I’ve been wanting to get a live taste of Bowen and Darryl’s work since at least 2010, when Jesse Johnson, Sam Shah, and Kate Nowak all came back from PCMI saying things like “that was the best math learning experience I’ve ever had,” and I started to have a look at those gorgeous problem sets. It was clear to me that they had done a lot of deep thinking about many of the central concerns of my own teaching. How to empower learners to get somewhere powerful and prespecified without cognitive theft. How to construct a learning experience that encourages learners to savor, to delectate. That simultaneously attends lovingly to the most and least empowered students in the room. &c.

I want to record here some new ideas I learned from Bowen and Darryl’s workshop. This is not exhaustive but I wanted to record them both for my own benefit and in the hopes that they’ll be useful to others. In the interest of keeping it short, I won’t talk about things I already knew about (such as their Important Stuff / Interesting Stuff / Tough Stuff distinction) even though they are awesome, and I’ll keep my own thoughts to a minimum. Here’s what I’ve got for you today:

1) The biggest takeaway for me was how exceedingly careful they are with people talking to the whole room. First of all, in classes that are 2 hours a day, full group discussions are always 10 minutes or less. Secondly, when students are talking to the room it is always students that Bowen and Darryl have preselected to present a specific idea they have already thought about. They *never* ask for hands, and they never cold-call. *This means they already know more or less what the students are going to say.* Thirdly, they have a distinction between students who try to burn through the work (“speed demons”) and students who work slowly enough to receive the gifts each question has to offer (“katamari,” because they pick things up as they roll along) – and the students who are asked to present an idea to the class are *only* katamari! Fourthly, a group discussion is only ever about a problem that everybody has already had a chance to think about – and even then, never about a problem for which everybody has come to the same conclusion the same way. Fifthly, in terms of selecting which *ideas* to have students present to the class, they concentrate on ideas that are nonstandard, or particularly visual, or both (rather than standard and/or algebraic).

This is for a number of reasons. First of all, the PCMI Developing Mathematics course has something like 70 participants. So part of it is the logistics of teaching such a large course. You lose control of the direction of ideas in the class very quickly if you let people start talking and don’t already know what they’re going to say. (Bowen: “you let them start just saying what’s on their mind, you die.”) But there are several other reasons as well, stemming (as I understood it anyway) from two fundamental questions: (a) for the people in the room who are listening, what purpose is being served / how well are their time and attention being used? and (b) what will the effect of listening to [whoever is addressing the room] be on participants’ sense of inclusion vs. exclusion, empowerment vs. disempowerment? Bowen and Darryl want somebody listening to a presentation to be able to engage it fluently (so it has to be about something they’ve already thought about) and to get something worthwhile out of it (so it can’t be about a problem everybody did the same way). And they want everybody listening to feel part of it, invited in, not excluded – which means that you can’t give anybody an opportunity to be too high-powered in front of everybody. (Bowen: “The students who want to share their super-powerful ideas need a place in the course to do that. We’ve found it’s best to have them do that individually, to you, when no one else can hear.”)

2) Closely related. Bowen talked at great length about the danger of people hearing somebody else say something they don’t understand or haven’t heard of and thinking, “I guess I can’t fully participate because I don’t know that idea or can’t follow that person.” It was clear that every aspect of the class was designed with this in mind. The control they exercise over what gets said to the whole room is one aspect of this. Another is the norm-setting they do. (Have a look at page 1 of this problem set for a sense of these norms.) Another is the way they structure the groups. (Never have a group that’s predominantly speed-demons with one or two katamari. If you have more speed-demons than katamari, you need some groups to be 100% speed demon.)

