Tuesday, July 20, 2010

The place for proofs

I have been thinking quite a bit recently about the place for proofs in introductory math courses such as trigonometry, calculus, and really anything prior to the "proofs" course.  As a mathematician, I realize the importance of establishing results rigorously.  My students, however, do not.  With the rare exception of the dedicated math major, most students would rather I just tell them the formula, let them memorize it, see a few examples of it in action, and move on.  Finding a balance is no easy task.  There are a few things to keep in mind that can make this challenging task easier.

First, we need to decide on the correct level or rigor in our explanations.  Freshmen will neither appreciate nor understand a complete proof of the mean value theorem, for example.  On the other hand, just stating the mean value theorem makes it unlikely that students will gain an understanding of the concepts contained in the result.  Some explanation is necessary, but that explanation should be used to illustrate what is going on in the theorem, not just to prove that it is true.  In the case of the mean value theorem, this might be to instead talk through a proof or Rolle's theorem (intuitively, there must be a max or min, and that is a place where the derivative is zero) and then maybe show how you can use Rolle's theorem to get the mean value theorem.  Of course, the mean value theorem can also be explained in terms of velocity and in terms of tangent lines, and these, although not proofs, are also important to help students understand the concept.  This is more proof than I would use on other topics, and part of the challenge is that each instance needs to be judged for itself.

Second, and the idea I've been spending most of the time thinking about lately, is where in the lecture to place the proof, whatever level of detail that might entail.  There seem to be three basic ways to go:
  1. The classic: you state the result, then ask why it is true.  Then give an argument to try to convince everyone that the result holds.
  2. The quest: you state what sort of result you are looking for.  For example, you want to find a derivative rule to help you take the derivative of a product of two functions.  You go through the "investigation" and derive the rule.
  3. The sneak: you do not say what you are looking for, or even that you are looking for anything at all.  Instead, you say that you want to "play around" with these formulas and see what you get.  Then you "stumble" upon some nice formula, and put a box around it.
For one reason or another, I have been using option 3 recently.  I don't know why I have fallen into this: maybe a book or lecture I saw used it and I picked it up.  In any event, I think it is a mistake.  A healthy mix of styles 1 and 2 are appropriate, but the third option should really be avoided.  At first glance, it appears that it would be a reasonable way to sneak in proofs under the objection radar of the students.  But students are much more savvy than that.  As soon as you put the box around the derived formula, they realize they've been had and respond with the likes of, "you mean the last five minutes were not the important part?"

I can't blame them for thinking this.  The "playing around" part often entails a lot of algebra.  Students have a tough time keeping up with the notes, let alone understanding it.  By the time I get to the formula they may be two or three lines behind, and miss my comments about the whole point of the exercise. 

Options 1 and 2 both have their merits, and using both is probably the best way to keep the class exciting.  In general, I think it is a good idea to state the result first when you can - giving your students a rough understanding of a proof is useless if they don't know what it is the proof establishes.  However, we have all been in a lecture where it was nothing but statement proof repeat and that can be very dull.  Additionally, searching for an answer highlights an important aspect of problem solving: we want students to try different things when they get stuck, so modeling this behavior has benefits beyond that of including proofs.  Similarly, option 1 impresses the importance of critical thinking.  We want students to check their answers, or even ask themselves if their answer makes sense.  This is what we are doing when we ask why the theorem might be true.

While many students might not be overly enthusiastic about seeing the proof of various results, I have found that if I am upfront with them about it, they will usually listen.  Maybe they will not all take notes when the proof begins, but then I would rather they sit and listen and think than mindlessly copy every line.

No comments:

Post a Comment