I have a small calculus class this year (only 6 students), and to be honest, I wasn’t sure what to expect at the beginning of the year. I have taught these kids for the previous two years and so I knew that none of them were exceptional math students. Great kids, who mostly don’t seem to care too much about math. Boy, have they surprised me. The kid who last year could hardly keep his eyes open in class is this year coming up with some of the best input of all. The one I would have considered the weakest student got the highest grade on the quarter test. I’ve been having a lot of fun with this group. Today we learned the trig rules for derivatives. This might have been my favorite day this year. I started by asking them to graph y=sin(x) and label all important points. Then we used a calculator to find the numerical derivative at several different points. We plotted these points on the same graph as y=sin(x) until they were all certain that the derivative was cos(x). Some people may disagree with me on this, but I didn’t attempt to prove this fact, we just accepted it based on our results.
Then, without any prompting on my part, someone asked “so, is the derivative of cos(x) = sin(x)?” And they all set off on figuring out if that was true. It didn’t take them long to figure out the correct rule. And now they had a graph of this derivative as well.
Next question, again without me doing anything, was “so does that mean that the fourth derivative of sin(x) is just sin(x)?” I asked why he thought that would be true and he explained that each successive derivative of sin(x) was a graph that was shifted left pi/2 units. Once you shift it 4 times, you will be back to a sin(x) graph. I have to admit, I had never thought of it that way. That’s one of the things that is so much fun about this class. They see things so differently than most. Naturally we had to verify that this student was right in his reasoning.
Then the next comment came. “Where’s tangent? Tangent has to be in there somewhere.” Okay, let’s talk about tangent. And so they derived the rule using trig identities. Then they wanted to know if the derivatives of tangent followed a pattern like the derivatives of sin and cos. Which naturally brought up the need to know the derivative of sec(x). And so it went for the entire hour. I got the ball rolling by asking them to graph sin(x) and find it’s derivative at several places. They took it from there, and went exactly where I was planning to take them. All I had to do was facilitate their discussion and ideas. It just may have been my favorite lesson so far.
On a related note. This year, as I have tried to step back and let kids take more ownership of their own learning, it has been a lot of fun for me to see the creative ways they approach things. Sometimes it is exactly how I would do it, but many times they come up with a process or a way of thinking that I was not expecting. I’m loving the opportunity to see how they are thinking.