This past week, one of my students stopped me on my first pass around the class with the exciting words "I think I discovered a theorem! In a right triangle, the angles the median makes with the hypotenuse are twice the other angles!" We had discussed the median to the hypotenuse in the previous class, but I'd never thought about the angles before. So I started doing algebra in my head, confirming the relationship he was describing:
Before I could get very far, Vik showed me the elegant diagram below:
With the inscribed angle theorem, the proof is immediate.
Although my department has a reputation for integrating all kinds of whiz-bang, high-tech devices into our classes, we've become increasingly excited by one decidedly low-tech tool: large whiteboards.
The model is one I learned from our AP Physics teachers several years ago. Contrary to my usual one-problem-at-a-time practice, I give students a handout with several problems, and instruct them to work on one that they find tough--so that they can build their mental muscles. Then, each group gets a large whiteboard, and writes up a good solution. Groups present their solutions and field questions from the rest of the class.
When I first heard about this technique, I thought it was a poor second-best to having every student work on every problem. But I tried it, and now I've seen some substantial benefits
Students get to see a variety of problems worked out carefully, and ask questions in a nonthreatening setting. And while they're not grappling with each individual problem, the more questions they ask and answer, the more they think through the problems presented--even ones they didn't work on.
Students work on communication skills. In other instructional models, it's rare to have an entire student response of this length subjected to a full-class critique.
Students focus on one problem for enough time that they can think deeply about it, and transfer the ideas to other contexts.
The class above is my AP Calculus AB class, working on optimization problems. After a brief introduction, groups chose between four: minimizing the distance from a point on a line to a given point not on the line, minimizing the surface area of a can with fixed volume, maximizing the area of a rectangular pen with dividers, and the traditional "box problem" shown above. In this case, I had two major goals:
Students needed to understand and be able to recreate the major steps in solving any calculus optimization problem: identify the constraints and the objective function, rewrite the objective function using a single variable, and apply calculus to find the maximum or minimum value.
Students needed exposure to standard problems, both in case they should encounter those very same problems later, and to provide foundational experiences from which they could build out to other, similar but perhaps more-complicated situations.
Whiteboarding was perfect: it was enough for kids to see and discuss the different setups--some of which they had encountered in precalculus--without having to struggle through creating each one themselves. (Assessment today: they were pretty comfortable doing another distance minimization with a harder graph. So it worked!) And the in-depth thought about the process really paid off: again, in today's followup class, students seemed able to articulate the major steps and follow them with little confusion about sequence or "what comes next?" kinds of questions. Finally, their writing and communication were much stronger than I would have anticipated: students presenting today wrote clearly and identified major steps and results along the way, and other students used vocabulary correctly without hesitation.
So there you have it: for less than $15 per board all-in, a mechanism to facilitate thoughtful collaboration, presentation, and discussion.
Postscript: if you're looking to acquire some, we get ours from whiteboardsusa.com--they ship quickly and have reasonable prices.