Date: Nov 11, 2012 10:16 AM
Author: Joe Niederberger
Subject: Re: How teaching factors rather than multiplicand & multiplier<br> confuses kids!

Excuse me, but I feel the need to clarify the post copied below. Clyde was saying his description of "zooming" and "convergence" (see below) was the "common sense view" of a continuous functions, to which I say WHAT?

The common sense view of a continuous function is as I sate further below -- the graph of which I can draw without lifting the pencil. I think the conversion of that intuitive view into a definition involving limits was a great achievement for mathematics, and took some time to hit upon, nothing common sense about it. One must first struggle with Zeno's paradox to appreciate it.

Joe N

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Clyde Greeno says:
>But the essence is that: (1) students first must grasp that the line of rational numbers is dense ... but also having a density of "irrational" holes, and (2) also perceive how the "zoom in" squeeze on a function, at each point within or outside its domain, converges to a (sometimes empty, sometimes single-point, sometimes otherwise) "vertical" interval.

I would say the common sense view is that a continuous function is one whose graph can be drawn without lifting the pencil off the page. What you describe is a mental picture to go with the more advanced understanding. All well and good and I wouldn't be surprised if someone has created a nice interactive computer animation to illustrate.

Joe N
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