If you ask them to do a multi-step problem it is an enormous struggle to get them to write down intermediate results. This, of course, makes it much, much harder to get to the final result. Trying to add two numbers together while remembering a third is so much harder than just adding two numbers. Why is this hard to learn?

I suspect it is evolutionary. Externalized memory is evolutionarily recent. As a survival skill it has been of no value until relatively recently (even Socrates despised writing as something that weakened the reason, rather like the current argument that Google rots the brain). Human beings like to take in the whole picture at once.

Given that this is a more or less universal problem in teaching math I think it’s reasonable to assume that this is a reluctance that is genuinely difficult to overcome. There have been plenty of great math teachers (not enough to distribute adequately, of course, but a large number nonetheless). If there was some teachable trick – teachable to teachers, that is – that could convince students, at an early stage, of the value of externalizing intermediate results, then someone would have figured it out. (This is an obvious area for empirical research.)

Instead we have generation after generation of math teachers, themselves all too often under-trained, and unaware of the existence or implications of short term memory limitations trying to browbeat children into following a rule none of them really understand.

Those who go on to advanced mathematics are generally either docile enough to have done what they were told, or had capacious enough short term memories to get by until they finally figured out the value of externalizing intermediate results. An article in the New York Times covered some interesting research on innate number sense and its correlation to achievement in mathematics – about as surprising as a correlation between hand eye coordination and achievement in sports.

Dijkstra, the programming genius, once wrote: “The competent programmer is fully aware of the strictly limited size of his own skull” This awareness generally involves painful experience, and more time than can be fit in a school math curriculum. It certainly involves far more comfort with error than most school boards have.

The Punchline

If I don’t know the answer to the teaching problem I do know an implication of this for the reporting of mathematics in the public sphere: make minimal demands on the reader’s short term memory.

People can compare two charts. They can’t remember one chart while looking at another. In fact most people can compare dozens of charts – as long as they can see them all together.

No one I know of does this better, on a regular basis, than Martin Wolf in the Financial Times. Most of his articles are illustrated by a number of charts, all beside each other, most with multiple lines or bars, often in different scales. (Here is a recent, depressing, but illustrative, example.) These are fairly extreme examples, written for a specialized audience, willing to devote considerable attention, and knowledgeable about the subject matter. But the principle holds for any audience. Teachers should keep trying to get the damn kids to record their intermediate results. Those trying to communicate to the public should know that their audience wont do that. The audience won’t do this work for you. Show them the working yourself.

There are two types of justification, the aesthetic and the practical. The aesthetic case is actually the easy one. Math is one of the least obvious, most complex, profound and uniquely creative things our species has done, right up there with music, painting and dancing. As a liberal art math needs no justification: if you want to understand the glory of what it is to be human then you need to be understand something of these spectacular creations.

But if that was all there was then we certainly wouldn’t be spending all this money teaching math. We had no problem dropping music from the curriculum. (Doubly unfortunate given that music lessons help math scores.) Math is taught for economic reasons: we think the return justifies the expense.

It probably does, but there are many unexamined assumptions here. Consider some justifications for teaching math.

• it’s useful
• it’s good for the brain
• it separates the wheat from the chaff
• engineers need it, and we need engineers

Usefulness

It’s only useful if you use it, or if being able to use it unlocks some new ability. (Being able to swim enables me to go on the boat, even if I don’t fall in and never have to swim.) But if you never use it, never consider it, are never empowered by it, can never do something only because you have the knowledge, then it is not useful. A lot of the math people are taught is like that.

Good Brains

Any kind of study is good for the brain. Latin and Greek are excellent. Memorizing decks of cards, wine regions of France, the plays of Shakespeare, all good. The brain is a muscle. Use it and make it strong.

That doesn’t explain why mathematics instead of Greek, but there might actually be an answer to this one. I believe that there are useful, brain reshaping ideas in math that none of us is likely to come up with in a single lifetime. Things like the reductio ad absurdum, like proof by induction. The kind of idea some human being comes up with every few hundred years. Things worth sharing. I believe that there are brain rewiring mathematical ideas that most people should be exposed to. But that needs to be proved. And which specific ideas should be prioritized and taught should be determined by something other than habit and inertia.

Math as Gatekeeper

If you need to narrow down the field then math works about as well as height. If you need to eliminate 50% you can choose one type of math and one cut-off score. If you need to eliminate 90% then, hey, let’s require calculus… If the math skills are needed then fair enough, otherwise this is as unfair as discriminating on any other irrelevant criterion. And yes, I’m talking about requiring nurses to know algebra.(this caused a stink) I want nurses to be bright. I’m not sure that requiring algebra measures the kind of smarts that really matter. I might be wrong, but it isn’t obvious. Using math as a proxy for an IQ test is just dishonest, and lazy.

We Need the Eggs

Suppose traditional math education ruined almost everyone’s chances of understanding or enjoying math. And suppose that the only way to train engineers, physicists, quants, etc, was to teach them that way from an early age, before they could even be identified as prospective quants. If that were true then it might well be worth teaching everyone that way. We do need the engineers. (Under the right circumstances even I would swap understanding how the defibrillator works for having a defibrillator.) But this is implausible. Colleges and universities need to be fed numerate students, and the burden imposed on them by the need to supply remedial math classes is considerable, but if the best answer we can come up with is to teach in ways that are known not to work for most people then we need to keep looking.

What To Do?

What is useful and necessary? It depends on personal accident, but there are probably lots of overlaps. This is the kind of thing a rational; research policy would research: what math gets used? What is unknown but of most value? We already spend an unimaginable fortune on math education (This is the good news. Can you imagine how jealous art teachers are right now?), we could spend a few more bucks on measuring needs and results. Every scrap of curriculum needs to earn its place. Historical importance is no more relevant in math than any other historical importance. The speeches of Cicero are of enormous historical importance, but we gave up teaching Latin. Quadratic equations are the Cicero of high school mathematics.

What actually is good for the brain? (This is a special case of ‘usefulness’.) And it’s an empirical question. We ought to be able to answer it and to agree upon the answer.

What do engineers need to learn, and when? In most things we learn what we need to know close to having to use it. That helps with motivation, understanding, and retention. On the other hand musicians, sportsmen and dancers, are generally considered to need an early start. That might also be true for mathematicians. This is an interesting and difficult question, but it’s the right question.

There are real questions here, and issues that should generate research instead of debate. Rather than argue about the facts we should find out what they are.