Meaning or Strength

When teaching first year calculus on a University level, the teacher faces the same old problem: How do we combine mathematical strength with the intuition the student brings from his school days?

E.g., the students have a very good working knowledge about  the trigonometric functions, or the power operator, when they come. Often, they completely loose this knowledge after first year calculus. At least, they start to get confused about the seemingly contradiction between their intuition and the stuff they are taught now.

Teaching all the concepts of calculus in the limited available time is not easy. So we often choose the seemingly fastest road, which is the axiomatic, set theoretic approach, forgetting about any geometrical or other previous knowledge. I have heard professors say in the first hour of the first class that the students should now forget all they know. Do we need a justification for our definitions, if we can derive all relevant properties from them? So we teach without respect for the history of our subject, or for any geometrical intuition.

The more I do this, the more I feel that it is insufficient. First of all, we throw away too much, which was built up during years of school. Secondly, we fail since the students will continue to use their previous skills nevertheless, because they do not appreciate the new methods, or our strength does not help them to solve problems. In fact, I feel that good mathematics can only be achieved with a good intuitive background. Then, I think we should present mathematics as a useful and beautiful extension of the previous knowledge, not as a replacement.

So we need to take the time to relate the axioms to the real life.

For an example I take the trigonometric functions. The geometrical definition is on the unit circle, using the two coordinates (cos x, sin x) of a point on the circle at a given arc length x away from (1,0). Everything can be derived from this. After all, the Greeks already knew about the trigonometric identities! However, this would take too long time and also miss the necessary strength. So we introduce the sin and cos functions by their power series, or by the exponential series. However, it is our duty, a must, a responsibility, to show to the students, that this definition is in fact the same as the geometric definition. It can easily be done by studying the path (cos t, sin t) and its curve length.

To point this more sharply, I would say that first year calculus should be taught to teach the student how his knowledge can be supported with mathematical strength. He should learn that the new, strictly analytical, logical way of thinking actually provides information beyond the previous knowledge. He should feel like understanding better what he already knew. Then he will not feel like thrown into a new world with no relation to his previous thinking.

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