Every so often, Twitter feels less like a service I use than a place where I hang out — and one of the users that I feel like I would love to hang out with (in that sort of detatched, ambient way one does in, say, a college dorm or TA office) is Tom Henderson, aka @mathpunk. At some point in the not-too-distant past, I found him or he found me. Yesterday, I was delighted to be pointed (also via Twitter, but not by Tom) to an interview he gave technoccult.net where he sketches a bit of what he’s about:
Many students want teachers to “show me the steps.”
They want a sequence of steps that they can perform that will give them an answer. This is not unreasonable; they know that their performance on exams, and therefore their performance on the All-Seeing Grade Point Average, is largely determined by being able to Do The Steps.
But “The Steps” are cargo cult mathematics.
The Steps are seeing the sorts of symbols that count as “right”, and trying to replicate that dance of steps. It turns out that the easiest thing in the world is to look at a student’s work, and tell the difference between “Knows what’s going on, made mistakes and dozed off” vs. “Can memorize steps, has no idea what’s going on.”
Now, the way that I explain mathematics, it sort of looks like I’m torturing the poor bastards. I handwave. I refer to certain groupings of symbols as “Alphabet soup” and write it down as a wild scribble with one or two symbols around it.
Because I’m trying to avoid showing The Steps and instead show them enough of The Idea that they can reconstruct what the steps MUST be.
Many students want to know the formulas, so that they can float them on top of their short-term memory, ace the exam, and then skim them off. Why do they want to know that?
Probably because, for their entire mathematical careers, math has been a sequence of Steps, and if they get them wrong, they get red pen, bad grades, No No No Look What You Did. Plus, bonus, there is no apparent relevance of these algorithms other than To Get The Answer.
What’s wrong with math education in the US? What’s wrong is, Whatever it is that makes my students uninterested in learning any more math than is required to minimize feeling stupid.
So that we’re clear, lots of my students are totally awakened to the interesting weirdnesses of mathematics. But, it takes some doing, and I can’t do it by myself. Hence the podcasts and the lunatic twitter stream and the plans for TV shows and online games and godknowswhat else.
I’m trying to get across that if you are highly motivating, if you have a high degree of fire and “Fuck yeah!” and “What, that’s impossible, but true!”, you can get students to express interest in theorems named after dead Hungarians.
I also love this idea, which seems important and true (particularly re: mathematics and its models):
Let me tell you a theory about math knowledge. A mathematical concept can be expressed in symbols (algebra), in pictures (geometry and diagrams), verbally, and numerically. This is a common theory; my additional spin is that math knowledge also exists as a performative concept. Like, the way that I direct the attention of the students (“If you ignore this alphabet soup for a minute, you can see it’s really just a product of two things…”) Or, the way I will use physicality. Like, the other week, I climbed onto the chair and then onto the desk while I was trying to explain slope.
ANYway, the theory goes that you don’t understand a mathematical concept until you understand it in TWO modalities. I do very well with visual knowledge, so my notes of understanding are full of color and pictures and mindmaps and arrows linking concepts, and I highlight the holy hell out of math books. However, I don’t believe I KNOW a concept until I can explain it verbally, because I can barely understand anything if someone just talks it at me.
First swipe is through my best modality, second swipe is through my worst modality. The whole “learning style” thing may be overstated, but it remains true that getting students to understand things in a variety of modalities seems like the way to go.
Maybe they don’t get the picture. So you ask them many verbal questions. (Questions, not explanations, 99% of the time.)