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.

Deborah Ball and Lucy West are F*cking Masters Sunday, Mar 31 2013 

I recently saw some video from Deborah Ball’s Elementary Mathematics Laboratory. I actually didn’t know what she looked like so I didn’t find out till afterward that the teacher in the video was, y’know, THE Deborah Ball, but already from watching, I was thinking,

THAT IS A F*CKING MASTER. I F*CKING LOVE HER.

It put me in mind of a professional development workshop I attended 2 years ago which was run by Lucy West. Both Ball and West displayed a level of adeptness at getting students to engage with one another’s reasoning that blew me away.

One trick both of them used was to consistently ask students to summarize one another’s train of thought. This set up a classroom norm that you are expected to follow and be able to recapitulate the last thoughts that were said, no matter who they are coming from. Both Ball and West explicitly articulated this norm as well as implicitly backing it up by asking students (or in West’s case, teachers in a professional development setting) to do it all the time. In both cases, the effect was immediate and powerful: everybody was paying attention to everybody else.

The benefit wasn’t just from a management standpoint. There’s something both very democratic and very mathematically sound about this. In the first place, it says that everybody’s thoughts matter. In the second, it says that reasoning is the heart of what we’re doing here.

I resolve to start employing this technique whenever I have classroom opportunities. I know that it’ll come out choppy at first, but I’ve seen the payoff and it’s worth it.

A nuance of the technique is to distinguish summarizing from evaluating. In the Ball video, the first student to summarize what another student said also wanted to say why he thought it was wrong; Ball intercepted this and kept him focused on articulating the reasoning, saving the evaluation step until after the original train of thought had been clearly explicated. Which brings me to a second beautiful thing she did.

Here was the problem:

What fraction of the big rectangle is blue?

What fraction of the big rectangle is blue?

The first student to speak argued that the blue triangle represents half because there are two equal wholes in the little rectangle at the top right.

He is, of course, wrong.

On the other hand, he is also, of course, onto something.

It was with breathtaking deftness that Deborah Ball proceeded to facilitate a conversation that both

(a) clearly acknowledged the sound reasoning behind his answer

and

(b) clarified that he missed something key.

It went something like this. I’m reconstructing this from memory so of course it’s wrong in the details, but in overall outline this is what happened –

Ball: Who can summarize what [Kid A] said?

Kid B: He said it’s half, but he’s just looking at the, he’s just…

Ball: It’s not time to say what you think of his reasoning yet, first we have to understand what he said.

Kid B: Oh.

Kid C: He’s saying that the little rectangle has 2 equal parts and the blue is one of them.

Ball [to Kid A]: Is that what you’re saying?

Kid A: Yeah.

Ball: So, what was the whole you were looking at?

Kid A [points to the smaller rectangle in the upper right hand corner]

Ball: And what were the two parts?

Kid A [points to the blue triangle and its complement in the smaller rectangle]

Ball: And are they equal?

Kid A: Yes.

Ball [to the rest of the class]: So if this is the whole [pointing at the smaller rectangle Kid A highlighted], is he right that it’s 1/2?

Many students: Yes.

Ball: The question was asking something a little different from that. Who can say what the whole in the question was?

Kid D [comes to the board and outlines the large rectangle with her finger]

Kid A: Oh.

I loved this. This is how you do it! Right reasoning has been brought to the fore, wrong reasoning has been brought to the fore, nobody feels dumb, and the class stays focused on trying to understand, which is what matters anyway.

0.99999… Tuesday, Oct 5 2010 

So I’m teaching this course this year. It’s for the math faculty of a high school. It’s called:

MA600 Algebra and Analysis with Connections to the K-12 Curriculum

I am unspeakably excited, and want to do the best job possible.

The class: 7 teachers, deeply committed to kids, serious, not real talkative, rightly protective about their time, which is in short supply, but eager to get sh*t done.

The content: Basically, all of mathematics, seen as a unified whole.

It’s met twice. The second class was last Thursday. I need to get my thoughts sorted out here. I’m expecting this to help me visualize the next moves more clearly, just by doing it, but I’d love your thoughts too.

I didn’t really know anything about the mathematical background of the group when I wrote the syllabus, so for the first class I gave them a getting-to-know-you problem set with a wide range of problems and just let them work the whole time. Magically the experience of watching folks work on the problems and then later looking at what they did on paper gave me just enough information to plan the direction of the class’ first unit. We’re beginning with analysis. My first goal: the \epsilon-\delta definition of the limit. (I.e., the definition of the limit, for the snobby among you.) My second: the completeness axiom.

The plan: generate the need to define the limit by working with 2 everyday concepts that are actually limits. Namely, infinite decimals, and instantaneous speed. My hope is that by pressing on these concepts, we’ll see that in spite of our familiarity with them, we don’t actually understand them unless we have a precise way to talk about limits. Then, develop the definition out of the need to fully understand the familiar. Then, develop the completeness axiom out of the desire to make sure infinite decimals have a limit.

Here’s what we did:

I opened class with a problem set designed to get them thinking about the meaning of decimals in particular, and various other contexts for the idea of limits. I shamelessly bit the format from PCMI. The problems span a wide range of skills and I didn’t leave enough time to do them all, so people could attack problems appropriate to their skill level. This is now my favorite way to differentiate problem sets, a propos of a) using it in some NYMC workshops last year, and b) hearing about how wonderful it was for everyone at PCMI.

Then, since we are all just getting to know each other, I did a short presentation on the mindset I wanted us to be in:

(Scribd did not handle slide 6 very well, which is too bad because I was proud of that slide. This is my first PowerPoint presentation ever. Actually I did it in Keynote.)

