## What She Said Monday, May 21 2012

Three weeks ago Sue VanHattum and Kate Nowak recommended Bob and Ellen Kaplan’s Math Circle Training Institute. If you are looking for a PD opportunity this summer and you are interested in cultivating students thinking for themselves, I strongly second their recommendation.

This is a weeklong training on the campus of Notre Dame in South Bend, Indiana where you learn how to run a math circle in the spirit of the Kaplans. What that means is that you ask thought-provoking questions and you facilitate students discussing them. Heaven, right? The setup is that in the morning, the Kaplans run a math circle on you, and in the afternoon they bus in local kiddies for you to try out your thought-provoking questions on, and watch others do it, and give and receive feedback. At lunch and at night you hang out with like minded educators talking about math and education. The \$850 includes room and board for the whole week.

I did this training in the summer of 2009 and it was a key step on my path to being the educator I am now. In 2007-8 I had come to the realization that my most central, pressing goal as an educator was to empower students to find their own mathematical curiosity, and I started stretching my pedagogical boundaries to find out what it would look and feel like to teach with this as the only goal. But I felt like I was reinventing the wheel. Reading the Kaplans’ book Out of the Labyrinth, I felt like I had found my comrades. Going to the Summer Institute, I felt like I had met them.

For example, Sue and Alex, and my fairy blogfamily Kate and Jesse Johnson. See what I mean?

Tangential to the math PD but also a wonderful benefit was the opportunity to spend a week on the Notre Dame campus. As a Jew I did not go into the experience expecting to be so moved by the shrines and sanctuaries of this Catholic institution, but I was. After my first experience with a labyrinth (the meditative kind), Alex McFerron said to me, “the Catholics really ace those sacred spaces.” True that.

## Dispatches from the Learning Lab: Partial Understanding Monday, Apr 30 2012

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 $y$-intercept of a linear function, in slope-intercept form, given that the slope is $4$ and the point $(6,11)$ lies on the line. You explain that the equation has the form $y=mx+b$ and that because we know the point $(6,11)$ is on the line, that this point satisfies the equation. Thus you write

$11=4\cdot 6+b$

on the board. At this point I recognize that we are trying to find $b$ 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 $y$-intercept, and since I recognize that this must be $b$ I calculate $11-24 = -13$ and raise my hand.

Meanwhile, I have only the vaguest sense of the meaning of the phrase “$y$-intercept.” I have literally no understanding of why I should expect the equation to have the form $y=mx+b$. I have a nagging feeling of dissatisfaction ever since you substituted $(6,11)$ into the equation because I thought $x$ and $y$ were supposed to be the variables but now it looks like $b$ 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 $x$ values into some equations and obtain corresponding $y$ 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 $b$ 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.

## Justin Lanier Thursday, Apr 26 2012

How did I miss that Justin Lanier started blogging (finally!) last August?

His blog is called I Choose Math. Keep your eye on this one, he’s the real deal.

## Best Calculator *Ever* Wednesday, Apr 18 2012

Somebody please tell me this is for real:

The QAMA calculator.

## Dispatches from the Learning Lab: Yup, Time Pressure Sucks Friday, Mar 2 2012

Continuing the series I began here and here, about snippets of new-feeling insight about the learning process coming from my new role on the student side of the desk…

This one is funny, because I knew it, I mean I knew it in my bones, from a decade working with students; but yet it’s totally different to learn it from the student side. I’m a little late to the blogosphere with this insight; I’ve been thinking about it since December, because it kind of freaked me out. Even though, like I keep saying, I already knew it.

Learning math under time pressure sucks. It sucks.

It sucks so much that I ACTUALLY STOPPED LIKING MATH for about 5 days in December.

I didn’t know this was possible, and I don’t think anyone who’s ever worked closely with me in a mathematical context (neither my students, colleagues, or teachers) will really believe it. But it’s true. It was utterly, completely unfun. There was too much of it and too little time. It was like stuffing a really delicious meal down your throat too quickly to chew, or running up the Grand Canyon so fast you puke. Beautiful ideas were everywhere around me and I was pushing them in, or pushing past them, so hard I couldn’t enjoy them; instead they turned my stomach, and I had the feeling that the ones I pushed past in a hurry were gone forever, and the ones I shoved in weren’t going to stay down.

