Date: Nov 14, 2012 3:25 AM
Author: Robert Hansen
Subject: Re: How teaching factors rather than multiplicand & multiplier confuses kids!

On Nov 13, 2012, at 11:03 PM, Joe Niederberger <> wrote:

>> That is like saying we grounded Einstein's special theory of relativity in the observation that Mercury was a fraction of a second late in its orbit.
> You have obviously not comprehended any of Einstein's own popularizations of Relativity. Nor of the history of ideas in mathematics, regarding continuity. The big discontinuity you experience between what you call "formal reasoning" (can you define that, by the way?) and common sense, is just that -- your experience.
> Joe N

What does it mean to comprehend a popularization?

Back to continuity.

What does it mean to comprehend "a function whose graph can be drawn without lifting the pencil"?

What if I said "a curve drawn without lifting the pencil?"

I think anyone can comprehend "a curve drawn without lifting the pencil", right? That is why it is a popularization.

But "a function whose graph..."? I think someone would have to understand a function first, right? I mean, if I just think about curves that can be drawn without lifting the pencil, then I can draw a step without lifting the pencil. Ah, but we said the graph of a "function", what ever that means.

I guess we could add "at no time are we moving the pencil strictly vertical" but what if I draw a circle? Assuming that "strictly vertical" means in the vertical direction for more than one point, the graph of a circle is never strictly vertical. But what does "for more than one point" mean?

So let's add that the pencil must move from left to right or right to left.

Ok, what about the graph of 1/x? Start at x=1 and make your way towards x=0. You will draw forever and never lift the pencil, but you will also never make it to x=0 either. Hmmm, maybe we should say "a curve that can be drawn in finite time from left to right or right to left without lifting the pencil".

What about Abs(x)? Ok, I can draw Abx(x) without lifting the pencil, but I had to stop at x=0 and start again. If I try to draw it continuously (not stopping) I don't get a sharp corner. What does that mean?

Common Sense: Draw a curve without lifting the pencil.

Formal Reasoning: What does that actually mean and why is it significant?

Bob Hansen