How Do We Support Students in Reflecting on Mathematics?

As I worked with teachers in classrooms this fall, the topic of how to help students reflect on their learning and the learning of others kept coming up. I’m still thinking about how to recognize, encourage, and promote student reflection about math ideas. What traits do reflective students possess? How can a teacher nurture a learning culture where reflection is a natural part?

When students engage in math experiences that include time to reflect on their reasoning and the thinking of others they are more likely to become self-reflective. They become better at thinking about thinking and making connections to other mathematical concepts and contexts. But what does it look and sound like when students reflect on mathematical ideas?

Consider the following discussion from a grade 1 classroom, where students are sharing their strategies for solving a problem about two groups of toy cars (6+7). After recording two strategies in which the students counted all of the cars, the teacher asks, “Can anybody see a way that Hunter’s strategy is similar to Danny’s? … What’s the same about them?”

Jamie: That um he both put numbers.

Teacher: So they both labeled with numbers? That is one thing that they did the same. Does anyone see another way that these are similar strategies?

Luke: Um well they were counting all of them after they put ‘em up? …  They just counted all of them.

Teacher: So in both cases, Luke, you’re right. They drew or used a finger to count every single car to get the total of how many.

Asking students to compare strategies and representations encourages them to reflect on their own thinking and the thinking of others. This teacher focuses students’ attention on how the two counting all strategies are similar. Students must first make sense of each strategy, both of which represent someone else’s thinking. Then, they must step back and reflect on the ways in which those strategies are the same.

Later in the discussion, the teacher focuses the discussion on a different strategy, which involves using known facts to solve the problem.

Elizabeth knows that 6+6=12, and uses it to solve 6+7=13. She is able to reflect on how she solved this problem in order to share it with the class, no small feat for a 6-year old. In addition, Derek sees a relationship between thinking about this problem as 6+6+1 and 7+7-1. The teacher supports such reflection with her comments and questions, which name and explicate the strategy and encourage students to make sense of it and compare it to their own.

How does one build such a classroom culture, where reflection is an ongoing, natural part of doing mathematics?  Teachers help such a culture develop when they:

• anticipate and plan for potential reflection opportunities based on the mathematics at hand
• actively listen to and observe students at work, and use the information gathered to inform discussions and reflection opportunities
• encourage reflection through comments, questions, and sentence stems such as:
  • I wonder what would happen if ____?
  • I noticed ______ and ______ are the (same/different) because…
  • As you’re working on _____ look for _____.
  • Compare your thinking with ____.
  • What are you noticing?
• facilitate reflective discussions grounded in a mathematical idea. Such a discussion might focus on:
  • comparing and contrasting several different strategies
  • making sense of a strategy that’s different from one’s own
  • using different representations and contexts to explore one strategy
  • investigating an error or misunderstanding

Building a classroom culture where students make connections to the thinking of others takes time. As teachers encourage and promote student reflection about math, thinking about their thinking becomes a natural part of how they learn and make sense of what they’ve learned.

I’m curious, what do you do to encourage students to reflect on what they are learning, and on others’ ideas?