I had some independent study projects to work on during winter break, and what was incredible was the way the day after my last final exam, math suddenly became delicious again. Engaging on my own time and on my own terms, that familiar sense of wonder was back instantly. All I had to do was not be required to understand any specific thing by any specific date, and I was a delighted, voracious learner again.

Now part of the significance of this story for me is just the personal challenge: most of the grad students I know are stressed out, and I entered grad school with the intention of not being like them in this respect. I was confident that, having handled adult responsibilities for a decade (including the motherf*cking classroom, thank you), I would be able to engage grad school without allowing it to stress me out too much. So the point of this part of the story is just, “okay Grad Program, I see you, I won’t take you for granted, you are capable of stressing me out if I let you.” And then regroup, figure out how to adjust my approach, and see how the new approach plays out in the spring semester.

But the part of the story I want to highlight is the opposite part, the policy implication. Look, I frickin love math. If you’ve ever read this blog before, you know this. I love it so much that most of my close friends sort of don’t feel that they understand me completely. So if piling on too much of it too quickly, with some big tests bearing down, gets me to dislike math, if only for 5 days, then the last decade of public education policy initiatives – i.e. more math, higher stakes – is nothing if not a recipe for EVERYONE TO HATE IT.

And, not learn it. Instead, disgorge it like a meal they didn’t know was delicious because it was shoved down their throat too fast.

In short. The idea of strict, ambitious, tested benchmarks in math to which all students are subject is crazy. It’s CRAZY. The more required math there is, and the stricter the timeline, the crazier. I mean, I already knew this ish was crazy, I’ve been saying this for years, but in light of my recent experience I’m beside myself. If you actually care about math, if you have ever had the profound pleasure of watching a child or an adult think for herself in a numerical, spatial or otherwise abstract or structural context, you know this but I have to say it: the test pressure is killing the thing you love. Its only function is to murder something beautiful.

If you teach, but especially if you are a school leader, and especially if you are involved in policy, I beg you: defend the space in which students can learn at their own pace. Fight for that space.

## Dispatches from the Learning Lab: Inauthentic Agreement Wednesday, Jan 25 2012

Here’s another one. It should be quick.

When a student says, “Is it like this?” or the equivalent, I used to err on the side of “yes.” I.e. even if I wasn’t sure exactly what they were saying, but I thought it sounded like it might make sense. I think this was somewhat a function of the fact that I adopted a generally encouraging posture (this is my personality but also a deliberate choice), but it itself was just sort of my reflexive response from within this posture (not a deliberate choice).

It never felt quite right, so over time I trained myself instead to say things like, “I can’t understand what you’re saying but I think you might be onto something, but I’m not sure.” I never had concrete evidence that my original response was doing something unhelpful though.

Now I do. In a recent conversation with one of my teachers, several times I said, “Let me explain back to you what I think you’re saying, and you tell me if it’s right…” And he said, “yes yes yes it’s like…” But I didn’t recognize my attempted explanation in what he seemed to be saying yes to. So, it’s official: this is TOTALLY UNHELPFUL. I’m disoriented; that’s why I asked the question. Unless I come away from your answer feeling sure that you understood me, your “yes” only serves to make me more disoriented.

Take-home lesson. Never say “yes” unless you are sure you have understood fully what the student is saying, and agree with it. As I’ve often discussed before, sometimes a “yes” is inappropriate even then; for example if there’s a danger that the student is trying to foist onto you the work of judging for her or himself. But if you have any doubt, then the “yes” is definitely inappropriate: the encouragement is fake, and the student is left being equally unsure as before, and now also having exhausted the resource of checking with you. Retrospectively the only student who even feels good hearing the “yes” in this situation is the one who is playing a Clever Hans game, and in this case it does him or her the disservice of encouraging the game.

## Dispatches from the Learning Lab: Why I Don’t Always Ask My Question Tuesday, Jan 24 2012

One of the many reasons I put myself in a math PhD program is that it is an intense full-time laboratory in which for me to examine my own learning process, and my experience as a participant in math classrooms from the student side. I hope to record many lessons from this laboratory on this blog. Here is one.

