I was having a conversation a few weeks ago with a computer programmer and math enthusiast whom I’ll call Dorian. He was arguing very passionately that talking about a square root of was the wrong way to introduce complex numbers. He recounted this moment in his own schooling: 16 year old Dorian, told by his teacher “we introduce a new number whose square is …,” asking, “but I can prove that the square of any number is positive, what about that?!” His teacher wasn’t able to satisfy his objection and made him feel that it wasn’t valid. He left the experience feeling angry and frustrated and that his question had been treated as a failure to understand.
Dorian later learned that complex numbers can be visualized as a plane containing the real line; that addition of points in this plane is just vector addition; and that multiplication is done by multiplying the distances from the origin and adding the angles from the positive real axis (see here for a brief explanation if desired). Here was a concrete model for the complex numbers, with concrete geometrical interpretations of the operations and . And it was clear to him that in this model, there is a point, in fact two points, whose squares correspond to the point on the real axis. But philosophically, this fact is a consequence of the concrete geometrical description of the operations in the plane, rather than an ontologically dubious starting point for the whole project.
Dorian concluded that actually this model, via the geometry of addition and multiplication in the complex plane, is a pedagogically superior introduction to the complex numbers. His argument is that it presents no ontological quandary. Nobody will object to a plane. Nobody will object, at least on philosophical grounds, to these new definitions of and , as long as you can prove they have nice properties and coincide with the old definitions on the real line. You’re not saying anything so wildly speculative as “postulate a square root of …”
I am not writing this post to get into the question of whether Dorian is right about this. I see lots to say on both sides. What I am writing this to say is that there is a lesson in Dorian’s story much deeper than the question of how to introduce the complex numbers. That is not the real question here as far as I am concerned.
The real question is this: when you’ve picked your approach and gone with it, how will you deal with the students it doesn’t work for?
Now you can always obsess about how to introduce a topic, and I believe there is basically always value in thinking and talking about the pedagogical consequences of different ways of looking at things. And I think some models for ideas are legitimately better than others. But no model will speak to every student. This point is so important, and was so lost on me as a young teacher, and is lost on so many (especially young) teachers that I have spoken with, so excited that they are about the way they have thought of to present negative numbers or whatever, as though miraculously everyone in the room will get it this time, that I need to repeat it:
There is no model that is the right model for each and every student, each and every time.
No matter how awesome your idea for how to think about XYZ concept is, there will be somebody in your class who will have no idea what you are talking about. To me, the big question here is, what are you going to do about it?
More specifically, how are you going to treat their thinking?
Now, I like to think that nobody reading this blog would be so callous as to intentionally make a student feel stupid for asking an honest question. But there are far subtler ways to do it. The one I most want to warn you against is the sin I know I’m guilty of: being so wrapped up in the awesomeness of your presentation that the kid who doesn’t get it does not compute to you. You say whatever you say out loud but in your mind you’re like, “wait – you don’t understand? Huh?” Or, you’re like, “oh my goodness can’t you just see it as I do?”
Regardless of what you say out loud, having such a response in the back of your mind invalidates whatever obstacle the student is facing. I want to suggest an alternative:
Take the case that any earnest failure of a student to see your point of view is actually coming from a legitimate mathematical objection.
This is how you treat dissatisfaction with honor.
I don’t care what the kid’s IEP says. Mathematical convention does not require us to check somebody’s Wechsler results before they are allowed to raise an objection. If they don’t buy it, they don’t buy it. Now it’s your turn to understand their objection and answer it.
“I don’t get it.” “I don’t buy it.”
A student I’ll call Manny, whom I had in my 2003-4 AP Calculus class, came to me around March and said something like, “this entire class is based on a paradox.” He objected to my (retrospectively totally hand-wavy) discussion of limits. It never gets there, so how can you talk about what happens if it were to get there?
I tried to answer Manny’s objections; I spent some time with him on it; but he left the conversation unsatisfied. Retrospectively it is clear to me that this is because (a) I didn’t get what the problem was, and (b) to my shame I didn’t consider the possibility that there was really much to it. Then, less than a year later, I read The Calculus Gallery, whereupon I learned that actually Manny’s objection was more or less exactly Bishop Berkeley’s famous objection that in due time forced mathematicians to invent real analysis. For a sense of the importance of this development, let me mention that I have read, though I don’t recall where right now, that the development of real analysis was really the event that led to the birth of modern mathematical rigor.
So, yes, I am on record as having treated as essentially invalid an objection that actually led to the creation of modern rigor. Don’t let that be you.
If they don’t get it, take the case that there’s a legitimate mathematical objection behind that. Treat their “I don’t get it” as “I don’t buy it.” Now getting them to buy it is your job.