How can we ensure access for all students in the math classroom?
Today on the podcast we’re joined by Amy Lucenta and Grace Kelemanik. Amy and Grace are co-authors of Routines for Reasoning: Fostering the Mathematical Practices in All Students. Amy and Grace are big proponents of helping students develop their mathematical thinking and reasoning skills, and say that building routines into math instruction is one way to do that.
We started out conversation by asking: “what does routine look like in a math classroom?”
Below is a full transcript of this episode!
Amy: So we make a distinction between just a classroom routine and an instructional routine. And the reason for this is there are a lot of routines in classrooms that make a classroom work smoothly for teachers, for kids, for materials, for logistics. And then there are routines that focus on instruction. And instructional routines have specific designs built into them, so kids know what to do, when, with whom, and how to engage in the math content as well. And repetition in a math classroom ... Well, when I hear repetition, I actually think memorizing facts, kind of drill and kill repetition, which is in stark contrast to the instructional routines for reasoning.
Grace: Right, so doing the same thing repeatedly over and over again, meaning the same kind of math problem over and over.
Grace: And so what's routine about instructional routines is not necessarily the type of math problem, but the way in which students interact with the problem, and with each other, and with the teacher.
Steph: So the routines of those thought processes?
Grace: Yeah the ways of thinking, certainly. And also the... am I going to have some time on my own to think about this? And then am I going to talk with a partner about it? And then we're going to share out in full group. And when I talk with my partner am I just talking about whatever I want to talk about? Or do I always have a set in stone to help me talk about it? And those kinds of design features is what stays the same each time.
Steph: And do you find that a lot of classrooms right now have those instructional routines in their math instruction?
Amy: Well the ones we go into do!
Grace: I think there's ... Maybe this is not the right response to this, but I think in the field we're mashing up a couple of different ... We don't yet have the language clear about what we mean by instructional routines or what we mean by task types, like kinds of math problems we ask kids to do. And I think we're trying to work that out.
Amy: A lot of teachers are taking up instructional routines for really good reasons. So when you go into classrooms, they're working on number talks. And kids know how to engage in that in order to learn computational fluency with conceptual understanding. People are up taking routines for reasoning. So you'll go into classrooms and see three reads being enacted. And when you go in classrooms and see instructional routines being enacted, one of the hallmark features is that you recognize it when you see it. And that means that in one classroom it looks kind of the same as the other classroom. And so if a kid were to switch classrooms mid-year, they recognize the routine also.
So we're seeing it more and more. And honestly teachers are really appreciating instructional routines because they're agreeing size that teachers feel comfortable test driving it in their classroom. And when they test drive the instructional routines in the classroom, they quickly see kids engaging in ways that maybe they didn't typically engage in. Or seeing how all kids have a voice in the development of mathematical thinking. So they see benefits really quickly and uptake them even more.
Steph: And so how can routines build space for students at all levels, of all abilities?
Grace: Yeah, so routines are incredible tools for helping a wide range of learners be successful in the mathematics classroom. In part because they're routine, they're predictable. So students understand the ways in which they're going to in their quotes, "do math today." How that's going to look in class and what that's going to sound like. And so because that's the same all the time and because routines are done repeatedly, it's just sort of a habit of this is how we engage in the math. So the routine in and of itself is a support. And then if you have routines, like routines for reasoning that have specific designs and strategies just baked into the flow of them, so strategies like ask yourself questions, turn and talks, sentence frames and starters, annotation, those kinds of supports for students, then that allows a wider range of kids into the conversation.
Steph: So they go in knowing what to expect.
Grace: They go in knowing what to expect and they have time and structures to process the ideas and the thinking. So individual think time to think about something, time to work with a partner to talk through the idea and work through the language. If they're sharing ideas in a whole group and the teacher is annotating while students are sharing, then students have this sort of visual that they can track while they're listening to their classmates talk about their ideas. And if the teacher is asking, then other students to repeat or rephrase what was being said, there's another pass hearing the idea, there's another pass at trying out the language, there's another opportunity to process what's happening. And the mathematical ideas and the thinking get increasingly precise for kids and understandable.
