Math Discourse as Formative Assessment

We know that teachers need to be aware of students’ strengths and needs to help them learn math. But if we wait until the End-of-Module Assessment or even an end-of-lesson Exit Ticket to gather information on student comprehension, we may already be too late. To generate the data you need for just-in-time teaching decisions, purposeful discourse throughout the lesson is key. Let’s look at how to prepare your class for discourse and take steps to ensure that you get useful data and act on it effectively.

Create a culture for productive discourse.

Talk less, and let students talk more. The more your students talk, the more data you gather to make informed instructional decisions. To get students talking, move away from a back-and-forth pattern in which you pose a question, a student answers, and you respond to the answer. Instead, encourage students to talk about math with one another as a way to both promote their math development and reveal where they are in the learning process.

To create a culture for productive discourse, consider the following strategies that worked for me. Tell your students that you expect them to respond to one another. During class discussion, use silence to indicate that students should be speaking. Students will often fill the silence without your prompting them. Instead of looking at the student who is speaking, look at the rest of the students, inviting them to respond instead of you. When necessary, encourage conversation by posing questions like “Who can add to what Max said?” or “Who can say why they agree or disagree with Emma?” To guide the conversation, provide students with sentence frames like the following:

• I agree/disagree with _____ because _____.
• I solved the problem differently. What I did was _____.
• I like how _____ solved the problem because _____.
• I don’t understand _____.
• I tried _____, but then I got stuck when _____.
• I know this answer is correct/incorrect because _____.

Have more conversations about strategies and mistakes.

You learn more about your students when you shift their focus from finding a correct answer to thinking about how they solved a problem. Show interest in how they arrived at their answer—right or wrong. Allow students to share their strategies and listen to, consider, and ask questions about their classmates’ strategies. Encourage students to try different strategies, especially those that they find easier or more efficient.

Establish the practice of talking about mistakes, not just about multiple solution paths. It’s important that students value mistakes as a crucial part of the learning process. The message we want all students to hear is that mistakes are expected, respected, and inspected. When students learn to embrace their mistakes as opportunities for growth, they are more open to sharing their thinking. Encourage students to share their mistakes and misunderstandings with their classmates, even if they’ve corrected those mistakes on their own. A favorite sentence frame in my classroom was “At first I thought _____, but now I think _____ because_______.”

Normalizing errors in math helps students feel more comfortable with mathematical discourse. As students open up, you get a better idea of what they know, how they’re thinking about math, and where they still need help.

Plan two or three high-order questions for your lesson.

As teachers, we ask our students many different types of questions. We ask questions to gather information, to probe thinking, to shine a light on the mathematics, or to prompt reflection. Some of our questions are planned. Others are spontaneous. Either way, asking questions is a great method for gathering data through discourse.

Eureka Math^® lessons have many higher order teacher questions built into the teaching vignettes and the Student Debriefs. When you prepare lessons, look for and make note of questions that can help you uncover student thinking. If the sample questions in a lesson don’t elicit the student data you need, then write your own. For example, you might want data on whether students are using efficient strategies. If you plan to discuss different solution strategies, you might ask which methods students prefer and why. Maybe you want data on how well students model a problem with a drawing. In that case, consider asking students why they chose to represent the problem the way they did and how they know the drawing accurately represents the information in the problem. Perhaps you want to know whether students understand a concept that underlies a procedure. Then you might plan a series of what if questions that modify a problem to see whether students know how to transfer the concept when the problem looks different.

Make instructional decisions.

Collecting data through discourse lets you adjust instruction in the moment. Rather than waiting until tomorrow’s lesson, tailor instruction in today’s lesson. Let’s consider a few common types of adjustments.

Help students adopt appropriate solution strategies.
If you notice that students use strategies that are inefficient or won’t transfer to other problems, adjust instruction to focus on how to solve problems. You might pose a problem with more challenging numbers so that students can see the inefficiency of their original strategy. You might also discuss different student solutions. When students see and talk about more efficient strategies their peers use, they may adopt those strategies themselves.  

Make connections to prior learning.
If students do not apply prior learning to a new problem, adjust your instruction to make the prior learning stand out. Consider asking students how yesterday’s learning can help them solve today’s problem. You might also prime students’ thinking by having them work a problem from a previous lesson that emphasizes certain concepts or procedures relevant to the current lesson.

Examine common errors.
Data often illuminate common student errors. When that happens, consider adjusting instruction to discuss the errors. Sometimes it helps to discuss the reasonableness of answers. It can also be helpful to look at two different solutions and discuss which solution is correct. Taking the time to inspect and discuss mistakes can help students make sense of the mathematics.

Students must engage in purposeful discourse if they are to develop deep mathematical understanding and begin to think like mathematicians. The way students talk about the math helps us, as teachers, discern where our students are in their development while also providing clues to what supports they need in their mathematical journeys.