For mathematics, it’s time to focus on individual student thinking

Students do better in math when their teachers have a positive attitude about math, a Stanford study says.

Allison Yin/EdSource

In a post-Covid world, we can no longer assume that a student’s age and mathematical experiences line up in previously expected ways. Indeed, they may not even be close.

It’s a tension that’s always existed, of course, but our current situation truly brings attention to this issue. Our curricular, assessment and accountability systems, along with our expectation-setting mechanisms, still operate on the assumption that standards for a given grade level are the appropriate content for all students at that grade level.

In recent years it’s become common to center curriculum around standards, which has been helpful in many ways. Standards focus the content through grade-level slices, provide a common, shared reference for the mathematics, and ideally reflect a coherence from year to year. They also set a baseline for expectations around mathematics that all students should have the opportunity to learn. This aspect is important, as history tells us that not all students have access to important mathematics content. Without shared expectations, there is a legitimate concern that too many students may fall through the cracks. Perhaps certain topics or courses would not be offered, or adults may set low expectations for students in their charge.

If we view mathematics as something other than a checklist of skills — which we should — we need to consider the progressions that coherently connect the ideas together. That is, we must reflect on how ideas build on ideas, much like a plot line develops in a novel. Good standards are based on such progressions. Standards writers imagine the mathematical progressions and figure out how to slice those progressions into grade-level-size chunks. A grade-level chunk is designed to be taught over roughly nine months. So, as long as the opportunity to learn (seat time) matches the grade-level chunk of time requirements, the standards may work as designed.

Textbooks, assessments and accountability are all designed around grade-level chunks. As for expectations, the notion of access to grade-level content becomes an efficient way to talk about the mathematics all students should experience. That is, it becomes easy to use the collective expectations of grade-level standards as a proxy for individual student expectations. To many on the outside, if your students are not meeting grade-level standards, the thinking would go, you are setting low expectations. If you are teaching a group of third-grade students, those students should now be working on third-grade mathematics. Expectations may too easily be driven by student age, not prior opportunity to learn, nor based on what was previously learned, nor how a student is currently thinking about mathematics.

There’s a well-intentioned desire to maintain the existing focus on age-based grade-level goals even as the opportunities to learn have become uneven. And with that, there’s much talk about accelerating learning to catch up and maintain high expectations. I think it’s time, though, to address some challenging questions:

• What happens when prior opportunities to learn no longer reasonably match the pre-set, age-based grade-level chunks of mathematical content?
• At what point does accelerating to reach grade level cause more harm than good? (Such a point, I would contend, must exist.)
• If not simply grade-level standards, what expectations should we have for our students?

The wonderful thing about mathematics is that it makes sense. It’s a discipline that is grounded in sense-making. An unfortunate irony is that it’s the subject where students often leave class believing it doesn’t make any sense. When math is presented primarily as a checklist of procedures to learn, such a belief is unsurprising. However, an emphasis on mathematical thinking, and specifically how students are thinking, promotes making sense of mathematics and highlights how procedures are only part of the experience.

Thinking transcends grade levels. Instead of focusing first on age-based, grade-level standards that assume a prior opportunity to learn, we might instead examine how we can set high expectations based on existing student thinking — where students are now, and where we want them to be. Such an approach would require a tremendous amount of thinking and effort to design and implement, but the outcome would be better for students.

Too many students already have a tense relationship with mathematics and see it as a collection of disparate facts and procedures rather than a coherent system that makes sense. Left unquestioned, the current situation could too easily further separate students from experiencing that connectedness and the beauty in mathematics. In a rush to catch up, or accelerate, the possibility of ignoring how students are thinking about mathematics and connecting ideas may be seen as optional, rather than essential. It’s time to find better ways to set high expectations for learners that are grounded more in individual student thinking than their dates of birth.

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Ted Coe is director of content advocacy and design in mathematics at NWEA.

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