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Comment of the Fortnight
19 June 1997

A couple of nights ago, I was listening to a local radio talk show host saying that he hadn´t learned anything useful in higher math courses, and that all his math courses past sixth grade were useless.

This talk show host said he hadn't needed any of the math in his career as a congressional aide, a teacher (and coach of a football team, I believe), or a talk show host. Here's the commentary that I sent him:

I think you have confused the vehicle with the destination. Let me explain what I mean:

When you were in grade school, you were taught the alphabet song. Why? I´m fairly sure you´ve never been asked to sing the alphabet song in a business meeting. So what was the use of learning the alphabet song? This is an obvious case -- the alphabet song is the vehicle to get you to the destination of "learning alphabetical order," a useful skill for using the dictionary or phone book, sorting lists, etc. It so happens that the vehicle was amusing in and of itself, and so you had no objection to it.

-o-
You´ve been a football coach. I´ve seen football practices where the coach has the players do sit-ups and push-ups and run through a set of tires laid out on the field. Why? I´ve watched plenty of football games, and I´ve never seen tires on the field nor a play involving the team stopping to do push-ups. So why does the coach insist that the players do those exercises?

Again, an obvious case -- the exercises are the vehicle to get you to the destination of "being in shape," a useful skill for winning football games. Although the vehicle isn´t very pleasant, the destination is something the players want, so they are willing to use the vehicle.

-o-
Now we come to geometry, where you learn to figure out if two angles are equal or not. Why? In all your career, you´ve never been asked to find the distance across a river or determine if triangles are similar. So why do we insist that you learn geometry? In this case, the vehicle is geometry, but the destination is not obvious.

Most students will either conclude that there is no destination (thus rendering the vehicle useless) or they will conclude that the vehicle is the destination, and it´s a useless destination. Because the vehicle isn't all that much fun, students naturally object violently to it. So, what is the destination?

When you do a geometric proof (or just about any higher math), you are learning to do these things:

  1. Assess a situation to determine what facts you know and what facts you do not know.
  2. Develop a plan for using the known facts to find out the things you don´t know.
  3. Follow the rules of logic when carrying out the plan.
Those skills are the destination, and geometry is a wonderful vehicle for getting there. A world of "pure" lines and angles lets you concentrate on planning and logic skills rather than having "real world" details and exceptions sidetrack you from those skills.
-o-
Now we are left with the question of whether those skills are useful or not. As a congressional aide, of course, those skills are marginally useful. [No Congressperson who wants to get re-elected would dare allow a factual assessment or a logical plan within 500 yards of the Capitol. <grin>]

As a talk show host, you seem to use those skills quite often. It is my contention that those skills are useful in the business world, or, in fact, any arena where there are problems to be solved.

-o-
The crime is not in asking you to learn geometry or algebra; the crimes are
  1. not telling you what the destination is
  2. pretending that everything in math has to be relevant
Relevant -- give me a break! Do you know why Isaac Newton developed the calculus? Was it to build better bridges? Was it to increase business efficiency? No. He had to develop the calculus to answer this question: Gee, there´s a practical, relevant question! (sarcasm intended). A lot of mathematical discoveries have been made by mathematicians just playing around with numbers and ideas to see if they could throw together a few simple facts to come up with some unexpected new result.

Insisting that math always be relevant is as ridiculous as insisting that students read "Moby Dick" so that they can be ready for careers in whaling. Instead, students read "Moby Dick" because it is a great adventure novel, fun to read just for itself, and, by the way, it happens to have a lot to say about the futility of obsession and vengeance.

I see nothing wrong with framing advanced mathematics as "let´s see what we can put together with these numbers and facts," and, by the way, it happens to improve your planning and logical skills.

In summary: geometry and advanced algebra, if learned purely for the mathematical techniques (the vehicles), are of little value. If you use the vehicles to get to their intended destination, they can be very valuable indeed.

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