Second Week of Instruction (1/10, 1/12, 1/14)

Monday (1/10)One student, one was only going to audit, informs me that her schedule is too busy and she will no longer be coming to class. This leaves me with 13 students which is a reasonable number though I think it would have been nice to have 20-24. That would allow for 5-8 groups and a good mix of students.

Before turning to specific problems or techniques I begin class by talking about the problem solving process. I classify the three different levels of problem solving: strategies, tactics and tools. I define the idea of a crux move as the significant step in solving a problem. I liken problem solving to finding a path from two points, where the terrain between them may be hilly, separated in places by ponds or moving water, valleys and forested areas. The path is made of individual steps though some particular steps may open up clear areas in which much progress can be made towards the goal. A crux move is one which, though not obvious or easy, when taken brings us on the final steps to the destination.

Following Polya, problem solving begins by getting oriented: familiarizing ourselves with the problem, its givens (hypothesis) and what it asks us to compute or prove (conclusion).

The next step is to formulate a strategy or plan. Some possible strategies (or heuristics) are: Do an investigation (or more colloquially, get your hands dirty, i.e. compute, do some examples), make things easier (maybe even engage in some wishful thinking, that is, treat the problem as if it were solved), look for a penultimate step. The make things easier and use wishful thinking are examples of psychological strategies. Other psychological strategies are: Be mentally touch, be creative, use peripheral vision, don't be trapped in a box, bend the rules.

I suggested two problems. One is the connect the boxes problem. The other is the monk problem. I then discussed the pigeonhole principle again and had them return to the lattice point problem for the remainder of the period. I also point the students to the monochromatic triangle problem and the divisibility by 20 problem. Later in the class I will not do this, that is, give hints as to a particular method, tool, or tactic but for now I want these students to have an experience of their own with the types of arguments I introduce.

There is not much time for groups and as it is pretty quiet during the first few minutes and there wasn't much more timeso not much gets done. I will have to try and talk less and leave more time for groups.

Wednesday (1/12)Today I talked in a little more detail about investigations, computing and getting our hands dirty. In particular, I draw the students attention to one of the problems on the sheet, an infinite series problem. We started work on this together: How might we attack it? One student suggested that instead of computing the general term we test some cases for n, starting with 1 and working our way higher. I made a table at the board suggesting a way to format numerical investigations like this. I then put up the individual calculations with n = 1, n = 2, n = 3, n = 4, n = 5 and had students call out the results. In the process I asked if we had to do the entire addition, say for the case n = 5 and got the response that no, we just need to add the new term to the result for n = 4. I didn't even have to ask if the students saw a pattern since an obvious one was suggested:

The sum of the first n terms is n/(n+1). Consequently, the sum of the entire series would be one.

I asked if we had really completed this problem, had we proved that the sum of the first n terms was n/(n+1). Everyone understood that we had not proved it, but rather we had a conjecture and that the conjecture needed to be proved. I asked how we prove it and several students answered: mathematical induction. I then turned the class over to groups to complete this problem, work on previous problems and also called their attention to the problem of ordered partitions. There was some confusion as to what ordered partitions are and so before we broke up I did the examples of ordered partitions for n up to 4.

The students seem to be getting more lively in their groups and talking to one another. There is a lot of buzz and animation and through the cacophony I hear the terms I have introduced being used: pigeonhole principle, what do we need to prove, induction. For nearly everyone, all too soon the time is up, in fact past the end of class time as the students are occupied with their problems they do not notice that they have worked five minutes more than required.

Friday (1/14/00)A couple of students say that they have an idea of how induction goes but they are not completely sure of how to set one up and carry it out. This leads me to spend about 15 minutes going over induction. I just pull an induction problem out of the hat (sums of squares) for illustration (I don't think that is one of my problems). After the induction I break the class into groups and let them work on problems. I suggest some more problems such as the collection of Fibonacci problems, the circular table problem, the chess player problem and some others. I don't say explicitly that these problems will be amenable to things we have already discussed but I think it is suggested by my statement that they have all the tools needed to do those problems. Lots of talking and animation. I drop in and out of groups to observe and sometimes ask questions, still finding it hard to not give hints. I think considerable progress is being made.

Overall, I think this has been a good week. The students are getting to know each other. They appear to like the groups and I believe they are making progress learning the tools I have taught them. Some have turned in problems which means I have grading to do this weekend. There are no classes on Monday (Reverend Dr. Martin Luther King, Jr. observance) and so I have at least to Wednesday to get them done.

Click here for second week problems.