Newsletter 116     Brown Bag on Math 310 (Introduction to Mathematical Reasoning and Proof) 3/04

One of the great sources of pleasure for me is a really well-attended Brown Bag. Since people vote with their feet, that is my best possible indicator that I have managed to choose a topic that really interests people. Last Tuesday's was such an occasion. I'm not sure there were any empty chairs left in the lounge. Most of those chairs were occupied by people with opinions and/or questions, so the discussion was lively indeed. The topic in question was Math 310, originally known as Math 300, entitled "Introduction to Mathematical Reasoning". Several of us have taught it, are teaching it or will be teaching it (or both, or all three), so we started off describing the course -- our goals and how we try to achieve them and what the probability is of doing so -- to the rest of the folks. It didn't take long before a really fundamental issue emerged: there is a considerable spread of goals. As far as the actual mandate of the course goes, I like Jerry Folland's description of it as "the Seinfeld of courses: it's a course about nothing!" That is, whatever you put into it is basically a vehicle for the teaching of reasoning and proof-writing. For me (probably representing the extreme of one interpretation) this translates to having a very large percentage of class time be spent on group work, either work solving problems or self-editing work on constructing a careful write-up of a solution they already understand. At the opposite extreme is Bob Dumas, who is struck by the course's potential for attracting bright young pre-majors into the field. This interpretation has caused him to spend a large amount of time producing what should be a really interesting set of notes replacing the text book (about which more later), and also leads him to feel that time constraints are too tight to permit the kind of inefficiency that in-class student work produces, because there are a number of really beautiful pieces of mathematics he is eager to demonstrate for them. Others have chosen various ways of making the course feel more coherent by selection of topics from within the book rather than by jettisoning it altogether. There was, however, a marked lack of enthusiasm for the book. It has some excellent aspects, which presumably is why it was selected in the first place. In particular, it has a large collection of really excellent problems, with a pretty good coding system to let users know whether a problem is routine, moderate, or tough, or involves some slightly unexpected turn of thought. It is also organized in such a way that one can do the first few chapters of really basic facts and tactics and then jump to any of several later modules. The snag is that the portion of each chapter intended for student reading is very dense and sometimes downright sloppy. Jerry took corrective measures with a set of notes which he kindly made available to the rest of us, but patching a garment never quite renders it as good as new, and some of the ragged edges remain a problem. One of the notable aspects of the course is that it involves a large amount of paper grading. If you are trying to teach someone to write good proofs, there is no way that the correcting of the paper they turn in can be delegated. And, for better or for worse, this means that relative to a standard course the instructor is considerably more in touch with the students' strengths and weaknesses -- especially, of course, the latter. Steve Mitchell spent last quarter tearing off pieces of syllabus and throwing them away, because if the students were struggling desperately with a basic concept he could see no benefit to dragging them through a sophisticated application of it. He wound up in something of a state of shock about the students' level of preparation and understanding, and has been adjusting his sights accordingly in preparation for teaching it again next quarter. As a result he greeted with slightly mixed feelings Jerry Folland's comment that in his second quarter his students had managed vastly better. This may have been due to his having informed them on day one that this course was going to require that they work their tails off, so that the faint of heart quietly disappeared (not a problem when there are 60 students for 40 slots), or it may have been random fluctuation. In any case, Steve might have to re-readjust his plans. Or then again, he might not. I'll finish with a remark that didn't come from the Brown Bag, but from one of my students. I make no claim that he is completely representative, but I think it is a pretty heartening chunk of evidence that some, at least, of our students are, in fact, tuned in to the issues at hand: After spending a long period of time attempting to solve the problems in this course, I realized a couple things. First, math (at least this course) requires rigorous reasoning, analytical thinking, and clear understanding of the definitions (and perhaps two observant eyes.) For many of these concepts, I had to re-read the definitions several times and try to figure out how I can use them to my advantage. Second, some of these problems require a long period of ``incubation'' to the uninitiated math students (like me.) I hope that this ``incubation'' period will be shorter as the course progresses. --

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