While this concern resonates with me (and I’m sure everybody who’s ever taught, esp. a highly heterogeneous group), I had not named it before, and I think I want to follow Bowen and Darryl’s lead in incorporating it more essentially into planning. In the past, I think my inclination has been to intervene after the fact when somebody says something that I think will make other people feel shut out of the knowledge. (“So-and-so is talking about such-and-such but you don’t need to know what they’re talking about in order to think about this.”) But then I’m only addressing the most obvious / loud instances of this dynamic, and even then, only once much of the damage has already been done. The point is that the damage is usually exceedingly quiet – only in the mind of somebody disempowering him or herself. You can’t count on yourself to spot this, you have to plan prophylactically.

3) Designing the problem sets specifically with groupwork in mind, Bowen and Darryl look for problems that encourage productive collaboration. For example, problems that are arduous to do by yourself but interesting to collaborate on. Or, problems that literally require collaboration in order to complete (such as the classic one of having students attempt to create fake coin-flip data, then generate real data, trade, and try to guess other students’ real vs. fake data).

4) And maybe my single favorite idea from the presentation was this: “If a student has a cool idea that you would like to have them present, consider instead *incorporating that idea into the next day’s problem set.*” I asked for an example, and Bowen mentioned the classic about summing the numbers from 1 to n. Many students solved the problem using the Gauss trick, but some students solved the problem with a more visual approach. Bowen and Darryl wanted everybody to see this and to have an opportunity to connect it to their own solution, but rather than have anybody present, they put a problem on the next day’s problem set asking for the area of a staircase diagram, using some of the same numbers that had been asked about the day before in the more traditional 1 + … + n form.

I hope some of these ideas are useful to you. I’d love to muse on how I might make use of them but I’m making myself stop. Discussion more than welcome in the comments though.

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I tutored a girl T a while back. A 3rd grader and a middle sibling of three. In the narrative her family passed on to me, she’s getting by in math class but math is harder for her than her siblings; she’s not as “quick.” All I have to say is that “quick” is not what it’s all about.

Here is a story from our work together, illustrating both

a) What my some of my best work looks like – how to get somebody to an ambitious result without any cognitive theft, how to utilize legitimate uncertainty to cultivate curiosity – and also

b) Some awesome mathematical thinking that doesn’t look like the stereotype. Not “quick” but something much deeper and richer. My dude T might not fit somebody’s image of the math whiz, but I am telling you, do not sleep on this kid.

At a previous session, I had asked T what she knew about multiplication, and she had told me, among other things, that is four sixes, and because that’s 24, she also knew that six fours would be 24. I asked why she said so and she didn’t know why. I asked her if she thought this would always be true for bigger numbers, or could it be possible that there were some big numbers like 30,001 and 5,775 for which 30,001 5775’s was different than 5,775 30,001’s. She wasn’t sure. I asked her if she thought it was a good question and she said she thought it was.

So this session I reminded her of this conversation. I forget the details of how we got going on it; I remember inviting her to wonder about the question and note that there is something *surprising* about the equality between four sixes and six fours. She could count up to 24 by 4’s and by 6’s and mostly you hit different numbers on the way up, so why do the answers match? And would it be true for any pair of numbers? But the place where I really remember the conversation is when we started to get into the nuts and bolts:

Me: Maybe to help study it we should try to visualize it. Can you draw a picture of four sixes?

T draws this –

Me: Cool! Okay I have a very interesting question for you. You know how many dots are here –

T: 24 –

Me: and you also told me that six 4’s is also 24, right?

T: yes.

Me: – so that means that there must be six 4’s in this picture! Can you find them?

T: I don’t understand.

Me [writing it down as well as saying it this time]: You drew a picture of four 6’s here, yes?

T: Yes.

Me: And that’s 24 dots, yes?

T: Yes.

Me: And you told me before that 24 is also six 4’s, yes?

T: Yes.

Me: So it must be that right here in this picture there are six 4’s!

[It clicks.] T: Yes!

Me: See if you can find them!

