This week, I brought a nice activity to my geometry classes. At least, I thought it was a nice activity, and almost every student was fully engaged. The task itself doesn’t seem too exciting, maybe. Students used a compass to draw a circle, and then measured the diameters with a ruler. Then, they carefully used twine. Yes, twine, left over from my graph theory unit last year. I couldn’t find my string or wire, so twine it was. Anyway, they used twine to carefully trace around the edge of the circle, and then straightened the twine to measure the circumference. Then, they got together in groups of 3-4 and tried to find a relationship between their numbers. It wasn’t long before they discovered a similarity between the ratios of the circumference and diameter, which they then realized was π.
Then, we went a step further, and looked at the class averages, and also looked at the % error. Students then started to make a conjecture that there was a correlation between size of the diameter and the % error. I indulged them in this direction that was much more interesting than the standard C=πd lesson that this was supposed to lead to. We used Google Sheets to analyze all of this data that we collected. (See the scatterplots below).
After this class, a student came to me. This student said, “What was the point of that lesson? I mean, wasn’t it just the formula for the circumference of a circle? The assignment was just about using that formula. Couldn’t you have just told us the formula, done some examples, and be done with it? That’s what teachers are supposed to do. The rest of it might have been interesting, but wasn’t it kind of a waste of time?”
So, where did I go wrong here? Or did I? I haven’t surveyed every student, so maybe the one student was an exception. I thought about this for a while today.
Then, tonight, I realized something. I’m a bit of a hypocrite. Not a complete hypocrite, but just enough to have an interesting contrast of lessons. Later today, during a trig class, after going through a few derivations of various identities (or telling the students that there’s an identity for a double angle formula, go ahead and find it), we got to a few identities where I told them, “I’m just going to give you these few identity formulas. We won’t go over the derivations, but we’ll practice using them a few times so you know how to apply them in different situations.”
Wait, what? That’s not who I am as a teacher, right? I thought I didn’t just give students the formula and tell them to use it. They have to work for it.
It all comes down to context, though. In some cases (Pythagorean formula, quadratic formula, exponential growth functions, angle sum identities), there are very, very rich and meaningful lessons, extensions, and mathematical conversations that can be generated. In other cases (half angle identities, product-to-sum formulas), maybe there are meaningful lessons that I just haven’t developed, but I haven’t developed them, and I do have limited time.
Common core has sometimes been referred to as a way to refocus mathematical education to focus on deeper understanding, and pull back from the mile wide, inch deep approach to math education in years past. I mostly agree with that, but every once in a while, it sometimes makes sense to just give a student a formula. Hopefully I can always do it only when there isn’t a rich concept or application that can come out of trying to derive that formula, though. Speaking of which, anyone have a really interesting lesson on half angle formulas that doesn’t sound completely contrived?