Wednesday, September 22, 2010

Separating proofs in low level classes

The semester has gotten off to a busy start, thus the lack of postings.  Anyway, this morning, I was thinking about a student who came to my office hours for some review on derivative rules.  He had forgotten how to take the derivative of $3^x$.  As he flipped through his notes, and eventually found the relevant page, he asked, "Oh right, is that where you sue the frog rule?"  You see, I had joked with my class that I call the logarithm rule for exponents the frog rule because the exponent jumps over the log.  Like a frog.  This of course has nothing to do with the derivative rule for $a^x$, except that when we derived the rule, we needed to use some properties of logarithms (we took the derivative of both sides of $a^x = e^{ln(a^x)}$). 

This got me thinking about a problem that I have noticed before.  Students seem to have a hard time distinguishing between the proof and the result.  This is not as much an issue in a higher level class, where proofs are common.  By then students understand what we are doing.  But in a first semester calculus class, many students are not aware of the important role of proofs in mathematics.  So what to do?

I admit that part of the blame lies with the way I present the material.  I try hard not to make my lecture into a list of theorems each followed by a proof.  Doing so makes the class rather dry, I think.  Plus, not everything we say in calculus comes with a detailed proof.  For the derivative rules, I think it is much nicer to start out with the desire to discover the rule, and then derive it.  That's what I did for $a^x$.  But with students furiously copying down everything I write on the board, the distinction between reason and result can be lost.

Here is my idea: tell students not to take notes.  I want them to see the reason something is true, and I want them to realize that finding this reason is important.  But I don't want them to miss the punchline.  In the end, a student who knows the derivative rules front and back is way better off than a student who has a book full of notes on how the rules were derived.  I think that such a tactic might actually improve understanding of the derivation.  If I tell my students that I just want to them to watch carefully what I am doing, then just maybe they will and see what is going on right then.  When I get to the result, I would tell everyone that this is the thing they should right down and memorize. 

I am sure that this would have improved my evaluations of trigonometry last spring as well.  Many students mentioned that they did not like all the time spent going over why something was true, they just wanted the facts.  I am not about to cut out proofs and derivations, but announcing ahead of time that it is not necessary to copy the whole thing down, might just do the trick.