Tuesday, May 12, 2009

Why Teach Mathematics Revisited

Why is Mathematics Taught Revisited

Why is mathematics taught? I ask myself this question from time to time but have come up with no universal answer.
Since mathematics is the language of science, the scientist must know mathematics but it doesn’t seem universally agreed on how much and what kind. Statistics seems a necessity for a variety of disciplines from business to biology and evolutionary genetics.

One answer is that mathematics teaches students to think rationally and critically.
I have espoused this raison d’etre to my students who have asked me why they are having mathematics inflicted upon them. I pointed out that the difference between people and the two dogs that lived next door to me, whose total activity was eating, copulating and sleeping, was that people thought rationally and critically.
A young man in the class remarked that if the dogs had a fast car their lives would be perfect.
He didn’t seem to put a high value on the human capacity for original thinking.

When I was in junior high school the cold war had yet to become an obsession and Strom Thurmond was just starting the Dixiecrats, but even the school hoodlums discussed these things. They went to the library regularly and read books about Black Hawk and Red Grange; they read about egocentric sports stars who learned about team work.
Cheyenne, Wyoming was not the intellectual center of the country in the late 1940s, it was a cowboy-railroad-military town, but I never heard a classmate say they hated mathematics or any of their courses as far as that goes.
I haven’t been back to Cheyenne since that time and I don’t know what it’s like now. I do know that the junior high, which was new when I started school there, has been torn down, sic transit gloria mundi, an expression I use a lot these days.
But, quoting myself, “Of all sad words of tongue or pen, the saddest these, I wish it was the way it used to be.”
But it isn’t the way it used to be and education in general has to be rethought.

I think students in the early grades can see that basic arithmetic is useful. The use of fractions is less obvious as is the technique of computation, but, the student is told, “Just do it and you will see why later.”
And then comes algebra. While the student could at least see how fractions could be used in dividing up a pizza, the uses of algebra were truly obscure. But, the student is told, “Just do it and you will see why later.”
For many students this is not true. It will be true for those who become scientists but I don’t think it is true for those who don’t. I don’t think most people, even scientists, compute how long it takes Ed and Bill to paint a house together.
I have thought about it and I can’t think of a time when I have used algebra outside of my profession in mathematics. Well, one time when in college I did a mixture problem making gallons of Manhattans for a party.
Even as a mathematician I have never done one of those long cancellation problems with fractional exponents.
I don’t think that the teaching of mathematics has really changed much since…forever. Probably the teaching of history hasn’t changed all that much either.

Maybe mathematics could be taught as topics arise instead of in some pseudo linear way. Why not introduce calculus before algebra and then calculus would supply a reason for considering algebra? Why not try to describe a damped-spring-mass system which would give a reason for considering calculus?
Why not try a different way of developing mathematics?

I will often ask a high school graduate if they took any mathematics and they usually admit that they have taken algebra. In New Mexico it seems that some algebra is required for graduation.
And then I ask them to tell me something they learned in algebra. The quadratic formula is the popular answer to my question but when I ask what the quadratic formula is, the fact that there is a square root in it is all they really remember. They don’t remember what the quadratic formula is used for.
Why is it that a student can take Algebra 1, Algebra 2 and Pre-Calculus and remember almost none of it? Why is it that mathematical amnesia doesn’t seem to bother anybody? Why isn’t this talked about?
Why don’t mathematics teachers discuss the fact that they spend hours and hours teaching kids things that go in one ear and, at the end of the semester, go out the other?
Why is it that if I ask a person who has passed a course in differential equations what a differential equation is, I receive a blank stare. (The derivative and differential equations describe physical systems. The integral computes.)

Why is it that nine out of ten people I ask what they think about mathematics say they hate it and aren’t any good at it. They often blame a teacher, most often their 7th grade teacher. Why isn’t this general dislike of mathematics talked about?
I recall that when a new book was chosen for algebra or calculus the procedure involved going through a lot of books but with no discussion about what should be taught. In retrospect I suppose this was a topic we would rather not talk about.
Why don’t mathematics teachers discuss the fact that they spend hours and hours teaching kids things that go in one ear and at the end of the semester goes out the other?
Why is it that if I ask a person who has passed a course in differential equations what a differential equation is, I receive a blank stare. (The derivative and differential equations describe physical systems. The integral computes.)

Why is it that nine out of ten people I ask what they think about mathematics say they hate it and aren’t any good at it. They often blame a teacher, most often their 7th grade teacher. Why isn’t this general dislike of mathematics talked about?

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