This is a followup to my last post, processing a provocative conversation I had with my awesome friend and colleague Justin Lanier.

**I want to try it vs. I want to hear how you did it**

In my previous post I was saying I’d kind of like to take an Algebra I class and alternate units between my profound homespun curriculum design (if I may) and carefully group-digesting the textbook, and then at some point start letting the class choose which approach to follow. This gets to something that Justin brought up, which broadens this question beyond textbooks:

You’re presented with a mathematical problem. Unless you’re on the research frontier (and often even then), there are always two things you can do:

1) Try to solve the problem

2) Find out how somebody else solved the problem

Both things are, to paraphrase Justin, essential parts of mathematical experience. Both are options that the two of us sometimes go for.^{1}

*When, if ever, do we give our students this choice?*

What percentage of the time do we make it for them?

Justin was struggling with this question (what percentage of the time am I making this choice for my students and consequently what percentage of the time am I letting them make it?). In particular, with the sense that the former percentage is quite high, and not feeling entirely right about that. Having now had a month to mull on it,

a) I dig that Justin raised it as a question to be concerned with because now we get to choose deliberately what we want the answer to be and then try to implement that; and

b) depending on the class, it’s fine to make the choice for your students a very high percentage of the time.

More detail about (b) – this is related to another thing on my mind a lot lately, partly due to the conversation with Justin and partly due to its obviousness as a thing to think about:

In what (pedagogically relevant) ways am I similar to my students as a math learner and in what ways am I different?

I have a partial answer to this question, in the form of a principle:

The most important difference among math learners is not who is visual and who is auditory, or who is linguistically-oriented and who is nonverbal, or who is algebraic and who is geometric, or anything like that. These are interesting, useful differences, but they’re not the most important.

It is *certainly* not who is “fast” and “slow” or any such meaningless and pernicious bullsh*t. (More to come on this topic.)

The most important difference among math learners is, when faced with a problem to solve, an idea to grasp, a theorem to prove, **who believes that they can they can solve/grasp/prove it and who doesn’t.**

Category I: People who believe they can solve/grasp/prove it.

Category II: People who do not believe they can.

I believe that the best thing math teachers can do for the world is take people in Category II and move them to Category I.

To repeat my question: in what (pedagogically relevant) ways am I similar to my students, and in what ways different?

My partial answer: One difference is that I am in Category I and most of my students are in Category II. I think this is probably the main difference.^{2}

Now, when faced with a math problem and thus the choice above between option (1) (try to solve it) and option (2) (find out how somebody else did), people in Category II always choose option (2). (Why? Clear: because they don’t really believe (1) is an option.) Meanwhile, in order to learn math, you need to be taking option (1) a big percentage of the time. This is true of everybody but *especially* true of the folks in Category II. How could you ever come to believe you can solve it except by solving it, a lot of times? And how could you do this if you don’t first try to solve it?

Now as I write this I can’t escape the feeling that I’m writing stuff down that is obvious to the point of being tautological. But I think it adds up to something useful in considering Justin’s question. (How often do we choose for our students between (1) and (2) and how often do we let our students make this choice?)

What it adds up to is this: if you teach a lot of Category II folks, you need to do a lot of choosing (1) for them, since they need it desperately and won’t be choosing it on their own. In fact, you need to be all up in their sh*t, not just at the moment they are faced with the choice between (1) and (2) but

*When they have a sound thought you need to point it out to them

*When they have a wrong thought that leads to a productive conversation you have to call their attention toward the productive conversation it led to

*When they see something from a new angle that illuminates it in a new way, you have to make sure the reflexive feeling of exhilaration doesn’t go unnoticed

*When they jointly solve a problem you need to make sure they noticed their own contribution to it

*Etc.

In other words you need to be in control of quite a few elements of their math learning process. Their choice between (1) and (2) is only one of these elements.

So under this circumstance (namely a room full of Category II folks), it’s really actually quite appropriate to be choosing for them – in particular, choosing option (1) for them.

What you *don’t* want to be doing is choosing option (2) for them when they were ready to choose option (1). I think one of the important sensitivities to develop as a math teacher, along with the ability to recognize at a glance the “I don’t understand but I don’t want to ask a question” face and the difference between somebody thinking about it and somebody being totally stuck, is a sharp ear for the quiet, tentative, tinkly sound of a Category II student who is about to choose option (1). If you hear it, do not intervene.

This all said, one of the things I took from the conversation with Justin was a reminder that although Category I is always a better place to be than Category II, option (1) is not essentially better than option (2). They are both legitimate choices. Category II people need a lot of option (1) and won’t choose it for themselves, so you gotta take control there, but in general there is a time and place for each kind of mathematical experience. Each one’s got something going for it the other one doesn’t:

By far, all the most exhilarating mathematical experiences of my life have followed the selection of option (1). These were the times there was adrenaline involved, so that my hand was shaking as I put pen^{3} to paper. Nothing in mathematical life can top the peak experience of *realizing* something.

By the same token, option (1) immediately preceded all my most profound experiences of mathematical *power*. (Empowerment, if you like.) I pretty much walk through life feeling like I can accomplish anything with enough patience, and solving mathematical problems *myself* has been maybe the biggest thing that’s made me feel that way.

However, by definition, I had chosen option (2) prior to every time I’ve had the privilege of seeing something beautiful that *I would never have thought of myself*.

