Re: [Phys-l] Work and Energy: which first?

[Hugh Haskell <hhaskell@mindspring.com>]

At 13:17  -0400 6/13/06, Robert Cohen wrote:
> 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?
I am highly in favor of introducing calculus ideas without the 
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.

Hugh
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Hugh Haskell
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<mailto:hhaskell@mindspring.com>

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