This was the week when I looked out over the class and realized I had lost them—they were not responding to lesson prompts, their body language spoke of anywhere-else-but-here.
It’s the fourth week of the quarter and for this course it may be the longest week of the quarter. We’re covering a chapter on sequences and proof by induction and the start-up material is vocabulary intensive, with multiple, short topics that need to be stitched together into theorems that are then proved by mathematical induction. Not everyone’s idea of fun.
Mathematical induction, by the way, is a deductive, not inductive, method of proof (in the natural science meaning of induction and deduction). The induction in mathematical induction refers to the inductive step in the proof method which shows that the assumption of the truth of, P(k), k an integer, induces, or leads to, the truth of P(k+1); P(n) is a predicate that falls true or false based on the integer n.
This week I was pleased to have students, in groups of two and three, ask several questions on difficult homework problems. This was good for several reasons: some students are clearly working on homework before the weekend, and, more importantly, small communities are being formed where students are discovering, together, both answers and dead-ends to mathematical questions. A little not-so-good was my discovery in last week’s homework that several students had the same, exact, almost-correct answer, to one problem—it seemed to be a transcription from a common earlier source. Apparently teaching now requires some skill in exegesis and hermeneutics. I wrote on each homework paper, the same, exact, almost-correct comment, to which I have been awaiting questions on interpretation.
One embarrassing moment was my forgetting a key insight needed to clarify one of the homework problems (a proof); it’s embarrassing since I solved all the homework problems before the quarter started. After looking at my notes at home, the insight returned, and it really wasn’t too difficult. It may or may not be true for everyone, but I find that I have a problem-solving state of mind that can only to be entered in quiet isolation. In that state of mind solutions to problems develop and outside that state of mind the solutions fade away. I’ve had the same experience with developing software. An algorithm can be constructed in a state of quiet concentration, and luckily it is written down and tested since it has to be used, but, as time goes by, the essence of the reasoning behind the algorithm fades and returning to it at a later date, for example to answer a colleague’s question about it, also creates embarrassing moments.
Next week we continue with induction, strong induction and the well-ordering principle. And then we move to recursion. Recursion is challenging and useful and can encode very terse, yet information-dense, definitions, theorems and software. I’m working on some small-group activities to help introduce recursion. Next week will tell if they work to make the week shorter than this one.