### Some disjointed thoughts on pedagogy

Jan. 23rd, 2015 04:34 pmI had an interesting conversation with my singing teacher this week that started off by her mentioning that she's been really trying to emphasize relaxing the jaw with all her students. It's a difficult thing for her to do, she said, because she naturally relaxes her jaw when she sings, and she can't remember a time when she didn't do this, and so it's hard for her to either remember to tell her students to do this or to figure out singing exercises how to help them do this.

I was interested in this because I have been having a lot of these "wow, how do I explain this?" moments with C. (*) When I told my singing teacher I was trying to teach someone with math difficulties, she mentioned that the math teacher at the school where she works emphasizes, among other things, understanding how to group numbers in different ways quickly and easily — the simple example she gave was that of seventy-five cents: immediately being able to group that as three quarters, or seven dimes and a nickel, or fifty-cents plus a quarter.

Something that's interesting about

*this*is that it's something that all the math geeks I have ever known do

*naturally*. No one had to invest in a fancy pedagogical system to teach us how to do this, although it's true that a lot of us had parents who did this kind of math familiarity as a matter-of-fact sort of thing. (My mom, for example, expected us to be able to compute a 15% tip by taking ten percent and then half of that, and to understand why this worked.) It was something one did, to factor one's address, etc., to understand and be interested in how numbers related to each other.

C. can't do this naturally — how much of this has to do with math not really being a thing with her mom the way it was with mine, and how much of it is her natural bent, is somewhat academic at this point — but it's clearly something that she would have benefited from learning carefully and thoroughly. (And still would, although neither of us have the time or inclination to do it at this point. If she were my kid, though, I would absolutely be investing in some careful math training. It's interesting that even the amount of number manipulation she had to do for physics has been very helpful for her. When I first took her on, she didn't know what a decimal

*meant*, whereas now she understands that 1.5 is the same as one and a half. Yes, she did not understand this in September.)

I do think some of this must be innate ability. The reason is that I think E. can do this kind of thing pretty easily, at least with smaller numbers, without our explaining anything in detail. But other things another kid would be able to pick up without explanation she needs spelled out carefully and thoroughly; I find myself frequently giving her detailed instruction about why another child might be feeling emotion X at a particular time, or why character Y in a book behaved the way she did, or careful specific enumeration of E's potential choices in a given situation.

Anyway… it's just interesting, the kinds of things we expect kids(/people) to be able to pick up immediately and the kinds of things that require careful pedagogical explanations and/or exercises, and it's particularly interesting to me how it can vary, and how teaching can differ a lot based on that ability. (And I wish so much that they gave classes in interpersonal relationships. That is what my child needs! More than math class!)

(*) …like the time when I was trying to make sure she understood the concept of dividing both sides of an equation by the same amount, and she said, "I know how it works with numbers! I just get confused when we do it with letters," and I realized — I'd had other clues as well -- that she fundamentally

*does not really get*the concept of a variable. Which, y'know, is a problem if you're doing physics where it's fundamentally assumed that you understand this. I don't know how to explain this! I feel like I've always understood this, and it's hard for me to figure out how to explain something like that. (I think I could do it with some research and a LOT more time than I actually did have, but I'd definitely have to put some work in it.)