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.
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definition of the limit
Research in Practice 