Sunday, July 6, 2008

Please visit an actual classroom before you make recommendations, II

Here we go again:

My cousin, bound for a top liberal arts college in the fall, was amused when I told her I was reviewing a book about big ideas in mathematics, from the classical to the contemporary. “Don’t they already know everything about math?” she asked. “You know, there’s algebra ... and then calculus ... and that’s it, right?”

Andrew Hodges, a fellow at Oxford and the author of the lively new book “One to Nine,” would have been horrified, but not surprised. My cousin, in his view, is a victim of the pedagogical tradition that presents math as an eternally fixed array of computations, to be memorized and repeatedly executed without motivation or explanation. The result, he writes, is a “legacy of fear and anxiety generated by schools, which leaves most of their victims with a lifetime of mumbling apologetically about ‘my worst subject.’”

[The opening paragraphs of Jordan Ellenberg's review of Andrew Hodges' One to Nine: the Inner Life of Numbers, which appears in today's New York Times Book Review.]

Both writers--each of them math professors-- also characterize classroom math as "abstract and remote."  To this, Hodges' book, in Ellenberg's words, is an antidote: "offer[ing] a different model for teaching math."

Hodges hails from Britain, which hasn't yet gone whole hog for Reform Math, but if either professor had visited any number of American elementary or middle school math classrooms, they would see that:

1. in place of an "eternally fixed array of computations to be memorized and repeatedly executed," we have math as a mess of multiple, ad hoc solutions that students are required to explain and motivate ad nauseam.

2. far from "abstract and remote," today's math marginalizes pen and paper calculations and relentlessly requires students to learn through hands-on activities and real-life applications.

3. all this has so watered down the material and so slowed the pace of learning that, objectively speaking, math is the "worst subject" of more students than ever, however much their fear and anxiety levels about so-called "math" (as currently defined) have diminished.

4. too many math classes are as conceptually disorganized (organized more by "topic" than by concept) and as fleeting and superficial in their coverage as Ellenberg reports Hodges' own "discursive rather than linear" prose as being:

...The book is composed of nine chapters, each focused — very, very softly focused — on one of the first nine natural numbers. Chapter Four, for instance, starts out with the observation that four is a perfect square, and from there skips along to the construction of Latin squares, the irrationality of the square root of two, the definition of the logarithm (whose relation to “four” never comes entirely clear), complex numbers, and the even more exotic quaternions (a number system in which “numbers” are actually strings of four integers, and the product of two numbers depends on the order in which you multiply them!), the theory of four-dimensional spacetime and Einstein’s equation E=mc2 (squares again) before finishing with a short and speculative account of the theory of twistors, one of many competing candidates for the universe’s underlying geometry.

As Ellenberg notes:

The overall effect is like that of a lecture by the type of professor who paces back and forth in front of the blackboard, with insistent voice and waving arms, and has trouble adhering to the ostensible syllabus for any extended period. Being this type of professor myself, I can attest that the style is popular with students. But it requires discipline to convey real information as well as enthusiasm.


Mathematicians are in a unique position to hold Reform Math accountable for its shortcomings. By the same token, when they make recommendations that are as out of touch with current practices in math classrooms as physicist Brian Greene's are with current practices in science classrooms, they have the potential to do real harm.

You can bet that Reform Math proponents will start citing Hodges and Ellenberg as their allies, thus ascribing more credibility to practices that most mathematicians would be horrified by.


vlorbik said...

here's part I.

i linked here yesterday; thanks.

lefty said...

Thanks, Vlorbik!

Who is Einstein disguised as Robinhood?

Anonymous said...

he was famous long ago
for playing the electric violin
on desolation row.

Myrtle Hocklemeier said...

A timely post.

I was just expressing disatisfaction in a blog entry about the disorganization of some high school algebra books.

The student is left with no indication about the object of study, the properties of the field of real numbers and how it's constructed, nor of the methodology of math--its axiomatic system, but rather than algebra is just this thing you do to solve for x and figure out how fast two pipes fill up a swimming pool.

So, yeah, I am tired of people talking as if pure math is taught in high schools. That is so far from the truth it's laughable.

lefty said...

Can you recommend any good algebra texts, still in print? Everything you say is more or less true of the various algebra texts I've seen, except for Gelfand. But Gelfand is a bit tough for beginners (and for parents like me!) I should check your blog...