This is a great first day of class activity, which works well with math phobic students (I use it in a math for elementary teachers class). Students make a list 1:00, 2:00, etc. on a piece of paper — however many times I assign — and then they have to make an appointment with a different student in each slot. Students move around the room making appointments and learning each other’s names. If there are an odd number of students, I passively participate, accepting appointments whenever students come and ask me.
At some point a student or a few students will announce that they are done, and I tell them they aren’t done until everyone’s schedule is filled in. It’s an interesting question to figure out a good number of appointments to give them so that they will get stuck, but not overwhelmed; I usually go for a few more than half the number of students (12 for a class of 20).
Occasionally students are able to make a schedule just by switching around a few appointments, so then I ask them to do it again with a few more time slots, but usually they get stuck enough that they have to start over. I sit near the back to make it clear that students can use the board at will, and usually a few students go up to the board and take the lead; if not, I put the students in small groups to brainstorm, but either way, many students participate.
The first time I taught this activity, I happened to have sixteen students in class, and they broke themselves into two lines and made 8 appointments by shifting one line and writing down the person across from them. Then someone who had used a reform curriculum in high school suggested breaking the problem into a smaller one — a great strategy that doesn’t always work very well for this problem, but works perfectly with 16 students. I saw before they did that they were going to be able to get all 15 appointments that way, and then after they got there (the next class), we looked at using their algorithm with different numbers of people, which related to number theory concepts later int the course.
After that first class, I rushed to the library (it was pre-WWW) to study more about the problem, which in the literature is under round robin tournaments. There is a standard algorithm, which amazingly, a few of my classes have discovered. This algorithm gives a different role for one person, so students are a lot more likely to discover it when there are an odd number of people in the room, and they see my role as distinct.
One of my students said that she had done this activity in elementary school, and that’s where the “Clock Buddies,” name came from; they would meet with, say, their 3:00 buddy to be partners for science on a particular day. She did not recall any conflicts making the schedules (probably fewer appointments).
This activity is particularly rich in representations. Students use a variety of pictures, graphs, tables, and notations to describe their thinking. It’s a great way to introduce the notion of an algorithm in a context that’s totally separate from elementary arithmetic.
Homework after the first class is usually to make a complete schedule of five appointments for six people and to represent it in two different ways (and to extend the problem to eight or ten people if they can). The activity lasts from two to four classes, depending on how engaged students are and what their strategies are, and along the way I can introduce them to the goals of the class and such.
Every class has come up with a good solution to the initial problem. This handout (Doc version, pdf version) illustrates some of the strategies and representations students have used, along with good follow-up questions for each strategy.
When we finish up the problem, especially in a class where many students have gone out of their way to tell me how bad they are at math, I ask a few questions. Was this a hard problem or an easy problem? Clearly it’s hard. Is it math? Clearly yes. Did you solve it or did I solve it? I am careful to hang back on this problem; I only facilitate when there are communication problems, so students own that they solved it. Then I say that they are telling me they’re terrible at math, but I have seen no evidence, as they just solved a hard math problem by themselves.
In order to teach this activity, especially on the first day of class, you have to be very comfortable teaching an open-ended class where students might come up with something you’ve never seen before. I love this style of teaching, and I find that the beginning of the semester adrenaline flow helps make me more present. In a way it’s fool proof, in that it easily gets students talking and moving — worst case scenario they learn names and that the class is active, and that’s not bad for the first day (although they always get further).
Several years ago I worked on a Mathematica simulation to try to decide what is a good number of appointments to assign. There are some assumptions that have to be made to model how the students interact when they are shopping for appointments. My programming skills are not the best, and I moved on to other things, but if someone reading this is a good programmer (or can recruit one) and is interested in working with me on this question, let me know (and if you just want to work on it without me, please let me know what you find).