When I was at Sarah Lawrence College, doing my first master’s degree in Child Development, I worked on a conference project where I wrote a book of ways that parents could talk to their children when they went to science museums. Questioning was a big focus, and the two works that inspired me the most were Vygotsky’s work (both his ideas of the Zone of Proximal Development, as well as learning as a social activity, and community’s role in making meaning) and Eleanor Duckworth’s piece called The Having of Wonderful Ideas (the idea that the student learns the most when they’re wondering something that they then answer – the “right question” to push student thinking is their own because it’s individual to each student as to what question will be best for them).
Somehow, despite my best efforts, when I went to grad school for math education, there was so much new knowledge in my head, that when I started my first year of teaching, I didn’t deploy much of my knowledge about how we learn through talking and in social situations. Now, however, I’ve dedicated myself to ensuring multiple opportunities for students to talk about their understanding out loud.
But it got me thinking: why isn’t talking emphasized more in teacher education, especially for math teachers? And I realized some of it is about what we’re talking about. There are the open-ended questions that allow us to make thinking visible and push student thinking, and there are the close-ended questions that don’t promote a DIALOGUE.
There, I’ve said it. Dialogue. It requires two people to engage in the conversation – talking AND listening. Instead, what I feel like we hear more often in math classes (and I’m guilty of this too!) is math monologues!
We know it’s important to get students to explain their work, so we get them up to the board and they present their work something like this: “First, I did this. Then I did this. I did these other steps until I got this answer. Here’s my solution.” We ask the other students, “Does anyone have any question for the presenter?” Crickets chirp and no one makes eye contact with you. You’re not even sure that anyone listened to the presenter. Eventually, you send them back to their seat, convinced that you’ve done your job – perhaps you even get the rest of the class to clap for them.
Now, this makes me consider: what is the GOAL of having students talk in math class? Is it to get them ready to do public speaking? Because I don’t think that’s the main objective! At least, not for me.
Rather, I think the talking part is where the sense-making happens. I think that as you’re resolving cognitive conflict in your mind, the only way to untangle all the threads and make sense of it is to get it organized – and for that, talking helps. Sometimes, the cognitive conflict might come as students are solving a problem (for a well chosen task!) or as their knowledge of mathematics expands (i.e. they think about how negative numbers fit into what they already know about numbers). Either way, I think students are not always fully aware that there IS a conflict until they begin to articulate their thinking – and someone else points out the conflict or has difficulty restating it and asks for clarification.
Which brings me to the next idea: talk moves! Why did no one teach me about those in college??? The ONLY one I ever heard about was wait time, and while it’s true that it’s the easiest to deploy (and the one that starts with you), I think revoicing and restating are even more valuable. Another teacher mentioned difficulty getting their students to listen to each other the other day, and I realized that the revoicing strategy, when incorporated into your daily math practices from the start, will begin to resolve that conflict because you change the math monologues into dialogues by adding an audience. We need to teach students how to listen and speak to each other, but also to think about whether or not what they’ve heard makes sense.
I think the purposes for talk are varied, but I think these are the main ones I would highlight:
- Helping a student to clarify and solidify their own thinking by talking aloud with a partner or to a group who offers revisions and asks for clarification.
- Practicing explanations before writing them down so they will become better writers about mathematics.
- Explaining not just the what or the how of solving a problem, but the why. Doing so in a way that convinces not only themselves and a “mathematical friend” but also to convince a skeptic.
- Hearing other ideas and strategies from students who may explain things in a way that makes sense to them (where a teachers’ explanations may not make as much sense perhaps).
I also think another issue teachers have with talking (especially in the math class) is equitable distribution of WHO is doing the talking. I think my biggest issue is that the only way I was taught in grad school to draw in the reluctant hand-raisers was to use cold calling or call students randomly, and I now know that the anxiety that causes can shut down working memory so students panic and freeze and are unable to answer. So I think we need to replace that with some other strategies for engaging students in dialogue.
First, I like to ensure I establish a culture of talking and listening early on. I think one way I do this is by asking students to make sense of each other’s ideas by asking someone to re-voice or restate what someone else said. I may call on a student who I’m not sure is understanding it as a check for understanding, but I also give them words to use if they’re not sure they do understand. I remind them that it is their responsibility to ask another student to speak up if they can’t hear them (as I often do at PDs when other participants speak too quietly for me to hear!), or to ask someone to repeat their idea if they didn’t understand it. I say, “You can ask student to repeat what they said if you couldn’t hear it or you are not sure you understand it yet. Sometimes, we need to hear ideas more than once before we’re really sure we’ve understood it.” I’ll model that myself, too, sometimes asking a student to repeat something they’ve said aloud for us as I make sense of it. If a student asks the other student to repeat the idea and they still can’t repeat it, I might say, okay, let’s see if someone else can restate it in another way so we can see if we’re making sense of this. First student, I’ll come back to you – I want you to listen carefully to what everyone’s saying about this idea, and see if we can make sense of it together. Then I might call on one to three more people (depending on the quality of the idea or the length of it) to repeat the idea before returning to this student to check for understanding again.