As a teacher I have always strongly encouraged people to pipe up when they’re confused, whether working in groups or (especially) at the level of whole-class discussion. To encourage this, I do things like:

* I leave lots of wait time.
* I respond to questions (especially those expressing confusion) with enthusiasm when they are asked, and after they are discussed I point out concrete, specific ways in which the questions advanced the conversation.
* I give (very deeply felt) pep talks about the value of these questions.
* Sometimes I directly solicit questions from people whose faces make it seem like they have one.

I am behind all of these practices. However, in every class that I have taught, whether for students or teachers, including all those of an extended enough length so that the practices would have time to shape the culture, it has always seemed to me that participants are often not asking their questions. This has puzzled me a bit. I’ve generally responded by trying harder: leaving longer wait-time, making more of a point to highlight the value of questions when they happen, giving more strident and frequent pep talks. This hasn’t resolved the matter.

Now I am not about to pronounce a new solution. But I have what for me is a very new insight. I imagine some readers of this blog will read it and be like, “Ben, I could have told you that.” I’m sure you could have, but this wouldn’t have helped me: retrospectively, students have told me it many, many times. But I didn’t get it till I felt it. This is the value of putting yourself in their position.

What I’ve realized since beginning graduate school is that I had an incomplete understanding of why students don’t ask questions. I believed that the only reason not to ask a question is the fear of looking dumb. My approach has been entirely aimed at ameliorating this fear and replacing it with the sense that questions are honored and their contribution is valued.

Now one of the great advantages of going to grad school as an adult, rather than going fresh out of college, is that I have very, very little fear of looking dumb. (In the immortal words of my friend Kiku Polk, you get your “f*ck you” at 30.) To all my early-20′s people: your 20′s will be wonderful but if you make sure you keep growing, your 30′s will be better.

And one of the great advantages of going to grad school after over a decade as a teacher, is that I have a strong commitment to asking my questions, stemming from the value that I know they have both for myself and the class.

Perhaps as a consequence, I found that in all four of my classes last semester, I asked more questions than anyone else in the room.

Be that as it may, I frequently didn’t ask my questions.

What’s up?

As it turns out (and now, okay, maybe this is Captain Obvious talking, but a propos of all of the above, somehow I’ve been overlooking it for a decade), not wanting to hold up class is its own reason not to ask questions! Maybe it’s a basic piece of our social programming. If things are going one way in a room of 20 or 30 people, it feels sort of painful to contemplate forcing them in another direction on your account. Especially if you’ve already done it once or twice, but even if not. And more so the further your question seems to be from what the people around you (esp. the teacher) look like they want to talk about. All this is intensified if you’re not sure your question is going to come out perfectly articulate – not (necessarily or only) because of how this will make you look, but because you know that your interruption is going to take up more mental and social space if it has to start with a whole period of everybody just getting clear on what you’re even asking.

There is an added layer that it is often perceptible that the teacher desires for everyone to understand and appreciate what was just said as clearly as she or he understands and appreciates it. Last night I was in a lecture in which I was hyperaware of not always asking my questions, and part of the dynamic in that case was actually the professor’s enthusiasm about what he was saying! I did ask a number of questions, but one reason I didn’t ask more is that I sort of felt like I was crashing his party! My warm feelings toward this professor actually heightened this effect: messing up someone else’s flow is worse when it’s someone you like.

As I mentioned above, students have been trying to tell me this for years. I never got it, because on some level I always believed that the real problem was that they were afraid to look dumb. I remember a conversation with a particular student who was my advisee as well as my math student. When I pressed her on asking more questions in class, she said something to the effect of, “you know, you’re doing your thing up there, and I don’t want to get in the way.” I literally remember the voice in my head reinterpreting this as a lack of belief in herself. Now I think that that was part of it as well; but my response was all aimed at that, and so didn’t address the whole issue.

Now my process of figuring out how to operationalize this new insight in terms of teaching practice has only just begun, and one reason I am writing about this here is to invite you into this process. I am certainly NOT telling you to withhold your enthusiasm on the grounds that it might make kids not want to interrupt you with questions. Furthermore, evidently when I describe experiences from my graduate classes, I am describing a situation in which the measures you and I have been taking for years to encourage question-asking are mostly absent. I doubt most of my professors have even heard of wait time. Nonetheless, I am sure that this new point of view is fruitful in terms of actual practice. Below are my preliminary thoughts. Please comment.

