Some Miscellaneous Awesomeness Wednesday, Jul 18 2012 

Just some awesome stuff I feel like pointing out:

Vi Hart does it again. That young woman has created a new art form.

Terry Tao’s airport puzzle. If you have to get from one end of the airport to the other to catch a plane, but you really need to stop for a minute to tie your shoe, is it best to do it while you’re on the moving walkway or not? (I learned this problem from Tim Gowers’ blog.)

Paul Salomon quotes Vi Hart quoting Edmund Snow Carpenter, and the quote is absolutely worth me quoting yet again:

The trouble with knowing what to say and saying it clearly and fully, is that clear speaking is generally obsolete thinking. Clear statement is like an art object: it is the afterlife of the process which called it into being.

Dan Goldner is doing my job for me. The original purpose of this blog was to read writing about math education, and to summarize and discuss it. I don’t do this very much any more (although expect summaries of a couple articles from the current JRME in the next few weeks months), but I do have a long list of things I wanted to read and discuss here but figured I’d probably never get to. On this list was the 1938 NCTM Yearbook, The Nature of Proof, by Harold Fawcett. But I’m taking it off; Dan’s got it covered.

Purging Thursday, Jul 5 2012 

This is an impulsive and probably self-indulgent post.

When I moved to New York almost 6 years ago, I stowed two crates of hanging files in my grandmother’s closet. They are artifacts of my 2000-2005 teaching career in Boston. One crate, curricular materials; the other, student work. They had made it past one round of purging – this was the stuff I chose to bring with me to New York.

But they’ve been gathering dust since 2006 and I figured I owed it to my grandmother to get them out of her hair, so I picked them up on Tuesday. They’re sitting on my living room floor. I have absolutely no sensible place in my apartment for them. I am next to them, on the couch, a bag of paper recycling at my feet.

I didn’t budget time for these guys, and the time efficient move is to not even think about it; just dump it all.

I can’t bring myself to do this. That said, knowing how I get, if I start going through it paper by paper then (a) I will be here till next week and (b) at least half of it I will not be able to throw out.

Maybe I can make this blog post some kind of middle ground.

* Here’s the Jeopardy game I played with my Algebra I and Calculus classes they day before winter break! Optimization for $300: This is the maximum amount of money you can make selling cookies if you know that you could sell 100 cookies for $1 each, and that every time you raise the price $0.25, you lose 10 customers. Final Jeopardy (Algebra): x, given that a=4, b=2, c=-1, d=37, and ax+b=cx+d. Mr. Blum-Smith trivia for $200: Mr. Blum-Smith’s grandmother was kissed by this former US president. (Same grandmother whose apartment has been housing all this sh*t! Correct response: who is Bill Clinton?)

* Here are my various attempts at teaching about proof in Algebra I! My first year, I tried to teach a “proof unit.” It culminated with a “proof project,” where I had students attempt to prove one of six eclectic elementary theorems (e.g., sum of first k odd numbers is k^2; any composite has a factor >1 but \leq its square root; …). I remember being essentially unsatisfied. In the notes I made to myself after implementation (ed note: HOW CAN I THROW THESE OUT! F*CK!) I was starting to realize the whole thing was ill-conceived. I was smashing together the problem of actually figuring out what’s going on (interesting, unexpected, no guaranteed outcome) with the formal process of making it into an argument. I was setting the kids up. In my fourth year, I revisited the idea except with more coherence because the whole thing was based on creating a “number trick” (“think of a number; add 6; multiply by 2; … ; you got 42!”) and proving it worked. Still, the proof aspect of the unit was stilted and poorly motivated because the kids couldn’t see the need for the amount of formality I was insisting on.

* Here is a unit I wrote my student teaching year, about tessellations and symmetry, based on Escher. Here are pages of transparencies with Escher prints and other tessellations. Here are the 5 envelopes of tessellating polygons (triangles, rhombi, a nonconvex quadrilateral, some special pentagons…) I designed on the computer and lovingly cut out of paper. I never taught this unit again.

* Order of operations. I used to use this activity I stole from my own 7th and 8th grade math teacher, Steve Barkin, an institution of the Cambridge public schools. Take the year (I used to use the kids’ birth year, or just make it 1994 if I wanted it to be easier), and using the digits in that order, put any math symbols you want between them to get as many of the numbers from 1 to 100 as you can.

* Ah! And an inheritance from Steve I never actually made use of: a kind of integer number sense activity where you label the vertices of a graph with integers so that the numbers on adjacent vertices differ by 10, or else one of them is double the other. Like this! Fill in the blanks: 12\leftrightarrow ? \leftrightarrow ? \leftrightarrow 13. Solution: 12\leftrightarrow 6\leftrightarrow 3\leftrightarrow 13.

