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.

I’ve tried to throw too many new things into the mix this semester (and some are already falling by the wayside). So I’m not sure I’ll be able to try this now. But if I don’t, I definitely want to try it next semester. I’ll have to figure out when it makes sense, and how to do it naturally. I’d love to see you in action, Ben.

Ben:

This was fascinating, and it’s something that I’m going to be conscious of, too. Many thanks,

Steve

Awesome indeed!

Reblogged this on Shade Tree Math Teacher and commented:

I’m using Ben’s technique.

Ben

I linked to this in my most recent post. I feel energized by incorporating this into my practice and casting a wider net in my classroom conversations.

Awesome.

I’ve heard this before, but I haven’t tried it. I like the way you have made it a part of the lesson process. Your students expect and participate without thinking. I am definitely going to use this. Thanks for sharing!

I LOVE IT. I’ve used summarizing before in small groups, like when a group asks a question, I’ll give an answer and ask another kid to summarize my answer. But I’ve never thought of making this a part of whole class discussions! I’m trying this Monday!

I tried this today in my college Algebra class. It was magical. I blogged about it at mathybeagle.wordpress.com. or http://wp.me/p3GVbH-F

Um

awesome.Stumbled on your post because of mathybeagle (above me in the comments). I’ve spent my summer thinking and reading about how to up the level of math discussions in my classroom. This is so simple and so excellent at the same time. Thanks for writing it up and sharing.

I love the idea of summarizing another’s ideas. Just curious, what were your lead up activities on primes to bring about such insight from your students?

Below are the notes I taught off of the previous day. I remember that in class, we spent a lot of time with questions like “can you find a number that is r mod a, b and c?” I had previously taught them what “r mod a” means. For about 20 min at the end of class I put them in 2 big groups with the question “What if there were finite list 2,3,…,P of ALL primes?” After about 10 of those minutes where they generated some ideas but none that I thought would yield an answer, I said something like, “is there a number that is 1 mod everything on the list?” They stewed for another 10. Right at the end of class, J made the connection. I was curious to see if he’d still remember it the next day; you can see from the post that he did! The notes:

Day 4 Infinitude

Do now: Is 733 prime? Is 737 prime?

* Frame yesterday’s events.

* Have N*** and S*** present existence of factorizations; honor our dissatisfaction.

* Sequence of questions to develop infinitude theorem:

First, frame the inquiry. Why there is a question to answer (if we need it).

1. Find a number greater than and not divisible by 2, 3, 5, 7, 11, 13.

2. Find a number not divisible by and greater than 2, 3, 43.

2a. 1 mod 2, 3, 43?

3. Can you write down an algebraic expression that is definitely not divisible by n? Greater than and not divisible by?

4. Come up with a general method for finding a number not divisible by and greater than a, b, c (primes)

How about 1 mod a, b, c?

(If method is to find the next prime after a, b, c, point out that the whole point of what we’re doing is to find out if there even is always a next prime. Can you come up with a *formula*?)

5. If the primes end, then there is a biggest one, and we can list them:

2, 3, …, P

(P is the biggest prime)

Can you write down a number that is bigger than all of these and is not divisible by any of them?

Found my way to your blog through #MTBoS. Glad I did – you’ve helped me solidify my thinking on a question that just came up for me last week. Referred to your blog from my post – thanks for the inspiration and validation!

http://blog-storming.blogspot.com/2013/10/not-calling-on-those-who-know.html