Tuesday, May 12, 2009

Foundation (Calculus)

Foundation
From Ideas Presented in The Calculus: An Opinion
(mathematicsteacher.org)


If I am faced with a Calculus I class on the first day of the semester, I have to decide what I’m going to base my presentation of calculus on and I have to decide what cohesive block of calculus I want the students to leave with.
It is important that the calculus is coherent so that the student has a solid foundation to build on, not the sand of unrelated techniques scattered here and there.
In what follows I present some of my ideas on teaching beginning calculus. Since my approach doesn’t get to the stuff engineering wants, or even the stuff that mathematicians want, I doubt that it will be tried in my lifetime.
None-the-less I present them. I think that a foundation that gives the student confidence in what she knows early on and then she could learn what engineering or mathematics or physics wanted her to know. It is my belief that without a solid understanding at the beginning, the rest of the calculus sequence is a waste of time.
What is this fascination that mathematics has with giving information to people who haven’t the faintest idea what you are talking about?
In an effort to get my first job I was giving talks at various universities and hoping my dissertation, which was the topic I chose, would dazzle my listeners. After one such talk a professor at the university I was visiting came up to me and said, “Your talk was great. I didn’t understand a word of it.”
I might say in passing that it is interesting to see what famous names did in their dissertations.

So, my goal is to provide the student with a coherent framework for mathematics. I want the student to feel comfortable with the ideas. More to the point, I want to erase their fear.
In one of the Castaneda books Carlos asks Don Juan how you become a man of knowledge. Don Juan replied that there are four enemies you must defeat. When you start to learn something, really learn something, it is totally different than you thought it was going to be. The familiar landmarks and toeholds are gone and you meet your first enemy, which is fear.
A student comes from high school thinking they have taken some mathematics, perhaps even some calculus. They think they know what mathematics is all about. They think that geometry is an answer on a multiple choice test and that algebra and calculus are formulas and rules to memorize.
And then they get me for Calculus I. It is nothing like they thought it was going be. If the student understood the ideas, my tests were trivial; if he tried to memorize his way to success, my tests were impossible. The students met their first enemy, fear.
Don Juan tells Carlos that you must take your first step into the face of this enemy if you want to eventually defeat it. Some students take this step, others drop the class.





Genesis

The first brick that I put in the foundation is ‘function’. The grist for the calculus mill is function; calculus is about functions; no functions, no calculus. Any understanding of calculus must be preceded by an understanding of ‘function’. And ’function’ must be understood well enough that the idea seems worth emblazoning on one’s very soul.
I say “worth” on purpose. If “functions” have no value to the student, then I can’t suggest a humane way to teach them. The first step is to convince the student that functions are worth understanding.
It could well be that this step is never achieved. I’m sure that many people live satisfying, productive lives without any knowledge of functions. But the students in front of me have voluntarily signed up for the course, usually to satisfy a degree requirement and their degree program must have some rationale for requiring calculus. This rationale provides a toe hold and if there is no rationale, I invent one.


I introduce functions by associating how far an object has fallen with the time it took to fall that far. I try to convince the student that it is reasonable to want to associate the numbers that describe one physical quantity with the numbers that describe another physical quantity. The problem is that I don’t know of any earthly reason why a person would want to make such an association. I can say why this process interests me and many of the people I know and hope that one of these reasons clicks with the student.
In any event, after having made the association of time and distance I slowly generalize the idea. I point out that Western Civilization is into single valued functions because the world we live in appears to our sense organs to be single valued. Dolphins don’t live in a single valued world and have only an academic interest in single valued functions; probability may arise more naturally in their environment.
If I watch a leaf fall from a tree, at every instant of time the center of gravity of the leaf is some distance from the ground. Thus I have a function that associates the numbers that give instants of time to the numbers that give the corresponding distances from the ground.
But while that function exists, its rule can at best be tabulated using a finite number of approximate distances at corresponding approximate times. Further, since every leaf falls differently, the act of tabulation is the only time that the function applies to anything.
Students get so used to seeing functions given by formulae that, if they aren’t careful, they can possibly come to think that formulae are the only functions there are. Through out the course I interject weird functions that can arise from experimentation.

I start out with functions defined by rules that arise from experimentation and the notation is all symbolic, either numerical tables or rules that use the symbols of mathematics, t, sint, cost and the like. (Beware the deadly symbol. I think that the confusion between a symbol and the concept, real or imagined, that it is supposed to represent is harmful.)
I spend a lot of time on the symbolic aspects of functions and I point out that functions whose rules can be expressed using the symbols of algebra and trigonometry, the standard symbols of mathematics, are relatively rare.

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