What to teach? What to teach?
Every time I think about how to teach undergraduate mathematics, say, algebra through calculus, I can think of ways to teach the standard stuff but I always get hung up on the question: Why am I teaching this shit?
The first topic that should be taught in school is “learning technique”. If a student knows how to learn, the life of the teacher is much more interesting. There is a big difference between talking to someone who knows how to learn and someone who doesn’t. It’s more interesting for the student too.
So the topic that should begin in pre-school is learning how to learn. It doesn’t make any difference what they start learning because the point of the exercise is the act, the process, of learning. The child should start learning something that that isn’t unpleasant for them.
When the child is young they will probably want to learn something that most adults know how to do and are quite able to show the child how to learn it. The first things I learned were taught to me by adults and their teaching technique was quite satisfactory.
I went to Junior High in Cheyenne, Wyoming and we lived on Warren Air Force Base. As soon as we moved in my dad gave a plane geometry book, an algebra book and a college algebra book; I was instructed to learn them.
This was OK with me. The winters in Cheyenne were long and cold and learning mathematics while listening to radio plays was as good a way to spend winter evenings as any. Well, there were also the Friday night fights to go along with factoring polynomials.
But the mathematics that I learned wasn’t as important as the fact that I learned how to learn.
From that point on, school was no problem. I didn’t necessarily want to learn everything but if I wanted to, I did.
People who know how to learn do it at different rates. Actually people who don’t know how to learn do whatever it is they do at different rates.
As I think back on my teaching career, the time restraint always bothered me but I didn’t stop to examine time more closely.
The time that it takes to learn something is a statistical distribution. There are some students that can learn the next thing in an instant, there are others that take a long time before they are ready to learn something else.
Because most of the students don’t learn the mathematics, we have tests to evaluate”¦ It is not really clear what is being evaluated. My class grades gave a bi-modal distribution, those that learned how to take tests and those that didn’t. Learning the mathematics was a sufficient but not a necessary condition to pass tests.
The more I think about giving a 50 minute test every 2 weeks the more bizarre it seems.
In my upper division courses the tests were all the same: “Write down what you know”.
If the class wasn’t too large, about 15, plus or minus, I would give an oral final. They would have to give an hour talk on, say, the proof of the Heine-Borel Theorem without notes. They would get as many tries as they wanted until the day before grades had to be in.
But in larger calculus classes there isn’t enough time.
I didn’t give numerical grades, just letter grades. I couldn’t tell the difference between a 71 and a 72. It’s pointless to make such distinctions. A teacher should be able to tell the difference between and A paper and a B paper.
It’s not that there weren’t students who could do well on tests without learning the material. A student who does well on tests but doesn’t know any mathematics, has learned a skill. The student has been given a problem to solve, get through school with good grades. The problem is not “to learn mathematics”.
I think there is a better appreciation of what learning how to make a guitar or to use a lathe (no, not the computer lathes), how to turn a cartwheel. There is an understanding that something more than memorizing instructions is requires. Learning how to play a guitar is more than putting your fingers on the right frets and hitting the right strings. I have heard a man play the piano and hit all the right keys at the right time and it wasn’t music. He hadn’t learned how to play the piano.
Perhaps it is music they haven’t learned. Memorizing which notes to play doesn’t make music and memorizing how to work selected max-min problems doesn’t make mathematics.
People say that music and mathematics are related and then start talking about octaves and fifths and Pythagoras and group theory. They miss the point.
You learn how to groove on both.
I think one problem with beginning mathematics is the teachers. I think most teachers like to read and their students see this. I think that a lot of elementary school teachers do not like mathematics and that shows.
I knew a grammar school teacher who told her students that she didn’t like mathematics either but that it is something you have to learn. And she really felt that they should. As distasteful as it might be, she thought that balancing a checkbook is important.
The conversation took place some years ago.
It is my opinion that once you know how to learn, you can learn anything. Well, anything within reason.
One of the problems is the amount of material that is shoe -horned into beginning courses. In the Calculus I that I took we spent three weeks on conic sections. In the last Calculus I syllabus I taught it was about three days. What I learned about conic sections has stayed with me for over fifty years. That’s what learning something does for you.
I liken the Calculus syllabus to driving down the freeway at night at 90 mph with your dims on. You read the green signs but you miss the off-ramp. You touch on everything and don’t get to really teach anything”; no time.
The “include everything” mentality leads to books that are too big, too heavy, too poorly written.
But if the student knows how to learn, all these problems dissolve in the mist. You teach the important basics, like what a function is and what its graph is, the pros and cons of continuity, what the derivative actually is and what an integral actually is.
A student who knows how to learn can learn the technique of differentiation in a few days. I had a friend who claimed that he could teach a parrot to take derivatives. After all, there are five basic functions and five ways to combine them; you have to know the derivatives of five functions and how the derivative deals with the five ways to combine them. End of story.
Doesn’t anyone ever wonder why more than a day is spent on the derivative of a product? Could it be because the wrong things are taught and that memorization is called learning?
We spend all this time thinking about ways to teach stuff that the students should be capable of learning on their own. The fact that they are not capable is the fault of education, not the student. If I were a conspiracy theorist, I might see a conspiracy to keep knowledge from the populace.
One might have thought that as the country progressed from being a manual labor economy to a more mechanized economy, the citizens would be educated to keep up. But there is a problem with people coming to this country, legally or illegally, and are taking manual labor jobs. Why haven’t we left manual labor jobs behind for people in countries on their way up?