# Math Discourse on a Plane and in the Classroom: How to Help Students (and Adults) Think Differently about Math

By: | 01/12/2021
Category: Instruction

“OH, I hated math class,” said every person I ever sat next to on a plane. I, then a middle school math teacher, would say something like, “Oh no!” or “I’m sorry,” and then they’d tell me why they hated what I’d dedicated my professional life to. They’d say math class was hard, or that they felt stupid, or that they weren’t a math person. And then the topic would change or the flight would end, and those nice strangers would go on to their vacations, or business trips, or reunions with their families without giving our conversation a second thought.

However, I thought about these conversations long after I reached my destination. Why did so many adults have such a visceral reaction to math class? There’s clearly something different about math class that sticks with people for years. I knew that my students thought my class was hard, but what would they say if they sat next to a math teacher on a plane?

I think part of the blame lies with the subject itself. In math (at least in the math students learn in K–12), the answers to problems are clear and indisputable. Some students find this certainty comfortingthey can check their work and be sure of their responses. Many students, though, find the stark right or wrongness of answers harsh. To these students, hearing that their answer is wrong can feel like a judgement or a rejection. It makes sense that students would internalize this and develop a dislike for the subject.

Another problem with math is that the apparent simplicity of the answers (e.g., x = 5) masks the processes, concepts, and understandings behind them. When studying literature, students know that an unsupported answer is just a thesis statement, not a complete response. In science, a hypothesis is one of many steps that will be evaluated as a whole. In math, students often see the answer as correct or incorrect by itself and don’t consider whether their reasoning, setup, or calculations were sound. The tempting clarity (and resulting focus on) of the answers to problems hurts students and teachers alike because it blinds us to the richness of students’ thinking and hides the years of learning that led us to the solution on the page.

So how do we get students, and the adults they grow into, to think differently about math? By changing the way we talk about math. If we shift the focus in class to students’ processes and understanding rather than their answers, we can create a more inclusive and generative math class. Students who come to value their thinking and recognize that, even in math, learning is a process of revising their understanding, will develop a healthier relationship with math.

“If we shift the focus of class to students’ processes and understanding rather than their answers, we can create a more inclusive and generative math class.”

For what it’s worth, I’m not saying that getting the right answer isn’t important. It is! It’s just not as important as being able to find the next right answer to a problem you haven’t seen before, knowing why your answer is correct, understanding what you’re doing while you find that answer, and becoming a flexible, lifelong learner. If x equals 5 today, great. It won’t tomorrow, so the fact that I’m right right now doesn’t count for much.

I know this change sounds like a big ask. Adopting a new perspective on your teaching isn’t easy. Thankfully, there are lots of great resources out there with concrete, actionable steps to help you shift the focus of math discussions with your students. In an article from Mathematics Teaching in the Middle School titled, “Rough-Draft Talk in Mathematics Classrooms,” authors Amanda Jansen, Brandy Cooper, Stefanie Vascellaro, and Philip Wandless describe three principles that resonate with my experience of making this change in my classroom and are a good way to start shifting your (and your students’) perspective on math class.

## Foster a culture supportive of intellectual risk taking.

For students to value mathematical thinking, they must feel comfortable participating in it. This involves building a respectful, safe classroom culture where students feel that their thinking is valued.

Here are some examples of things you can say to put the focus on students’ thinking, rather than right or wrong answers:

• “Tell me how your thinking changed by filling in this sentence: ‘I used to think _____, but now I think ______.’”
• “Turn and talk to your neighbor about how to solve this problem. When you’re done, I’m going to ask you to share how your and your partner’s thinking was similar or different.”
• “We voted on this at the beginning of class. Does anyone want to switch their vote?”

These prompts show your interest in students’ sense making and promote a growth mindset. Once students volunteer their thinking, keep them invested in it by initially only offering nonevaluative feedback. For example:

• “Can someone explain why they agree or disagree with Lily?”
• “What questions do you have for Gelson about his thinking?”
• “Did anyone else notice what Antoly brought up?”

Knowing whether something is right or wrong is such a powerful, overriding idea that when students hear it, they can tune out the crucial conversation about mathematical reasoning that we want them to participate in. Like saving the student’s name until the end of a cold call (e.g., “When is the product of two factors less than either factor . . . DeAnna?”), giving students more time when they have to consider whether an answer is right or wrong on their own keeps them more engaged.

## Promote the belief that learning mathematics involves revising understanding.

Reframing new information as “revised thinking” lets students see their learning as discovery and growth. For example:

• “Yesterday in class we defined exponents as repeated multiplication, but these exponential expressions have teeny-tiny answers. Is there anything you think we might have left out of our definition from yesterday?”
• “This idea seemed good on this problem, but now, looking at the next problem, I’m not so sure. What do you think?”
• “Why does this strategy work here? How could we know if it will always work?”

I always loved letting students persuade me. They had fun disagreeing with me, and I got to model a growth mindset and healthy academic discourse. Do be careful to make it obvious if you’re pretending to adopt an incorrect position, though!

## Raise students’ statuses by expanding what counts as a valuable contribution.

Praising students for a behavior is one of the most effective ways to ensure you’ll see it again. When process- and thinking-oriented contributions get positive attention, students catch on. Here are some student behaviors you can praise in your classroom to promote a broader range of responses:

• Changing their minds or revising their thinking
• Offering respectful critical feedback to peers
• Explaining or showing an idea with new representations or justifications
• Being a good listener
• Giving a persuasive argument
• Thinking creatively about how to approach a problem

Affirming more diverse student contributions has the double bonus of making class more inclusive. Students who may not be able to find the answer still have a lot of ways to be engaged and valued. Teachers’ targeted praise lets students know they belong in their math classrooms. It also gives them the confidence they need to move forward in their personal math journeys.

So, in the end, I’m not sure what my students would say to a teacher on a plane. There’d probably be a mix of bellyaching, fond reminiscing, and neutral shrugs. What I do know is that my students—and students who come from classrooms that value discourse, growth, and change—will be prepared to do something the strangers on my planes weren’t ready to do: talk about math.

Integrating Effective Teaching Practices: Teacher Moves That Engage Students in Discourse and Mathematical Thinking

Recognizing Misconceptions as Opportunities for Learning Mathematics with Understanding