At this point, I go wash my hands. An essential tool that has developed in my tutoring practice is to give the student the social space to feel not-watched while they work on something requiring a little creativity or mental looseness, or just anything where the student needs to relax and sink into the problem or question. The feeling of being watched, even by a benevolent helper adult, is inhibitive for generating thoughts. Trips to wash hands or to the bathroom are a great excuse, and I can come back and watch for a minute before I make a decision about whether to alert the student to my return. I also often just look out the window and pretend to be lost in thought. Anyway, on this particular occasion, when I come back, T has drawn this:

T: I found them, but it’s not… It doesn’t…

I am interrupting because I have to make sure you notice how rad she’s being. The child has a sense of mathematical aesthetics! The partition into six 4’s is uglifying a pretty picture; breaking up the symmetry it had before. It’s a kind of a truth, but she isn’t satisfied with it. She senses that there is a more elegant and more revealing truth out there.

This sense of taste is the device that allows the lesson to move forward without me doing the work for her. Her displeasure with this picture is like a wall we can pivot off of to get somewhere awesome. Watch:

Me: I totally know what you mean. It’s there but it doesn’t feel quite natural. The picture doesn’t really want to show the six 4’s.

T: Yeah.

Me: You know what though. You had a lot of choice in how you drew the four 6’s at the beginning. You chose to do it this way, with the two rows of three plus two rows of three and like that. Maybe you could make some other choice of how to draw four 6’s that would also show the six 4’s more clearly? What do you think? You wanna try to find something like that?

T: Yes! [She is totally in.]

At this point I go to the bathroom. I hang out in the hall for a bit when I get back because she seems to still be drawing. Finally,

Me: Did you find out anything?

T: I drew it a lot of different ways, but none of them show me the six 4’s…

She’s got six or seven pictures. One of them is this –

Me: Hey wait I think I can see it in this one! (T: Really??) But I can’t tell because I think you might be missing one but I’m not sure because I can’t see if they are all the same.

T immediately starts redrawing the picture, putting one x in each column, carefully lined up horizontally, and then a second x in each column. As she starts to put a third x in the first column, like this,

she gasps. Then she slides her eyes sideways to me, and with a mischievous smile, adds this to her previous picture:

The pieces just fell into place from there. Again I don’t remember the details, but I do remember I asked her what would happen with much bigger numbers – might 30,001 5,775’s and 5,775 30,001’s come out different? And she was able to say no, and why not. Commutativity of multiplication QED, snitches!

What’s the point here? Well, there are two. One is that I’m proud of this, and I’m excited to show you what I did that was awesome. So to that end I’ll highlight my (awesome ;) ) operating principles:

A) It was important that T did not know what the outcome would be, but wanted to know. She indicated at the beginning that she was familiar with multiplication’s commutativity (not in that language), but to cultivate uncertainty, I asked her if it would always be true (and she was legitimately unsure). To cultivate curiosity, I asked her if she thought it was a good question (and she did). This is the setup I was arguing for last time. If she hadn’t been unsure, or hadn’t been interested, I would have turned to a different question.

B) I was aiming to create a situation for T that directed her attention to profitable seams in the substrate of our inquiry without taking away any of her ownership over any of the important steps that led to the conclusion. Therefore, I was committed to *not* at any point drawing an array of dots. That picture had to be hers. And I was committed to not explaining how the array picture leads to the conclusion that a b’s is b a’s. That train of thought had to be hers. Finally, if she had come up with a different picture that revealed the six 4’s in a generalizeable way, that would have been just as good. My specific moves and choices were governed by the twin questions “what is really mathematically at stake here?” and “how can I utilize her curiosity to get her attention closer to it without visibly pointing it out?”

But the other point is that I want to show you what *T* did that was awesome. T is one of many, many students who get dismissed mathematically because they don’t produce *fast* and they don’t get right answers all the time. But she is not to be dismissed. Notice that no amount of speed or right answers would have served the function that T’s curiosity, interest in the question, and sense of mathematical aesthetics served. No amount of awesomeness from my end would have gotten her anywhere either. T is an eight year old who has proved a theorem about every one of the infinity of pairs of natural numbers. She didn’t do it with speed or accuracy, she did it with depth of engagement and her developing sense of whether ideas are fitting together snugly or awkwardly.