If I want to really encourage question asking, what I have been doing (aimed at building a culture of question-asking) is necessary, but insufficient. It is also necessary to think about lesson structure with an eye to: how do I design the flow of this lesson so that (at least during significant parts where questions are likely to arise in students’ minds) asking their questions does not feel like an interruption? One model, which is valuable in other ways as well, is to have students’ questions be the desired product of a certain segment of class. For example, when the lesson arrives at a key idea, definition, or conclusion, ask students to turn to their neighbors and discuss the key idea and try to produce a question about it. Then have the pairs or groups report their questions. This way, the questions cannot be interruptions because they are explicitly the very thing that is supposed to be going on right then.

I like this idea but it has limited scope because it requires the point in the lesson at which the questions arise to be planned, and of course this can never contain all the questions I would want to have asked. Another thing to think about is the matter of momentum. I think my discussion of enthusiasm above really revolves around momentum. Enthusiasm generates momentum, but momentum is actually the thing that it hurts to get in the way of. Therefore I submit a second idea: the question of managing my/your own and the class’s momentum. Having forward momentum is obviously a big part of class being engaging, but perhaps it also suppresses spontaneous questions? Or under certain conditions it does?

(In a way this reminds me of the tension – one I am much more confident is an essential one of our profession – between storytelling and avoidance of theft – I discussed a particular case of this tension in the fourth paragraph here. Momentum is aligned with storytelling: a good story generates momentum. Avoiding theft is aligned with inviting questions.)

A last thought is that in a class of 20 or 30, having the class engage every question that pops into any student’s head at any time is obviously not a desirable situation. You might think I thought it was desirable based on the above. But the question is how to empower students to ask questions when we want them. I know that I for one have often known I wanted some questions so I could be responsive to them, and they weren’t forthcoming. The question is about how to change this. Part of the answer is about the culture, valuing the questions, encouraging the risks, and making everyone feel safe; but it’s the other part – how to structurally support the questions – that’s the new inquiry for me. As I said above, please comment.

## Never Be Wobbly Friday, Sep 16 2011

I spent at least 9 hours today thinking about squishing baloon-shaped surfaces into other shapes. This is what a PhD program in math is doing to me.[1]

Having learning as a full-time job is really, really delicious. But tonight when I stopped mathing and engaged the edublogosphere it felt like a relief to read about classrooms, populated by humans. (To my fellow humans: I love the differentiable manifolds but I love you more.) Thanks Jesse Johnson, Dan Goldner, and Kate No Wackness[2] for your continuing dedication to learning (your kids’, yours and ours).

Holy Crap, a Three-Legged Table Can Never Be Wobbly.

This is my favorite thing ever.

[1]The particular thought that was driving me crazy at least from 7pm to 10pm, not that you care, was: if $f:X \rightarrow Y$ is any surjective continuous function between topological spaces that maps open sets to open sets, then I can prove that the inverse image of a compact set is compact. I studied a converse if $X$ and $Y$ happen to be smooth manifolds and $f$ happens to be an injective immersion. But these are very very strong assumptions. How much can they be weakened?

[2]Kate, this is A’s name for you. (An homage to your no bullsh*t approach.)

## What can you do with this? Saturday, Jul 16 2011

We interrupt our regularly scheduled long patch of radio silence to share with you an arresting, mathematically rich visual:

[source]

Thanks to Josh Kershenbaum for the tip.

UPDATE 7/17:

To clarify something: this image was originally created as an answer to a question, but I didn’t create a WCYDWT tag for that angle on it. To me, the image lands as (a) very beautiful; relatedly, (b) haunting, hard to get out of my head; therefore, (c) incipiently highly narrative – it is asking us to surround it with story, whether the original story conceived by the maker of the image or some other; and, lastly, (d) unavoidably mathematical. I don’t have a clear sense of the next move, but I do think this image opens a rich vein of something for a math teacher to use. The question “What can you do with this?” isn’t rhetorical: what’s your next move?

(See exchange with Dan Meyer below.)

