On Jun 1, 2013, at 4:06 AM, Richard Strausz <Richard.Strausz@farmington.k12.mi.us> wrote:
> The link below is to an interesting discussion, coincidentally about volume of cylinders, that deals with how teachers deal with misconceptions. > > http://blog.mrmeyer.com/?p=17148
That's funny, not just the teacher's original mistake, but posters' responses as well. A couple come close but the majority look like they have just saw a radio for the first time and are trying to figure out how it works.
First off, thinking that Tuesday comes after Wednesday is a misconception. This is a mistake in mathematical reasoning. Ultimately, this is a poor understanding of algebra and the ability to apply it. And naturally, I am not surprised since this is essentially an anti-algebra site. I make the point about "misconceptions" because labeling this a misconception is like thinking that algebra and mathematics is all about knowing the right facts or formulas.
I like #23's response, it says it all ...
"Is this not a delightful case of letting the algebra getting in the way of the understanding."
How does algebra get in the way of understanding when the understanding in this case is algebra?
In any event, Lou already addressed the crux of these comparison problems previously, using algebra. Lou showed that if you are making a comparison between two cases then you introduce a multiplicative constant "k" and then describe k, which is the ratio of the two cases. The same gist applies here except that you will show (algebraically) that the ratio of r1^2*h1/r2^2*h2 does not equal r1*h1/r2*h2. In fact, it is different by a factor of r1/r2 which destroys the idea that one comparison can imply the other.
What is ironic is that every teacher that saw the mistake in the original teacher's algebra did so because they have been exposed to algebra. No one, including myself, Dan or anyone here, would have ever tested that conjecture using trial and error or pouring popcorn into cylinders except for the fact that the ALGEBRA LOOKED WRONG. Yet, do they proceed to devise lessons to teach algebra? No.