Don’t sleep on the Divergents.

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The last 6 or 7 years in public education policy seem to have been characterized by the following trends:

1) Tying teacher evaluation, hiring, firing, and teacher pay to student standardized test results.

2) Relatedly, using value-added measurements in making these decisions.

3) School closings and state takeovers.

4) Using VAM in making decisions about those too.

I.e. Stressing all the adults who work in schools the f*ck out.

Also:

5) Subcontracting to charter networks.

6) Direct funding cuts.

I.e. divesting from education as a public trust.

All in all, these trends, spearheaded by the US Dept. of Ed. under the leadership of Arne Duncan, but with numerous assists from other folks, representing both public and private interests (being in NYC, I’m lookin at you Mike Bloomberg), seem to me to have an obvious common theme:

*Making public schools shittier places to work.*

Recently, both the NYT and EdWeek have reported a national teacher shortage as enrollment in teacher training programs has dropped precipitously for several years in a row. Even TfA is having trouble recruiting.

*Motherf*ckers, what did you think was gonna happen?*

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(Cynical voice at back of head: Ben, you are so effing naive. That’s what they *wanted* to happen. What better excuse to hire un-credentialed people to teach poor children? Me: No! I don’t believe it!)

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This is something I was repeatedly and inchoately hollering about five years ago.

As usual I’m grateful for Dan’s cultivated ability to land the point cleanly and actionably. Looking at my writing from 5 years ago – it’s some of my best stuff! totally follow those links! – but it’s long and heady, and not easy to extract the action plan. So, thanks Dan, for giving this point (which I really care about) wings.

I have one thing to add to Dan’s post! Nothing I haven’t said before but let’s see if I can make it pithy so it can fly too.

Dan writes that an approach to proof that cultivates doubt has several advantages:

- It motivates proof
- It lowers the threshold for participation in the proof act
- It allows students to familiarize themselves with the vocabulary of proof and the act of proving
- It makes proving easier

I think it makes proving not only easier but *way, way* easier, and I have something to say about how.

**Legitimate uncertainty and the internal compass for rigor**

Anybody who has ever tried to teach proof knows that the work of novice provers on problems of the form “prove X” is often *spectacularly, shockingly* illogical. The intermediate steps don’t follow from the givens, don’t imply the desired conclusion, and don’t relate to each other.

I believe this happens for an extremely simple reason. And it’s not that the kids are dumb.

It happens because the students’ work is unrelated to their own sense of the truth! You told them to prove X given Y. To them, X and Y look about equally true. Especially since the problem setup *literally informed them* that both are true. Everything else in sight looks about equally true too.

There is no *gradient of confidence* anywhere. Thus they have no purchase on the geography of the truth. They are in a flat, featureless wilderness where all the directions look the same, and they have no compass. So they wander in haphazard zigzags! What the eff else can they do??

The situation is utterly different if there is any *legitimate uncertainty* in the room. Legitimate uncertainty is an amazing, magical, powerful force in a math classroom. When you don’t know and really want to know, directions of inquiry automatically get magnetized for you along gradients of confidence. You *naturally* take stock of what you know and use it to probe what you don’t know.

I call this the **internal compass for rigor**.

Everybody’s got one. The thing that distinguishes experienced provers is that we have spent a lot of time sensitizing ours and using it to guide us around the landscape of the truth, to the point where we can even feel it giving us a validity readout on logical arguments relating to things we already believe more or less completely. (This is why “prove X” is a productive type of exercise for a strong college math major or a graduate student, and why mathematicians agree that the twin prime conjecture hasn’t been proven yet even though everybody believes it.)

