So here’s another one that I suppose is kind of obvious, but nonetheless feels like big, important news to me:

It’s possible to only partly understand what somebody else is saying.

Let me be more specific. When you’re explaining something to me, it’s possible for me to get some idea from it in a clear way, to the point where my understanding registers on my face, but nonetheless the other 7 ideas you were describing I have no idea what you’re talking about.

<Example>

I am a 9th grader in your Algebra I class. You’re teaching me about linear functions. You are explaining to the class how to find the -intercept of a linear function, in slope-intercept form, given that the slope is and the point lies on the line. You explain that the equation has the form and that because we know the point is on the line, that this point satisfies the equation. Thus you write

on the board. At this point I recognize that we are trying to find and that we have an easy single-variable linear equation to solve. My face lights up and you take mental note of my engagement. Maybe you even ask for the -intercept, and since I recognize that this must be I calculate and raise my hand.

Meanwhile, I have only the vaguest sense of the meaning of the phrase “-intercept.” I have literally no understanding of why I should expect the equation to have the form . I have a nagging feeling of dissatisfaction ever since you substituted into the equation because I thought and were supposed to be the variables but now it looks like is the variable. Most importantly, I do not understand that the presence of the point on the line implies that its coordinates satisfy the equation of the line and conversely, because on a very basic level I don’t understand what the graph of the function is a picture of. This has been bothering me ever since we started the unit, when you had me plug in a bunch of values into some equations and obtain corresponding values, graph them, and then draw a solid line connecting the three or four points. *Why am I drawing these lines? What are they pictures of?*

Occasionally, I’ve asked a question aimed at getting clarity on some of these basic points. “How did you know to put the 6 and 11 into the equation?” But because I can’t be articulate about what I don’t understand, since I don’t understand it, and you can’t hear what I’m missing in my questions because the the theory is complete and whole in your mind, these attempts come to the same unsatisfying conclusion every time. You explain again; I frown; you explain a different way; I say, “I don’t understand.” You, I, and everyone else grow uncomfortable as the impasse continues. Eventually, you offer some thought that has something in it for me to latch onto, just as I latched onto solving for before. Just to dispel the tension and let you get on with your job, I say, “Ah! Yes, I understand.”

</Example>

This example is my attempt to translate a few experiences I’ve had this semester into the setting of high school. The behavior of the student in that last paragraph was typical of me in these situations, though it would be atypical from a high school student, drawing as it does on the resources of my adulthood and educator background to self-advocate, to tolerate awkwardness, even to be aware that my understanding was incomplete. Still, often enough I ended up copping out as the student does above, understanding one of the 8 things that were going on, and latching onto it just so I could allow myself, the teacher and the class to move on gracefully. Conversations with other students indicated that my sense of incomplete understanding was entirely typical, even if my self-advocacy was not.

The take-home lesson is two-fold. Point one is about the limitations of explaining as a method of teaching. Point two is about the limitations of trusting your students’ (verbal or implied) response to your (verbal or implied) question, Do you understand?

The basic answer (as you can tell from the example) is, No, I don’t.

Now I myself love explaining and have done a great deal of it as a teacher. I fancy myself an extremely clear and articulate explainer. But it couldn’t be more abundantly clear, from this side of the desk, how limited is the experience of being explained to. I mean, actually it’s a great, key, important way to learn, but only **in small doses** and **when I’m ready for it**, when the groundwork for what you have to say has been properly set.

I am somewhat chastened by this. I am thinking back self-consciously to times when I’ve explained my students’ ears off rather than, in the immortal words of Shawn Cornally, “lay off and let them fucking think for a second.” It’s like I was too taken with the clarity and beauty of the formulation I was offering, or in too much of a hurry to let them work through what they had to work through, or in all likelihood both, to see that more words weren’t going to do any good. Beyond this, I’m thinking back on the faith I’ve put in my ability to read students’ level of understanding from their faces. I maintain that I’m way better at this than my professors, but I don’t think I’ve had enough respect for how you can understand a small part of something and have that feel like a big enough deal to say, and mean, “Oh I get it.” Or to understand a tiny part of something and use that as cover for not understanding the rest.

Thanks, Ben—this is hugely validating. I’ve been explaining as little as I can in my calc class and explaining very nearly nothing in my pre-calc class. As a result I’ve gotten to listen to students do their own explaining. Result: I can see the yawning chasms in their models. I don’t yet know how to efficiently address the gaps–way too often, the moment that I perceive the void is the moment I start trying to explain, and, of course, it doesn’t work.* It’s very disheartening.

I still feel encouraged by what you’re writing, though, because when I explain over them, papering over those gaps that would otherwise be exposed, none of us are satisfied, the whole experience feels like a sham, and most importantly,

the gaps are still there.*I’m excepting the small doses when students are ready that you mention. Those really do work.

Thank you Ben. Wouldn’t teaching and learning be so much more of everything we hope for if there were more safe places to reflect on our experiences and design improvement based upon those discoveries? I can imagine how powerful it would be to have parents, students, and teachers share in that work.

Looking forward to following your blog.

I think the biggest issue here is that we are trying to explain something that is essentially abstract in nature – that’s why students want to know ‘what is it a picture of?’ – they want to turn it into something concrete and that they can understand.

In many respects trying to teach abstract concepts such as these to students at this age is always going to be somewhat fruitless. Many students will never grasp what it is that they are doing or learning about. I guess, this might sound like a sad thing, and if that is all we are focused on, well it really would be! As educators we are also promoting a love of learning, and a love of maths (of course).

All I want to see is as many students as possible continuing to learn more about these abstract concepts, when their brains are ready for it, and for some, that won’t be until they are in their thirties. But that’s ok – let’s just not ever turn them off with poor teaching – thanks Ben for this post – great to see such dedicated teachers out there.

You know I appreciate your engagement and interest, and I pretty much completely disagree with your point! I don’t think there’s any age-bound issue of cognitive development at play, and I don’t think there’s anything in the HS curriculum that students are not developmentally ready for if it’s taught in a way that engages their inquisitiveness and empowers them to think for themselves.

To make sure you have a piece of context I’ve only talked about in previous posts, I’ve been a teacher and teacher-trainer for a decade, and just this year I’ve gone back to school to get a PhD in math. This post is actually about

myexperience as a mature mathematical thinker (in my thirties!) learning new things in my PhD program. In other words, the student in the example is actuallyme, if you take out the discussion of linear equations and replace it with a discussion of the representation theory of finite groups. Any of us, from a preteen student to a professional mathematician, needs a place to start that is concretefor us, and we can understand ideas at an arbitrarily high level of abstraction if we get to those ideas by progressing through questions and ideas that are compelling to us.