Date: Nov 13, 2012 11:52 AM
Author: Joe Niederberger
Subject: Re: How teaching factors rather than multiplicand & multiplier<br> confuses kids!
>There seems to be much confusion and quibbling about the meaning of "common sense" ...
I certainly didn't mean to quibble about the meaning of common sense, contrast it with a keen developed intuition, or even imply what is common for one is always the same for all.
I agree that most algebra/pre-calc students if willing and led by a competent teacher can appreciate the mathematical concept of continuity. I also realize now that was your position - you weren't saying the picture you presented was a staring place, it was the arrival place.
I'll just add a few observations. I'm not pretending to speak as an expert educator - but this is just my view on this.
* The mathematical concept certainly does arise from our common experience.
* Discontinuities would seem most naturally to be the exception rather than the rule -- Leibniz thought there were not any actual discontinuities in nature.
* I wouldn't discuss continuity of functions without grounding the discussion in common experience and at least a few remarks about the rich intellectual history - perhaps starting with Zeno.
* Assuming mostly continuous function, approaching a jump discontinuity from left and right (let's call them left and right travellers) leads one to easily see that what characterizes the discontinuity is that the travellers remain > some fixed distance apart as they approach the jump point. Continuing as above, explore the other varieties. Obviously, single point types (singularities or removable discontinuities) don't fit this mold, (and in my viewpoint, for good reason).
* I think the notion of distance apart at a discontinuity is more intuitive at first than convergence at a continuity. The sort of reversal of viewpoint that leads one to define "continuity" rather than discontinuity is complicated to motivate, however, the eventual definition as given by Clyde or variants thereof I think is easier to see if one has travelled the more historic or natural path. For precalc, I wouldn't spend a whole lot of time justifying it.