But novice provers don’t know how to feel that subtle tug yet. If you say “prove X” you are settling the truth question for them, and thereby severing their access to their internal compass for rigor.

Fortunately, the internal compass is capable of a much more powerful pull, and that’s when it’s *actually giving you a readout on what to believe*. Everybody can and does feel this pull. As soon as there’s something you don’t know and want to know, you feel it.

This means that often it’s enough merely to generate some legitimate mathematical uncertainty in the students, and some curiosity about it, and then just watch and wait. With maybe a couple judicious and well-thought-out hints at the ready if needed. But if the students resolve this legitimate uncertainty for themselves, well, then, they have probably more or less proven something. All you have to do is interview them about why they believe what they’ve concluded and you will hear something that sounds very much like a proof.

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Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength. – Jordan Ellenberg, *How Not to Be Wrong*

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Dear Ben,

J’s class is learning about volume in math. They’ll be working with cubes, rectangular prisms and possibly cylinders, but that’s all. He asked his teacher if he could work on a “challenge” that has been on his mind, which is to find a formula for the volume of one of his favorite shapes, the dodecahedron. He build a few of these out of paper earlier in the year and really was/is fascinated with them. I think he began this quest to find the volume thinking that it would be pretty much impossible, but he has stuck with it for almost a week now. I am pleased to see that he’s not only sticking with it, but also that he has made a few pretty interesting observations along the way, including coming up with an approach to solving it that involves, as he put it, “breaking it up into equal pieces of some simpler shape and then putting them together.” After trying a few ways to break/slice up the dodecahedron and finding that none of them seemed to make matters simpler, he had an “ah ha” moment in the car and decided that the way to do would be to break it up into 12 “pentagonal pyramids” (that’s what he calls them) that fit together, meeting at the center of rotation of the whole shape. If we can find the volume of one of those things, we’re all set. A few days later, he told me that he realized that “not every pentagonal pyramid could combine to make a dodecahedron” so maybe there was something special about the ones that do, i.e., maybe there is a special relationship between the length of the side of the pentagon and the length of the edge of the pyramid that could be used to form a dodecahedron.

He is still sticking with it, and seems to be having a grand time, so I am definitely going to encourage him and puzzle through it with him if he wants.

But here’s my question for you…

I sneaked a peak on google to see what the formula actually is, and found (as you might know) that it’s pretty complicated. The formula for the volume of the pentagonal pyramid involves (or something horrible like that) and the formula for the volume of a dodecahedron involves or something evil like that. In short, I am doubtful that he will actually be able to solve this problem he’s puzzling through. What does a good teacher do in such a situation? You have a student who is really interested in this problem, but you know that it’s far more likely that he will hit a wall (or many walls) that he really doesn’t have the tools to work through. On the other hand, you really want him to find satisfaction in the process and not measure the joy or the value of the process by whether he ultimately solves it.

I certainly don’t care whether he solves it or not. But I want to help him get value out of hitting the wall. How do you strike a balance so that the challenge is the right level of frustrating? When is it good to “give a hint” (you’ve done that for me a few times in what felt like a good way… not too much, but just enough so that the task was possible).

In this case, he’s at least trying to answer a question that has an answer. I suppose you could find a student working on a problem that you know has NO known answer, or that has been proven to be unsolvable. Although there, at least, after the student throws up his hands after giving it a good go, you can comfort her by saying, “guess what… you’re in good company!” But here, I’d like to help give him some of the tools he might use to actually make some headway, without giving away the store.

I think he’s off to a really good start — learning a lot along the way – getting a lot of out the process, the approach. I can already tell that many of the “ah ha” moments have applicability in all sorts of problems, so that’s wonderful.

*Best, O
*

Dear O,

Wow, okay first of all, I love that you asked me this and it makes me really appreciate your role in this journey J is on, in other words I wish every child had an adult present in their mathematical journey who recognizes the value in their self-driven exploration and is interested in being the guardian of the child’s understanding of that value.

