Some disjointed thoughts on pedagogy

[cahn]

C. had her physics exam today. I am DONE with tutoring for a while, possibly for a very long time while I recover from this round of it :) I *crosses fingers* seem to have succeeded in the primary (for her) goal of getting her to pass physics, while mostly failing (with some very partial success, see below) in the secondary goal of improving her fundamental math skills, and utterly failing at getting her to see physics as anything but a kind of pointless torture. Thanks a lot, California school board! Oh well.

I 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.)

Comments

[luzula]

I think the ability to manipulate and understand numbers has seriously suffered from the availability of calculators. They're a great tool! But if you don't understand numbers then you can't (for example) judge whether your answer to a problem is reasonable or not.

Right now I have students who are going to be math teachers. I don't allow calculators on exams. One of them, on my recent exam, made this calculation: 0.001/2 = 0.002. To me, this is ridiculous on the face of it because the answer is larger than the number that you just divided by two. But obviously they did not react to it.

Variables, hmm. I would explain it this way: "We want to have a number in this equation, but we don't just want to have a specific number. We want to be able to put any number into it. So we put a letter instead that stands in place of the (unspecified) number we will eventually put there."

[thistleingrey]

Maybe? My father is old enough to have used a slide rule unironically, but he had a calculator as of the 1960s in Canada (not LCD, mind). That's about three generations' worth of adults with some calculator access in North America, plus or minus socioeconomic reach. Simple ones are somewhat cheaper now than they were during the 1980s (for me in California), though they weren't "expensive" then either: much more within household reach than the cheapest mobile phone plan is now.

I agree with all of the rest of your comment--I'm only reluctant to lean on tech availability as a premise.

[luzula]

I do think that the ability to do calculations in your head or with pen and paper is something you get good at by practicing it. Obviously if you don't do much math at all it doesn't matter whether you have access to calculators or not. But if you do math and depend on a calculator, then I think you will be less good at doing calculations by hand (and less good at analyzing and thinking about the size of numbers) than if you hadn't used a calculator as much. I'm sure there has been research about this, though, and I am open to changing my mind if the research results say otherwise. : )

Slide rules made you aware of powers of ten and the size of numbers in a different way than a calculator, I think, since you have to determine yourself where the decimal point should lie. Not that I want to go back to slide rules. *g*

[cahn]

I do think that calculator use is a lot more prevalent now in academic settings than it used to be. When we were in school, one didn't really use them except in science classes (not in math classes -- I remember it being very controversial when an algebra teacher I was TA'ing for at summer camp encouraged calculator use). Now they're allowed in the SAT, math classes...

That being said, like I was saying to luzula, I agree that calculator use isn't the entire explanation, although I would totally buy that it faciliates being lazy about understanding numbers.

[thistleingrey]

Ah, well, it varies: from sixth grade my middle school permitted calculators in math class. In high school honors classes there was a class set of graphing calculators--you checked one out for the year, and if you broke it, you paid a replacement fee--or you could use a regular scientific calculator if you owned one and preferred it. We had the set of Ach tests (subsequently SAT II) without calculator, and likewise no calc for AP Calculus, but the teachers were cool with it because they wanted us to be able to do things both ways: prepare for the standardized exams without, prepare for imminent college homework with.

Then too, it was one high school in a small public district with a gifted-program coordinator who was very good at writing grant proposals. I think the larger takeaway is that our overlapping youth years were a time of transition. :)

[cahn]

Ah, yes, I did have a graphing calculator -- in fact my TI-85 from those days is still my calculator of choice (if only because I don't own another calculator, these days). So we must have been permitted to use them in some classes -- and yet I remember everyone being up in arms about that one Algebra I teacher. So yes, I have memories that conflict even within my own head :) and I like your point that it was a time of transition.

[cahn]

Owwwww. And this sounds like something C. might do... except... she's not going to be a math teacher when she grows up. Ouch.

I've actually tried explaining variables this way. I think what C. needs to do is lots of manipulations with variables alongside manipulations with numbers, which didn't really occur to me until way too late in the game. (We've done both, but usually disconnected from each other.)

About the calculators: I agree, and yet... my coworker's wife, who taught math for a while, noted that she would routinely get students who divided numbers by ten by going through the entire long division process. So it's not just the calculators, although I agree that calculators make it very easy not to have to understand numbers.