* CAN THEORY. This was the name of my linear-equations-in-a-single-variable unit, the core topic of my Algebra I class. I took the name and the idea from Maurice Page, then the math coordinator of the Cambridge Public Schools. The unit became what it was in my classroom in collaboration with my awesome colleagues Jess Flick (then Jess Jacob) and Mike Jenkins. The whole unit was based on physically modeling the equations with plastic cups and poker chips on a table; I put a piece of tape down the middle of the table and the rules were, all cups have to hold the same number of chips and both sides of the table have to have the same number of chips total. You figure out how many chips go in the cup. I beat that model to death every year. I tweaked the model in various ways to accommodate negative and fractional coefficients and solutions. That was the one topic I would have counted on nearly all my students still having mastery of the following year.

* Qualitative graphs! One of the years of my collaboration with Jess and Mike, we implemented an idea Mike brought to the table of a unit in Algebra I that was about interpreting qualitative features of cartesian graphs. The culminating project was, you picked a container (we had all kinds of shapes – beakers, vases, wine glasses, etc.), you filled it steadily with water and measured its height against the amount of water it contained, and you drew a graph of that. Before you did the experiment you predicted what the graph would look like. Afterward, you wrote an explanation of the features of the graph (changes in slope; concavity; inflection points) and discussed how they related to the shape of the container. My experience of the unit was that it was very difficult for kids, but it definitely felt like some proto-calculus skills.

That was the easy stuff. (I know; I’m being dramatic.) STUDENT WORK:

No, I can’t even open this up. GRRR. To every student I taught in 2000-2005: I am about to dump into a bag of paper recycling a whole lot of both your and my blood, sweat and tears. RRRRR okay. I have to immortalize a few memories. This will be spotty and haphazard, please forgive me. I am leaving most of you out in the below, but to all of you let me say that I hope you learned half as much from me as I did from you.

W: Best handwriting ever. Every homework assignment literally looked like the inscription on the One Ring. May you bring that level of love to everything you do.

D and M: The two black women in a calculus class I had allowed to be dominated by the personalities of cocky (mostly white) boys, you had the courage, and the respect for me and my potential for growth, to tell me what this felt like. I am grateful you did and sorry you had to. You are both rock stars and I regret that my class wasn’t a better environment for expressing that.

N and M: You stand out in my mind in your willingness to put in time and effort to understanding what you didn’t before. You put in after-school time to the degree it could have been a part-time job. That kind of commitment got you past hurdles higher than many adults I know have ever had to face. In my life I have come to understand that anybody can learn anything, and you guys helped teach me that.

C and M: I was a rigid grader. I put the numbers in the computer and whatever came out, that was your grade. I used an old-fashioned grading system that punished missed work harshly and made it very hard to climb out of a hole. Knowing I hated grading, this rigidity was how I protected myself: I didn’t have to make judgement calls, I just put the numbers in the computer and didn’t think about it. I didn’t allow myself to imagine what receiving the grades felt like. Both of you were students who had some bad student habits but showed tremendous growth over the time we worked together, stretching yourself to contribute positively to both your learning and the class community. I gave a lot of F’s that in retrospect I regret. Yours are the two I regret most.

W: As a math student you were an amazing combination of depth of thought and engagement, on the one hand, and desperate difficulty mastering computational techniques, expressing yourself in writing, or doing anything at all in a subinfinite amount of time, on the other. You asked some of the most thoughtful and interesting questions in class that I have ever heard. You practically never finished a test, even if you came after school for 3 hours to work on it. You were uniquely gentle and generous with myself and your classmates at all times. Rest in peace, W.

Another One to Keep Your Eye On: Anna Weltman Saturday, Jun 23 2012 

Here’s another blog to keep an eye on:

Recipes for Pi, by Anna Weltman.

I know Anna IRL. In fact, both of us have seen the other one teach. Thus prior to discovering her blog I already knew her as mathematically thought-provoking, endlessly creative, and deeply tuned in to student experience, not to mention a total sweetheart.

So I was excited to learn that she had started blogging in February, and her writing hasn’t disappointed. It’s sporadic, but who am I to complain about that, and more importantly it’s characterized by that same deep thinking about math and student experience that marks her teaching. Check it out.

Aside: Anna teaches at St. Ann’s School, along with Justin Lanier, Paul Salomon, and Paul Lockhart.

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.


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.”


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.

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