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.

to teach mathematics at a high school level, a teacher has to have a knowledge of university level mathematicsCannot agree more.

However, it is worth mentioning that the “university level mathematics” necessary for the high school teachers, should be different from the “general” math curriculum: agility with reducing matrices to the Jordan form adds nothing to the teaching skills, while good proficiency in working with “infinity” (limits, asymptotic behavior – things sometimes left aside in the common calculus courses) is a vital necessity.

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Sergei Yakovenkoon May 17, 2011at 7:19 am

@Sergei: of course, teachesr should be taught “teacher friendly” mathematics.

This is an abstract of my talk that I will soon make at 30th MATHEMATICS TEACHERS AND ADVISERS CONFERENCE in Leeds, https://www.ncetm.org.uk/cpd/25090

“The eternal cycle of mathematics”

ABSTRACT. Our civilisation owes its existence to the eternal cycle of reproduction: Autumn’s harvest is the next Spring’s seed; a disciple

steps in for his teacher. In my talk, I will look at the cycle of reproduction of mathematics as a cultural system and a professional

community at all its levels:

school – university – teacher training – CPD – school

and will try to address the question: what kind of mathematics and how should be taught at school and at university in order to facilitate the

smooth working of the cycle? I will argue that potential future teachers should be exposed to, and get first taste of, “pedagogically oriented”

mathematics during their school years and that this currently neglected stream of mathematics should recieve full attention in university level

mathematics education.

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Alexandre Borovikon May 17, 2011at 7:26 am

@Alewxandre: recently I got involved in a pilot program aimed at instilling an appropriate feeling of mathematics in the selected high school (Israeli) math teachers, which prompted me to rethink the realistic math curriculum which could be taught in the two-years Masters-granting degree (without thesis).

I tried to experiment with the “blog-like” platform to involve students-teachers more actively in discussion of the topics, but failed. It turns out that the teachers (mainly ladies in their late thirties till late forties) proved to be too conservative to play these games.

The outcome (mostly in English for technical reasons, only occasionally in Hebrew) is available at http://en.wordpress.com/tag/rothschild-course/ . Any comments, corrections or remarks are most appreciated.

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Sergei Yakovenkoon May 17, 2011at 7:46 am

“a more general principle: in order to secure a certain level of mathematical skills, the learner has to learn the next, higher level.” I think it is because the unagitated “matter of course”-ly way in which something is used in articles/texts/lectures at a higher level works a bit like hypnosis and liberates the mind from previous blockages, where one had automatically expected that something new and sophisticated to be difficult to grasp.

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Thomason May 17, 2011at 8:58 am

@Thomas:

A very good point, and I agree with you. Moreover, my own childhood recollections (a highly unscientific source of evidence, I admit) suggested that my mathematical development much benefited from gloriously long summer holidays, three full months: June, July, August.

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Alexandre Borovikon May 17, 2011at 10:41 am

An interesting reportabout an experiment in physics teaching. It shows (in this case) that “traditional lecturing” works much less good than usually expected. (I guess lecturing derived from middle age universities, where books were rare and expensive and many students entered it with deficient reading skills.)By:

Thomason May 24, 2011at 8:14 am