I had a really good day in both of my IBL courses today. The key was getting out of the way.
This started in Math in Decision Making. We have had a difficult week due to bad weather. Lots of snow, ice and nasty temperatures have kept many students from getting to class. So on Monday I only had 15 out of my 30 students make it, and Wednesday only 20 students. Worse, it was many of the same students both times. Almost all of what happens in this course happens through our classroom discussions, so I am worried about those students. We are hitting the main points of the beginning of the first unit, and preparing for the true weirdness to come, and lots of people are missing out.
So I took my foot off the gas. I introduced no new stuff today, and I asked the students to make groups that were mixed of “present and not present” from earlier this week. I gave them twenty minutes to sort through everything and discuss one more time. This way, students who had been in class got to practice their reasoning on new people, and those who had missed got to check their individual make-up work against the understanding of their classmates. (What do you mean, “I doubt they did the make up work”? My students are very hard working people!)
Then we started a class discussion. It was awesome. I said as little as possible. I tried to just direct traffic. There is a big ugly box in the front of my classroom that I am told contains a presentation station. It has been sitting there since August taking up space. Today I made use of it and hid behind it a bit. I just let the students talk. Eventually, they came to a good place on their own. It was glorious.
Just to share the joy of it, I had given them this:
Let be the set of all mathematical words, and let be the set of all mathematical words that begin with the letter A. Show that .
One student came to the board and explained quite clearly that is a proper subset of , so of course is bigger. With just a bit of prodding, she even used those words.
Then other students came to the board, and we eventually got to the point where the students had constructed a bijection (we don’t use that word) between the two sets, so they must have the same size.
And then they saw. I pointed out that we had seen the example of the natural numbers and the evens. They smiled. They groaned. They shook their heads trying to make the pain go away. But they all saw. It is possible for both of those statements to be true at the same time if you have an infinite set.
In fact, I think we will stop using the word infinite. I explained the ancient Greek concept of infinity as “too big to comprehend,” and pointed out that _non-finite_ might be better terminology so we don’t get sloppy.
In Euclidean Geometry, it was different material, but the same important dynamic. I put the question for the day on the board along with the relevant definition. I asked them what they thought. And after they started talking, I walked out. I felt like they had been looking at me too much lately, so I got out of the room. I walked around a bit. I went and washed my hands and fussed over my hair in the mirror. (Stupid stocking cap cramps my style.) When I got back, my fear was confirmed: the student who was speaking caught my entrance out of the corner of her eye and abruptly stopped talking. So I hid in the cutout for the door where the students can’t see me. I slowly reentered the conversation over the next ten minutes to guide it where it needed to go by directing traffic. It didn’t seem like it took too long, but soon the hour was almost up and the class had sorted things out, proved a new theorem, proposed a new definition and learned a lot. I was so giddy that I swore enough to be embarrassed about it afterward. (I have a bit of a potty mouth :(. )
Anyway, I have to keep this in mind: sometimes the right classroom environment is only possible if I am not in it. The students will naturally see the instructor as the center of attention, and it may be the only way to avoid that is to get out of the room.