Demystify student thinking about math

Two math teachers walk into a break room and proceed to have an accessible, fantastically math-nerdy chat about what it means to understand students more deeply. The result? A snack break–length video that celebrates and demystifies hosting a formative math conversation: a discussion where the teacher’s role is to listen, probe, and uncover deeper conceptualizations in the minds of their young mathematicians.

The math teachers in question? Ted Coe, director of Content Advocacy and Design at NWEA, brings 25 years of experience as a teacher, professor, department chair, administrator, and nonprofit director to amplify and elevate what it means to teach and learn math successfully. Basically, he knows what he’s talking about. And Scott Adamson, content designer on the Professional Learning Design team at NWEA, is an award-winning mathematics professor devoted to making math about sense-making and deep reasoning, redefining the idea of simply “getting it right.” His passion for math is formidable; just watch his TEDx Talk!

In a 12-minute conversation—which we were smart enough to record—these two math phenoms sit down to discuss not lofty geometrical proofs but, instead, how to host formative conversations that illuminate the vast web of student understanding. After watching this video, I thought, “I wonder what they would have said in two hours instead of 12 minutes.” As a math teacher myself, I wanted to know more. What do they mean by “ways of thinking” versus “ways of doing”? What does Ted mean when he says “packing the standards”? I invited them to elaborate. In other words, I listened and probed to uncover their conceptualizations. Did I just host a formative conversation? (Not exactly. For more information about how to host a real formative conversation in the math classroom, check out our grade-leveled, guided math formative conversation starters.)

First of all, how did you come to sit down to discuss formative conversations together?

Ted: It grew out of thinking that was spurred by the COVID-19 shutdown. It’s always been true that a good curriculum is developed around telling a mathematical story. Ideas, like plots, are developed in a way that makes sense to learners. With COVID, though, those storylines were interrupted. Knowing that strong teaching and learning requires both doing and thinking, we thought about how we might be able to bring to the forefront key ways of thinking that transcend grade levels. The NWEA math formative conversation starters (FCS) provide a means to do just that.

You’ve both been in math education for decades! When do you remember the notion of formative conversations entering the scene?

Scott: Decades indeed! Nearly four of them! While I don’t remember when I first encountered the phrase “formative conversations” (maybe Ted invented it), I do remember encountering the idea of listening to student thinking in the 1990s. As a high school teacher, I was involved with an innovative curriculum called the Interactive Mathematics Program (IMP), during which we were encouraged to prioritize student thinking over teacher instruction. This is where I first heard of the teacher’s role as “guide on the side” rather than “sage on the stage.”

Students are motivated to learn and have something meaningful to talk about if the task is interesting, challenging, and well designed.

Later in the 1990s, I joined a group known as the Maricopa Mathematics Modules in Arizona, where I live. We developed a series of 15 curriculum modules also focused on a student-centered classroom environment where students were actively engaged in making sense of mathematical ideas and solving realistic problems. We recognized the importance of listening to student thinking and worked to create activities that would reveal more about it.

I’m sure that early in my career, I viewed teaching as the ability to tell students what to do. That was about it! Now, I realize and value the importance not of students mimicking teachers, but of providing the opportunity for them to engage in the thinking and reasoning that might lead to understanding.

Ted: Well, Scott has been at this a bit longer than me, as I’m a bit below the three-decade mark. My evolution was much like his, where my focus early on was much about emphasizing procedures—or “ways of doing.” I imagine my students may have left my classes feeling as if math was much like collecting a box of puzzle pieces, knowing that they might make a cool picture when put together, but more or less merely rattling around at the moment.

Hopefully, formative conversations have been in play in classrooms all along, but the vision we have for this iteration is grounded in the types of questions that mathematics education researchers ask students as the researchers try to make sense of student thinking. With the FCS, we try to create situations for teachers to do similarly with their own students. Our focus is not on whether students can “do it,” but rather on how students might be thinking, which reveals how they connect important ideas that continue to evolve from grade to grade.

The formative conversation starter “starts with a task.” Can you explain why we start with a task, and what kind of task we might start with?

Scott: There is certainly a philosophical conversation to be had here. On the one hand, some people might believe that students must be taught ideas, concepts, or procedures first and then they can apply them on their own as they engage in a task. Kyle Pearce and Jon Orr call this “pre-teaching.” On the other hand, others believe that when students are presented with a meaningful task, the important mathematical ideas that are needed to solve the problem presented in the task may emerge.

Students are motivated to learn and have something meaningful to talk about if the task is interesting, challenging, and well designed. In the context of formative conversations, the task allows for the following:

  1. It sets the stage for students to have something to talk about. Formative conversations will hit a dead end quickly if you want students to talk about rationalizing denominators! But a rich task will promote student thinking and allow for conversation.
  2. It allows big mathematical ideas to emerge and helps students connect new learning to existing learning, which promotes a richer formative conversation.

Take, for example, the three-act task format coined by Dan Meyer. The task is quickly presented via video, picture, and/or text, which hooks students and gets them talking. This leads to formative conversation as students decide what is needed to solve the problem. As students request information, the teacher decides what information is provided. Students take the newly discovered or shared information and pursue a solution strategy. Throughout the process, the teacher listens to student thinking.

