On Monday, all 3 classes had homework that included the bit I mentioned for Calc II: First, find 5 problems you can do, and do them. Then find 5 problems you can't do and write them up to share with your group. So on Tuesday, in both Pre-Calc and Calc II, they worked for part of the time on the problems they had brought in. I gave them a sheet titled homework sharing that told them to put any problems no one in the group could do onto the board. And then "If you see a problem on the board you can do (or offer ideas on), go on up and do it."
After about 30 minutes, there were about 7 problems on the board, 2 or 3 of them solved. We discussed 2 or 3 of the unsolved ones, and still need to discuss the last few.
Then we worked on the problem Kate brought up a few days ago, Follow that Diagonal, except that I called it Crossing Tiles. They have not finished solving it yet; I plan to come back to it.
At the end of class, I asked them to let me know what they think of the way we're working together. As students were leaving, W and D asked me if I had worked with CPM. I replied, "That's a high school program, so I haven't worked with it, but I have heard of it. I'm guessing you didn't like it. Can you email me to tell me what you didn't like? That will help me improve what I'm doing."
Shortly after class, another student came to my office, and with a shaky voice asked if I was planning to lecture. She said she needs lecture. I felt for her! She seemed so scared about telling me she wanted me to do something different. I thanked her and said I would sometimes lecture. I know that in the business world it's accepted that each complaint you hear about represents possibly hundreds of complaints you didn't hear. So I figured lots of the students were probably nervous and wanted some lecture.
Having them work in groups got them to know each other more, but they still didn't have much sense of who I am. So I planned a lecture (sort of) for Wednesday. As I came in, I asked them to sit in the same group areas, but to push the desks into pairs facing forward. I started with a request on the board: Write your definition of a circle. I talked about the fact that I want them to feel safe. Even though I know that group work is what will get them the most engaged in mathematical thinking, I want to give them what will help them see that they're safe with me. (It was a good moment to point out the retest policy on the syllabus for those who hadn't noticed.)
Definitions of Circles
I love performing, so Wednesday was my favorite day so far. I gave them one minute to finish up their definition, and two minutes to share with their partner and revise. I got volunteers to offer their definitions, which I wrote on the board. There were 4.
Then I asked for votes on whether a definition was a good one or not. If not, did a circle not fit the definition, or could things that are not circles slip in? I asked those who voted no to volunteer to come up and show us a counter-example. From those we fixed the definitions. We ended up with 4 very different definitions of a circle, all pretty much working. Except that I thought there might be a counter-example to one of them.
Here they are:
- N's definition: A moving point travels around a fixed point, always at the same distance.
- A's definition: A shape with no beginning and no end, that's symmetrical.
- X's definition: The diameter stays the same in every direction.
- Y's definition: A shape with all points equidistant from the center.
I had students google (on their smart phones, of course) "rolls smoothly, not a circle". We didn't find anything. I said I'd get back to them. Later in the day, I was able to google a better phrase: 'constant diameter', and Cut the Knot came through. I'll show this to them today.
I had lots more planned, but I loved where we went with this inquiry into definitions. Maybe today, when I talk about distance and the Pythagorean Theorem (leading into equations of circles), I can help them get a better feel for the difference between definitions and theorems. It doesn't sound like there was much lecture, but I did talk way more than the first two days.
My Calc II students were happier about working in groups, because that Calc I review sheet* was full of problems they wanted to get more solid on. We also worked on a problem I found at the Exeter site, Turning Two Squares Into One. I had forgotten scissors and rulers, and had to run up to the office to get them. We used this as a springboard to discuss the Pythagorean Theorem. Last semester I found out my Calc II students were terribly weak in trig, so today we review that.
On Tuesday and Wednesday, I work from 9:30 to 3, plus what I do at home. On Monday and Thursday, I work from 9:30am to 8pm, a very long day. I ought to get out and do something physical in the late afternoon, so I don't get exhausted in my evening class. Hmm...
OK, I've made a simple, boring handout for my Pre-Calc class, so they won't need the textbook to work on problems (first week of class, expensive,...).
I'm writing these rambling here's-what-I-did posts more for myself than for my readers, but I'd definitely like to know whether the handouts I'm posting are of any use to you. And I'd love your thoughts about what I'm doing.
*What I uploaded yesterday left out the equations. :^( I just uploaded a pdf of this handout. Or email me if you want it editable.