You might look at our below average scores on international mathematics tests and conclude that American students struggle in math, but the latest research suggests they actually don’t struggle enough.
Sure, they struggle with math content, in part, because the secondary mathematics curriculum is jam-packed. Like 40 others, my state relies on the Common Core standards. The last to be written, the secondary math standards were particularly rushed and did not get nearly as much review as the elementary grade standards. The goal was to create a Common Core—a few key standards that all states could agree on that would define what it meant to be “College and Career” ready. This “core” turned out to be a fantasy: there are actually more standards in secondary math, not fewer.
And struggles for traditional accountability surfaced because math is a highly tested, high stakes subject. Success on tests like the SAT and state assessments is critical for schools and students. After initially trying more integrated approaches to math, the high school where I teach, like other innovative schools such as the Met and High Tech High, returned to a more traditional course sequence. This is in part due to the need to help students meet the demands of state testing and college placement, the perennial post-secondary tail that wags the curriculum dog.
But here’s the thing I want to highlight: these kinds of curricular and accountability mandates are often counterproductive to the kinds of deep, conceptual learning our students most need if they’re to be successful after graduation.
Our students need to engage in “productive struggle” in which they wrestle with doubt and confusion until they experience the joy of understanding—that “a-ha moment.” Too much content leaves scant time to productively struggle with deeper conceptual and thematic inquiries.
Productive Struggle Leads to Deeper Learning
In our efforts to “cover the curriculum” we essentially cover over those juicy, meaningful, difficult problems in which struggle is required. Productive struggle is the popular term for developmental psychologist Lev Vygotsky’s concept of a “Zone of Proximal Development” where students are met with the just-right level of challenge: outside of their current knowledge or skill, but close enough to be within reach with guidance from a teacher. The National Council of Teachers of Mathematics (2014) described this state as pushing students “more deeply into understanding the mathematical structure of problems and relationships among mathematical ideas, instead of simply seeking correct solutions.”
In his recent book, Mathematics for Human Flourishing, Francis Su describes the benefits of productive struggle this way:
In the struggle, we acquire competence to solve new problems, and fortify an expectant hope that we will one day solve them. And when we struggle, and at last succeed, we build self-confidence. Over time, through incremental and hard-won victories, this leads to mastery.
It’s no coincidence that productive struggle helps to develop the very “knowledge, skills, and dispositions” that make up “deeper learning” that is the goal of student-centered pedagogy. Students might pass a test or succeed in memorizing steps with more efficient forms of instruction but deep conceptual understanding requires struggle. As Jo Boaler reminds us in Limitless Mind, “If you aren’t struggling, you aren’t really learning.”
With that idea in mind, more and more secondary mathematics teachers are building the right kinds of struggle—productive struggle—into their classes. I wanted to learn about how good math teachers were using productive struggle in their classrooms, so I interviewed a sample of math teachers who had been identified as innovative by my colleagues in the Student-Centered Learning Research Collaborative.
The teachers I interviewed suggested that there are two primary strategies for supporting students in productive struggle: rich content and collaboration.
Rich Content and Tasks Initiate Productive Struggle
Rich content is essential for supporting productive struggle. To put students in their zone of proximal development requires going beyond the tried and true “I Do, We Do, You Do” model of math instruction. Instead, rich tasks often start with ‘You do,” allowing students to puzzle over what Dan Finkel calls “challenging, meaty and authentic” questions” and then share their ideas with classmates. Dan Meyer describes this approach as “giving students a headache” so that they realize that “math is the aspirin.”
This kind of rich content is becoming more standard in mathematics classrooms. The teachers in my sample emphasized finding “worthy,” “open-ended” problems with multiple entry points and “low floors and high ceilings,” so that anyone—whatever their typical level of success on traditional math tasks may be—can find a way in, and where they can demonstrate their knowledge in many different ways. One teacher described her class work as having “multiple solution pathways.” The conversation in her classroom focuses less on the answer than the process: “I did this. You did that. Are they the same? Did we get the same answer? Why did they both work?”
Collaboration Keeps the Productive Struggle Alive
This kind of conversation requires intentional development and practice and is why collaboration goes hand in hand with rich content as a key practice in developing the productive struggle that leads to deeper learning. This may be because rich content engaged in groups helps students use their “social brain.” Patricia Kuhl, the co-director of the Institute of Learning and Brain Sciences at the University of Washington, describes the social brain—the human hardwiring that makes infants gaze at faces or mimic expressions—as the “gateway to cognition.” We are social animals: optimized learning happens in relationships and it begins in infancy. Building opportunities for students to collaborate engages their social brain and deepens the connections they make to new concepts, new skills and new friends.
The innovative teachers with whom I spoke all used group work extensively. They require students to take on different group roles, develop group norms and give students “participation quizzes” where they are assessed solely on their contributions to the group process. These innovative teachers also take themselves “away from the front of the room,” by positioning themselves as facilitators, coaches and environment designers.
With group structures, roles and norms, students can express their math ideas and contribute to others’ understanding of math concepts. As one teacher told me, “A computer can solve every problem that we see, but it can’t explain its ideas or interact with other people. As an engineer or a scientist, if you can’t explain them and hear another idea, you are not going to get very far.” It’s important to note that communication and collaboration are among the competencies most frequently described as essential by employers.
Personalizing Math Means the Right Kind of Struggle
Taken together, rich content and collaboration intentionally disrupt what often bestows status in math classrooms: quick computation and algorithm memorization. Instead, rich content rewards perseverance and engagement, and collaboration iteratively deepens that engagement and provides students the opportunity to see mathematics from beyond the algorithm. Students may struggle with a concept, idea or task, but they will realize they do not struggle alone.
So, let’s resist making students struggle in secondary mathematics for the wrong reasons. Whenever we can, secondary mathematics teachers need to step off the curriculum train, veer from the accountability track and engage students in productive struggle. The beauty of mathematics isn’t necessarily revealed to the student who follows the steps the best, nor does the highest SAT score go to the students whose teacher “covered” all the material in their district’s curriculum framework. To personalize secondary math and prepare ours students for post-secondary success, we must deepen their learning through productive struggle with rich content and collaboration