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Topic: Re: Electronic Math
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Richard Sisley

Posts: 4,189
Registered: 12/6/04
Re: Electronic Math
Posted: Jul 21, 1998 9:53 AM
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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

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
anti-differentiation. 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 anti-derivatives 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 anti-derivative expression for
the integrand they chose, and then correctly compute the integral from
the anti-derivative. Again, there is not much overtly conceptual about

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, hand-held computers on the
market which could readily do any of the rigged anti-derivative 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

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.


Richard Sisley

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