Then, we got to business. I put this up:

I asked them to talk about it with their tables. (I had them in 2 pairs and a group of three, in three tables in a horseshoe shape in front of the board. I like this and think I’ll keep it. Easy transitions from pair/group to whole-class; tables feel separate enough so you don’t feel like your conversation with your partner is in front of everybody; but everyone’s close enough so we can all talk. On day 1 I put us all around one table, for a sense of collegiality and common purpose, but it was too close; you couldn’t discreetly check in with your neighbor, for example.)

There was a widespread sense of mathematical discomfort, and rightly so. Infinite decimals enter most people’s math educations with no attention to the fact that they actually violate everything you’ve learned about math up to that point. You don’t get the full story until analysis, but unless you really get intimate with and own that content, you probably don’t connect what you learn there to what your teacher introduced without comment somewhere between 3rd and 7th grade, as though it weren’t a mind-boggling idea. “When you expand 1/3 as a decimal, the 3’s just keep going.” Or, “3.14159… It never ends or repeats.” Um, excuse me? It NEVER ENDS?

So it’s no surprise everybody has an underdeveloped idea of infinite decimals, and therefore that objects like 0.99999… cause some dissonance. This is very productive dissonance. I’m hoping it carries us all the way to the completeness axiom; we’ll see.

One of the three tables produced the standard argument that if x = 0.9999…, then 10x = 9.9999…, so 9x = 10x – x = 9, so x = 1; but even this table found this conclusion unsatisfying. I asked them why. The table that had produced the argument said, “usually this method gives you a fraction.” I asked for an example. They produced one from the problem set:

x = 1.363636…
100x = 136.363636…
99x = 100x – x = 135
x = 135/99 = 15/11

I asked how many folks found this argument convincing. 7 out of 7. (Well, one raised hand was kind of hesitant.)

I asked the same question about the same argument with .9999…. 4 out of 7. Then I dropped this:

How many people found this one convincing? 0 out of 7.

Reasonable.

Right?

Then what’s the difference?

At first, they cast about a bit, but then one of them said, “1.363636… has a finite limit, but …9999.0 doesn’t.” Their ideas began to coalesce around this type of language. Another one said, “we can actually estimate 1.363636…, for example we know it’s between 1 and 2.”

From the point of view I am ultimately heading for, this is the rub. Infinite decimals suggest convergent series, and the standard way to give them meaning as real numbers is that they are equal to the limit of the convergent series they suggest. …9999.0 suggests a wildly divergent series, so it cannot become a real number in the same way. (To bring home that convergence is the heart of the matter: there is an alternative way to define distance between numbers, the 10-adic metric, according to which it is actually …9999.0 that has the convergent series, and in this alternative system the above proof is valid and it actually does equal -1.) What I’d like us to do is a) define limits precisely; b) use this to prove that when a series has a limit, you can do the above type of manipulations to find it; c) try to prove that the series suggested by an infinite decimal always has a limit; d) realize that we can’t prove this without articulating the completeness axiom; e) articulate the axiom; and f) prove from the axiom that any infinite decimal has a real number limit. (Somewhere along the line, produce an \epsilon-\delta proof that 0.9999… = 1.) Now, how to orchestrate this…

For next time I told them to try to craft a definition of the meaning of an infinite decimal 0.abcd… I gave them a few minutes just before the end to discuss this with their groups. I’m expecting to learn a lot about their thinking from what they come up with, but I’m not counting on anyone to have a mathematically satisfying answer. I’ll be pleased if somebody does though.

As I think about next class, here’s what’s on my mind:

1) When we develop the \epsilon-\delta limit, what I’m going for is for this definition to feel like a satisfying relief. I know how easy it is for this definition instead to feel like a horrible monstrosity designed to oppress analysis students. I think what I have to do is keep them thinking about the reasons why anything less than this definition is too vague, which means I need to keep coming up with objects and problems that throw monkey wrenches into whatever more naive definitions they go for. (Of course, if they come up with something equally precise as the \epsilon-\delta limit but different, that would be amazing.) I feel like we’re off to a good start on this, but I want a fuller catalogue of head-scratchers (like …9999.0 = -1) to push the level of precision higher.

2) Relatedly, I sense a danger that the “real answers” will be unsatisfying because it’ll feel like “wait, I already said that.” For example, the participant who said that the difference between 1.3636… = 15/11 and …9999.0 = -1 is that “the first one has a finite limit”… I mean this is basically the answer. But it’s not based on a precise definition of limit yet, so it’s not what I want yet. I’m afraid of a “what was the big deal?” moment when we’ve got the real sh*t up there. I think the way to avoid this lies in that catalogue of head-scratchers I need to develop, so that nothing less than the real thing is satisfying. What do you think?

3) Where to go immediately next. Basically the question is: stick with decimals? Or change gears completely and press on the notion of instantaneous speed? Most (not all, I think) of the teachers have had a calculus course, but think at most 1 or 2 of them have internalized the philosophical lesson that instantaneous speed needs to be defined as a limit in order for us to even access it. I’m attracted to the idea of switching gears because I’m drawn to the connection between the disparate realms: two highly familiar, but totally different, objects – infinite decimals and speed in a moment – both getting pressed on to the point where you realize you never fully understood either one, and then you realize that the missing idea you need is the same thing in the two cases. (A precise way to talk about what number some varying quantity is “heading toward.”)

Actually as I write this out, it seems clear to me that switching gears is the way to go. I think it’ll give us a clearer understanding of what we’re missing with the decimals. Also, it’ll allow us to access all this rich historical stuff around the development of calculus. For example, maybe I’ll share with them some choice quotes from Bishop Berkeley’s The Analyst, to help articulate why the 18th century definition of the derivative was inadequate.

Anyway. Very excited about all this. Will definitely keep you posted.

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