Ted: The tasks also serve as a way to engage and organize the conversations as they situate a conversation at a grade level and aim at a particular standard. Through this, we hope that teachers can see immediate connections to curricular work. That tight alignment is short-lived, however, as we focus not only on a single standard, but also on the many types of thinking around big ideas that are relevant in that space. In that sense, we are trying to compel a big-picture approach.

You both discuss “ways of thinking” versus “ways of doing.” One is easier to grade than the other! As a teacher who is expected to deliver grades, sometimes it’s daunting to explore the less quantifiable realm of student thinking. What advice can you give to a teacher like me?

Scott: Yes, it is daunting! It is challenging! It is difficult! How do we assess student thinking as opposed to assessing student ability to regurgitate algorithms? Let’s look at an example.

If students are asked to compute 8 – (-4), we don’t find out what they understand. Rather, we find out if they can remember how to do something. Perhaps students remember an algorithm, such as “add the opposite,” and write 8 – (-4) = 8 + 4 = 12. If that is the end of the assessment, we only know that the student knows how to obtain the correctly computed answer. If getting the right answer is the only goal, then tell students to ask Siri, “What’s 8 minus negative 4?”

So, what is the alternative? Consider these ideas:

  • Explain why 8 – (-4) = 12 using the chip model and the number line model.
  • Write a real-world problem situation that would require the computation 8 – (-4) = 12 to solve the problem.
  • Explain why 8 – (-4) is equivalent to 8 + 4.

Is it harder and more time-intensive to grade student work on items that assess ways of thinking rather than just getting the right answer? Yes! But is it necessary? Yes! We can ask fewer questions on an exam when the focus is on student thinking, and this can help with time management.

For me, I often reflect on my teaching and wonder if students are having a full mathematical experience. What do I mean by “full”? If students are primarily engaged in remembering what to do to complete a procedural or computational process, then the experience is not full. To be full, using very broad descriptions here, students should experience mathematics in the way I expressed in my TEDx Talk. Mathematics includes:

  • Making sense of problems, asking the right questions, determining what information is needed to solve the problem
  • Being able to model/mathematize the situation
  • Computing or engaging in procedural work (which likely will involve a computer)
  • Verifying, justifying, analyzing, and communicating the results

Unfortunately, many students spend a lot of their learning time on learning to compute or complete procedures—something that computers can do. They spend little time on the other things—things that only humans can do!

Is it harder and more time-intensive to grade student work on items that assess ways of thinking rather than just getting the right answer? Yes! But is it necessary? Yes!

Can you explain what you mean by “packing the standards” rather than focusing on just the standard at hand?

Ted: It follows from what I said above with reference to standards alignment in a task. There’s a temptation to focus on a single standard, but mathematics is so much more than that. Mathematical episodes combine multiple ideas. Your task might be about dividing fractions. But that deeply involves what you bring to the table about the meanings of division and how you think about fractions. So, where much of the standards-based work over the past decade has often focused on “unpacking the standards”—which is important work—I sometimes like to think about the FCS as “repacking the standards.” How are all of those big ideas working together in someone’s mind?

I loved what you said about inviting students to share at the end by asking, “What were you wondering during this? What was uncomfortable for you?” Do you recommend any other post-conversation reflection strategies—either for teachers or for students?

Scott: I love the 7 math exam questions from Francis Su. They are extremely helpful in revealing student thinking. Of course, the prompts would have to be adjusted to meet the needs of the audience for which it is intended.

Maybe we can remind students of the work of Al Cuoco, et al., in his 1996 article “Habits of mind: An Organizing Principle for Mathematics Curricula.” According to research, students should be pattern sniffers, experimenters, describers, tinkerers, inventors, visualizers, and conjecturers. Ask students if they felt like any of these during the FCS!

Ted: I also like the notion of students reflecting on their own points of dissonance. You might ask if there were points where they realized that answers to the different questions in a formative conversation weren’t quite fitting with each other. If so, are there goals they would like to set to help make more sense of those things? Also, we can use FCS to reinforce the idea that mathematics is about thinking, not just about doing, and how it’s OK to be vulnerable and share our ways of thinking. In fact, that’s one wonderful way to get stronger.

What advice would you give to teachers to build the rapport necessary to host a productive formative conversation?

Scott: It is worth taking time at the beginning of the semester to use team-building tasks to do this. Taking time early to build rapport, trust, and good relationships in the classroom allows for the opportunity to challenge students and expect them to engage, share with, and challenge one another.

Monty Williams, the Phoenix Suns’ head coach, says that to build this trust, rapport, and team environment, he doesn’t call players out; he calls them up.” It’s all about setting expectations, looking students in the eye when you talk to them, earning trust and respect, and then holding them accountable for engaging in mathematical discussion.

Ted: And, I would add, truly sticking to the design. For that moment, it is not about getting right or wrong answers. It’s about talking through how we are thinking and reinforcing the notion that we, as teachers, really do value thinking as much as doing.

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