Here's some of what I've been reading for the past few months.
Computation in Math
Robert Talbert talks about computational thinking and computation in math class. He admits none of this is particularly novel, but I like how he expresses these ideas. One key quote:
"Computation is computation no matter where it’s found. Learn it by hand to the extent that it helps you reason about it, and come to a strong understanding of the algorithms that govern it. But then let the computer do it and focus on the human part instead."
And one fun quote:
"But nobody expects computer scientists to do this sorting by hand except for fun; that’s why God made computers."
Math (and Teaching Math) as Nonlinear Discovery
Junaid Mubeen writes about how math happens versus how we teach it. I agree with the main point of this, that there's value in telling students how long mathematics took, how many directions people tried that didn't lead to what we now take for granted.
But some of the examples are weirdly chosen, I think. In particular, I've never met anyone whose reaction to learning about different infinite cardinals and how sizes of different sets of numbers related to each other was a sense that this was inevitable. Mine definitely wasn't. I've seen awe, amazement, excitement, and plenty of skepticism. (I still remember feeling like I'd failed in math communication on one occasion when I couldn't convince someone that the cardinality of the reals was larger than that of the naturals and never had the chance to circle back to that conversation with her.)
Procedures and Concepts
This is Michael Pershan writing about procedural vs conceptual knowledge and how sometimes, starting with procedure works. But a day or two later, I also read this by Sarah Caban. It's about procedural and conceptual knowledge as well. I think I'd say the two posts come to different conclusions or take different stances, but to me, the most important part of both is their exploration of particular kids, their thinking, and how to interact with those kids. Around the same time, James Tanton wrote this post about memorization in math that also adds to this conversation, I think.
Magnifying Inequities
Mark Chubb writes about the Matthew Effect, where small initial differences among students grow in the classroom. In that sense, it reminds me a little of a chaotic system; a small perturbation in initial conditions leads to diverging longtime behavior. (It's not a perfect analogy in that in the classroom this needn't happen, whereas it's part of the definition of mathematical chaos.) The question is then how do we control this system, prevent this divergence from happening. And that's really, really hard. Like Chubb points out, two practices that seem opposite each other can both promote this kind of inequity. And even more than that, there's plenty outside the classroom that contributes to the divergence.
Later in the month, I read a journal article about the effects of peer instruction in a physics course at a two-year college, and one of the things that article pointed out was that the gap between students who scored below versus above the median on the initial concept inventory widened more in the course using peer instruction than the one using only lecture. Peer instruction led to significantly more gains for both groups, so the conclusion is that it's the better choice, but the widening of the gap seems worth thinking about.
Abstract Algebra, Rings vs Groups
Beth Malmskog talks here about teaching abstract algebra in a few weeks and covering rings before groups. I have a known bias towards rings over groups, so I was really intrigued by this! It's definitely a structure that would have resonated with me, and I can see why it was good for students with a number theory background. But then this tweet (and those following it) from Brian Katz had me thinking about the ways in which groups are embedded within rings and how that can help students approach rings more independently.
Fractions are Really Cool
Dan McQuillan talks about how you get from a pretty elementary question about fractions to a Putnam question. It's really interesting to see the steps he takes, how he builds to a place where the Putnam problem doesn't seem intimidating. (For what it's worth, I think there's a lot on Putnam that is approachable while being difficult.)
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