“Subtraction!” most of the 3rd graders responded in chorus when asked what operation they needed to use to solve a problem. I noted that 100% of students were engaged and I gave the teacher an approving nod. It was 2018, and I was working as a math coach in a charter network. I’d seen many classrooms that year and I was excited to see that these students were so engaged.
At least, that’s what I thought then. In the last few years, I’ve come to think differently about what it means for students to be engaged in math class.
My thinking shifted when I picked up psychologist Daniel Willingham’s book Why Don’t Students Like School? In the book, he shares an anecdote about a seemingly engaging history lesson on the Underground Railroad that had students making biscuits, a staple food of the clandestine slavery route to freedom. The students were engaged in the activity, but Willingham notes they probably spent about 40 seconds thinking about the relationship of biscuits to the Underground Railroad and about 40 minutes thinking about their baking techniques. Learning to make biscuits wasn’t the goal, but it is what students learned.
“To teach well, you should pay careful attention to what an assignment will make students think about,” Willingham advises, “because that is what they will remember.” It’s a sentence that has made me rethink my whole approach to teaching.
Doing isn’t enough
Engagement isn’t just about the degree to which students participate in activities. It’s really about what students are thinking. When 100% of the class called out “subtraction,” they were engaged on the surface. But now I wonder: What were they thinking?
To elicit real engagement in math class, teachers have to ask the right questions or assign meaningful tasks. Consider how you’d ask students to solve the following word problem: Stacey sold her famous cupcakes at the school fair. When the fair was over, Stacey had 9 cupcakes left. She began with 25 cupcakes. How many cupcakes did Stacey sell at the fair?
In one approach, a teacher might ask, “We have the whole and one part. What operation do we use to find the missing part?” The students will then answer “subtraction.”
But it’s a leading question. You’ve done half the thinking for the kids and narrowed down the possible answers. What if instead you ask, “In this word problem, what do we know? What are we trying to find out? How might we do that?”
This way, students have to think through the questions and the math needed to solve the problem. What you’d probably hear from students is something like: “Hmm ... I know the end amount and the starting amount and I’m trying to find how many cupcakes Stacey sold. To find the missing part, I could draw 25 cupcakes, circle the 9 remaining, and count the ones she sold. Or I could subtract 25 minus 9. I might even count up from 9 to 25 using friendly numbers (10, 20, 25).”
In both instances, students might look engaged. But in the second example, students are demonstrating a true understanding of math concepts they’re learning. It’s thinking that will allow them to tackle more complex problems later.
What are they thinking about?
I can’t go back to that moment when the kids yelled “subtraction,” but I’m still in a lot of classrooms these days, primarily to support school and district leaders in understanding how math materials are being used. I always ask, “What did students think during today’s lesson, and did those thoughts reflect an understanding of the core concepts in the day’s lesson?”
Recently, I watched an entire class of 5th graders subtract fractions by using drawings to find the common denominators. It felt like a great moment until I asked one student to explain their thinking when solving 2/3 - 1/7.
The student explained, “I drew a rectangle and then three lines this way (horizontally). Then I shaded two rows. Then I drew lines this way (horizontally). ...”
Hmm, I reflected. The student was not describing math concepts. I asked other students, and they responded the same way.
Afterward, I stood in the hallway with another coach, and we shared an “aha” moment: Yes, these students were seemingly engaged and getting the right answer. But were they thinking about math concepts? No, they were thinking about rectangles, lines, and shading, not about the need to find common denominators. It reminded me of the students Willingham referenced who were thinking about mixing ingredients instead of the Underground Railroad.
A measurable impact
For an internal study to assess the impact of fostering deep thinking in math classroom, my Great Minds colleagues and I visited hundreds of math classrooms across 25 elementary schools in a large district in the Houston area in the spring of 2024 to observe how students expressed their mathematical thinking in speech and writing. We wanted to know whether fidelity of implementation and student engagement (as defined by our curriculum-specific implementation support tool) resulted in student learning (as measured by state assessments).
The results were compelling: schools where most or all students were engaged in lesson-level mathematical thinking—for example, solving multiple problems during a fluency activity instead of only one or two problems or explaining why a particular strategy works instead of describing the steps to execute the strategy—showed significantly higher growth: The classrooms where we observed this mathematical thinking on display saw a 10.6 percentage point increase in 3rd to 5th grade students meeting grade-level expectations on end-of-year state achievement tests that year compared with just a 0.5-point increase districtwide.
Whatever students think about—really think about—is what they will remember and learn. As math educators, it’s our job to make sure students are truly thinking about the concepts and skills we want them to learn. And that, in turn, will help them become confident and successful math thinkers and doers. Participation, and even getting the right answer, just isn’t enough.
