received to advice I've given to students in the past.

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