Teaching and the Philosophy of Mathematics

Teaching and the Philosophy of Mathematics


I will first assume that I am talking about students who are being introduced to mathematics (which doesn’t mean that they haven’t had previous courses that were listed in the catalog under mathematics) and want gain facility with mathematics as a tool.
I think that students who feel a calling for the Ideal world early in life should be aided in their studies in ways that are appropriate for them.
Since ‘continuity’ is an Ideal World concept, beginning calculus is a course in elementary-calculus-game-playing. Continuous functions are probably the most popular game pieces although their use in the game is often poorly taught.
But I am getting ahead of myself. There are those who don’t consider calculus to be in any “Ideal World”. They think of it as the stuff Reality is made of.
Now, I do not judge lest I be judged. While I don’t believe that f(x) = x defines a function in the Real World, I would defend to the very death a person’s right to have such a belief.
On the other hand I would judge a teacher whose students couldn’t model a simple physical system using calculus.
I don’t have a problem with a teacher that bases the use of the calculus tool, or even as far back as the arithmetic tool, on magic, mystic runes and spells. Well, yes, I do have a problem with that. Indeed, that’s the way mathematics is often taught; consider the magic word, “FOIL” and the mystic incantation, “Invert and Multiply”. (This is a little known passage in the Old Testament that has been much over shadowed by the more familiar, “Be Fruitful and Multiply”. “Being Fruitful” was a not unpleasant activity and quickly caught on; people have been following that exhortation with great energy ever since. “Inversion” was a short-lived fad and faded quickly from the scene.)


Everybody seems to have a different idea about how to teach mathematics, indeed, a different idea about what the tool even looks like or what it’s used for.
My stab at teaching beginning calculus has a link on this site. I think I would make modifications if I were to write it again. In 37 years of teaching calculus, I don’t think I ever taught it the same way twice.
My general idea, to which I remain faithful, was to motivate the ideas behind the making of a mathematics-game by looking at a specific example; in the book I study the motion of a falling rock.
The method in my madness derives from watching people work on automotive engines. For example, if you can work on one motorcycle engine, you can work on them all. I heard this some years ago when motorcycle engines, even car engines, could be worked on in your garage.
But the mathematics in Calculus I was known by Newton. Calculus is not a new tool on the cutting edge, so to speak. Teaching Calculus is analogous to teaching backyard mechanics to repair lawnmower engines.
Calculus is an Ideal World game but in many cases models the Real World very closely. In fact the calculus-game was designed with modeling the Real World in mind. You play the game and see where an Ideal cannonball lands in the Ideal World. Then you shoot a real cannonball and see how good a prediction the game made. Well, the calculus-game makes really good predictions; so good that classes in major universities are given in playing calculus. There are even advanced courses in the calculus-game.
A course in calculus is a course on “The Construction of Simple Ideal World Games for Fun and Profit”.
My approach in the calculus book was to construct a game in the Ideal World that I could use to predict the motion of the rock when it was dropped. Proofs were aimed at making the results seem reasonable, both as to why they were true in the Ideal World, using the close relationship between the Real and Ideal World physics, and why anybody really cared.
In the book I stress the difference between the Real World and the Ideal World. I do this because of a personal opinion that it is important to distinguish between what is Real, what is Earnest, and what is sublime, what is a game. I think that this distinction should be made generally and not only in mathematics; and I believe in “teach the whole child”.
So, what I teach is affected by how I look at the world generally and I would suppose that this is the case with teachers generally. I don’t see how it could be otherwise.
I couldn’t keep a personal philosophy out of the calculus courses I taught, I can hardly expect others to rarify their courses.

The whole point of calculus instruction is to, at the very least, turn out good mechanics. Metal Shop 101 doesn’t start with metallurgy; it starts with making a chisel. Starting calculus with epsilons and deltas for the general run of people, like myself as a student, is so unrealistic as to be stupid. I can remember going to an office hour in an effort to get some inkling as to what this ε, δ stuff was all about. I left the office of my teacher no wiser than when I entered. I couldn’t see the problem that ε and δ were supposed to solve. I had no doubt that the function, f, given by f(x) = sin x was continuous, any damn fool could see that by looking at the graph. Archimedes found the area under a parabola without the help of ε and δ.
During a time of mental illness I tried to teach myself the C-Language using the pamphlet written by the guy who invented it. After many frustrating hours I saw why I was having trouble. I didn’t know what problem the inventor was trying to solve. ‘Pointers’ seem to come out of nowhere but I supposed that there were people who saw ‘pointers’ as a clever solution to a problem that other programming languages had.
I was in the same quandary with ε and δ. The people who developed the calculus I was learning didn’t use them, so they clearly weren’t necessary.
When I started playing advanced Ideal World games, it became obvious why you needed them to get to the next level. I had expected that insight into ε and δ would be an epiphany but instead it was obvious and not all that exciting. “Was it for this I screamed and cried and kicked the stairs?”
It was so obvious that I couldn’t understand why I hadn’t seen it before. Now I do understand. It was because ε and δ had no place in the Calculus I game.
I had the same problem with the exercises at the beginning of Lara Croft: Tomb Raider.

Newton was a natural scientist, not a pure mathematician. Newton constructed his game to be as close to the Real World as he could make it. Calculus instruction should start in the Real World and with Real World intuition. If the first-game in the series, which is Real World based, is mastered, then the advanced versions of the game are easier to learn. Some of the advanced games, while enjoyable to play, give little or no insight into the Real World nor does the Real World help much in playing the game.
(I have found that mechanics are quite interested in the 2nd law of thermodynamics and pressure/volume graphs showing how the various internal combustion engines work. I conclude from this that mechanics had some interest in advanced game theory.)

If a course talks about the general structure of the games of mathematics and then studies one of the games in detail, it doesn’t really make a difference which one, then the student is being taught how to learn any mathematics-game with Real undertones.
It seems strange that students, for example engineering students, who will spend much of their lives learning some new mathematical technique, are not taught how to “learn” mathematics on their own.
So often I hear, “I want to learn about something, I think I’ll take a class.” It doesn’t seem to occur to them that it is possible to learn things on their own. The musicians that I have talked to started out by trying to play a tune on a piano or a guitar, whatever. I read “Audel’s Handbook of Auto Mechanics” but I learned about engines from taking one apart and putting it back together.
When Pete, a friend of mine was twelve, his brother got him an old engine from a junk yard and put it in the front yard. Pete spent the summer taking it apart and putting it back together again, over and over; kind of like Zen archery.
The same thing occurred the next summer at the end of which an Oldsmobile engine could have been seen sitting on the grass in Pete’s front yard and running cheerily.
I think these examples have relevance to learning mathematics, actually, to the learning of anything. Well, maybe not everything, but I can’t think of a counter-example at the moment. You teach how to learn. What’s so hard to understand about that concept?

So what is my opinion about the appropriate place of Philosophy in the teaching of mathematics game-play? (I don’t think cheat codes, like FOIL, are useful. I think that authors who put shaded formulae in their game-play-books should be subject to criminal prosecution.)
I don’t think that philosophical biases should displace the study of good, solid game-play. But if someone converts a student to a particular ism as well as facilitating the student to become a competent game-player, so be it. I suppose that an excellent teacher of game-play could be as crazy as a March Hare and that some of his students might join the tea party; intellectual growth does have its risks.
As long as good gamesmanship is taught, I would not limit the free speech of the teacher. Personally, I try to put forward the idea that mathematics games are good examples of using critical thinking and reason. Oh yeah, and that there is a difference between the Real World, where things exist, and the Ideal World where they don’t.