We just had two spectacular Brown Bags in a row. It's probably just as well it will be three weeks until the next one--that's a hard act to follow!
For a start, last week Sive Athreya talked about the projects he assigned in Math 308 (Linear Algebra with Applications.) He brought a collection of the projects themselves, including some good, some OK and some not-so-OK. He also brought a cumulative collection of all the hand-outs he had given the class--an impressively thick stack. His general idea was to get the students really to dig into one application and see the linear algebra at work, rather than just being told about it. Students had six different topics to choose from, ranging from game theory to genetics, and five weeks to work on the project. Siva made the guidelines clear, but carefully avoided leading the students by the hand--autonomy was one of his specific objectives. He also gave a few points for originality of presentation, which was a well-rewarded decision--how many times have you had a student turn in a silver shopping bag, split down the sides so as to show a collection of pipe-cleaner genes in pertinant patterns?
The project replaced the second midterm, which worried Siva a little, since the mathematics in the projects was carefully selected not to be too heavy-duty (difficulties with the mathematics would distract from the point, which is how it connects to the other fields). His solution was to increase the amount and challenge of the homeworks--and the students clearly rose to the occasion. Siva's description was that the energy level stayed high for the whole quarter, which sounds to me like a modest way of describing a class that was engaged and learning at a high level straight through.
Project details are attached to Siva's homepage. The address is
http://www.math.washington.edu/~athreya/308/
Appropriately enough, this week's Brown Bag then dealt with mathematics as applied to biology. Tom Daniel discussed mathematics as it comes into his life as a research zoologist and into his life as a teacher of both undergraduate and graduate students. At the research level, I found his basic analogy really interesting: biology, he says, is now going through a development that physics went through several centuries ago. Both started out as descriptive sciences, and both reached a level where increased knowledge and a wider vantage point made seat-of- the-pants operations inadequate--and that's where mathematics became essential. He gave us some examples which made sense to me, though not to the point where I can reproduce them!
On the teaching level the issues became really interesting-- especially the ones that extended beyond the immediate questions at hand. The first that arose was the obvious question from us: "In an ideal world, what would come to you knowing of mathematics?" To which his answer was very firm and clear: "I don't care whether they can carry out the computations. That's easy to fill in one way or another. I want them to be able to set up the problems. They HAVE to be able to set up problems." They need, in short, to know what the mathematical concepts mean, not just how they work. Certainly very much an objective in our teaching--but how does that get taught? It is very difficult to come up with a treatment of it which does not flunk the "Will this be on our midterm?" test. This led to the familiar topic of students' motivations and their laser-like focus on grades, and the demoralizing impact that has for teaching. Michael Keynes came up with a heartfelt defense of students which deserves our awareness: whether or not, individually or departmentally, we subscribe to it, we are inalterably part of a system in which it is in fact true that the difference between a 3.28 GPA and a 3.3 GPA can determine a student's entire future. It is perhaps not unreasonable that students should worry about that.
Though being aware of that pressure does not mean simply giving in to it--I maintain firmly my right to respond to "Will this be on our midterm?" with a look that freezes the student's blood in his/her veins!
The other major issue that arose in a generality that went beyond the context was retention. It came up from the fact that typically a biology student will encounter the need for calculus for the first time in a senior level course--having taken calculus as a freshman. How are we going to teach them in a way that bridges that chasm? Well, we're not. Not with a direct bridge, at any rate. We've all gone through the experience of calling for students to use material from a preceding course--even one taken the previous quarter-- and drawing a complete blank. At which point we mutter under our breath about what the instructor of the preceding course covered. Or,if we happen to have taught the course ourselves, we mutter under our breath about how students aren't learning anything. Some of the muttering is even justified. But definitely not all. Jerry Folland brought that out by reminding us all of how often we run into something we need to know and know we once knew, and nonetheless have to go back and look up all over again. What the previous learning does is make the new one easier and swifter and more likely to stick-- but it doesn't make it unnecessary. I think that fact is something else we all have to re-learn from time to time!
Not all of the high points were beyond the biology context, though. I shall treasure Tom's description of the ideal mechanism for a student to get into the graduate or medical school of his/her choice. The thing to do is get started working in a particular lab as a freshman, and stick with that lab straight through. "Because that way", says Tom, "the professor in charge of the lab can write a letter of recommendation pointing out that not merely do they walk on water, but the very universe glows at the thought of them!" I think I've got to work on my letter of recommendation prose.