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Question #19
Posted:
May 24, 1997 11:01 AM
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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.
Richard Sisley
keckcalc@earthlink.net
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