Science isn't dead, but it is ailing - part 4 [Jay Guevara]

Previous installments:

Science isn't dead, but it is ailing - part 3
Science isn't dead, but it is ailing - part 2
Science isn't dead, but it is ailing - part 1

In the most recent installment I was ripping on chemistry instruction, which God knows deserves it. This time I'm going to rip on mathematics instruction, which really deserves it, in my opinion.* There are good textbooks and lecturers out there, but they're kind of finding patriotic Democrats.

Just as in chemistry instruction, the problem arises from intellectual pretension. Now you would think that impressing teenagers that you know more about a subject than they do would be rather pathetic, and you'd be right, but many succumb to the urge to do so.

Pretension takes the form of math instructors and their textbooks making a fetish of "rigor." This is bogus for a variety of reasons. First, it's pedagogically unsound; students don't give a rip about rigor, they're just trying to understand the underlying concept. The only person in the room who gives a toss about rigor is the instructor, i.e., the least important person in the room.

Second, historically most mathematics was devised to solve practical problems. "Geometry" literally means "measurement of the earth." Trigonometry literally means "measurement of triangles." Making such practical treatments rigorous oftentimes only took place well after the event. Even great mathematicians sometimes whiffed badly on what today is considered straightforward material. For example, no less a mathematician than Friedrich Gauss - one of the best in human history - famously thought that the infinite series formed from (-1)n converged. C'mon, Fred, just add up a half dozen terms and then tell us that the sum is going to converge!

A more sound approach pedagogically, I believe, is to follow the historical development and introduce concepts on an intuitive basis that is easily grasped on a visceral level, and then clean up with rigorous reasoning later. Key to this approach is to interweave the algebraic picture with the geometric one (i.e., a pictorial one), which being visual is typically much more accessible intuitively.

As a simple example, if you ask someone to define the tangent, 99.9% (including me, until quite recently) will tell you it's the quotient of the sine and the cosine, i.e., they'll approach this from an algebraic perspective. But that's is not the definition; that's a consequence of the definition. And why is it called the tangent, anyway? Shifting to the geometric (that is, pictorial) perspective makes the answer obvious. To spare the Horde the math, I've broken this out into a PDF.

The point is that weaving between geometric and algebraic approaches is very powerful, and very efficient in conveying concepts on a visceral level. Yet math instructors typically eschew the geometric approach in favor of the oftentimes more obscure algebraic approach, and in pursuit of "rigor." And that is a pity. They fail to grasp that they - and what they want - are less important than the student, and what he wants. After all, the whole exercise is for the student.

*Curiously, physics texts, in my limited experience, do not suffer from this problem. I had Halliday and Resnick (quill pen edition) and the Berkeley physics course, and both texts were superb, and began at the beginning. What a concept!

posted by Open Blogger at 07:45 PM

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