As those who take my summer institutes know, I do not spend any class time going over mathematics. No vectors, no trig, no unit conversions, no significant figures. Physics teaching literature has for years disputed the efficacy of the mathematical prelude; experience and anecdotal evidence also leads to the conclusion that such preludes are worse than useless. You can find my diatribe about the first day of school in this post.
Of course, the begged question becomes, do I ever go over mathematical methods? Well, never in a way that is divorced from actual physics. I teach juniors and seniors. They've been taught how to convert units and how NOT to write down 15 digits from their calculator, and they've probably been taught these things multiple times. I have more important things to teach.
Yes, when a mathematical issue comes up in a real physics problem, I might mention a technique; for example, I'll show on the board how know to move the decimal 6 times to convert milliliters to cubic meters. I'll go through the solution to a two-simultaneous-equation problem IF the problem springs naturally from a quantitative demonstration. And I always demonstrate an appropriate use of significant figures, occasionally with a brief comment.
Last week at my AP Summer Institute at North Cobb (Georgia) High School, a participant shared perhaps the most ingenious approach I've ever heard of to significant figures and numerical precision. Tiana Stroud, an Atlanta-area physics teacher, suggested a reference to money owed.
Say a student in the first week asks breathlessly whether he got the right answer to a problem. "You said the answer is 50 cm, but I got 49.8 cm. Aren't I right?"
Tiana's answer: "Okay, kid, pretend you owe me fifty bucks. And you pay me 49 dollars and 80 cents. Is that okay for you? Are you going to risk our relationship over those missing 20 cents?"
I love that reasoning...
* It works for the students who complain about estimating g as 10 N/kg. (Okay, I owe you $9.80 and I give you a $10 bill. What would you say if I stood on principle and demanded my 20 cents of change?)
* It works for students who write down every digit on the calculator. (I owe you $49.891202. I give you 49 dollars and 89 cents, and ask for .1202 cents on credit toward our next transaction. You say I'm a bloody idiot.)
* It works when an answer is off by a good bit. (Sure, you calculated 0.15 m, but I measured 0.10 m. That's more than a 30% difference. Let's say I owe you $150, but I give you $100. Are you okay with that, or are you going to send Vinny and Guido to ask my kneecaps for the remaining $50?)
Love it. Thanks, Tiana. I'll possibly update you throughout the next school year as I use this analogy.