Joshua Zucker and David Klein have exchanged the following thoughts about one of the sample multiple choice questions provided by the college board in a book in which the new syllabi are presented:
> Another new question, #19, in my opinion asks for lack of > understanding. They give a table of values of f(1.7), f(1.8), f(1.9), > f(2.0). They tell you f is differentiable on [0,3] and ask for the > best approximation for f'(1.7). The correct answer SHOULD be that we > have no idea, because we don't know whether f is oscillating rapidly > between the given points. We need more information about f! But of > course, the AP people just want us to plug in delta-f over delta-x > without thinking about these issues.
Another excellent point. I think we see here the negative influence of Harvard Calculus which has questions of this type. I agree with you that this question asks for lack of understanding.
> And not only that, but the wrong answers they give are foolish! They > do give the answer choice that would come from just taking delta-f > without dividing by delta-x. But they do NOT have the wrong answer > that would come from using f(2.0) - f(1.7) / .3 instead of > f(1.8) - f(1.7) / .1 ... that is, they don't test the one idea that >> they should be, which is that the best approximation to the derivative > comes from using the point that's CLOSEST. They do have the answer > that comes from using 1.9, though.
It's hard to say that f(2.0) - f(1.7) / .3 is necessarily wrong (as you point out above).
They are of course correct about the technical flaws of the problem. None the less, I have a great deal of enthusiasm for what I think is the underlying point of the problem, so I have given some thought as to how it might be amended.
As stated, the table of values implies that the derivative of the function is not monotonic in the interval from 1.7 to 2.0. Suppose this were changed so that the values were consistent with strict monotonicity of the derivative and the additional condition were given that the second derivative is defined and has no sign changes over the interval? Would this then make the computation of an approximation of the derivative value using the nearest point the best approximation?
But even if some esoteric example can be concocted which defeats this attempt to rid the problem of technical flaws, I am still sympathetic to the point of the problem. Here is my reason a tangent line to the graph of a function at a certain point, do we not make a sketch showing that the the graph of a function which is monotonic to the right of the point of tangency? Do we not do this in connection with a discussion of looking at a succession of approximations? If we use a standard asymmetric difference quotient, do we not imply with our sketches and words that the goal (a slope which defines a tangent line) is approached more and more closely as the denominator of the difference quotient approaches 0? How many of us then take the time to show that these implications need not be true in every case?
The derivative rules we expect the students to know are usually presented and justified by an analysis of a difference quotient. Whether or not students follow the details of the discussion, they know it is essential for doing well in the course to learn the bottom line and to practice its use. In fact, in my view, a good deal of mathematics instruction encourages students to wait for the bottom line because "that is where the money is."
To me the question under discussion is asking students to show some numerical and geometric understanding of some of the intermediate lines. This is why I am so sympathetic to this question and to reform calculus in general. I have been teaching calculus for more than 20 years and I recall having the experience of incidentally discovering that even students who were masters of bottom lines sometimes had little feeling for what the story was all about. This is in spite of the fact that I tried to say some clever and meaningful things to them on the way to the bottom line.
When computers (including hand held calculators) can do the bottom line work for us, we are free to concentrate more on what the story is all about. To me a bottom line calculus course is colored only in black while a reform calculus course, well done, is in full color.
I recently gave a talk in which I told about some preliminary steps I have taken recently with students to whom I intended to introduce the Chain Rule. I have them think about a sine wave position function and its associated velocity function. Then I ask the students questions inserted into several daily assignments to get them thinking about the effect, if any, on the model velocity function if the fundamental period of the position function is decreased. If the right questions are asked, students can come to the view that in such a case, it is reasonable to expect the amplitude of the velocity function to increase.
One person attending the talk wondered how I would have time to teach a complete calculus course if I took so much time just setting up a march to the bottom line of a derivation (or proof) of the Chain Rule. I think that a rush to the bottom line without any sense of why it might be reasonable is rather empty intellectually.
A reform calculus course certainly need not be less engaging intellectually than a traditional type calculus course. Quite the contrary. We who want to share a technicolor version of calculus with our students face a great challenge. They are very accustomed to bottom line mathematics courses. We want them to open their eyes and minds.