Three things I wish about calculus

  1. I wish that students who are interested in math could learn something other than just calculus during their first year. Calculus is just so unrepresentative of mathematics as a whole that it gives students a misleading first impression about the discipline. I wish we could do number theory or geometry — something simple that doesn’t depend so much on a person’s high school background and which quickly gets students involved in the really important questions of the discipline. Or maybe some sort of Putnam-exam-type course where students hone their intellectual skills on really complicated problems. As it is, it’s like having all first-year football players play only the wide receiver position for the first year.
  2. I wish that calculus could somehow be decoupled from algebra, if only for the first part of the semester. Calculus is really a simple subject to describe, and getting the main questions of calculus out there is really easy from the prof’s perspective and students can pretty easily identify with the concept of using slopes to find a rate of change. But then all hell breaks loose when the moment algebra shows up. Don’t get me wrong — you need algebra to really get a full understanding of calculus — but most students can’t remember half the stuff they learned in high school algebra, and so they are in the position of taking essentially two classes simultaneously, one in calculus and one in remedial algebra. The main ideas of calculus get lost immediately and irrevocably. If it were me, I’d do the entire calculus course from front to back using only graphical techniques for the first half of the course, and then go back to the beginning around midterms and do it all over again with algebra.
  3. I wish we taught more about the limitations of calculus, and what calculus will not do. In particular, there’s no distinction made to students about the discrete versus the continuous, and the deterministic versus the probabilistic. Having calculus as (in many cases) one’s only math course leaves one with the impression that everything is deterministic and continuous. In fact only a very small portion of real problems are like this.

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One response to “Three things I wish about calculus

  1. I agree with the number theory bit. Teach them some fun math that is also useful. Of course when you get to proofs by induction half of their heads will explode anyway.