Second: no matter what happens, you have access to the “guess what… you’re in good company” response, because the experience of hitting walls as you try to find your way through the maze of the truth is literally the experience of *all* research mathematicians, *nearly all* of the time. If by any chance J ends up being a research mathematician, he will spend literally 99% or more of his working life in this state.

In fact, I would want to tweak the message a bit; I find the “guess what… you’re in good company” a tad consolation-prize-y (as also expressed by the fact that you described it as a “comfort”). It implies that there was an underlying defeat whose pain this message is designed to ameliorate. I want to encourage you and J both to see this situation as one in which a defeat is not even possible, because the goal is to deepen understanding, and that is definitely happening, regardless of the outcome. The specific question (“what’s the volume of a dodecahedron?”) is a tool that’s being used to give the mind focus and drive in exploring the jungle of mathematical reality, but the real value is the journey, not the answer to the question. The question is just a tool to help the mind focus.

In fairness, questing for a goal such as finding the answer to a question and then not meeting the goal is always a little disappointing, and I’m not trying to act like that disappointment can be escaped through some sort of mental jiu-jitsu. What I am trying to say is that it is possible to experience this disappointment as *superficial*, because the goal-quest is an exciting and focusing activity that expresses your curiosity, but *the goal is not the container of the quest’s value*.

So, that’s what you tell the kid. Way before they hit any walls. More than that, that’s how *you should see it*, and encourage them to see it that way by modeling.

Third. A hard thing about being in J’s position in life (speaking from experience) is that the excitement generated in adults by his mathematical interests and corresponding “advancement” is exciting and heady, but can have the negative impact of encouraging him to see the value of what he’s doing in terms of it making *him* awesome rather than the exploration itself being the awesome thing, and this puts him in the position where it is possible for an unsuccessful mathematical expedition to be very ego-challenging. This is something that’s been behind a lot of the conversations we’ve had, but I want to highlight it here, to connect the dots in the concrete situation we’re discussing. To the extent that there are adults invested in J’s mathematical precociousness per se, and to the extent that J may experience an unsuccessful quest as a major defeat, these two things are *connected*.

Fourth, to respond to your request for concrete advice regarding when it is a good idea to give a hint. Well, there is an art to this, but here are some basic principles:

* Hints that are minimally obtrusive allow the learner to preserve their sense of ownership over the final result. The big dangers with a hint are (a) that you steal the opportunity to learn by removing a part of the task that would have been important to the learning experience, and (b) that you steal the experience of success because the learner doesn’t feel like they really did it. These dangers are related but distinct.

* How do you give a minimally obtrusive hint?

(a) Hints that *direct the learner’s attention* to a potentially fruitful avenue of thought are superior to hints that are designed to give the learner a new tool.

(b) Hints that are designed to facilitate movement in the direction of thought the learner already has going on are generally better than hints that attempt to steer the learner in a completely new direction.

* If the learner does need a new tool, this should be addressed explicitly. It’s kind of disingenuous to think of it as a “hint” – looking up “hint” in the dictionary just now, I’m seeing words like “indirect / suggestion / covert indication”. If the learner is missing a key tool, they need something direct. The best scenario is if they can actually ask for what they need:

Learner: If I only had a way to find the length of this side using this angle…

Teacher: oh yes, there’s a whole body of techniques for that, it’s called trigonometry.

This is rare but that’s okay because it’s not necessary. If the teacher sees that the learner is up against the lack of a certain tool, they can also elicit the need for it from the learner:

Teacher: It seems like you’re stuck because you know this angle but you don’t know this side.

Learner: Yeah.

Teacher: What if I told you there was a whole body of techniques for that?

Okay, those are my four cents. Keep me posted on this journey, it sounds like a really rich learning experience for J.

All the best, Ben

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It needs crowdfunding to cover publication costs. I am about to help out and I invite you to do so too!

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