by Brian Katz
I’ve been struggling with motivating calculus students this term, so this issue has been in my mind constantly. I’ve found two different techniques for improving student engagement in the course, and I wanted to share. Also, I sincerely hope that you will post other ideas that have worked for you.
Application Project: I have asked my precalculus and calculus I students of the last few years to seek out an expert in their field of interest (which is almost never mathematics), articulate the skills learned in our course, and find an application in their field that could not be understood without calculus. Almost every one who has done this assignment has realized why they are being asked to take calculus, and a few of them even caused faculty in their field to discover why they were asking their students to take calculus. In my experience, you need to ask student to write 2-3 pages about their interview or they will not get as much out of the experience. (Be prepared to ask for a rewritten version.) I also think this works better as an extra credit project than as an assignment, but both have worked for me. (For the record, I got the idea for this from Adriana Salerno, now at Bates College.)
Challenge Problems: I give a lot of quizzes in my calc courses, and I like to write a challenge problem on the board during the longer quizzes that the faster students can think about when they’ve finished. These work best with the students who could be math majors if they only knew the true beauty of mathematics. Here is one of my favorites:
Consider the integers, {…,-2,-1,0,1,2,…}. Notice that:
(i) The integers come with an operation, addition, that takes two integers and produces a new integer, and that operation is associative. This means that (a+b)+c = a+(b+c) for any integers a, b, and c.
(ii) There is an integer, 0, so that a+0 = a = 0+a for any integer a.
(iii) For each integer, a, there is another integer, -a, so that a + (-a) = 0 = (-a) + a.
Now consider the set of strictly increasing, differentiable functions, with the operation of composition. Translate each of the sentences above into this parallel case and then try to argue that they are true. (You are proving that these sets are “groups”; look this term up on Wikipedia after class.)
Question: What happens if their field is, say, English literature?
Response: This is a fantastic point. Notice that I use the language of “expert in their field”, not “prof in their major”. Even some of the students in majors that obviously use calculus choose other interests. And the point of this assignment is to improve student affect, not really to learn that much calculus, so I’m comfortable with topics that are a bit of a stretch if it helps them engage.
This term, I had a math education major who wrote about the physics in the sports he hopes to coach in his career, a history education major who got excited about atomic bombs that led him to write about radioactivity and carbon dating, and a vocal music education major who was inspired to think a little more carefully about the kinds of maximization problems that he and his colleagues engage in. I had a pre-physical therapy student this term who was not interested in writing about the calculus-based physics side of her training, so she ended up writing about the business side of her hopeful career. Also, many of my past pre-pharmacy students have talked to working pharmacists, not their biology professors, which was a step towards a more mature relationship with their schooling. And for the student who really can’t find a topic of interest to them, I offer a few suggestions. Two that have worked for me are investigating the predator-prey (Lotka–Volterra) differential equations and proving that cannons fire furthest at 45degrees above horizontal without regard to gravity or initial speed.
In short, calculus is the analytic study of change, and I do believe that there is at least one area of interest for each person that can be better understood using the tools calculus provides. (Whether we should require so many of them to take it is another point altogether.)
(btw) Those strictly increasing differentiable functions had better be surjective!!!
Probably the most important thing is your own attitude. A teacher who really loves the subject and everything it is and really encourages students to think will certainly attract the attention of the BEST students, but as for the rest, you know, I’m not convinced that to some degree education doesn’t take certain kinds of people and put their heads in to boxes.