Date: Jan 14, 2007 10:45 PM
Author: Michael Paul Goldenberg
Subject: Re: If you really want math reform....

A couple of points on induction, etc.

I've posted here before my experience last spring co-teaching a  
discrete math for teachers course (the instructor of record being a  
mathematician has published a discrete math text and research  
articles in the field). He decided about four weeks or so into the  
course to teach mathematical induction (with a few standard examples  
of the sums of the first n integers, first n squares, etc.). The  
students appeared almost without exception to be hopelessly confused  
both as to the method and the concept they were being shown (and  
though this mathematician generally teaches a lot by what he believes  
to be a guided discovery approach) in this case he was pretty  
straightforward with the class, but as I've said, to no apparent avail).

Eventually, I introduced the "Gauss Trick" as a way to help see it  
from another viewpoint, and to help increase their "belief" in the  
formula. I was successful with these limited goals, but that was as  
far as it went: they still found induction a mystery. My colleague  
became increasingly frustrated, as did the students. Few did at all  
well on the midterm, and particularly they bombed on induction. This  
led to some things of interest only to those who care about pedagogy,  
so I'll save it for my memoirs. But the relevant point here is,  
rather, a question:

For those who believe that the principal of induction is an important  
(indeed, vital) mathematical tool (or technique, method, concept,  
etc.): any thoughts on how to better get folks who aren't  
mathematicians or future mathematicians to "get" it? (And trust me,  
we looked at the domino metaphor and my colleague and I probably  
waltzed around this in a variety of other ways. No dice.)

Any suggestions other than what boils down to "louder and slower"  
would be appreciated. :)