I came across this from an interesting blog on teaching Calculus:
There are people who have taken Calculus one or more times and still don’t have a feel for what it is they know. They can solve specific problems in context. They have learned which word problems are like which of the completely solved examples in their text. They know how to curve sketch because the rule says that if in this region the curve has a positive first derivative and a negative second derivative then the curve must look like this.
This reminds me of the time when I was doing Calculus in secondary school. For some reason, I still remember my first Calculus lesson where the teacher explained how differentiation works, and I distinctly remeber not understanding what he meant when he talked about “the limit to zero” this and “the limit to zero” that. Everyone else was equally lost, but no one was about to ask for a clearer explanation — the teacher’s body language made it clear that he was just going through the motions of covering the introduction, and it that wasn’t anything particularly important.
And we were right. Although the class never really understood how Calculus works, it made no apparent difference. We knew how to apply differentiation and integration to standard problems, we knew how to mechanically manipulate the numbers to arrive at the final answer, and that was good enough. In fact, most of the class managed to get top grades for that subject the following year in our GCE ‘O’ level exams. All this while without a feel for Calculus.
A year or so after that, I unearthed from a bookshelf dad’s ancient dust-coated Feynman Lectures on Physics. I liked Physics, so I flipped open one of the books in the series (if the title was Lectures on Calculus, things would have been quite different). The page I looked at was an explanation on how Differentiation works, using a distance-time graph of a car to find out its average and eventually instantaneous velocity at certain times, and how the method of Differentiation could be derived from there (are you still with me?).
I was amazed. It was less than two pages of explanation, but things finally clicked. Now, “limit to zero” made perfect sense, and I could see the elegance of alculus. I feel Calculus.
Unfortunately, countless students have gone through Calculus (and life) never having a feel for it. It’s not just Calculus, but most of Mathematics.
Math, for too many, is not much more than a mechanical manipulation of figures — learning and doing what the computer can do much faster and infinitely more accurately. Is there a problem with math education if our students go through their school careers with such an idea, and having no feel for all but the simplest mathematical operations? Most certainly.
I wonder what could or should be done to improve the situation.