Two Essays on Math Education

Note: These essays were originally written as editorials for the CMS Notes

JUMP And Deliver?

You may have watched the inspiring film "Stand and Deliver", based on the real-life teaching successes of Jaime Escalante, and thought "yes, but what we need is something that everybody can learn to do". Canadian author John Mighton's book The Myth of Ability; Nurturing Mathematical Talent in Every Child (Anansi,2003) may be a big step in this direction.

In this book, he sets out and describes in detail a program, JUMP ("Junior Undiscovered Math Prodigies"), that he has developed for teaching the basics of mathematics, especially at a remedial level. It has been tested in many places, with many different instructors, and it seems fairly clear that (a) it works, and (b) anybody can do it. The basic principles are straightforward, and can be summarized in the phrase "Students must be allowed to succeed". Learning is broken down into tiny and carefully-structured chunks, that any student, working with any tutor, can learn thoroughly. A tutor, who does not need any great level of training, follows the child's work carefully. That's pretty much it, at least on a first pass. And it seems to work brilliantly.

But today's high school textbooks are full of topics such as graph theory, statistics, geometry, and linear algebra; the methodology stresses exploration, discovery, and constructing one's own mental picture of the subject matter. Many professional mathematics instructors will object that the JUMP approach leaves little room for developing creativity, or even self-guided learning skills. These are valid criticisms: or at least, they would be if Mighton or anybody else was putting this forward as a complete and self-contained mathematics curriculum; and I don't think that Mighton is doing so.

It is true that, at times, it seems that while he avoids making claims for his program that go beyond what he has actually observed, he is reluctant to suggest any limitations on it. This may be partially explained through Mighton's own anti-elitist agenda. The title of the book makes the reader wonder whether he really means that there is no such thing as mathematical ability; and he does make it clear [p.21] that he personally does not believe that there is any genetic component to intelligence - a radical view that many won't share, or at least will consider unproven. Does his work with the JUMP program qualify him as an expert on the question of the distribution of mathematical genius potential? With great respect, I would have to say that it does not; it is an area with few experts.

Mighton has, I think, demonstrated (as others have done before) that with proper instruction almost everybody can learn elementary mathematics to a level well beyond the contemporary school norms; and thus he has earned the right to be heard as an authority on that question. But the use of the word "prodigies" in the title of his program must, I think, be interpreted as a deconstruction: "we're all prodigies so nobody's a prodigy in the old-fashioned elitist sense".

Alas, "the gods have seen it otherwise". If the JUMP program could ever train a significant proportion of its students to the level of (say) today's CMO contestants, let alone Fields Medalists, then Mighton would have demonstrated that the average child in a classroom today is indeed an "undiscovered math prodigy". It seems unlikely that this will happen, and Mighton has never really claimed that it will.

It is true that in the chapter on "Logic and Finite-State Automata", he does demonstrate that some material usually considered as "advanced" can be taught by this method. But there is a big difference between teaching a few isolated skills and teaching the entire subject. It's easy to learn how to count the holes in a pretzel and decide that it has "genus three", hard to finish even the first week of a course in algebraic topology. It's not at all clear that Mighton's methods are sufficient to teach basic theorem-proving, or even the techniques for solving those elementary puzzles that are not amenable to systematic searching.

But that is not the point. Let us consider Mighton's work for what it is - a powerful (if slightly labor-intensive) way to stamp out mathematical helplessness in most children. Is this something today's schools and society need and can use? I think so. Can the program teach skills that we have observed our own students to lack? Definitely.

If we don't have unreasonable expectations, and don't let a few unimportant details of the packaging put us off, this is overall an excellent, thought-provoking, and important contribution to mathematical education. With methods like this to bring students up to speed on the fundamentals, the dream of being able to teach exciting, advanced mathematics to most students could become a reality. Mighton himself makes it clear (see, for instance, page 62) that this is his vision as well.

Just Imagine...

A few months ago, the President of Harvard University caused considerable controversy, and some anger, by speculating that innate differences between males and females might go some way to explaining the comparatively small number of women in mathematics. Many groups responded publicly, in some cases stating as fact the diametrically opposite view that there could be no such differences.

President Summers' statements were made in a conference not open to the media, and no exact transcript of his words appears to exist; therefore it is difficult to know exactly what he said, and in particular whether his words could reasonably have been interpreted as claiming the existence of such differences. In this column, I would like to argue that, at least now and in the immediate future, any definite statement on either side of this argument is, at best, theorizing in advance of the data.

Last month, we discussed John Mighton's JUMP program, as described in his book "The Myth of Ability". Mighton appears to have shown not only that almost every child has the potential to perform above current grade level expectations in mathematics, but that almost any older volunteer can carry out the necessary tutoring. An easy corollary of this is that, in general, children who have difficulty with basic mathematics are not being limited by their own ability. Another is that we can no more observe the real limits of children's ability in today's classrooms than we can study the relativistic "lightspeed barrier" by watching cars driving down the highway.

At a higher level, anybody who follows high school mathematical competitions in Canada will notice that there are a few schools that regularly provide one or more CMO finalists and members for the IMO team. Perhaps some of this can be explained in terms of ambitious parents of gifted children arranging to have their sons and daughters attend a certain school, and it is true that many of these students are first-generation immigrants whose earliest math education took place in (say) China or Rumania. However, it appears that there are some schools that have math programs that can consistently train a few students out of every year's intake to the level of the national elite. Moreover, in international competition, these students perform at a level comparable to the elite of their native countries.

This sort of training is still at the "Jaime Escalante" stage (see above). Evidently a few people can do it, but we don't yet seem to know how to make such teaching into a transferrable skill. Is is too much to dream that some day we will know how? And while it is true that mathematics competitions are not all there is to mathematics, it seems likely that if that specialized subset of mathematical skills can be taught so much more widely than it is now and to such a level, so can others. Has anybody got any good ideas?

Until we can do everywhere what a few pioneering teachers have shown us is a possibility, speculating about differences in innate limitations seems pointless. Not only have we little evidence to go on, but there are far, far more important and exciting things about to happen - if we make the effort. Imagine...

Imagine a world in which almost everybody really understands high school mathematics and, when in school, exceeded our current system's graduation expectations. Imagine a world in which the top few students in every high school have realized a mathematical ability comparable to the top few in Canada today. Imagine a world in which subjects like statistics do not have to be divided into a rigorous discipline studied by a few and a soft, math-free version for the masses.

Imagine a world in which most people enjoy mathematics.

Just imagine.

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