Ratzel: Teaching mathematical discourse

Marsha Ratzel
Marsha Ratzel

The following article by Marsha Ratzel was published in a December issue of "Education Week". Ratzel is a National Board Certified teacher from Kansas where she teaches middle school math for Blue Valley School District.


What's on the horizon for my young learners? I can't predict the future, but I know this much is true: Performing basic computational tasks won't be a gateway to a well-paid or long-term career. My students will need to be adept at locating information, analyzing it, and synthesizing it into something useful. They will have to be able to think, reason, and communicate to solve complex challenges.

This has big implications for how we teach math.
The Common Core State Standards, of course, highlight the importance of "mathematical practices." The idea is that if you can't talk about or explain the math you're doing, you don't know it well enough.

Middle school students are fairly accustomed to making educated guesses and talking in science class about how something works or will turn out. But it isn't something they are used to doing in math class. I realized that my middle schoolers needed to start discussing their math ideas in a logical way: forming conjectures, then using evidence and logic to "prove" their ideas. So I set out to get them talking.


My students had never heard of "mathematical discourse," so first we had to define it. Being typical middle school students, they liked the idea of arguing, but needed to learn the difference between arguing and discourse.

We started from the premise that a "conjecture is a statement for which someone thinks that there is evidence that the statement is true. The main thing about a conjecture is that there is no proof." That is, there's no proof at the time, but mathematical thinkers can create a process by which we test and generate proof, learning that our conjectures are (or are not) accurate.


I wasn't exactly sure how to accomplish this kind of conversation, so I went to my Twitterverse friends and colleagues. Many math teachers I follow seem to be encouraging mathematical discourse effectively. I feel lucky to be able to read about how other educators have done this with their students before trying it with my own.

My students first worked on this kind of thinking/reasoning when I adapted an activity created by Malcolm Swan and shared by my Twitter friend Fawn Nguyen. I presented students with 20 equations that they had to classify as being Always, Sometimes, or Never true. The results were mediocre the first time, but as we tried versions of this activity again and again to work on different kinds of problems, students got better and better.
What's especially amazing: Students liked this approach and asked that we do something similar again. And let me tell you, when 8th graders ask to do an assignment again, it's a real victory!

These initial experiences helped students as we tackled lessons about linear and nonlinear equations and models. We worked on different versions of Dan Meyer's 3-Act Math Tasks, including Split Time and Leaky Faucet. In each case, students considered the information at the beginning of the problem, offered a conjecture, figured out what else they needed to know, and set about testing their ideas. They compared notes with each other to identify what was—and wasn't—working.

We're nearing the end of the first semester of the school year, and lately I've been noticing that students are approaching problems in more systematic ways.

In a recent series of lessons, we were studying functions and trying to figure out what the domain of a function might be.
Mind you, most of my students are still trying to amp up their number sense. Thinking about functions requires students to have a working knowledge of how numbers are strung along the number line and why numbers fall into different categories. This goes well beyond identifying and understanding odds, evens, composite, or prime numbers—I ask students to build on that knowledge but to consider bigger sets/categories.

For example, students had to decide if you could divide by zero. I asked them what kinds of numbers have square roots—do positive numbers, negative numbers, fractions? And what about zero? Students couldn't find a domain if they didn't have a grasp of those kinds of number sense questions.

After further developing their number sense, students began building conjectures. It's critical for students to have the opportunity to exchange ideas and figure out how to test them, working alongside classmates. I stood nearby, occasionally offering questions or encouragement—more like a sports coach than a traditional lecturing math teacher. If you're anything like me, you may find it tough not to jump into the conversations, but it's so exciting to hear students stretching their thinking!
I had the delight of watching students work and work on finding the domain of a function, figuring out whether a table was a function, and similar problems. Calculators in hand, they'd test out a bunch of ideas, then check in with other groups to compare notes. They'd be excited and hopeful about an initial answer … only to have that idea go, "POP!" Then they'd regroup and try another approach.

They didn't give up. With time, they pieced together a common understanding. And they drew upon the language and concepts we’d been building, lesson by lesson, throughout the semester.


For homework one night, students answered this question: "What is the set of routines that defines how you approach testing your ideas?" They shared their responses with partners in class the next day, and then the class collaborated to identify a common working process. Students defined a set of steps they felt were handy for sizing up a problem, logically working through possibilities, and (after the testing) crafting a general statement.

This is abstract stuff. Open-ended educational experiences can be tough for students (especially if they haven't learned this way before), and I see how they struggle to hang in there. But when I continuously build this kind of thinking into daily lessons, I see students becoming more confident.

Here's my status report, midway through the school year: In a class of about 30 students, I'd wager that at least half look forward to tackling open-ended questions. About a quarter are enjoying the experience and can function well within groups, but struggle with individual work. And a quarter are frustrated—they just want the right answer.

What does this mean for me as a teacher? I provide additional support, prompting, and encouragement to students who don't feel comfortable with offering their guesses about math ideas. It's a delicate balance; I don't want to do the work for them, but they sometimes need specific direction to keep them from giving up.
Reflecting on where we are and how far we've come this semester, I see great progress. Progress that I'm not sure we'd have accomplished without incorporating mathematical discourse and conjecturing.

Sometimes I think it's the actual math skills where I see the most build-up of students' proficiency. Other times, I believe where they've made the most progress is in the act of talking math with each other.

When I observe my students, I see future architects, engineers, accountants and computer app developers gaining critical skills in analyzing, creating ideas, testing them out, and then defending them. I also witness students taking risks and supporting each other in being mathematical thinkers.

And as these kinds of assignments become routine, I see our classroom culture shifting. My classroom is becoming more like the collaborative, challenging work environments my students will face in the future—whether or not their careers have anything to do with math.

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