Yesterday, I found myself surprised while sitting in a math department in-service. Not only did the training extended beyond our contract day, but it also overlapped the time of a funeral many teachers desired to attend. Even the location of the meeting changed at the last minute from our convenient high school campus to the downtown district offices. Oh, and did I mention that we were going to be "taught" how to "teach" to the TAKS test by someone who has had proven results "TAKS teaching."

With all these seemingly negative circumstances going into the meeting, you would think that I would be silently resistant and mentally aloof through it all (my MO for such circumstances), but I was surprisingly open-minded and receptive to the presenter's well-researched and proven insights. Because I teach the PreAP and AP math students, the TAKS test really isn't part of my solicitude. Of the juniors I do have, I expect all of them to pass (and to get commended). Seniors don't even take the exam. Instead, I concern myself with my curriculum and in preparing my students for the rigors of the AP exams and universities.

Perhaps is was the combination of the presenter's professionalism, her sense of humor, and awareness that she was "preaching to the choir," she did a great job of disarming us, rather than angering us, and I was drawn it. In fact, in all my years of teaching, hearing about the TAKS test, and listening to all the petty squabbling over ratings, subgroups, data this, AYP that, etc., yesterday was the first time I got a comprehensive, carefully dissected view of what ALL that means. I even learned something new about the way the test was actually constructed, with the first and last quarter being the easiest questions to answer. This is in stark contrast to the SAT, which gets progressively harder with each increasing integer problem number, and the AP Calculus exam, which has no pattern of level of difficulty.

This knowledge alone arms teachers with a great strategy to pass on to students (without actually having to "teach" specifically to the test questions--of which I am trenchantly morally opposed.) Even the best students tend to work in numerical sequence and can get bogged down, frustrated, or even disinterested near the end of a 72 question test. This means they could potentially shut down before reaching the tantalizingly easy ones on the last 25% of it.

So for the first time in a long, I felt that I actually learned something valuable at an in-service. But just as I thought about calling the local newspaper with a headline, the presenter posed a question to the group: "What makes special triangles so special?" Having prefaced the question by saying she asks that ONE question to all prospective math teachers in her district, I knew she would be looking for an answer that went beyond to obvious or trivial. As many chimed in with what I thought were great characteristics of the special triangles i.e. consequences of them being special. As expected, the presenter quickly dismissed each of the responses. "Thank God this isn't a job seminar," I thought.

I felt that I was in a unique situation to answer the question with what I thought she wanted to hear. Because I have taught geometry, trigonometry, and calculus, I have seen and used many principles that are direct applications of the two special triangles (30-60-90 & 45-45-90) including the unit circle and trigonometric identities. I'm also an avid reader of math history, so I felt I could add a little of that in as well. I thought carefully how I could summarize the underpinnings of my colleagues responses in as concise a statement as possible. Never liking to "sound off" in a group setting at the risk of appearing almighty, self-righteous, or idiotic (besides, I enjoy concealing my knowledge a bit), I had mentally selected my words. I spoke with quite clarity and self-assurance: "It's because they are two building blocks, from which we can find information about any angle in increments of fifteen degrees when used with trig identities."

Proud of the fact that I didn't get into the details, but only alluded to them, I awaited my accolade . . . . only to be off-handedly dismissed as the others had. I immediately got angry. "She must have not heard me," I thought. "Perhaps she didn't realize the depth of my statement," I rationalized. I thought that perhaps I should have conveyed that since the measure of a full rotation is 360 degrees, preserved from the ancient Babylonians sexagecimal (base 60) number system, the angles of 30, 45, and 60 degrees were factors of 360, so that combinations of their angles can be used to develop other angles, though not all. It is for the same reason we only memorize multiplication tables up to, say 12. At some point, for other products, we develop algorithms that allow us to use our knowledge of the "fundamental" products. Sure, we can develop unit side lengths for other right triangles, but it is mentally and mathematically more efficient to use the fundamental "special" angles to develop the other angles as we need them. In essence, they are easy to remember.

Anyway, my anger subsided as I realized there was no "right" answer, any more than being told that vanilla is not my favorite flavor of ice cream (which it is, even if you say "no"). I sat quietly, calming myself, shoring up my argument mentally, and eagerly awaiting the "correct" answer. Second seemed like minutes. The presenter went on and on about "thinking outside the box," "creative answers," and other platitudinous commentary. When she drew a breath, I awaited the answer. Out came information about bugs and spiders--she had moved on to an unrelated activity about scatterplots. Never mentioning why she thought special triangles were special, she left the entire group hanging in mathematical suspense. But to my surprise, no one else WAS! I figured she must have answered it while I was preoccupied with mentally beating myself up. At the risk of making a bigger fool of myself, I bit my tongue and tried to forget about it, figuring I'd ask a colleague later.

Well, she NEVER told us. It wasn't just my imagination. I was apparently the only one that took offense to being "shot down." I wanted to know. That's why I read those damn boring math history books to begin with. If I think some little insight, regardless of how small or insignificant, will make me understand the big picture better, I know it will ultimately make me a better teacher--I want to know. Now I NEEDED to know.

I emailed the presenter the next morning (this morning) requesting the answer, to which she replied that I would have to wait until our next seminar with her in late February. She only hinted that it had something to do with "where" they came from, rather than "how" they are used. My suspicions exactly.

I guess I have no choice but to be content for now with my over-analysis argument. But as my wife reminded me, in today's educational setting, it's really unfair to single out any one group. ALL triangles are special, not just the two. To treat any triangle any differently from all the others is to show favoritism. She's right. In just a few years, every triangle will be "college-ready."

## Thursday, January 31, 2008

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