## Friday, 22 May 2009

### More About Understanding Math, Teaching Math, and Understandiing Teaching Math

A while back I snitched part of a post to the AP Calculus EDG from John F Mahoney.
He wrote, "Friends - Here are some topics I've used in pre-cal and calculus for 'debates' with students..." and then he lists five items, all of which are good, some of which are potentially great because you can talk about them over and over.

I wanted to amplify his remarks a little, so here is his list intersparsed with my comments, questions, concerns, and .... well, other stuff. Anyway, pick your favorite topic, and tell me what you think with a comment.

1) Should rectangles be considered to be trapezoids? Argument for: The trapezoid rule is based on trapezoids. If the function has a piece that is constant, wouldn't the trapezoids under that piece be rectangles?"

I wrote recently about the choice of inclusive and exclusive defintions in geometry, and particularly whether a trapezoid should be defined as "at least one pair of parallel side" vrs. "exactly one pair of parallel sides." I tried to engage another teacher who answered the discussion by looking in the glossary of his book... exclusive definition, end of discussion... teaching is like that for some folks. So are squares a subset of rectangles which are a subset of parallelograms which are a subset of trapezoids, which are... (ok, I'll stop now)..what is your answer, and why?

2) What is 0^0? Since 3^0 etc = 1, it should be 1. But since 0^3 = 0, it should be 0
Remember when you answer.. you are talking to High school Calculus and pre-calculus students.

3) Consider a circle of radius r. Draw a chord at random, what is the probability that its length is > r?
Yikes, this one can get real deep real fast...

4) What if you could take logs of negative numbers? How should they be defined?
This one is good for discussion because the technology is such that some of them have played around with their computer/calculator/W_alpha and actually have an inkling.

5) Circular functions are based on the unit circle. How should we define similar functions based on the unit square - or the unit diamond? I have never tried this one, but it seems like a great project: lots of things to consider.. like what is the "unit"?, and what is the period... and what about a "unit triangle"?

A unit circle has a formula x^2 + y^2 =1... what if we used a unit hyperbola, x^2-y^2=1, How could you make definitions for sin and cosine. What would it look like

I would add a one more that I ask, and students sometimes surprise me with, and I think have a lot to do with how they think to solve problems:

6) A cube is usually what people say when you ask them what is the thing like a square but in three-space... I still ask, but they usually give that answer, or can't explain much about why they gave the one they did. But then ask them what is the three-space equivalent of a triangle. I usually ask this with the instructions to write down the name, or draw it if you can't name it... that gives everyone that long thinking time to pick their own uninfluenced by a snap answer from a classmate. I get several different responses. Of course the follow-up is why do you think that is the best one. You can send me your choice and a justification if you would.

What would you add to the list above? What is your favorite discussion opener to make kids think? Share, Please.