At the NCTM Institute last month, we broke into task groups to discuss reasoning and sensemaking (the conference themes) in content focus groups. I slipped into Geometry a little late and found a seat. The group was discussing approaches to this problem:
This was the session immediately following my keynote and the difference between the tasks I had described and the task they had just finished was stark. Someone asked, "How would we apply Dan Meyer's approach to this problem?"
It isn't fair. It's apples and oranges. Paper is a great medium for a lot of math problems. Paper is a terrible medium for representing how people apply math to the world outside the math classroom. My techniques for one problem type have limited use for the other. My enthusiasm for one problem type shouldn't be mistaken for a lack of enthusiasm for the other.
That said, I don't find myself terribly enthusiastic when I think about assigning this problem to Geometry classes I have taught. As a challenge problem or extra credit, sure, but in its current form — with the abstract mathematical language and symbology smacking you right in the face — students are going to wonder, "Who comes up with these problems, seriously?"
If we make a better first act, though, we can engage, I dunno, 17.2% more students without any cost to the math. That's empirical, friend.
Here's the redesign:
- Show how this new, difficult problem arises from an old, easy problem.
- Make an appeal to student intuition.
- Introduce abstraction (labels, notation, etc.) only as a necessary part of solving a problem that interests us.
- Start with a square.
- Draw the diagonals of the square.
- Ask students to tell you what percent each of those regions is of the whole. This is insultingly easy and that's the point.
- Drag the endpoint of one diagonal halfway down the side of the square.
- Ask them, "How about now?"
- Ask them to guess the percents again.
Watch the video. Basically, we're applying pressure to their confidence, which is how I try to approach pure math problems. Start from what they know. Then mess with it in some trivial way (eg. we just dragged the endpoint down a little) that requires math that is anything but trivial.
You and your students will begin to find it very difficult to talk about all these different segments and regions without labels. So add them. A recurring point around here is that if you want to disengage a lot of students who might otherwise be engaged in the math, simply start the problem with as much abstraction as possible. If you want to engage those students, don't introduce that abstraction until students know why they should care about it.
You've been walking around and taking note of different solution strategies, right? Have students come up and explain those different strategies. Then show use this Geogebra applet to show the percentages changing, in case anyone still needs convincing.
The sequels here are really, really great.
Suppose M cuts side CD so that MD = n • CM. What are the ratios of the areas of the four regions?
Send n to infinity and watch the fireworks.
Again, though: print-based media require you to keep everything on the same page — the sequels in the same visual space as the original problem. I realize that math teachers by nature don't mind that. Do students?
J Michael Shaughnessy, President of the National Council of Teachers of Mathematics and designer of the problem under discussion in this post:
Not all math problems have to be posed everytime in a a high tech environment. Sure, it’s ‘cooler’ that way, but i completely disagree with your comment on this one, about ‘how the problem was posed.’ It’s only boring in the beholder’s eyes, depends on how it’s pitched to a group.
2011 Aug 29: My response here.