## The Inconvenient Truth Behind Waiting for Superman and other stories Monday, May 23 2011

I just recently learned of an organization in NYC called the Grassroots Education Movement, which last Thursday premiered a documentary film with the awesome title The Inconvenient Truth Behind Waiting for Superman. They will apparently send you a copy for free; I just ordered mine.

Meanwhile, the city of New York continues to besiege its own public schools with budget cuts, looming layoffs, and a multi-year hiring freeze. (Having spent the year training 12 new teachers, let me not even get started on the hiring freeze.) Another thing that happened on Thursday was that East Side Community High School, a wonderful school on the Lower East Side where I used to teach and where the math teaching is strong enough that we placed four student teachers there this quarter, had its first fund-raiser. Like, big event, speakers, performances by students, pay to get in, as though it were a non-profit, carrying out its own civic mission and in need of private funding to do it, rather than a public school, charged with a civic mission by the state, which no longer sees fit to pay for it.

I missed both the documentary premiere and ESCHS’s fund-raiser because I was teaching the final class of a 3-session minicourse at Math for America on the fundamental theorem of arithmetic. Let me do a little reflecting on the execution:

At the end of the 2nd session, I gave participants about a half-hour to try to figure out something quite difficult. I attempted to scaffold this with some unobtrusive PCMI-style tricks in a previous problem set: sequences of problems with the same answer for a mathematically significant reason. It turned out not to be enough. There was high engagement the whole time, but no one seemed to be headed in my intended direction after that half-hour. On the other hand, that half-hour had made the group into a legit mathematical research community. What was afoot was a live process of trying things out, questioning, pressing on others’ logic, and generally behaving like research mathematicians. I was left with a dilemma. I had one session remaining. I wanted to protect that process, meaning I did not want to steal from them any of the deliciousness (or pain – also delicious) of the process they were in the middle of by offering them too much direction. But at the same time I felt I needed to guarantee that we would reach resolution. (Storytelling purposes.)

The solution I went with: I had them pick up in the final session where they left off, but I brought in a sequence of hints on little cut-up slips of paper. I tried to call them “idea-starters” as opposed to “hints” to emphasize that the game was you thinking on your own, and this is just to get you moving if you’re stuck, rather than I have a particular idea and I want you to figure out what it is, but I don’t think I was consistent with this, and I think they pretty much all called them “hints,” and I don’t think it really mattered. They were in an order from least-obtrusive to most-directive. None of them were very directive. Most importantly, I told the participants that if they wanted to get one, they needed to decide this as a table. (There were 6 tables with 3-4 folks each.)

How this went: a) it preserved the sense of mathematical community. I do not think there was much of a cost to participant ownership of what they found out. b) People were actually pretty hesitant to use the “idea-starters.” Most of them went untouched. This would probably be different with a different audience. (High schoolers instead of teachers?) c) The “idea-starters” worked great, but very slowly. I planned to spend 45 min letting them work in this arrangement, but after 45 min, most of the groups were still deep in the middle of something. After over an hour, I asked two groups to present what they had, however incomplete, for the sake of a change of pace and the opportunity for cross-pollination of ideas between the tables. I had actually meant to do a lot more of this but had forgot to mention it at the beginning. I let everybody work for another 10-15 min while these groups laid out their presentations. By the time they presented, I realized that there wasn’t enough time left for everyone to really get back to work afterward, but in any case their ideas had gotten more fully developed in that 10 min. so they actually had pretty much figured out everything I had wanted them to. I presented the final link in the logical chain, just to fill in the picture, in the last 5 minutes. It was pretty satisfying to me to watch the presentations, except that it happened so late in the session. This for two reasons. One was that I would have ideally liked to have time to encourage the participants to interrogate the presenters more, but there wasn’t time for that. The other was that I had intended to spend the last half-hour with the participants consolidating their understanding of the argument by applying it to a new situation in which they didn’t know the outcome and it would tell them; but we didn’t have time for that either. I really feel a loss about that.

If I were to repeat it I think I would interrupt much earlier to have people present partial work. The cross-pollination of ideas might or might not accelerate the figuring-out process. Either way I think the change of pace would have been good for concentration. Also, I could have put some of the questions I used as “idea-starters” into the Session 2 problem sets, trying to move some of the combustion I got in session 3 into session 2. But these would both be experiments as well. I hope I get a chance to try them.