Amy: So as Grace was saying there are a lot of baked in supports. Our four essential strategies really support kids in the thinking. And the predictable design is an incredible support kind of implicit the fact that it's a routine. One example is say in the middle of connecting our presentations, that the way the discussion is facilitated is supportive of all kids. So kids work together to make a connection and then we share the connection out in the full group and students come up in partners to share their connection. One student does the talking, the other student points and gestures to the representations that are being talked about.
So kids in the classroom know this is the structure and they can watch the student who's pointing and gesturing as they're listening to the student speaking. And then the idea gets rephrased and refined by students in the classroom. And then that same idea gets annotated and color-coded so the thinking's organized and there's residue of the thinking. And all of that builds towards structural thinking in that routine. And all of those supports engage a wide range of learners. So for students who maybe there are students in the classroom who might lose focus for a minute or two. There's residue of the conversation.
Or students who are learning math and language, they're seeing the pointing and gesturing as they're hearing the language. And then they're seeing some of the language written as residue. So it's supporting them as well but for totally different reasons.
Steph: Right, so kind of multiple representations that can speak to different kids in the class. When we're thinking of these inclusive routines, like you were saying we want to make sure we are kind of centering those students who may have a harder time or may have specifically students with learning disabilities or sometimes also learning the language that the lesson's being taught in. So how can these routines specifically help those demographics?
Grace: So the way in which routines for reasoning support a broad range of learners, one, we're focusing on mathematical thinking. And I think we send a message in the routines that thinking takes time. And it takes time to think and reason through something. And it takes different structures to help you do that. Just like writing. You don't put pen to paper and write and you're done. It doesn't just flow right out of your pen onto paper and it's perfect, right? There's outlines and there's rough drafts and second drafts and third drafts. And conversations with folks. And it's the same with mathematics. So mathematical discussions in the classroom is that sort of rough draft thinking where students are working out ideas with a partner and in a full group.
So thinking takes time and the routines for reasoning sort of create that time and space and structures for students to process and form ideas and reform ideas and refine them.
Steph: I think that's great because I think sometimes kids can feel pressured to have the answer immediately and to automatically know and to offer that time to really think about your reasoning I think is great.
Grace: And I think there's a place for that in the classroom, right? There are certainly skills and there's fluency that we want to develop in students. So there are types of questions and times where you're looking to be able to get an answer and get it quickly and accurately. But mathematical thinking is not that. You don't think fast. You think slow and it marinates. And you look at it from different perspectives. And that just takes time to build. And so that's a shift in the classroom for students and for teachers.
Amy: One hallmark of routines for reasoning is the amount of mathematical discourse that happens during the enactment of an instructional routine. And for us as teachers, that's incredibly helpful because we elicit student thinking and then have the opportunity to hear it and make decisions around it and develop it. And it's super helpful for kids to be able to talk through ideas for all the reasons Grace was just mentioning. To be able to talk through an idea, refine the idea, do the rough draft, thinking and verbalize it. We also actually think about ourselves and adults and how often we have to verbalize things to remember or to make sense of.
And so we're looking to give kids those opportunities in classrooms through mathematical discourse as well. And ultimately we'll talk through things before we write them. We learn to talk typically before we learn to write.
Amy: So I'm giving kids that opportunity to build mathematical thinking through the discourse and with the discourse.
Steph: In your work together, you've identified several mathematical practices that match up with these different classroom routines. And one that I was really interested in was the decide and defend routine. Did I get that right?
Grace: Mm-hmm (affirmative).
Steph: Yeah, could you talk a little bit more about that?
Grace: Sure. The decide and defend routine was designed to help students develop the capacity to critique the reasoning of others and construct viable arguments. Which is the math practice three in the standards for mathematical practice. Amy calls it MP3-ing. Learn how to construct viable arguments and critique the reasoning of others. And in the upshot of it is, students are presented with a worked example. Someone else's work, not theirs. And their job is to make sense of it first. What was the question that was being answered, what did this person do, what did they find?
