I reproduce a post from my defunct blog of 2007 — simply because the issue raised remains out of mainstream educational discourse. Recently I mentioned it in my talk to a meeting of HoDoMs — organisation of Heads of Departments of Mathematics in British universities. It did not came as surprise to my audience — by default, highly experienced teachers — but it was also obvious that they hardly ever discussed it in public before.
I came across the following quote from Ralph P. Boas:
… a phenomenon that everybody who teaches mathematics has observed: the students always have to be taught what they should have learned in the preceding course. (We, the teachers, were of course exceptions; it is consequently hard for us to understand the deficiencies of our students.) The average student does not really learn to add fractions in an arithmetic class; but by the time he has survived a course in algebra he can add numerical fractions. He does not learn algebra in the algebra course; he learns it in calculus, when he is forced to use it. He does not learn calculus in a calculus class either; but if he goes on to differential equations he may have a pretty good grasp of elementary calculus when he gets through. And so on throughout the hierarchy of courses; the most advanced course, naturally, is learned only by teaching it. This is not just because each previous teacher did such a rotten job. It is because there is not time for enough practice on each new topic; and even it there were, it would be insufferably dull.
I believe the reason is not lack of time for practice (although this is an important contributing factor), but a manifistation of a more general principle: in order to secure a certain level of mathematical skills, the learner has to learn the next, higher level. Indeed, mathematical objects, concepts, procedures are interiorized in good working condition only if they can be assembled into a higher level mathematical constructs. Ability to solve routine, rote learned problems at certain level L is not a proof that one understands mathematics at level L; but ability to apply L level mathematics within routine problems at the next level L+1 is a proof that one has mastered level L.
Thus there is an important difference between mathematics and most other human activities.
To drive a car, one does not have to be trained as a Formula 1 racer; but to teach mathematics at a high school level, a teacher has to have a knowledge of university level mathematics. The same principle applies throughout the entire range of application of mathematics. Investment banks hire people with a PhD in mathematics or physics for jobs which require just a good knowledge of university level mathematics and statistics.
This also means that the work of a mathematics teacher should be assessed not by the exam results of his students, but by their success at the next level of education. In terms of English education system, the success of a GCSE level mathematics teacher should be measured by the number of his/her students who take mathematics at A level, and by their performance there. Similarly, the best measure of a work of an A level teacher is the number of his/her students who chose to pursue a mathematically intensive degree at an university, and by their performance at the university.