Monday, August 08, 2005

Calculus Problems

My schedule is still up in the air, as the honors section of the gen-ed class does not have enough students to run and will be cancelled. Does the problem lie with students who lack mathematical confidence? Or with a freshman advising office that lacks mathematical confidence? Hard to tell. At least it's certain that my calculus class will run: we try to keep sections of calculus capped at 30 students; mine is already at 36.

We've gone to a new edition of the book for calculus (One of the Stewart books. Will we stay with Concepts & Contexts, as I hear that it no longer does trig-subs* and cylindrical shells** in the main text of the book? Tune in later to find out whether the Calculus Textbook Committee has considered the full implications of the new edition.), so I can't do very much about making a calendar of the semester's topics. I will need to wait for the Calculus Topics Committee to hand down their ruling of which sections are to be covered and what exercises they recommend.

I've been mulling over some ideas for the first few days of the semester. Our calculus students are some of the brightest students in the freshman class, but they struggle with calculus. I'm pretty sure that in a typical semester, only about 2/3 of students enrolled in calculus will pass with a C or better. Some of them it's because they took a course called "calculus" in high school and expect that this will be the same course again. Others were the best student at their troubled high school and will expect math to still be effortless. Most are enrolled in the engineering program where they face a certain amount of academic hazing and more work than they have ever been expected to do before; they have trouble devoting the necessary time to all their classes. I need to get across the idea that this is a hard class, but one that they can be successful at if they devote the requisite time and effort.

The two tasks that I'm working on right now are a pre-test and the first problem set.

The main difficulty with writing the pre-test is that I'm not entirely sure what my goals are. I'd like to find out whether their algebra backgrounds go beyond competence at symbol manipulation and actually include some understanding of the ideas. I'd like to know how much calculus they already know. I'd like to hint to them that this is a hard class and that they won't be able to coast. I'd like to be able to give this test back to them mid-semester and show them how much they've learned. And I'd like to do it with just a few questions. So far I'm thinking that one of them will be about setting up a max-min problem (because the hardest part for many students is turning the words into a function in one variable) and asking, "What is the quadratic formula used for?" But beyond that? Not sure.

The other issue is the first problem set. I want to keep the workload steady in my class, so I want to assign a problem set at the very beginning of the semester. Unlike the exercises, the problem sets will include those tricky and clever problems from near the end of the problems in the book; however, if I assign this sort of problem on the first day that the students see the material, the chance of them being able to do it is about nil. I'm hoping to offset the problems by about a week from when the material was first covered -- which leaves me unable to assign "calculus problems" during the first week. As I'm on a bit of a quadratic formula kick, I'm thinking of asking my students to derive the quadratic formula. Maybe I'll also have them prove the Pythagorean Theorem. But I need more problems.

*Being able to do trig-subs allowed me to break the curve on one of my high school physics tests; none of my classmates could do trig-subs.

**The shells are my favorite. I took to them right away when we were taught them. Since the shells are so obviously the best, I was surprised that my students HATED the shells. I think it's because they struggle to visualize 3D objects.