Emely, a second grade girl in a pink sweatshirt, with wisps of brown hair framing her face, sits at her desk, her body poised in concentration over a small personal whiteboard. She looks at the classroom board, her eyes moving slowly over the words of a problem that her teacher projected there as the students came in from recess and sat down for their math lesson:
Mateo spent 14 minutes reading this morning. After lunch, he read for 28 minutes, took a 5-minute break, and then read for 16 more minutes. How many minutes did he read all together?
Emely glances absentmindedly at the busy classmates at her table before returning to her own work. On her board, she has written "10+20=30" and "4+8=12." She begins a third number sentence, carefully printing the number 20. Dissatisfied, however, she erases it. Then, bringing her face very close to the whiteboard, so close that she has to brush aside a strand of hair in order to write, she tries again: "30+12=42." She pushes the hair behind her ear as she reviews her work, and then, with a sudden rush of confidence, starts the next step of the problem with fresh momentum: "42+10=52."
As a visitor to Emely's classroom that day, I saw several children tackling the same problem in different ways. Emely's strategy, breaking the two-digit numbers apart and adding tens and ones separately, then recombining them in a series of addition sentences, was valid, efficient, and logical. It made sense to her. Other students in the room used methods that made sense to them:
- Brandon drew visual representations of base 10 blocks for each addend.
- Felix applied the traditional algorithm for finding sums of two-digit numbers.
- Jamina counted up on an open number line.
Their teacher, Mrs. Tambor, gave them a few minutes after independent work to share their methods in pairs before they gathered at the carpet to discuss the problem as a whole group and evaluate some of the different methods they'd used to solve it.
The Productive Struggle
The format of Mrs. Tambor’s math lesson reflected her desire to build productive struggle into her students' daily educational experience. To ensure plenty of time for puzzling and reasoning, she started her lesson with independent work time, moving into the teacher-centered portion of the lesson only after students had been studying the problem, first independently and then in pairs, for more than half of their math block.
Why would a teacher decide to structure a math lesson this way? Here are a few reasons that teachers have shared with me:
1. It prioritizes the student-centered portion of lesson.
If time runs out, the students' time to explore isn't cut short or eliminated.
2. It builds authentic engagement.
As each student confronts the problem and attempts to solve it, there is a feeling of mounting suspense. What is the question that I need to answer? How will I go about solving this problem? Will my strategy work? Will my classmates solve the problem in different ways? By the time the students gather in a group, they have a rich context for the problem at hand, and are genuinely curious about its solution.
3. It emphasizes that math makes sense.
Students are encouraged to seek solutions that are grounded in logic and prior knowledge and that make sense to them, instead of imitating methods used by their teacher or peers.
4. It creates ample opportunity for assessment, intervention, and feedback.
During independent time, teachers can work with struggling learners or circulate, making observations about student strengths and weaknesses. By the time students come together to discuss the problem, the teacher is well informed about the successful and unsuccessful strategies they have attempted, and can provide sturdy feedback about their work.
5. It builds perseverance.
Faced with a challenge, students experience the discomfort of not knowing. However, especially with practice, they become more comfortable with enduring this tension and working through it. Eventually, they will also experience the incredible personal satisfaction of solving a challenging problem.
When Challenge Gives Way to Frustration
After her first attempt at incorporating productive struggle, Mrs. Pierce, a fourth-grade teacher, reported, "We weren't even two minutes in when one of my students burst into tears. He had no idea where to start."
Instead of over-scaffolding or giving "hints," many teachers try to provide alternate points of entry when they spot an unproductive struggle. In one third-grade classroom, for example, students were asked to find ways to make 36 cents. When one student was confounded, her teacher suggested quietly, "Start by writing down the values of each coin. Remember, we discussed them in morning meeting yesterday." (She could also have suggested that the student start with only pennies or with three dimes.) Another teacher, discovering that she had overestimated her students' readiness, quickly replaced the original problem with a simpler one. Later, she wrote alternate problems on slips of paper ahead of time for students who got stuck.
It's also important to demystify the process so that students will understand how their initial uncertainty is a natural part of the learning cycle. One teacher encourages his students to talk about their strategies for breaking into a problem. Not only can they learn from one another, but also hearing that others experience the same tension can relieve students who internalize the discomfort.
As for Mrs. Pierce, when a colleague asked how she planned to rescue her student in distress, she replied, firmly and cheerfully, "We go again tomorrow!"
How do you build math positivity in your classroom or school?