Lessons from Bowen and Darryl Thursday, Jan 28 2016 

At the JMM this year, I had the pleasure of attending a minicourse on “Designing and Implementing a Problem-Based Mathematics Course” taught by Bowen Kerins and Darryl Yong, the masterminds behind the legendary PCMI teachers’ program Developing Mathematics course, with a significant assist from Mary Pilgrim of Colorado State University.

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.

Some work I’m proud of Friday, Aug 14 2015 

T was a third grader when all this went down.

At a previous session, I had asked T what she knew about multiplication, and she had told me, among other things, that 4\times 6 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 –

Tessa 4 6's cropped

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:

Tess 4 6's recircled cropped

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 –

4x6 cropped

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,

neat one cropped

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

final cropped

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!

Uhm sayin Saturday, Aug 1 2015 

Dan Meyer’s most recent post is about how in order to motivate proof you need doubt.

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:

  1. It motivates proof
  2. It lowers the threshold for participation in the proof act
  3. It allows students to familiarize themselves with the vocabulary of proof and the act of proving
  4. 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.

Sue’s Book Is Ready for Press and Needs Crowdfunding! Friday, Jun 20 2014 

Hey y’all, I am incredibly excited about Sue’s book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. If you have been around the math education blogosphere for more than a short time, you probably are too.

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

body_Book_cover_for_upload

Kids Summarizing Sunday, Sep 8 2013 

Back in the spring, I resolved to make a practice of having students summarize each others’ thoughts whenever I have classroom opportunities. This summer, I got the opportunity to give this technique a sustained go, when I taught at SPMPS (which was completely awesome btw). And:

It is an effing game-changer.

This summer, when I or a student put forth an idea, I regularly followed it with, “who can summarize what so-and-so said?” Or (even better), “so-and-so, can you summarize what so-and-so just said?” Following the models of Lucy West and Deborah Ball, I carefully distinguished summary from evaluation. “Not whether you buy it, just the idea itself.” When dipsticking the room on an idea, I would also make this distinction. “Raise your hand if you feel that you understand what was just said; not that you buy it, just that you understand what they’re trying to say.” Then, “leave your hand up if you also buy it.”

These moves completely transformed the way whole-class conversation felt to me:

* Students were perceptibly more engaged with each others’ ideas.
* The ideas felt more like community products.
* Students who were shy to venture an idea in the first place nonetheless played key roles as translators of others’ ideas.

Furthermore, for the first time I felt I had a reliable way past the impasse that happens when somebody is saying something rich and other people are not fully engaged. More generally, past the impasse that happens when somebody says something awesome and there are others for whom it doesn’t quite land. (Whether they were engaged or not.)

A snippet of remembered classroom dialogue to illustrate:

Me: The question before us is, do the primes end, or do they go on forever? At this point, does anybody think they know?

(Aside: This was after a day of work on the subject. Most kids didn’t see the whole picture at this point, but one did:)

[J raises his hand.]

J: They don’t end. If they ended, you’d have a list. You could multiply everything on the list and add 1 and you would get a big number N. Either N is prime or it’s composite. If it is prime, you can add it to the list. If it is composite, it has at least one prime factor. Its factor can’t be on the list because all the numbers on the list when divided [into] N have a remainder of 1. So you can add its factor to the list. You can keep doing this forever so they don’t end.

Me: Raise your hand to summarize J’s thought.

(Aside: although J has just basically given a complete version of Euclid’s proof of the infinitude of the primes, and although I am ecstatic about this, I can’t admit any of this because the burden of thought needs to stay with the kids. J is just about done with the question, but this is just the right thing, said once: the class as a whole is nowhere near done. This is one of the situations in which asking for summaries is so perfect.)

[Several kids raise their hands. I call on T.]

T: J is saying that the primes don’t end. He says this because if you have a list of all the primes, you can multiply them and add one, giving you a big number N. If N is prime, you can add it to the list. If N is not prime, and its prime factors are not on the list, you can add them.

Me: J, is that what you were trying to say?

J: Yes.

(Notice that a key point in J’s argument, that the factors of N cannot already be on the list, was not dealt with by T, and J did not catch this when asked if T had summarized his point. This is totally typical. Most kids in the room have not seen why this point is important. Some kids have probably not seen why J’s argument even relates to the question of whether the primes end. All this has to be given more engaged airtime.)

Me: raise your hand if you feel that you understand the idea that J put forth that T is summarizing.

[About 2/3 of the room raises hands. I raise mine too.]

Me: Leave your hand up if you also find the idea convincing and you now believe the primes don’t end.

[A few kids put their hands down. I put mine down too.]

N [to me]: Why did you put your hand down?

Me [to class]: Who else wants to know?

[At least half the class raises hands.]

Me [to T]: Here’s what’s bugging me. You said that if N is not prime and its prime factors are not on the list, I can add them. But what if N is not prime and its prime factors are already on the list?

T [thinks for a minute]: I don’t know, I’ll have to think more about that.

[J’s hand shoots up]

Me [to T]: Do you want to see what J has to say about that or do you want to think more about it first?

[T calls on J to speak]

J: Can’t happen. All the numbers on the list were multiplied together and added 1 to get N. So when N is divided by 2, 3, 5, and so on, it has a remainder of 1. So N’s factor can’t be 2, 3, 5, and so on.

T: Oh, yeah, he’s right.

Me: Can you summarize his whole thought?

[T explains the whole thing start to finish.]

Me: Do you buy it?

T: Yes.

Me: Who else wants to summarize the idea that J put forth and T summarized?

Unexpectedly, this technique speaks to a question I was mulling over a year and a half ago, about how to encourage question-asking. How can the design of the classroom experience structurally (as opposed to culturally) encourage people to ask questions and seek clarification when they need it? The answer I half-proposed back then was to choose certain moments in the lesson and make student questions the desired product in those moments. (“Okay everyone, pair up and generate a question about the definition we just put up” or whatever.) At the time I didn’t feel like this really addressed the need I was articulating because it had to be planned. Kate rightly pressed me on this because actually it’s awesome to do that. But I was hungering for something more ongoingly part of the texture of class, not something to build into a lesson at specific points. And as it turns out, student summaries are just what I was looking for! The questions and requests for clarification are forced out by putting students on the spot to summarize.

A last thought. Learning this new trick has been for me a testament to teaching’s infinitude as a craft. Facilitating rich and thought-provoking classroom discussions was already something I’d given a lot of thought and conscious work to; perhaps more than to any other part of teaching, at least in recent years. I.e. this is an area where I already saw myself as pretty accomplished (and, hopefully with due modesty, I still stand by that). And yet I could still learn something so basic as “so-and-so, can you summarize what so-and-so said?” and have it make a huge difference. What an amazing enterprise to always be able to grow so much.

Notes from the Learning Lab: How to Dull My Curiosity Friday, Dec 14 2012 

I know I say this kind of thing a lot but I’m sitting here studying for a final, and this truth is just glaring and throbbing at me:

If you want to dull my curiosity, tell me what the answer is supposed to be.

If you want to make my curiosity vanish completely, do that and then add in a little time pressure.

There is nothing as lethal to my sense of wonder as that alchemical combination of already knowing how things are going to turn out (without knowing why), and feeling the clock tick.

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.

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