October 10 still marks the best moment of the year. Pre-AP class was rocking at a perfectly quick pace (just barely beyond the comfort of previous understanding), it was mostly student-guided, and the few words I spoke met their mark perfectly.

Students worked through a packet that I stole from the 8th grade math teacher, who stole it from some other teacher. The packet guides students toward discovering the Pythagorean Theorem. By the bottom of the second page, students come up with "a squared plus b squared equals c squared" on their own, without yet realizing it's significance.

I basically stayed out of their way, except to give them the amazing push of time constraints ("Work with your group on questions 9 and 10. I'm setting the stop-watch at 3 minutes. Go."), and to come back together occasionally and summarize and synthesize their progress ("So if you're telling me this square has an area of 25 units, what would we have to do to find out the length of one of its sides?").

It was challenging, but not so much that students shut off. The lesson progressed beautifully and I think the students could feel the excitement culminating. At one point I even said, "I don't know if you guys can tell, but I'm loving this." Right when I had them on the hook, based solely on truths that they had discovered, I pointed to the equation that they came up with and unveiled it as "Pythagorean's Theorem."

"So you've shown me that whenever we have a right triangle, we know that the sum of the squares of the two legs equals the square of the hypotenuse... This is what Pythagoras discovered twenty-five hundred years ago, and you discovered it in 40 minutes today. This is a big moment. You opened a door today that you will be able to use for the rest of your lives. And once you see the things we can do with it, it's going to blow your mind."

There must have been an urgency in my voice because the students were beaming and burst into a round of applause. It couldn't have been scripted any better, and it all came together organically, almost magically. As challenging as teaching has been, these little rewards are amazing.

A couple weeks ago, I was having a cup of coffee before school and working out a problem that I was going to give to my students. I found myself doing some long division - dividing by 7, to be exact - when all of a sudden it hit me: that sweet, sweet sensation of uncovering a universal truth.

Since the number I was dividing by 7 wasn't a multiple of 7, my answer wasn't going to be a whole number. So as I added a decimal point and kept dividing, I realized (to the Patty Mayonnaise sound of angels singing) that I could sit in that chair and keep writing all day because the numbers after the decimal were never going to end. Not only that, but it would be true any time I divided by 7 and my answer wasn't a whole number.

It might seem trivial (or, at the very least, dorky), but the moment was euphoric. I'm sure hundreds of thousands have made the same useless realization before me and have since taken it for granted, much like I take 7th grade knowledge for granted every day. But I've forgotten how exciting and eye-opening those light-bulb moments can be. Though they happen less and less frequently to me, they are happening to the students all around me every day. I need to do a better job of celebrating them.

Written on 1/5/10 by a 7th grader, to be returned on the first day of 8th grade.