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Re: Electronic Math
Posted:
Jul 21, 1998 9:53 AM


When I suggested that Wayne Bishop might post a "conceptual calculus test" for a first college course in calculus of the type given fifteen years ago, he responded
"Good suggestion, Richard. I'll see what I can dig up. Please do the same."
If Wayne is asking me to post a "conceptual calculus test" from fifteen years ago which I might have given to my AP students, I do not think I could do that. I would not characterize the type of calculus tests I gave then as "conceptual." Rather, what I see myself as doing in those days was asking the students to show ready facility with the array of facts and special techniques which I tried to present to them carefully and systematically.
As an example, I might put a section on a test from those days on antidifferentiation. One problem would be rigged to invite a double use of parts method. Another might be rigged to invite a use of a partial fraction decomposition. Another might be rigged to invite the use of a trig substitution. Yet another might be rigged to yield to a student who could recognize an instance where a reverse use of the chain rule could be applied if a scalar adjustment were made. The students would then be asked to represent the family of antiderivatives in each case and show their work. I do not see much "conceptual" about this. It did take skill and those who did best were those who worked diligently and consistently during the time preceeding the exam and who were good at remembering clues and details of techniques.
As another example, I might describe a solid of revolution and ask the students to compute its volume. They would need to decide which of the three forms to use, correctly write an antiderivative expression for the integrand they chose, and then correctly compute the integral from the antiderivative. Again, there is not much overtly conceptual about this.
I could continue with such examples but anyone on this list who has taught calculus for two or three decades could do the same. By, the way, as we all know, there are now small, handheld computers on the market which could readily do any of the rigged antiderivative problems I described above.
As the my current testing, there is a change to questions which might be described as more conceptual. The best calculus test my students took this past year was the AB level AP exam. I was especially happy with the extended answer problem which probed the depth to which students understood that acceleration numbers were rates of change for velocities. I was also happy with the question which required the students to think carefully about setting up an integral expression to represent the accumulated cost of running an air conditioner when the temperature was above a certain level.
I see a big difference between what I used to test students on a decade or more ago and what I hope to test them on now. Now I want to focus on the idea of derivatives as rate of change functions and all that that implies, and on integrals as measures of accumulation and all that that implies.
I still look forward to seeing Wayne's example of a "conceptual calculus test" for a standard first course in calculus from fifteen years ago.
Sincerely,
Richard Sisley keckcalc@earthlink.net



