Avery started us out with a discussion of the Habits of Mind summary he's working on. I felt like we did some great thinking together. Here are my notes (which are only on the things I was intrigued by):
Avery mentioned a few distinctions he wanted to clarify:
- Problems versus exercises was familiar to me. I think most people reading this blog agree that we need to spend more time with students working on real problems, and help students to see the difference between problems and exercises. (Yes, exercises have their place, and can help us solidify our understanding.)
- Problem-solving versus habits of mind. I had thought of these as the same thing, framed just a bit differently, and didn't really understand the distinction Avery was making. He sees the habits of mind as broader than just problem-solving strategies.
- Avery: "There's an overemphasis on math as a noun. We need to get past defining content, so we can treat math as a verb."
- Someone else said: "Students think that guessing is the opposite of math." We talked about how to encourage guessing and then checking whether your guess makes sense.
- In our discussion of pattern finding, I mentioned Ben Blum-Smith's collection of pattern-breaking problems, and promised to link to it here. It's in two parts, here and here.
- Christine (I think) mentioned reviewing a book on math anxiety titled Managing the Mean Math Blues, by Cheryl Ooten. I've seen the book, but already had two favorites like it (Overcoming Math Anxiety, by Sheila Tobias, and Mind Over Math, by Kogelman and Warren), and never really looked it over carefully enough to compare. Christine thinks it's better than either of my favorites. I'll look all 3 over carefully this summer. I've been buying lots of used copies of Overcoming Math Anxiety, and selling them to my students. Maybe I'll switch.
- Avery: When we're estimating what reasonable answers a problem might have, establishing upper and lower bounds is a great technique. Students can feel some success even when they can't finish a problem.
Devising a Measure of Squareness
Our first step was to label the picture we'd been given, so we could discuss it more easily. I've included our labeling below. Before you read any further, would you please rank the pictures from squarest to least square?
We all came up with pretty much the same ordering. I thought a few pictures looked about the same, but still wrote the equal ones in the same order everyone else did. I wrote E before B, and A before D. As we discussed whether they were the same, Gretchen pulled out a ruler and checked. Yep. But everyone else had thought E was 'squarer' than B, and I had noticed it first in my quest for 'squarest' rectangles. We wondered whether excess horizontally 'bothers' us less than excess vertically. It was fun wondering how visual perception works.
When we got down to creating a measure, there were basically two proposals:
- Long side over short side (L/S)
- Long side minus short side (L-S)
Either measure created the same ordering for a particular group of boxes. The subtractive measure depends on the units used to measure the sides. So a 3 inch by 1 inch rectangle has a measure of 2, meaning 2 inches off square, I guess. But if that same rectangle were measured in centimeters, it would become 5.1 cms off square. The ratio measure is 'unitless', and gives 3, meaning one dimension is 3 times as big as the other, regardless of the units used to measure the sides.
I gave 102cmx100cm = 2cm off, though it seems square, versus 1cmx2cm = 1 cm off, as an example of this measure not 'working' to my satisfaction. Katie suggested adjusting for total area, by dividing by L*S. I was still thinking about the resulting units and suggested using the square root of L*S. It's a bit ugly now, but I can't see a real problem with (L-S)/sqrt(L*S), and I'm intrigued that we have two different measures of 'squareness', with no clear benefit to the simple ratio version.
You can check out the other 6 problems at Avery's blog.
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