Framing the lesson
Carrie was recently in a 5th grade classroom where she planned a mini-lesson with a group of teachers. In this lesson, students were asked to reason using the relationship between factors and products. We put up two multiplication expressions and asked students two questions: Which has the greater product? and How do you know? The students were asked to answer these questions without paper and pen and without computing. We built in these constraints because we wanted to shift the focus from computation (which these students were quite skilled at) to reasoning with relationships. Our goal was to help students understand that you can use a reasoning strategy to figure out which multiplication expression would have the greater product. To do so, however, they would need to think first about the relationship of the factor pairs to each other and how that relationship would impact the product.
What happened was what we expected: the students groaned loudly when they heard they could not use paper and pencil to compute.
Pause and reflect:
Please reflect on the following questions before reading on:
- Why would this happen?
- What are we doing in K-4 education that by fifth grade, students protest when asked to use a thinking strategy rather than a computing one?
- Would this happen in your classroom?
What happened in the lesson:
During the lesson, we named and charted the strategies students used to determine which set of factor pairs had the greater product. We modeled the use of if/then language and wrote it on the board so that students could refer to it while working independently on a game. The students used the if/then language from the comparing routine to create If/then statements to prove and explain their thinking in a game. As you can see from the illustrations below, the if/then language became a way to prove their thinking.
As we were doing this mini-lesson, one child stated loudly, “This is not math!” Because changing children’s understanding of what it means to “do math” is a critical part of this type of reasoning mini-lesson, we explored the question in a whole-group discussion. As children grappled with their disequilibrium over what they felt math was, they offered several interesting observations. Here are two of them: (1) “Aren’t we cheating if we don’t compute?” (2) “How do you show your answer without a computation?”
Here’s why we at Meta are focusing so much on using reasoning routines in our work with teachers and students: Math isn’t only about computation; it’s about knowing when to compute and when not to compute. For students, knowing when and if you have to compute is a very powerful tool for their problem-solving toolkit. When students develop this kind of ownership, they acquire an unwavering confidence in the power of their own ideas. When they have this kind of self-awareness, they decide if and when to compute.
What are some things we can start doing in math class to shift the focus from answers to reasoning?
We’d love to hear what you have to say in the comments section below.
We will also be exploring this question in future blog posts, so stay tuned.
 Cameron, Antonia, Reasoning Routines, 2016
 Multiplication Compare from Puzzles, Clusters, and Towers, Unit 1, TERC Investigations 3 in Numbers Data and Space, 2016.