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.

Another great post Ben. I remember going through something similar to what you describe with a student when I was teaching repeating decimals. The student couldn’t accept the fact that .9 with a vinculum (hardly anyone knows that word) was 1. He also didn’t buy the usual method of showing this by multiplying both sides of an equation, one of which was an infinite repeating decimal by 10. I never did make him a believer.

You have just taught me the word vinculum 😉

This is great. I’ve been taking the “How To Learn Math” MOOC from Dr. Boaler at Stanford and it’s very heavy on students developing a growth mindset and allowing them to feel comfortable making mistakes. I feel like having students replace their expression of “I don’t get it” with “I don’t buy it” flips the script. They can reach an obstacle without feeling like the cause of the obstacle is their perceived lack of mathematical brains. It puts the onus on you to find a different approach that suits them.

I completely wanted to take that course! I’m currently teaching at SPMPS and it’s as intense as I anticipated, so I decided I didn’t have time on top for a MOOC; but I totally want to know what I’m missing!

Very thoughtful and interesting post, ben

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?

This applies not only to content, but also to assignments and assessments. We try (hopefully) to design our courses so that 1) the activities we have our students participate in lead to meaningful learning and so that 2) our grades reflect accurately what our students know/understand/etc. But just as not all of our students will “buy” the same explanation, as you described so nicely in your post, 1) not all students will substantively benefit from the activities we design for the class as a whole and 2) not every student will have will have their understanding/knowledge adequately assessed by the homework/projects/exams/etc. that we use to determine grades.

The past few times I’ve taught an undergraduate course I’ve tried to build in a some flexibility regarding assignments and assessments. This coming semester, in the Calc 2 course I’ll be teaching, I plan to be very clear on this issue from day one. If any student thinks they would be better served by different assignments I would be happy to talk with them about that. And if any student thinks they would be more accurately assessed by a different grading scheme, I’d be happy to talk with them about that. I’ll need to be particularly clear that for a student to earn a certain grade by alternative means will require convincing evidence on their part. But if they can clearly show me what they know, understand, etc., I’ll be delighted to give the appropriate grade.

Aaron, how has this gone? Have students taken you up on it? What’s the negotiation like? This is a beautiful vision and I’d love to know more about what you’ve found to be involved in making it practical.

Thanks, Ben. This is a very thoughtful and interesting post. A small distinction, but an important one. I was talking to a teacher the other day who said they were lost the first time they had to teach geometry because they had never been asked to prove anything mathematical in school. It may be good to replace the “get” word with the “buy” word in students’ vocabulary and not just our interpretation of their words. From an early age if teachers are trying to prove things to students, it may change the school math culture for the better. Something to think about this semester.

I totally agree. The more entitled to their dissatisfaction students feel, the better.