On Mon, 27 May 2013 19:54:08 -0600, Wayne Bishop <email@example.com> wrote:
> If they really know the formulas, they would start by using the given > information to compute the volumes based on the evidence presented.
How very interesting.
Wayne prates incessantly about the necessity of avoiding math avoidance by teaching kids the power of algebra. And then he suggests the weakest possible mathematical strategy for arriving at a correct solution to this problem. And it's a solution that uses algebra in name only, substituting numbers into a memorized formula in order to compare numerical results. But let's give him some credit: This is consistent with his rote approach to word problems ("word problems by type"). You get a few points for consistency, Wayne, but none for mathematics---which looks at the *context* of a problem, and not just its answer.
If Wayne's approaches aren't algebra avoidance, I don't know what is. And Robert acquiesces, suggesting that "a student of algebra" would never avoid algebra---leaving us to guess that he agrees that Wayne's strategy is Real Algebra.
(In fact, students of algebra generally avoid algebra as much as they can: Learning new ways of thinking involves work that they'd rather avoid.)
Here's a *real* algebraic approach: The volume of a cylinder is Pi r^2 h, where r is the radius and h is the height. Let's begin with a sheet of paper of length L and width W = k L, where k is some positive real number that's at most one. If we roll the paper up along an axis parallel to the L side of the paper, the radius of the resulting cylinder is k L/(4 Pi), so the volume we've formed is
V_L = Pi [k L/(4 Pi)]^2 L = k^2 L^3/(16 Pi).
If, on the other hand, we roll the paper up along an axis parallel to the W side of the paper, the radius is L/(4 Pi), and the volume of the second cylinder is
V_W = Pi [L/(4 Pi)]^2 k L = k L^3/(16 Pi).
So V_L = k V_W.
Now we've answered the question---but we've done much more than that. We've explained the answer in a way that applies in more general circumstances. Moreover, we're building an understanding of the relationship between measurements of length, area, and volume. Substituting the given numbers into the cylinder's volume formula accomplishes none of these things.
What Wayne's sermons and Robert's acceptance of them give us is something less than a half-measure.
But the two of them are right in one respect. Without what I've just done here, the activity isn't complete. That doesn't mean that it's useless, though. Understanding of a phenomenon is based on repeated examination of the phenomenon in a variety of circumstances, of which this activity provides one. It should be part of a progression that leads students to ask "Why?" And that leads, ultimately, to the analysis I've given above.
- --Lou Talman Department of Mathematical & Computer Sciences Metropolitan State University of Denver