I’ve been doing a bit of ‘light’ reading in the whole area of cognitive science and it really is an interesting subject. Reading in this area makes me realise how many of us don’t really know what the hell we’re talking about when it comes to T&L matters.
One thing that especially interested me is the idea that our ability to solve problems is limited by the capacity of our working memory. However, we can limit the load on working memory by having a store of relevant knowledge in our long-term memory. In mathematics especially, problem solving ability is maximised if that knowledge is known to the point of automaticity. For example, a student will become proficient at solving calculus problems only if he/she has automatic recall of basic mathematical rules including simple rules of algebra and rules relating to logs, exponentials, powers etc.* That is not to say that understanding is not important as well. It is impossible, for example, to develop mathematical models of engineering systems unless you actually understand calculus, as opposed to being proficient at the mechanics of differentiation and integration. However, instant recall of basic concepts and techniques seems to be a key step on the road to being able to advance in mathematics. It makes one wonder about the whole Project Maths initiative.
This need for prior knowledge is surely true in other subjects as well. Think writing an essay and being consumed by rules of grammar rather than the points you’re trying to make.
It seems that there is only one way to reach automaticity and that is practice. And this leads me to what I suspect has gone wrong with the education system and the teaching of Mathematics in particular. Students are not reaching the point of automaticity with basic rules and techniques and are unprepared to deal with the next concept that inevitably comes along. They are not rote learning when they should be and rote learning when they shouldn’t have to.
It is often proposed that a solution to inappropriate rote learning is to ‘teach’ problem solving and critical thinking skills. This is a bit like trying to optimise the performance of a car by fine tuning the engine – and then forgetting about the fuel. I think the solution to the education system, especially the maths deficit, is to focus a lot more on getting the fuel into the tank. This means a renewed emphasis on practice. The real challenge is to make that practice as interesting as possible. We don’t want to stifle all enthusiasm for the subject.
*Whether the student really understands the laws of algebra is not really important. He just needs to know them. Here’s a nice quote from Timothy Gowers (Fields Medalist) in his book, Mathematics: A Very Short Introduction:
“It is quite possible to use mathematical concepts correctly without being able to say exactly what they mean. This might sound like a bad idea, but the use is often easier to teach and a deeper understanding of the meaning, if there is any meaning over and above the use, often follows of its own accord”.