I got into teaching to work with struggling kids. I’d enjoy working with an entire class of motivated and able kids, but it would also come as a shock. Most of my time I spend pushing kids up the hill and praying they don’t roll backwards.

But I do have exceptionally bright kids in my classes, too. My geometry classes in particular are a joy, since even the low ability, low incentive kids know they’ve finally escaped algebra and aren’t eager to repeat that experience by repeating geometry. So it’s a well-behaved group with decent motivation. On the other hand, 80% of my Algebra II classes scored Far Below Basic or Below Basic in Algebra two years earlier and a few of them are much harder to keep in line, even though the bulk of them are well-behaved and want to learn, not out of any love of math, but because they understand, thanks in part to my frequent rants, that they will be taking a placement test in a year or two and the only good thing about my class is that it’s free.

But I do have some students who chose an easier course because they are athletes, or because they struggled in geometry, or just because someone somewhere made an odd placement decision, and that are in fact quite strong in math—and of course, there are always a few students who finally “get it” and who actually start to grasp the material (this is extremely rare, and one major reason why value added is a problem in high school teacher evaluations).

Anyway, on Thursday, I had two interactions with bright students that stay with me because of their infrequency. I’m not complaining. It was just fun.

**Incident #1**

A solid geometry student had done horribly on her last test. She doesn’t often engage with me and remains a bit distant, but after I turned back the tests I checked in with her.

“I do not get special right triangles. I don’t understand how they work. I get similar triangles, but I can’t ever remember the ratios and I can’t see how it works.”

I drew a right triangle on the board with the two legs labelled “x” and the hypotenuse labelled “hyp”. “What kind of triangle is this?”

“Isosceles right.”

“Okay. Create an equation to solve for the hypotenuse using the Pythagorean Theorem. I’ll be back in a bit.”

I came back, and she was working through it, a classmate tossing in advice and argument.

“No, it’s square root of two! You took the square root of both sides!”

“Oh, that’s right.” and she had it solved, the hypotenuse equal to x * sqrt 2.

I pointed to the class notes which were on the board. The isosceles right triangle was labelled x, x, x sqrt 2.

“Oh! I get it.”

I then sketched out a 30-60-90, asked her what it was and she correctly said it was a half of an equilateral triangle. I told her to label the sides, making sure she put the x opposite the 30. Then I told her to solve for the second leg. When I came back, she’d finished that up and was in a great mood.

Had I taught the Pythagorean method before? Yes. Several times. Sometimes they just aren’t ready to get it until they’re ready to get it. But only a strong student can grasp the algebra of the Pythagorean proof and see how that knowledge can help her remember the ratios.

**Incident 2**

One of my strongest Algebra II students was struggling with synthetic division, which I introduced as a method of testing quadratic values (more on that later). She asked me to explain it to her again—not just the how, but the why and the what.

I began from the top and ran through it all. When she grasped that the division process revealed not only the quotient but that the remainder was the equivalent to evaluating the function at the divisor value, she said “Wow. You might almost think that was planned.”

I laughed in unexpected pleasure. I am not a believer in God nor a mathematician, nor am I a proponent of the “math is everywhere, math is beauty” propaganda that true math lovers preach. But the Remainder Theorem, like the Fundamental Theorems of both Algebra and Calculus, is indeed enough to make even me wonder if there’s some Grand Design. That a student of mine should reach that conclusion after a largely utilitarian but comprehensive explanation by yours truly was an unlooked for joy.

I was telling this to a colleague, and he reminded me of the famous quote: “God exists because Arithmetic is consistent. The Devil exists because we can’t prove it.”