> > 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. > > Bob Hansen
Bob, if you were leading such a workshop and someone made that mistake how would you respond?