And then decide if they agree with it. So do they think the work is correct, do they think the answer makes sense, do they think the approach makes sense? And then once they've made that decision, defend that decision to the class. So I think this work is correct because ... And then be able to craft an argument that convinced a skeptic and then has students in the class, their classmates play the role of skeptic and push on their argument to make it tighter and tighter, and more and more convincing.
Amy: Yeah some unique features about decide and defend include the fact that students are annotating their thinking. Annotating to make sense of the worked example. And annotating to help them craft their argument. And learning to really communicate like a mathematician. Mathematicians don't always write five paragraph essays. Actually, they might never write five paragraph essays. But they do use symbols and color and annotation to communicate a whole lot. So kids have the opportunity to really build that practice in order to both critique the reasoning of others and to construct their argument.
And another unique feature of decide and defend is this iterative process of making sense of someone else's work. Crafting, making a decision about the work. Crafting an argument about your own thinking and then analyzing classmate's arguments about the thinking. So there's layer upon layer of thinking going on.
Grace: We're positioning students to consider each other's ideas, take them up, push on then, pull on them, see how far they ... Do they have legs? See how far they go. Make them better. And I think you're right, if I'm a student and my idea gets served up to the class, and kids take it seriously, my classmates take it seriously, and add to it, or push back on it and clarify it, it makes me feel good, sure, yeah.
Amy: I think one of the essential strategies I'd really love to underscore that's in all of our routines is this thing called an ask yourself question.
So, so often when we help students learn in mathematics, we sort of tell them how to do something. We show them how to do something. And we might give them some sort of graphic organizer or that they can bring to this type of problem. And a different kind of organizer to bring to another type of problem. And when it's this situation, do this. And it becomes this overwhelming set of if this, then that. And so we like to sit in this idea of an ask yourself question. The kinds of questions that mathematicians ask themselves when they're making sense of a new situation or working through a problem.
Grace: And so if kids can learn to ask themselves a lot of these questions, it gives them avenues into problem solving. So sort of a classic example is students are given a word problem to work on. And you'll often hear teachers say, read the problem, circles the numbers, underline the key words, put a box around the question as a way to make sense of it. And we would argue that while it's important to notice the numbers, what you really want to do when you see a number and what a mathematician would say is, is that number telling me something about a quantity, or is it describing a relationship between quantities in the situation?
So what are the important quantities in this situation and how are they related? And I change the form of this to make it easier to work with. The kinds of questions mathematicians ask, if we can ask those questions to students and get students habitually asking those questions every time they work on mathematics, then they, as Amy likes to say, develop an internal compass for mathematical problem solving. And so they don't have to depend on some external GPS walking them through every step of a problem to get to an answer.
And in that way it sort of fights against learned helplessness and it just builds their sense of self as a mathematician and a powerful math thinker.
Amy: The essential strategies that we bake into our instructional routines, routines for reasoning serve to support students in the thinking, engage them in the thinking and give them a running start sometimes. Like sentence frame, sentence starters, actually orient their thinking. It almost steers their thinking in order to develop a specific line of thinking. Quantitative reasoning, structural thinking or reasoning through repetition. And those four essential strategies that Grace was talking about, the sentence frames and starters, annotation, ask yourself questions, and the four R's support kids, give them entry footholds into the thinking and also develop the thinking.
So they're all specifically steered toward the avenue of thinking that's being developed.
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Grace Kelemanik has more than 30 years of mathematics education experience. A frequent presenter at national conferences, her work focuses on urban education, special populations, and teacher training. She is a former urban high school mathematics teacher and Project Director at Education Development Center. Grace has also worked extensively with new and preservice teachers through the Boston Teacher Residency program.
Most recently, Grace is the coauthor of Routines for Reasoning: Fostering the Mathematical Practices in All Students. She is also coauthor of The Fostering Geometric Thinking Toolkit. She is a mathematics education consultant and professional development provider.
Follow Grace on Twitter @GraceKelemanik
Amy Lucenta has extensive K–12 mathematics experience with all students, including a focus on special populations. She is a frequent professional development provider who helps teachers implement the Standards for Mathematical Practice. Amy is the coauthor of Routines for Reasoning: Fostering the Mathematical Practices in All Students.
Follow Amy on Twitter @AmyLucenta