I also do a lot of turn-and-talks with the students (again, from very early on – it’s usually one of the things I do on the first day!). I find giving a specific short prompt, and only 30-sec to 1 min (depending on the topic) is best, and I loved the suggestion I saw this summer of specifying person A and B, and telling them who starts and when to switch, to ensure equal distribution. I think (next year, with 6th and 7th graders), I might want to “fish bowl” what this should look like/sound like when it’s about math, because I’m not sure that I ever really modeled it well to my 8th graders. I think this technique has a lot of different uses, and I think the number one best use is to get all students practice talking about their mathematical ideas aloud. I think it’s one that also allows me to circulate, listen to student ideas, and select/sequence those ideas in the share out. One way to leverage this technique is to ask a student who you’ve heard say something great to a partner “Would you be willing to share that with the whole class?” and then calling on them in the larger share-out. I like this technique of individually inviting quieter students into the conversation by letting them know I’ve heard their voice and affirming that they have something important to contribute to the conversation.
I very much like this technique for engaging the quiet students – telling them I hear them, I see them, and I want their voices to be heard, whether it’s sharing good ideas, sharing important questions, etc. I think it’s much more effective than the cold calling or popsicle sticks, especially because you can get at the root of WHY the student is quiet and setting a goal with them. I tell my shy students, while it’s not realistic that you’re going to raise your hand 5 times per day (if you never do it now!), saying you’re going to raise your hand twice per week from now on is a good step. Also finding out that a student is very shy may help me decide that I’m going to call on them for a question when I see their hand raised for a shorter answer because they don’t want to talk for a long period of time yet. Or I might discover that a student is very nervous about the accuracy of their work, and would prefer to share ideas when they already know if they’re on the right track – and would rather be informed ahead of time if I’m asking them to share a mistake so they can present it as “I originally thought this, but now I know…” as opposed to presenting it as fact and begin corrected publically. You don’t get any of this information by using popsicle sticks!
Another way I’ve engaged students in talking in math class is to have them anticipate and predict or have them notice and wonder – first with a partner perhaps and then a whole class share out (to make public the ideas from the groups). I find that these types of topics again can be safe for students to begin feeling comfortable sharing, and can also promote students realization that everyone has something to contribute (when a student who has low social standing shares a unique idea for example). I also like how this generates a lot to talk about, because they may have focused on different ideas.
Another thing I’ve been grappling with about talk is WHEN to do it. I often have a lot of student talk (and teacher talk!) at the start of class – going over the do now, launching a problem, etc. I circulate while they work, and perhaps they’re talking in partners or groups of 4, and sometimes I engage in discussion with small groups or partnerships. But by the time we get to the “end” of the problem, and we get ready to share, I feel like sometimes students are “over it” and they’re NOT listening to each other and they’re no longer engaged with the discussion – because it’s reinforcing the notions they have of themselves as math students already – either they already know the answer and they’re bored OR they don’t know the answer and they feel dumb because everyone else already knew the answer. I don’t know how to change this perception in the students’ minds, but I wonder if having the conversation happen SOONER (or perhaps in a different format) would be better. I think sometimes if students had conversations WHILE solving the problem, they might be better served by seeing and hearing other people’s ideas, as they get stuck, etc. This idea isn’t fully formed in my head, and I don’t know how to deploy it exactly, but I think my main vision about this is a whole-class mid-share (because I’m already having students talk to each other mid-solve in partners – but not in a structured way, admittedly).
One thing I want to work on more this year is getting students to ask each other questions. I think one thing I need to do is give suggestions (like on a poster with sentence starters), but I also think I need to do a better job modeling how I do it, and being transparent about asking them the types of questions I’d want to see them ask each other.
Another thing I grapple with is about when and how to share out our problems and thinking:
- If there was a problem set of practice for students to deploy a new skill, how should we check it over at the end? Should I go around to individuals and groups during work time and check everyone then? Should I get students to put their work up on the board (who then presents it – me or the students?)? Should I just show the answer key and ask them to check their papers and tell me the ones they want to go over? Should I choose ONE problem with the most potential misconceptions to go over and discuss it?
- If there was a problem-solving activity that we were investigating, how do we discuss the math involved? How do we draw out the BIG IDEAS while also paying attention to the specifics of today’s problem?
I think some of this goes back to the idea that I might need to reread the book, 5 Practices for Orchestrating Productive Mathematics Discussions. I know I’m doing a pretty good job with the anticipation skill, and to some degree the monitoring skill I’m doing much better at, but I feel like the selecting, sequencing, and connecting I don’t do as well at on a regular basis, depending on what the task that I selected was and what my goal in teaching that task was.
I also wonder if there are good videos of this practice being enacted in teachers’ classrooms. I remember the thing that transformed my mind (and practice!) about talk in the math classroom was an intervisitation where I actually went to another school and saw it in action in a teacher’s classroom!
This post is a bit of a winding ramble, and I think I still have a lot more to say about talk in the math classroom, but it will have to wait. My birthday is today (now that it’s after midnight!), and I need to go to bed so I can get up and celebrate!