Originally published on the Illustrative Mathematics blog (October 13, 2019).
We want students to think about math deeply. Creatively. Analytically. Instead, what often happens is that students race towards quick solutions. So what can we do to support this other kind of thinking in class—the slow, deep kind?
One way is through instructional routines like “Which One Doesn’t Belong” and “Notice and Wonder.” These routines give structure to time and interactions. Within the structure, there are opportunities to have time to think deeply and a predictable way to share and deepen thinking with partners and the whole class.
My colleagues and I experienced this first hand recently. We are participating in the alpha pilot of the Illustrative Mathematics K–5 curriculum, which means that we are among the first teachers to teach with the program. On the first day of the first unit in grade 5, students were asked to engage in a “Which One Doesn’t Belong” task. The routine is as follows:
Which one doesn’t belong? Students think quietly for 1 minute, and then share their thinking with a partner. A whole-class share follows, and teachers annotate student thinking on the board. Every time we do a Which One Doesn’t Belong, we follow this same structure.
Here’s the image from lesson 1:
Grade 5, Unit 1, Lesson 1, Warm Up
IM K-5 Math, alpha version, certified by Illustrative Mathematics
While planning for the lesson, several team members and I spent time anticipating what students might say. There are so many ways to interact with this image! For example, “Figure A doesn’t belong because it is the only one that forms a larger cube. It’s the only one where all the edges are the same length.” What we didn’t anticipate is how deeply students would look at the cube in the top left.
Students had a fresher take than we did.
In the first classroom, a student said that that the cube did not belong because it was the only one that was missing information. Huh. I asked the student to explain further.
“It could have seven cubes. We don’t know if there’s a cube missing on the bottom in the back,” she said. I did my best to record her thinking on the board. Meanwhile, the classroom teacher turned to me, and whispered “wow” with wide eyes. Students in the class visibly shifted in their seats. Some of them indicated that they had a similar idea, by using the ASL sign for “me too.” Other students squinted, and reconsidered the image.
“But… I don’t think that’s right,” said a student across the room. “I disagree because if there was a missing cube back there, the one on the top would fall down.”
I wrote the sentence stem I disagree because on the board, and encouraged other students to use this phrase. It’s one way to disagree respectfully. (I was thrilled that this student had internalized the phrase last year. We worked so hard on it in his grade 4 class.)
“Maybe the cubes are glued together,” another student suggested.
The classroom teacher smiled. Is there any greater joy that being surprised by the depth of student thinking?
I assumed this was an unusual line of thinking. However, the idea of “missing information” came up again in the second classroom—and then the third, and the fourth. Sometimes this line of thinking emerged during the Which One Doesn’t Belong warm up, and sometimes it didn’t surface until the next activity in the lesson, which asked students to determine which of two figures is larger. Surely it wasn’t an accident! Students were thinking creatively and critically about the image, every single time.
We were able to return to this thinking several days later, when students were being asked to focus on rectangular prisms. We talked about the kinds of mathematical assumptions we make when calculating volume for right rectangular prisms. I am not certain we would have had such clarity about the figures had we not already wrestled with the idea of hidden cubes.
The structure of the Which One Doesn’t Belong routine intentionally makes space for deep thinking. Starting the routine with a minute of quiet think time allows students the chance to experiment with ideas. All too often, students are only given a few seconds to process a task during math class. If we truly value deep thinking, we need to give students the time it takes to generate. Students might try out a few different lines of thinking during that minute. Some of this thinking might dead end in an idea that doesn’t work—which is fine when there’s still time to try out a new idea.
Then, students engaged in a partner share, which allowed them space to exchange and workshop their ideas. Students rehearsed ways to communicate their thinking to someone else. They also had the opportunity to listen to ideas from their partner, and to build off of them in new directions before the class share.
Then, during the class share, students were able to share their ideas with a larger audience, and to interact with ideas from classmates across the room. As a teacher, I knew my role was to respect, record, and probe student thinking. These three phases—self, partner, class—let students go deeper and deeper. Without these structures, I am not certain we would have had students so reliably engage with images like the cube, which can feel commonplace.
The Illustrative Mathematics K–12 math curricula embed these routines. In fact, there are several routines that are featured in both elementary and high school, like Number Talks. As Kristin Gray wrote in her blog post “Designing Coherent Learning Experiences K–12,”
Instructional routines provide an opportunity for all students to engage in, and contribute to, mathematical conversations. The predictable routine for the interactions between teachers and students allows each student in the classroom to focus on the mathematics rather than the actions that need to be taken. And the more experience students have with the routine gives them greater ability to focus on the mathematical content.
Instructional routines are more than just a nice addition to classroom time. They are essential structures that communicate our values. For example, in our classroom, we engage with ideas. We listen to one another. We communicate our thinking. In our classroom, we think deeply.
What instructional routine would you like to experiment with in your own practice? How do you envision it supporting student thinking?