I teach Calculus this year to this really great group of kids. It’s a very small class, only 9 students, and they are so eager to learn and work really hard.

They have their mid-year exams this week, so I know most of them spent a good part of the weekend preparing for it. When my students came in today, a group of three girls in the front had these big grins on their faces. One of them said, “Guess what we figured out this weekend?” She seemed very excited.

“What did you guys figure out?”

“Well, we were studying yesterday, and we figured out that if you take the integral of the circumference formula, you get the formula for area of a circle – and that makes sense because the integral gives you the area under the curve.”

“Are you serious?” I said it with all the amazement I could, and I have to say, I was thrilled.

Another of these girls wondered why we don’t teach these formulas this way. I explained that the idea of the integral is a big concept for a sixth or seventh grader, and that’s probably the reason. I also reminded them, especially the students I had last year, that sometimes they ask me about why we are doing something or what it connects to, and I find myself giving vague explanations and promising them that it does connect to something they will see. I ask them to trust me, and usually they do.

I also pointed out that, if I had told them this a long time ago, they never would have had that moment of making that connection on their own. I would never want to take that away from them. I remember these moments myself, as a student, when I would make these kinds of connections. It made me feel pretty brilliant, I must admit. The fact that these students made this connection to something they learned a long time ago was only beat out by the fact that they were so excited that they made this connection on their own. What more could a teacher ask for?