Question, QuestionsApril 30, 2010
I don’t know why it’s taken me so long, but I’ve decided to start teaching the art of inquiry to everyone I tutor. Step 1: all my private students will have the homework of bringing in a question to every tutoring session. I’ve run classes on that principle, but for some reason I never applied it to my tutees. I’ve finally started to catch up to myself.
I’ve been reflecting on the difference between teaching high school students and elementary school age kids. There’s a way that I have an affinity for the older kids sometimes: they’re able to connect dots that are further apart, and there’s a very clear arc to the subjects that they’re working on. The younger kids are great too, though. One thing I’m considering is how best to structure my time with them, since they often don’t have curricular need from their student.
One parent offered me some worksheet books. I haven’t looked at these yet, but I think I will. Often, math workbooks are stilted or problematic in different ways, but there are usually great things in them, and younger kids often need more structure. Totally open questions don’t always occupy them in the same way, or for as long. (The first time I taught fourth grade, coming from high school, the difference in attention span was a huge challenge. You’ve got to break things up into smaller bites.) And because there isn’t as much mathematical infrastructure built up in their minds already, there are fewer places to latch things on. And while I have great curricular arcs, it might be time for me to start adapting some preexisting work to my (and my students’) needs, so I can give them worksheets (as opposed to open questions) to take home with them. Part of it depends on whether they need/want homework from me at home anyway.
Questions, of course, are the best medicine for kids of all ages, and I developing the habit of articulating questions will help them prep the ground for the more major architecture to latch on. The math on it’s own, of course, is always so compelling. Yesterday evening a student ended by saying, “It’s really great to actually understand this stuff rather than just memorize it.” For so many kids, learning math is like memorizing monologues in foreign languages; the passion that originally motivated the work is thoroughly obscured. We (the student and I) had started with the law of sines, but quickly worked backwards to definitions of trig functions via circles and then the Pythagorean Theorem. And some remarks about the square root of 2, and how it upset people. How they called it “Alogon, the unutterable.” How they said:
It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man.
For the unutterable and the formless must needs be concealed.
And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves.
For just a breath of that passion today. (Not too much… I don’t need to see people killing each other over these things either. As they have in the past.)
The good news is that once you reveal to the student that all this work was motivated by questions (how long is the diagonal of a square?) in the first place, the field opens to them, and in most cases becomes captivating pretty darn quickly.
What about the younger kids? I met with a second grader today and told him he’d need to bring me a question each time we met from now on, and asked him if he had any for me today. He asked how kaleidoscopes make triangles (and is that all they make?). So we talked about projective geometry, as in the shadow cast by different shapes. Then I pointed out that if you want to know what types of shadows a circle cast, it should be the same as looking at sections of a cone. And then we talked about the reflective properties of those shapes, and how you can build an elliptical pool table with a hole at one foci and the ball at the other, and any direction you it it, it will go in the hole. And then we talked about lasers bouncing around inside circles (which you could imagine as circular billiards), and, as an immediate connection, star polygons, and finally, as a direct consequence, prime numbers. This gives me about eight directions to go for next week. If I brought in a curriculum, I might be able to guide it in the way I thought best. If he brings in a question (or several) and we let that question guide us, I think it will be best of all.