Why mathematics?

When I was in grammar school, 1941 to 1947, I was told that I should learn arithmetic so that I could make change and balance my check book. I already knew how to make change and I couldn't balance my check book until the bank did it for me and I read it on line. Fractions and long division were fun and I can still do them but the hand held calculator, which I don't use as a matter of principle, have made my knowledge obsolete.

In Jr. High School I learned about logarithms so that I would understand the slide rule and compute with trig functions. Alas, students of today have never heard of a slide rule.

In High School I learned how to take square roots by hand. I learned about Des Cartes rule of signs and arcane methods of factoring polynomials. I don't think they cover these topics in high school anymore.

I do however remember these topics and techniques. Well, I don't remember how to take square roots with pencil and paper.

Recently I asked a student who had graduated from high school a couple years ago with A's in algebra what he had learned. He couldn't think of anything he had learned. I asked another student what he had learned in algebra and he said he knew the quadratic formula. I asked him to tell it to me. He finally said that was a square root in it.

Both of these young men said they had a good teacher. What does it mean to be a good teacher if your students don't carry anything away from your course? What good does it do to pass tests and get a good grade and then have what you learned evaporate?

What do all the tests that students take today mean?

I helped a friend grade tests from a high school geometry class. The questions were by and large multiple choice and fill in the blanks. They had to know volume and area formulae, the volume of the frustrum of a pyramid comes to mind. Those students that remembered the formulae couldn't do the arithmetic. As I marked problem after problem incorrect I wondered if the few students who correctly found the volume of the frustrum would know any more after the final exam than the students who missed the problem. There were no proofs required although there was a two column proof on the test with one of the reasons missing; fill in the blank.

When I took geometry in 1950 I was told the study of geometry would help my ability to think clearly and precisely. I was told that arguments should be based on axioms and the rules of logic. I was told that the terms used in discourse should have clear definitions. I was told that these ideas could be used in any area of human endeavor.

Now, this made sense to me. It was not a particular proof that was important, it was what constituted a proof, what did it mean to prove something. It was not the conclusion but the means by which the conclusion was reached that was important because it was the means that gave the conclusion validity.

It is true that my sample size is small but in my small sample logical thinking has been replaced by memorizing and multiple choice tests.

What is the point in teaching things that are forgotten as soon as they are learned...if they are learned at all? And if they are remembered, seldom if ever used?

What is the point of covering material so quickly that by the end of the course the students are lost? In college calculus courses the rule seems to be a section a day. At the conclusion of these courses the passing score is set so that half of the class passes.

My viewpoint:

Teach fewer topics and cover them deeply so that the students understand concepts and how to approach learning mathematical concepts. It is my opinion that learning high school algebra is not particularly different than learning abstract algebra; or learning any mathematics. How to learn mathematics is the skill that should be taught and it doesn't really matter which mathematics course is used. If the student knows what the point of mathematics is and how to learn it, the student doesn't need a teacher they just need a book.

Of course, teachers have to know how to learn mathematics.