Wednesday, December 30, 2009

Math Abuse

For it seems to be precisely that science known by the barbarous name Algebra, if we could only extricate it from that vast array of numbers and inexplicable figures by which it is overwhelmed, so that it might display the clearness and simplicity which, we imagine, ought to exist in genuine mathematics. - Rene Descartes

A mathematician who is not also a poet will never be a complete mathematician. - Karl Weierstrass

I have lost count of the number of times asked, a variant of "why care about math?"  It seems few presentations provide good answers.

Most I have seen artless collections of exercises, possibly adequately prepared, sometimes with interjections or side boxes about the application or history of the material under discussion.  It's math neutered and overcomplexified by dense, obsessive prose

Consequently, those interested in it tend to favor gym class for the mind, unless maybe they got lucky enough to have a good teacher, are innately talented enough to see beyond the presentation, or are future scientists learning tools of their trade.

And masochists, who will almost invariably become sadists (perhaps after securing a teaching position.)

That leaves out a whole lot of people -- a bad idea when math is compulsory.  But what if math was presented widely, as a complex endeavor with thousands of years of history, diverse application, and open future?  Would it frighten children to learn that math models reality?  Would it pervert them to learn that math is the study of abstract relationships, a growing edifice built by the application of intuition to a standard of proof?

Maybe this is the first you have heard, and you don't believe me.  Try this, or this.  Both books are literary presentations of mathematics, recognizing that math is art, philosophy, and evolving.  The only thing missing are exercises, but getting through either book provides ample exercise.

Consider Calculus, a word which seems to summon the same uncomfortable awe as "jazz" or "murderer."  It is the art of things arbitrarily small, invoking infinity as a tool, and the mathematics this constructed/discovered  them vast, consistent, and pretty.  As a  means to model change, Calculus has been a cornerstone of physics for centuries, providing scripts for theories of motion, electricity, magnetism, light and heat. 

It was primarily invented, strangely, independently by two people in different countries at the same time.  One was Isaac Newton, who was creating solutions to physics problems.  The other was Gottfried Leibniz, who came at it more philosophically.  Neither was deliberately creating headaches for hapless youngsters.

Yet Calculus students will  take hundreds limits, derivatives, and/or integrals in a typical semester.  Wouldn't it be better to spend more time considering what the objects and operations actually are?   And how their existence depends on a dialogue between deduction, inspiration, and efforts to solve specific physical problems?

I knew a professor who taught Calc 1 with minimal exercises, assigning mostly proofs for homework.  He spent most of each lecture explaining the objects under consideration, and lucidly proving results.  Homework was entwined tightly with class material.  His students understood the subject on a level so far beyond the usual.

It was much more than mechanics, so students got a greater return on their investment.

Current standards create, at best, people who use math but don't understand it, and therefore don't necessarily use it well.  At worst, entire classes of students are alienated.  A solution to this problem is to mix exercises, intelligently, between proof and good writing. 

For example, consider Why Math?, an elective for anyone who has mastered current societally-necessary arithmetic.  This class presents the history of mathematics, it's development and use. Students walk away with some understanding of proof, major fields, looming concepts and problems, and how math hooks up with the natural world and psyche.

The core text is this, and Pi is recommended viewing.  Exercises are chosen to reinforce understanding of material from all presentations, not as abstract practice with no context.

New technology is not necessary for this course, and may be counterproductive.  The point being:  if you continue to spend time on nothing but the surface, you will still create drones.

PS.  Consider 2. That's the symbol for the number "two."  But where is the number?  What is the number?

Brought to you by the State Vector.

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