Picking up where I left off… Now looking at the “What Can I Do?” section.
“Go on tangents”: I agree that occasional purely random tangents are fun and worthwhile. But on the whole I prefer being able to make meaningful connections to the core material of whatever class I’m teaching. I don’t want to make math seem like a random jumble of cool, but not particularly understandable, facts. So the kinds of tangents she describes need to not take over the flow of the course. And of course the specific example she describes (introducing the hyperreals of nonstandard analysis) is not an option for most teachers.
“Foreshadow”: I definitely agree with this suggestion, as it fulfills two goals: first, to ask challenging questions that motivate later material, and second, to make the connections that show students the coherence of the subject.
But it does bring up a tricky point. For example, we spend a good deal of time in calculus using derivatives (and algebra) to find the maxima and minima of functions. If that were the first time we ever tried to do such a thing, it might seem that we have found a “killer app” for calculus. But it’s definitely not the first time we find maxima and minima, since those are important, intuitive questions about a function, that are easily answered (roughly) by looking at a graph or (pretty accurately, but not exactly) with a few buttons on a graphing calculator. So the problem is, how do we motivate using calculus to do max/min?
The essential point is that the teacher needs to understand why it really is useful to use calculus to do max/min. Given that calculators and computers exist, it’s not because calculus will find the max of a single explicit function. It’s because calculus gives you a way to understand max/min which leads to general formulas (for whole families of functions and examples), to important (and often simple) theorems and scientific principles (e.g. Snell’s Law), and to generalizations where calculators are less useful (e.g. higher-dimensional max/min). These are all higher-order reasons, and require a fair amount of knowledge on the part of the teacher; they are also challenging to communicate to students, but honesty requires us to try.
“Relate material back to previous classes”: more coherence in the curriculum is always desirable. And isolated review problems, disconnected from the current topic, are not a good way to do that (even though they are easy to plug in without much effort). One of my goals is to keep refining questions to bring in, in a natural, interesting way, previous material that isn’t getting reviewed already. It does involve real work to do that!
“Be less helpful” (a la Dan Mayer): Absolutely. I got this notion first from Dan Teague at NCSSM, who didn’t use this phrase, but the “less helpful” motto is short and sweet. Don’t sell students short. Don’t hold their hand when they don’t need it. Teach higher-order thinking by requiring higher-order thinking before they even start the problem. Teach formulating problems, communicating math, and using common sense. All by just not laying it all out for them.
“Teach an elective”: This goes back to the dilemma I mentioned in the previous post about highly advanced students. I’d love to teach a discrete math course, for example (and I may get to do that someday in independent study settings). Our school does teach stats, which is absolutely right on target, and which many students take. So we do some of this, but it is hard for students to make time. What I really don’t want to see is students spreading themselves too thin and getting a superficial treatment of too many subjects. I’m a big fan of depth over breadth, so I think students (particularly at a place like where I am, where we have many, many electives when you include all disciplines) take too many courses, and get frazzled, when they could be calming down, and focusing on fewer things.
For example, we have had a logic elective in the past, and it was a great course, taught by an expert in the subject. But recently we needed to bump it as a result of a decision to do a slower, more thorough treatment of multivariable calculus, which ate up another semester. Alison Blank might decry this choice as favoring a “sequential” subject in favor of a more diverse elective, and I can see her point. But multivariable calculus is amazing math, no less than is logic, and we wanted to make sure that students taking the multivariable class would actually be able to learn it—something that few students do in a one-semester college course in the subject. And while the mode of thinking you get from a logic class is very valuable, there aren’t many courses that treat it as a prerequisite, as opposed to the many that require multivariable calc. So the calc class opens more doors for further study—not “further progress along a boring track”, further really cool math.
I don’t think that either way is necessarily obviously best, and I really do see the appeal of broader electives, where possible. But there are arguments both ways.
“Start a math circle”: Absolutely. I love helping out with the local equivalent of a math circle, and I actually would like to see it move in a more diverse and eclectic direction. I also help students practice for math contests, which, while it has its own narrowness issues, does expose students to real problem-solving and different ideas from the regular curriculum. If it were purely up to me I would move that in a more “circle” direction as well, but the kids really enjoy doing well on contests, and I don’t mind supporting that.
“Talk about math”: I wish I did this more in casual conversation. I love being asked about math by someone with genuine interest in it, and I’ll take the slightest excuse to go on and on. But I don’t usually bring it up unbidden, since it can make eyes glaze over. I really enjoy giving talks to community groups (mostly senior citizens) about math, and there are many people out there who are very interested and really appreciate it. So that’s one way to “talk about math” in a somewhat structured setting, but outside of school per se.
Whew! Hopefully I’ve guaranteed that no one will read this but me. Congratulations if you read the whole thing. Comments?