To any of my students who find this over the summer:

You will get extra credit in any math class I teach next year if you:

1. Read this article about using something like stone-knives-and-bearskins calculus to find the value of pi
2. Try it yourself with shapes with different numbers of sides
3. Write a short paragraph (or more if you need it) about what you saw and found out

Due on the second Monday of semester 1.

For those who aren’t my students, read the article anyway because it’s cool.

At the moment, I’m mainly studying proof by finite induction (when my brain fries on that I switch it up by working on number systems, which I’m taking later this semester). This stuff isn’t easy but I’m getting it.

The main step of it seems to work by saying two things are equal and then simplifying until it’s obvious. My Trig TA, two summers ago, once showed us something to do with systems of equations by substitution, where if you plug the equation into itself you get something like 1=1, which got him to say “well that’s true” in kind of a goofy “duuuuh” way. As it turns out, getting to “well that’s true” is exactly what you need to do in induction, to show that something works.

Also, I found this a few Wikipedia clicks away from something I was looking up for number systems. Never tell me that math and art don’t go together.

I found a way to construct a square with compass and straightedge.  I think I see a way of making other shapes too but I’m still working on how to generalize it.  Have a look:

Darned if I know why; if anyone happens to know, feel free to comment.

I’ve gotten a few calculators lately, one of them a Casio fx-9750G Plus. The 7400 that I’ve had since a teenager will always be special of course–it got me through everything up to Trig, as well as giving me something to play with during Frank’s Fake Science, but the 9750 does polar coordinates and conics. Shiny.

I found a PDF of Casio programs, which I recognized as some of the last several pages of my 7400’s manual. I then took my old favorite, Prime Factor Analysis, and punched it into the 9750…and got a syntax error. One lucky guess (prompted by half a leftover memory from when I was trying to figure calculator programming out) later, I figured out that somewhere between the 7400 and 9750, Casio changed their syntax to get rid of the : at the end of most lines. Now it works…and I can let the chips do the work on factoring. Whoooo!

…get you a job

Happy pi day!

Just now, for the first time, I found a derivative without checking an example step by step. This is undeniably calculus. And I saw…it all links together. The same math I taught the other day in Friday School, finding differences between terms–constant first difference is linear, constant second difference is quadratic. The second difference is the derivative or the slope of the derivative or something like that. I’m still sorting out the words for what I see.

Holy…something.

(Note: I did not type this while on the clock. Just jotted it down to type later. )

A test that includes a LOT of calculus is the main thing standing between me and a full high school math license (I’m reasonably sure that the rest of it is just procedure). I have bright ninth graders that just might be ready for calculus or at least precalc in a few years if we catch them now and give them something to shoot for. And quite frankly…I’m not ready for either of those because elementary ed majors don’t take calculus. Therefore, I’m hitting the books.

Those are the practical reasons, and they’re valid as far as they go, and they have the advantage of implying a timeline, which might keep me on track through brainfry.

Beyond that…with math, the universe can be beautiful.

Where algebra and geometry collide, the graph of a linear equation takes an infinite number of right answers and turns them into the mark of a pencil and a straightedge. Two equations share one solution or none or all of them, depending on their slopes and, if their slopes are the same, their y-intercept. Which I keep thinking of a “base” because it’s the b in y=mx+b. Conveniently, not by coincidence but because it has to work this way, if two equations share a solution, their graphs cross, and the point where they cross gives the x and y value for the solution.

The Pythagorean Theorem isn’t just for triangles; it works, in a slightly rewritten form, on distances between any two points in 2D space, and with an extra layer it works in 3D space too. Spherical coordinates (which feel like polar coordinates squared to me, although polar coordinates feel wrong because dagnabbit theta should come first, not the radius!!) can be rewritten into 3D rectangular coordinates, so the Pythagorean Theorem could be applied to that too. If that distance is a chord, there must be a way to find the length of the subtended arc, which could be a way to find distances between places on Earth. Nothing in there is recent research–the Pythagorean Theorem has been known the longest, I suppose, although how many millennia you want to count depends on whether you mean a^2+b^2=c^2 or why. Probably that’s not even a new way to put it together, but seeing it moves it from black box magic to something that can make sense.

The chaos of the world is simplified to a row of numbers that can be turned inside-out in different ways to find out different things.

If you think that’s not beautiful, go look at the derivative of a sine wave until you get it.

