In my Euclidean Geometry class, everything depends on the students working hard and having ideas. Without it, we would have empty, dull meetings. I never lecture. I don’t even plan activities. The students are responsible for solving problems I set and then presenting to the class. My whole semester plan is bundled into the sequence of problems I have set for them to solve. Mathematics gets done outside of class, and then critiqued during class.
Even if the students are working hard, we can sometimes have a day where not many people are ready to share those ideas, and we can have a dull meeting time. I have developed a few strategies for handling these situations, but it is important not to prejudge what is going on. This post is about my class meeting on Wednesday, where I just decided to see where things would go, and it worked gloriously.
When I asked for volunteers on Wednesday, I only got two, and they were both reluctant and half-hearted. Our general rule is that you shouldn’t go to the board unless you think you have a solution, but the students didn’t claim any progress for a minute or two, and these two both said they had ideas, but didn’t feel finished. I chose to let them go with the incomplete idea. Both students were working on challenging tasks, and I hoped that the discussion we got might be instructive for everyone. In the end, we only discussed one problem because we ran out of time.
The task under consideration is this:
Find a reasonable definition of the term “The point X lies inside the triangle ABC.”
The presenter drew a few figures and talked about how he wanted to measure some angles in those figures and use the relative sizes of those angles as the determining factor. A class discussion got started without any input from me, a tangent was explored (GEOMETRY PUNS!) and then abandoned. Things got messy and confused and I lost track of what was going on. So I asked the presenter to write down the angles which were important to him. He did.
Now we are about 10-12 minutes into our class meeting. At this point, I made the only comment of any substance I did that day. This encapsulated the main part of my “teaching” for our meeting. I said,
It seems like you are measuring two different collections in your two different figures. Can you find a way to describe a single set of things you want to measure and then describe how different values of that measurement allows you to make a determination about inside versus outside?
I enjoyed the little lightbulbs turning on.
Everyone paused, stared, and thought. After a minute, another member of class got up, took the chalk and proposed a way to relabel the angles measured in one figure and use that collection of angles. Class discussion started again.
Things got hashed out (with two students standing at the chalkboard). All looked decided. I waited. Then a third student raised his hand and started describing a figure he was worried about. I asked him to go to the board. He did and redrew a diagram we had erased for some reason. He sat back down. The class paused, stared, and thought.
A minute later, a fourth student stood up, took the chalk and described how this diagram fit into the structure already decided, because he could prove the relevant angles had the right property. His proof rests on a lemma which he stated clearly, and clearly believed, but confused some classmates. I asked them all to bracket off the lemma and just work through the rest of it. This went successfully.
Then we turned back to the lemma. Another student raised her hand. (We are on presenter number 5!) She came up, and gave a proof of the lemma.
So, 45 minutes into class, with everyone engaged and working, we came up with a way around it. Their work looks like this:
Definition: Let ABC be a triangle and let X be a point. We say that X lies inside the triangle if the angles XAB, BAX, XAC, CAX, XBC and CBX taken together make 2 right angles. We say that X lies outside the triangle if those angles taken together make more than two right angles.
I am looking forward to asking them to prove this is equivalent to the standard definition some time next week.