Down At The Bottom Of The Garden

What with the outstanding early returns in Calc I, I’m even more convinced that “students working up front in the classroom” is something I should be doing in every class (at least once per quarter; I’m far from saying “every class meeting” [unless in a class of about three]). So the plan is to make part of the “quiz” grade depend on doing a presentation.

Today I drew a circle centered on a co-ordinate frame and filled it in. “Use calculus to find the area bounded by the unit circle”.

That is, though I didn’t say so, find the area of the unit disk. Since I’m feeling expansive and don’t always dare to introduce such sidetrackers in lecture—or even in “problem solving” sessions—the unit circle is known in the pros as S^1 (the “One-Sphere”… the n-sphere generally is the set of \langle x_1, x_2, ... , x_n\rangle such that x_1^2 + x_2^2 + ... + x_n^2 = 1 [i.e., points at distance 1 from 0 (the zero vector]). If the unit ball has a cool symbol of its own I’ve evidently forgotten it but of course it’s the set of points at distance 1or less from zero. Back to today’s class.

Our first volunteer then wrote out y = \sqrt{1-x^2} and observed that since the bit in the first quadrant was obviously a quarter of the whole, we were looking at 4\int_0^1  \sqrt{1-x^2} dx. Okay, good work… how do you do the integral? A brave stab at a “u-substitution”; I jump in (much too soon I imagine): “We want a trig substitution here…” and, when he doesn’t know right away how next to act, I allow as how it’d be okay to pass the chalk (really whiteboard marker; ugh) to somebody else. Who works through it pretty well, though with a few gentle reminders from me (again probably premature… of course I want corrections to come from the other students… and there were some of these) about, in particular, changing the limits on the integral when integrating with respect to the new variable. So things are going swimmingly.

“Now somebody do it in polar co-ordinates!” (Along with a little mini-rant by me to the effect that I should’ve already—in the first lecture about polar co-ordinates—have mentioned that [of course] “radius equals constant” gives the equation of a circle [centered at the origin] and “theta equals constant” [theta—again of course—here denotes the angle in a point given by polar co-ordinates (r, \theta)] is the equation of a line through the origin [one is supposed to say “pole” here and probably soon I’ll have that habit]. I’d like to be able to take it for granted that these are glaringly obvious [when pointed out] but actually they’d make darn good quiz questions in pre-calculus and might slip few minds even around here. To continue.)

And somebody does. For better or worse, it’s the student doing much the most participation during the others’ presentations (so this is probably way too easy for him); I’d hinted that I intended to make the problems get harder to encourage early volunteers (this appears to’ve been a tactical error). Anyhow, he knocks it off flawlessly… one could wish for a teacher-like running commentary I suppose but anyhow the written work required no corrections (and he wasn’t entirely silent as students often will be with boardwork). Part of the point here is how much easier the calculation is—\int d\theta could make a pretty good claim to be the world’s simplest integral (\int 0 d\theta is of course its only rival).

The work has essentially been done by choosing the right notations (and having already developed the “formula” A = {1\over 2}\int_{\theta_1}^{\theta_2} r^2 d\theta; I took the opportunity to remind everyone—anything you only say once in a lecture series, you might as well never have said at all [or so it sometimes seems]—that the “one-half” comes about because of triangles in a certain drawing… and that remembering the reason it’s there is a good way to remember that it is there).

Then a textbook exercise: slope of a parametric curve at a point. More good student work; some more hints from me about connections with stuff we’ve looked at before. Next exercise, the area bounded by the same curve gets no takers and I’m content to look the others in the eye and say they all owe me boardwork.

There’s plenty of other stuff I want to do in our two-hours-and-change today; more than I’ll ever be able to do… and this is progress. Still, it’s beginner’s progress indeed for me. I’m so doggone uncomfortable asking anybody to do anything they’re uncomfortable with… and it’s so easy to just wimp out and just lecture…I’d be deeply ashamed of this if so many of my colleagues weren’t even worse (at drawing students out by voice… at insisting on homework and times and dates and whatnot I’m probably one of the biggest wimps there is).

So the rest is mostly me talking about the other mid-course correction I’ve just instituted: “Project” style homeworks. I’d mentioned a couple problems as potential “project” problems; today I opened it up and said “pick any two from the ‘P.S.’ sections of chapters 9 and 10″… and went on to talk about what I‘d done with the two I’d singled out on Monday (without giving too much away; these can still be finished for credit).

Then went on and (quickly) worked on still another topic (Taylor Series… the centerpiece for half the course). But the point is that with the student boardwork and the substantial problems, I feel I’m one step closer to the course as it ought to be.

At which point there ought to be a rant about the textbook. I’ve got quite a bit of material prepared. But that’s it for tonight. Thanks for your kind attention.

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A Drop In The Ocean

Free online Calc III notes by Paul Dawkins (spotted at Lane Vosbury’s Calc III Exam notes). This stuff is mostly Calc IV at Midstate Community College but looks like a pretty good guide for me: how to do it right.

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I Can’t Deny/This Feeling That I…

Deep Down Inside at y of x. Random Dylan links from last week. Oh. The exams are great; mostly A’s and B’s. Homeworks too. Lots of gaps to plug of course…

The Best Lack All Conviction

“The Pattern Does Not Hold”. A classic example (taught to me one-to-one at the board in the coffee-break room of my home department [of course I mean Grad School] by… I think… George Minty… funny the things you remember… I’m sure of the room… coulda been Lenard… one of the break-room regulars for sure [here, found mid-sentence, is Lenard’s obituary of Minty]), presented in fine style by Kate (f[t]) Nowak. Fun comments too.

Kate’s subtitle of “DIY Professional Development” describes almost all the “work” I do outside the classroom (like blogging) on mathteacher stuff… and she’s taking it real seriously. This “paradigm shift” post (the last paragraph) is an invitation to “2.0” inspiring even to a long-time skeptic (me) and the related conversations around her neck of the mathblog woods have become quite the timesink for me. These folks are doing some amazing work.

Reports Exaggerated

The 51st Carnival of Mathematics (at squareCircleZ).

“Where is math 2.0?” by MariaD. Hey look. It’s my old sparring partner Kirby (Bizmo Diaries) Urner.

Your Vigor For Life Appalls Me

Miss Cal Q L8 has been at it since February. I finally spotted her today at Tom DeRosa‘s IWtTF. Then there’s Learning in Mathland, by Colleen King (spotted at Math Mama Sue’s (a site I first noticed not long ago).

MT@P

Well I tried to differentiate a constant
But I never got a whole lot outta that
That’s ’cause differentiation gives the slope
And the slope is nothing if the line is flat.
If a function only has a single value
Then its dee-dee-ex is zero where defined
And conversely if why-prime is always zero
Then why’s the same for all ex in the line.

You can sort of fit it to “Red River Valley” if you like. Not that I ever did. Till now. Alas.