[luzula]

my coworker's wife, who taught math for a while, noted that she would routinely get students who divided numbers by ten by going through the entire long division process

Ouch. Yeah, I guess you can do hand calculations by rote and lack critical thinking, too. Do you know of any research on the impact of calculator use?

I try to teach a critical attitude to problem solving--like, thinking about whether your answer is reasonable (for example, if you're calculating an area and get a negative answer, that is not reasonable). And for example, if you're solving a differential equation, you may fail to solve it but you have no excuse to give the wrong answer, since you can check your answer by plugging it into the equation. And then I get a student who tries to do that on the exam, only they don't know how to differentiate a product, and also they think that 1/(a+b) = 1/a + 1/b. So...kudos for trying to think critically, but they lacked some basic tools. /o\

Okay, I'll stop babbling about teaching now. : )

[cahn]

I don't know of that research, no, but it does seem like there ought to be quite a bit of it!

I wish more teachers were like you. My high school physics teacher, who was one of the best and most thoughtful teachers I've ever known, and who taught me basically everything I know about physics/math pedagogy, was a stickler for thinking about whether one's answer was reasonable. He would give partial credit if you got a completely illogical answer and pointed out that it was illogical but that you couldn't figure out where you'd gone wrong.

Oh, ouch, trying to solve differntial equations and not knowing fractions! That sounds even more painful, or perhaps as painful, as trying to do physics without knowing algebra....

[thistleingrey]

One of my math teachers in secondary school taught that kind of self-check, too. So valuable, at least IMO!

[thistleingrey]

Yesssss to the manipulation of parts. Someone must have told me about it, but I was doing that for MathCounts in seventh grade (and used to have some sort of awkward triangular trophy); unlike most of my fellow former literature majors, I don't fear Common Core for that reason, only want it to be taught properly. And I agree that innate tendency/inclination, at least, factors in.

(I am a weirdo lit major who wanted to study ophthalmology right up through first semester of college, when I realized that my grades wouldn't be high enough, so sciency kid but not sciency adult by formal training.)

[cahn]

Ha. Yes, the more I learn about Common Core, the more I think that in principle it's a good idea. But, like you, in practice, I worry about it being taught properly.

Hee, I didn't know about the ophthalmology! Although I was definitely under the impression that you'd been a math-geek sort in your past :)

[thistleingrey]

Yeah. This post is very cool, on one hand, and on another, that parent is also able to muster stuff like this, so it's not exactly representative.... ETA In case the posts don't say, SteelyKid is six going on seven, and The Pip is three--same ages as a close friend's children, so it's easy to remember.

I've also taken the LSAT and (though I did well) decided not to go to law school, but that's more common for lit!

[cahn]

...I'm not sure why I'm not following Chad Orzel's blog! (I have actually been following Kate's LJ for a while, so am vaguely familiar with the doings of SteelyKid and Pip, although had forgotten their precise ages.) That is really cool. I've also been looking at the Common Core standards for fractions (nominally for C., although we never did get to going over them in detail) and very much like the idea of being taught why math is the way it is. If C. had been taught Common Core correctly, I think it would actually have helped her a lot. (Who knows whose fault it was, but she was certainly not taught correctly or at least well.)

[thistleingrey]

There's a big gap in teacher training, I think, for various reasons. That's not the only reason, but certainly in some districts the "new methods" were pushed from the top down without any support for recalibrating teacher expectation or retraining.

[cahn]

*nod* A lot of this looks like it could involve quite a lot of teacher (re)training, more than could conceivably be feasible to do either in terms of budget or capability. A friend of mine teaches at a private school where they do Singapore Math (which has a lot of similarities with Common Core) and she says there is a lot of training involved. The private school can handle it, but I can see where it could easily get prohibitive.

[ase]

My inclination is to lean toward experience, building on some aptitude, for math at the middle to high school level. Partially by experience: my applied math was spotty in high school, but got a lot better with routine use.

[cahn]

Yes, I agree with you. There's an aptitude component which is liable to reinforce itself, since one naturally tends to want to do things one is good at, but yeah, I could see improvement in C.'s facility with numbers even after doing a semester's worth of number-crunching with a calculator.