A growing “Science of Math” movement has begun to take shape, drawing on cognitive science to reexamine how students acquire mathematical knowledge, writes Danielle Hankins, an educational psychologist and faculty member at Alliant University in California, in an Education Week essay.
The Science of Math core premise is straightforward: Reasoning, problem-solving, and mathematical flexibility depend on knowledge that must first be explicitly and systematically developed.
I have seen educators recognize the need for more explicit instruction, but they are often expected to follow the same instructional sequence, pace, and teaching methods across classrooms. This limits their ability to modify instructional sequences when students need more support. These constraints are not insurmountable.
Math classrooms shifted over time to the discovery model. There was a deliberate move away from rote procedures toward reasoning, discussion, and problem-solving. The instructional system began to reflect a shared belief that students learn best by figuring things out on their own.
This emphasized thinking, engagement, and independence. Those are good goals, but the problem was the instructional sequence used to achieve them. Students were often expected to struggle with new ideas before they had been taught the underlying concepts. Instead of building understanding first and then applying it, students had to try to apply what they did not yet know.
Addressing this problem does not require new programs or a wholesale replacement of curriculum.
State frameworks shifted to emphasize inquiry-based and exploratory approaches. Curriculum developers designed materials to align with that vision for the materials to be widely adopted. Many popular programs share common design features — an early emphasis on exploration, structured discussion, and delayed explicit instruction. In practice, this can place responsibility on students to construct meaning before they have sufficient background knowledge.
Research in cognitive science has consistently shown that novice learners benefit from explicit instruction and guided practice before independent application. When students are asked to solve problems before they understand the underlying concepts, they must rely on trial and error.
“Productive struggle” – carefully designed challenges — can be a meaningful part of the learning process. In practice, though, too much struggle can hamstring students.
Teachers and districts may recognize the need for change, but they cannot simply abandon the curriculum already in place. Addressing this problem requires a handful of practical changes to instructional sequencing and delivery:
1) Move instruction to the front. Before students are asked to explore, the teacher models the concept clearly and directly. This ensures that students begin with a foundation rather than confusion. Explicit explanation supports learning more effectively than unguided discovery.
2) Use worked examples. Instead of asking students to immediately solve unfamiliar problems, present fully solved examples and walk through them step by step. Students then attempt similar problems with guidance still in place. This helps students focus on understanding the structure of the task.
3) Structure practice. Structured guidance for problems helps students internalize procedures more efficiently. Open-ended prompts such as “figure out a strategy” can be replaced with clear, explicit steps that guide students through the process and help with accuracy.
4) Delay independent problem-solving. Independent problem-solving and real-world application should occur after students have developed initial understanding and fluency. The National Mathematics Advisory Panel emphasized that students must develop foundational knowledge and procedural fluency to a level of automaticity before engaging in more complex mathematical tasks.
5) Limit the number of strategies during early instruction. Research on cognitive load consistently shows that reducing the number of elements learners must process at once improves learning, particularly for novices. Focusing on one clear method first allows students to build stable understanding. Later, teachers can introduce alternative strategies to build problem-solving capacity.
These recommendations require reordering what is already in place: Teach before explore; model before solve; guide before release; and apply after understanding. The materials already in classrooms can support this work, but only if they are used in a way that aligns with how students actually learn.
Educators should not have to choose between adopted curriculum and applying what we know about learning. The two are not inherently incompatible. Students should absolutely reason, explain, and solve problems. But first, they need conceptual understanding to make that reasoning possible.
Education Week