Relatedly, most of those experiences of empowerment I mentioned above involved some option (2) that came before the option (1) and empowered my actually solving something. In an abstract algebra class last year, my professor said, “now I’m going to solve a cubic… with my bare hands.” I didn’t get that sense of power till I did it myself later that summer, but I probably wouldn’t have done it myself if I hadn’t watched him do it.

So Justin’s train of thought feels like it’s raised my consciousness a bit. This is something to be deliberate about. For students to appreciate both options (1) and (2) and have the choice as much of the time as possible feels like a beautiful ultimate goal. But the main thing is just to keep in mind that not only **[option (1) vs. option (2)]**, but also **[our choice vs. students’ choice]**, are pedagogically significant arenas, and we should stay aware of this as we navigate them.

One other thing I recollect from the conversation with Justin that I want to record before wrapping this up:

I really love the act of explaining things I think are awesome. But this is not a reason to do it in a teaching situation. The idea is to be pedagogically deliberate about making the choice between options (1) and (2) for your students. If I decide you’re going to listen to me share a solution, I should be doing it because I think it’s what’s best for you, not because I love doing it. I will repeat myself:

The impulse to share something awesome needs to be *entirely repressed* while teaching. Sharing something awesome should come from a judgement of the pedagogical need, not a desire to share.

(We are allowed to love it once we’ve decided it’s pedagogically called for; but the desire to do it can’t be driving this call.)

Notes:

[1] We didn’t talk about it at the time, but there is a third very important option, which is to work on the problem with somebody else. This is sort of “between” the other two and also sort of qualitatively different. But the two choices above work to define the extremes of a spectrum, so I’ll stick with them for the purpose of this discussion.

[2] It is not the subject of this post but I have to acknowledge what I think is the *second* most important difference among math learners: when faced with a problem to solve, an idea to grasp, a theorem to prove, who sees *value* in solving/grasping/proving it and who doesn’t.

[3] Yes, pen. I do all my math in pen. I am not going to tell you to stop telling your kids they have to use a pencil, but honestly I don’t like this pencil-only doctrine. “Put that pen away right this minute! If you don’t use pencil, you won’t be able to *hide your mistakes and pretend they never happened!!*” This is the message we’re going for?

(If you want to tell me you have a particular kid who really should be using pencil, I believe you. It’s the general principle I’m objecting to.)

I totally agree that it’s important for a teacher to confront “category 2” students with problems and activities that will stretch, engage, and empower them by getting them to figure something out on their own. But I’ve begun to think that it’s also important to empower these and all students in another way–to put them in touch with a world of mathematical resources. “Category 2” kids may only look to their teacher for the answers, while “category 1” kids have their own internal resources, too. Even the textbook for the class often seems out-of-play. But I value the fact that when I’m confronted with a problem or mathematical idea, I have people I can talk to and email, books and textbooks to peruse, an internet to search, online journals and articles to explore, and software to assist me. In tapping into these resources, I make connections and discoveries that wouldn’t happen anywhere along the “by your lonesome” to “someone tells you” continuum. This kind of active stance toward mathematics–even if it means I’m just ingesting something that someone else has already figured out–is what I’d ideally want for my students, equally as much as making them engaged problem solvers. I’d be just as ecstatic about a kid coming into class enthusiastic about a new proof she researched online as one she figured out herself, because they both show mathematical initiative.

WORD I was hoping you’d weigh in.

Thanks for being so articulate about the vision of “put[ting] them in touch with a world of mathematical resources.” Beautiful. I meant, in a vague sense, to include this inside what I called “option (2)” but I wasn’t articulate about this and I think you’ve developed this vision more clearly than me so I’m glad you committed it to print here.

This also gives me a context for clarifying something I now see I was incipiently trying to say above: in putting students in touch with the world of mathematical resources, I think the difference between “Category I” and “Category II” students is very important. To make the kind of empowered use of these resources that I hope for, a student has to be relating to them a certain way, one that will require a lot of very controlled training for Category II kids in particular. Specifically, the student has to go to the resource a) with the desire and intention to understand, and b) with the sense that what the resource has that they don’t is nothing more powerful or mysterious than some specific knowledge. In contrast:

When I took that complex analysis class last fall, a major tool for some of the undergraduates in the class for doing the homework was to get to a point where they wanted to be able to assert something in a proof, and then search wikipedia to see if they could find a theorem that would give them this step. There was no attempt to understand the result; they would just quote it and move on. UGH.

What I’m getting at that to make use of an internet resource, or the library, that I would be excited about, a student has to have as a prerequisite an understanding of her own capacity to be a mathematician. (Engage in mathematical thinking.) The kid you imagined does understand this. (That’s why she was enthusiastic about the proof.) A student who isn’t aware of her own mathematical capacity doesn’t naturally engage mathematically when she reads or hears about something somebody else figured out. This student needs explicit training in how to do this.

It adds up to the need pedagogically to make the choice between option (1) and (2) a lot of the time with kids in category II, if I can revert to my oversimplified terminology. We need to choose option (1) for them a great deal of the time, and control the setting under which they engage the problem, so that they can come to know themselves as a mathematician. Meanwhile, we need sometimes also to choose option (2) for them (thank you for helping me see this more clearly) and again, we need to keep tight control over the setting in which they engage the resource (classmate, teacher, book, internet…), so that we can train them in how a mathematician engages a resource. (I feel pretentious with all this talk of “mathematician” but I think it’s getting at what I mean.) The weight of what I’m saying is that while all this care and control needn’t be taken with category I students, for category II it’s very important.

I will be thinking about this as I teach next term. Thanks!