January 19, 2010, 9:22 pm
This is probably the last of three articles on how piecewise-linear functions could be used as a helpful on-ramp to the big ideas in calculus. In the first article, we saw how it’s possible to develop some of the main conceptual ideas of the derivative, without much of the technical notation or jargon, by using piecewise-linear functions. In the second article, we saw how to use the piecewise-linear approach to develop an alternative limit-based definition of the derivative of a function at a point. To wrap things up, in this article I’ll discuss how this same sort of approach can help in students’ first contact with integration, again by way of a hypothetical classroom exercise.
When we took this approach with derivatives, we used the travels of three college students from their dorm rooms to the cafeteria. Each student had a different graph showing his position as a (piecewise-linear)…
January 12, 2010, 10:45 am
This is the second post (here’s the first one) about an approach to introducing the derivative to calculus students that is counter to what I’ve seen in textbooks and other traditional treatments of the subject. As I wrote in the first post, in the typical first contact with the derivative, students are given a smooth curve and asked to find the slope of a tangent line to this curve at a point. But I argued that it would be more helpful to students’ understanding of the derivative to start with a simpler case first, namely to use only piecewise-linear functions at the beginning. This way, as we saw, we can develop some important core ideas about the derivative without resorting to anything more than pictures and an occasional slope calculation.
But now, we need to deal with the main problem: What happens if we do have a smooth curve, not a straight line or…
January 11, 2010, 7:30 am
Last semester I stumbled upon an approach for teaching the concept of the derivative, and later the integral, that worked surprisingly well with my students. It stems from a realization I had that much of what students see when they first learn about derivatives has very little to do with understanding what a derivative is. The typical approach to introducing the derivative throws students directly into the trickiest possible case: a smooth nonlinear curve, and we want to calculate the slope of a tangent line to this curve at a point. To do this, we have to bring in a lot of “stuff”: average rates of change, tables of sequences of average rates of change, and in a vague and non-rigorous sort of way the notion of a limit. It’s this “stuff” that confuses students — not because it’s hard, but because maybe it’s not suited for their first contact with the idea of the derivative. Maybe we…
May 12, 2009, 1:02 pm
I’ve been teaching calculus since 1993, when I first stepped into a Calculus for Engineers classroom at Vanderbilt as a second-year graduate student. It hardly seems possible that this was 16 years ago. I can’t say whether calculus itself has changed that much in that span of time, but it’s definitely the case that my own understanding of how calculus is used by professionals in the real world has developed, from having absolutely no idea how it’s used to learning from contacts and former students doing quantitative work in business amd government; and as a result, the way I conceive of teaching calculus, and the ways I implement my conceptions, have changed.
When I was first teaching calculus, at a rate of roughly three sections a year as a graduate student and then 3-4 sections a year as a newbie professor:
- I thought that competency in calculus consisted in…
February 26, 2009, 12:12 pm
In the Stewart calculus text, which we use here, the first chapter is essentially a precalculus review. The second chapter opens up with a treatment of tangent lines and velocities, with the idea of secant line slopes converging to tangent line slopes and average velocities converging to instantaneous velocities taking center stage.
Calculating average velocity is just a matter of identifying two time values and two position values and then performing two subtractions and a division. It is not complicated. Doing this several times for shorter and shorter time periods is also not complicated, and then using the results to guess the instantaneous velocity is a little complicated but not that bad once you understand the (essentially qualitative, not quantitative) idea behind shrinking the length of the interval to get an instantaneous value out of a sequence of…
May 5, 2008, 1:48 pm
Asking the following question may make me less of a mathematician in some people’s eyes, and I’m fine with that, but: How do you explain the meaning of the differential dx inside an integral? And more importantly, how do you treat the dx in an integral so that, when you get to u-substitutions, all the substituting with du and dx and so on means more than just a mindless crunching of symbols?
Here’s how Stewart’s Calculus does it:
- In the section introducing the definite integral and its notation, it says: “The symbol dx has no official meaning by itself; \(\int_a^b f(x) \, dx\) is all one symbol.” (What kind of statement is that? If dx has “no official meaning”, then why is it there at all?)
- In the section on Indefinite Integrals and the Net Change Theorem, there is a note — almost an afterthought — on units at the very end, where there is an implied connection between…