After my classes this morning, I had some students come in to ask a question from their Calculus class. They were not my students but their instructor was busy helping some other folks so they asked for my assistance. Here was their problem:
Find the volume of the described solid S:
The base of S is a circular disk with radius . Parallel cross sections perpendicular to the base are squares.
I began explaining where I’d start with this problem, draw a picture that helps us to understand the volume. I noted that if you look at the base you see a circle of radius whose equation is clearly, . I also pointed out that if we drew vertical slices across this circle you will see an line segment that represents the side of the square cross sections perpendicular to the base. If we could find the length of this side, we could find the area of the cross section (side-squared) and integrate this over all of these vertical line segments.
In an attempt to use a fairly general method for finding the length of the line segment, I suggested they take the “top curve minus the bottom curve”. It was at this pointed I noticed that we had attracted the attention of a colleague from a neighboring office.
I think at this point, I am intentionally avoiding a shortcut that gets me the length of this side. Apparently my colleague notices this. Continuing down the road of this method, we solved for , noting that the top half of the circle is and the bottom half is , subtracting the two we get that the length of the side is . Thus the area of the cross-section is . We can integrate this from to and find the volume of the solid.
Here’s the reason I bring this up: The comment made by the other professor was, “Man, that’s a long way to do that problem.” As far as I can tell there are a couple of shortcuts I missed (intentionally or unintentionally). Finding the length of the vertical segment does not have to be done by subtracting the top curve minus the bottom curve. My reason for doing this was because this is a procedure they have been using for finding areas between curves and they will use it again when they see multiple integrals. So the pedagogical question is whether I should take the easiest possible route to the solution when helping students with their homework, or do I take longer routes to help them see generalizations? Honestly, from my Analysis courses all the way down to my Intermediate Algebra course, I generally opt for the “long way” of doing things. It helps them see the patterns we are studying, the “why” of what we do. And yet, because of this, very rarely do my students get to see the way I actually solve a problem, i.e., taking the shortcuts.
From hearing my colleague when he works with students, it seems that he tends to stand back and first ask, “What is the best way to solve this problem?” Whereas, I dive in and ask, “What does our derived method tell us to do next?” Maybe I am stuck in the mode of “procedural” mathematics, even though I preach to my upper-level math majors that the further they go in their mathematics the more “creative” and “artistic” their approach to mathematics becomes. I don’t mean to imply that their aren’t still rules of logic that we must play by, but that, when stumped on a particular proof or other homework problems, we rarely have a set of “steps” to go back to and solve the problem. Such problems require a significant level critical thinking.
By the way, the second shortcut, that I should have mentioned but missed, is that instead of integrating from to , we can integrate from to and double the result.