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At 13:17 -0400 6/13/06, Robert Cohen wrote:
I am highly in favor of introducing calculus ideas without the
I am also intrigued by your statement that the area-under-a-curve method
"[brings] in calculus ideas without the formalism." Doesn't the
average-force method also bring in the same calculus ideas?
formalism. In fact, I never mention the word "calculus," or related
terms, when I am showing the students how to calculate the area under
a non-linear curve, or how to use a spreadsheet to, in effect, solve
a differential equation. But they get to see a lot of calculus
without realizing it, and since it isn't called "calculus," they
don't get freaked out by the fact that they are supposed to be in a
"non-calculus" course. Later, when they study calculus, and see the
ideas they saw before introduced in an explicitly calculus context,
they are ready to accept the development without the panic that would
ensue if they realized that I had been "sneaking" calculus in earlier.
But I would argue that using the average doesn't bring in calculus
ideas to the extent that using the area under the curve does,
especially for a non-linear curve. Using the area to find the average
for a non-linear curve enables the students to see that, for a rising
curve the later values have more weight than earlier ones, and for a
descending curve the converse is true. For an arbitrary curve, they
will see that a large value of the ordinate will count more that a
small value, over the same range of abscissa.
Once they understand that, it isn't much of a jump to the realization
that when you draw a horizontal line at the value of the average, the
area between the average line and the part of the actual curve
*below* the average must equal the area between the average line and
the part of the curve *above* the average, if the average is chosen
correctly. That is easily seen with a straight line graph, so that is
a good place to start.
The point is that both ways of visualizing the average should be taught.