Now that the WGA strike is over, shows other than reality TV are going back into production. We won’t be doomed to an eternity of Are You Smarter than the Girl who Wants to Marry a Bajillionaire? crap. YAY YAY YAY YAY YAY!!!!! Among a laundry list of shows that I don’t care about (I just don’t get into most stuff any more…in fact, I don’t even bother to watch most stuff any more), House is coming back on April 21 at 9 PM. Note the timeslot change–it’s on Mondays now, contrary to House’s line in a recent episode to the effect that broadcast TV would be good enough to watch on Tuesdays. At least it’s not a Friday Timeslot of DOOOOOOOOM.

While I’m on the subject*, what’s up with TV these days? I mean I get that TV networks would go broke if they planned their entire lineups to appeal to the IQ range that my friends and I inhabit, but on the flip side, all this crappy reality TV exists for two reasons: it’s cheap to make and it’s freaking insanely popular with The Masses. So, it would be nice if they would at least use those cheap profits to pay for the production of something of value…although really it would be more of an investment than a subsidy, since over time smart programming will earn more than Reality crap. Do the math…a Survivor season (whatevertheheck it’s called) can run each episode maybe 2-3 times, and then the tapes are worth their weight in film. Shows that can stand up to repeated viewings and that last a few seasons so that enough episodes are made can keep making money forever. A constant trickle over a long enough time can add up to more than even a huge single burst. I could say it in statistics notation with an inequality, which would be cool because any geekery involving Greek symbols cannot possibly be less than amazing, but I’d need another Wordpress plugin for that.

*note: this means “Avrila’s been waiting to rant about this for a long, long, LONG time”

So, I say to…well, everybody, though mostly the TV studios…think long term. Long term wins in the long run, and the nature of time will put us all into the long run eventually.

I can’t say very many good things about my fourth grade year. In fact, even the activity I’m writing about left me thinking “so what?” about what we were supposed to learn that day, though I think I learned more than I was intended to.

Ms. M. handed out strips of paper and tape, and told us to twist the paper once and tape the edges together. Then, she asked us how many edges the new shape had. I remember realizing that it only had one edge, because when I used my finger to trace the edge, it went all the way around on both sides.

I still don’t know what the Möbius Strip is to me, or I am to it, that I should care that it has only one edge, nor do I know what a fourth grader was supposed to get out of that. But I do remember having the idea that someone must have been the first to do that, and to realize what they’d done, and that on that day I’d done the same thing. So that’s what school is doing, I thought, taking us down the same path of figuring things out that other people have already been on. Although I didn’t tie that idea in with anything else for the next few years, it was in the back of my mind.

By sixth grade, I had picked up pi=3.14 from somewhere. I didn’t yet know that even that was an approximation, so I thought I was hot stuff when we measured diameters and cirucmferences and I was a step ahead when someone said, at the teacher’s prompting, that going from the diameter to the circumference was like multiplying by 3. But then I realized that what we were doing in that classroom had been done before by some ancient people, to figure out first that it was close to 3, then that it was 3.14. I found out later by playing with a calculator that there were more digits; later still, I found out that the digits went on forever. Once I knew that, it wasn’t any real surprise to me that people are still calculating the digits of pi, though I hope they’ve moved beyond measuring some round object and dividing by now.

Not long ago, I asked my at-the-time students to write a page starting in “Math is…” or “Math is good for…” Most of them couldn’t do it. They’ve been trained to do pass tests by doing mathematical tricks, but not taught to think about math. These were smart kids, but abstract thinking–anything that can’t be measured by a fill-in-the-bubbles test–isn’t in their curriculum any more.

I’m not ranting about NCLB or the testing systems. That’s been done, and anyway, the alternative (e.g. Oregon’s overall sub-40%, and lower in high school, pass rates before NCLB threatened the districts’ wallet) was worse. I’m talking about the best way of getting things done, which is a conversation that should be separate from “how do we beat the tests?”

Pure direct instruction does not work. There, I’ve said it. I can’t prove is, because as far as I know we’re not measuring the kids on the important stuff that direct instruction fails at–the easy things to measure, like whether the kids can do something, are all well within the range of things direct instruction is good at. Whether the kids can think something through, though–that’s not so easily measured, and they don’t learn it from being told and imitating.

I’ve gone back and forth on this one, because of some of the nonsense that sneaks in disguised as “reform” math, and I do think that there’s a place for direct instruction, contrary to what the radical philosophical constructivists say. However, I don’t think it should be considered the be-all end-all of teaching. Sometimes, yes, it works best for the knowledge to go from the teacher to the student, but if the students are to be active learners rather than passive receptacles, the emphasis needs to be on the teacher setting up the activity so that the student will figure it out. This will give students a reason to think that it matters if they try, and a sense of their own ability to figure things out.

I may not have cared about the one-edged Möbius Strip, other than as a curiosity. But because of what I saw in the Möbius Strip, I cared about learning.