One of the many things that makes math difficult to learn is the seemingly universal reluctance to be comfortable with incomplete thoughts. In math terms this translates to “write down you’re intermediate results”. Kids hate to do this.
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.