What is the point of presenting a proof?
I teach an advanced—I like to call it “University-Level”—geometry class for 11th and 12th grade students who have finished Calculus, or who are gluttons for more punishment than a single math class a day affords. And while much of their work is either a group discussion, or quasi-independent—they spent most of this week working on proving concurrences and collinearities they selected from the book 99 Points of Intersection—every so often I find myself doing what was done unto me in my college math courses: presenting the proof of a theorem.
I like to think of myself as an engaging presenter, and today’s example—a version of the trigonometric proof of Brahmagupta’s Theorem from Zuming Feng and Titu Andreescu’s book 103 Trigonometry Problems from the Training of the USA IMO Team—was one of my best. I put the trigonometric steps first to motivate the brutal algebra, and many of my students were able to stay the half-step-ahead that’s necessary to fully understand a proof. (That is: they understood not only why a particular step was justified, but why it might be desirable.) Cheers came at the end; one student stood up and shouted “That’s freakin’ awesome!” A good class, right?
I’m not so sure. I mean, it was fun, and the result extended the guided proof of Hero’s theorem they worked on yesterday. And as a capstone to a unit on advanced theorems and proofs, Brahmagupta’s formula is hard to beat. But what did students actually take away from the activity? The fact is, I don’t even know. They saw a new-to-them idea about trigonometry in cyclic quadrilaterals, that you can use the law of cosines on opposing angles to relate sides and angle measures.1 A few saw ahead to the fact that factoring complicated polynomials is almost always more useful than multiplying them out. Frankly, there was no real assessment, so I can’t even say with confidence what any particular student took away, other than awe. Really, at base, they watched me produce a proof.
And yet I think the experience was worth it. The students got to see an argument too elegant and too long to generate on their own. They saw a proof use four different ideas from Algebra, Geometry, and Trigonometry in combinations ordinary classes would avoid. And they experienced mathematical beauty.
As teachers, we typically entangle two distinct tasks: “generate X” and “comprehend X”, even when only one of those tasks is actually useful. For example, we spend as much or more time teaching language students to write correctly in the target language (not very useful) as we do teaching them to read it (absolutely critical if you want to understand the street signs, find out what a newspaper says, or experience literature). But if one reason for this entanglement is lack of clarity about what’s important; another is that sheer comprehension is difficult to assess. We teachers don’t have tricorder-like comprehend-o-meters: the way we figure what students have learned is by getting them to produce something. And it’s hard to create tasks that assess whether a student understands X without asking her to generate X. Finally, from a philosophical perspective, educational positivists might ask what “comprehending X” really amounts to besides the capacity to generate X or something like it.
I’m not yet sure how (or even whether) I’ll get my students to show me what they learned today, whether they comprehend the proof or can generate a similar one or…. But I stand by my teaching. I could trot out the old saws about preparing kids for college, in this case, college-style lecture classes, but I won’t: the pervasiveness of terrible math teaching in higher education is a poor argument for bringing that same so-called pedagogy into the lives of high school students.
Instead, I would argue that the experience of following a complicated argument and seeing different ideas interlock is a valuable one, if only because it expands one’s conception of what is possible. Being excited about math, and being in a group of people excited about math all at the same time, is also important, maybe just as important as anything else. And the proof was really cool.