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

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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  1. Nora 3 months ago3 months ago

    Testing in CA started to be pushed to find where each child had gaps in learning in order to fill those gaps. This was distorted into standardized testing for all to the advantage of testing companies. In my student teachng we employed individualized instruction. Very intensive involving more time, no scan sheets. Much more effective. More money allocated for instruction is needed.

  2. Sergio Flores 3 months ago3 months ago

    Do you feel the same way about reading? Many translate Lexile levels to some kind of grade level equivalency to determine if a student is behind their peers (the norm). There are a lot of intensive reading programs out there to get students to “grade level”. Do you see it differently for reading and math or does your opinion differ between the two?

  3. Erik Kengaard 3 months ago3 months ago

    Two pages of generalization, not one example, and no specifics? I believe he is correct in that "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." "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." What is the … Read More

    Two pages of generalization, not one example, and no specifics? I believe he is correct in that “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.”

    “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.”

    What is the author’s recommendation with regard to specifics?

    For example, what is his view on showing the validity of the Pythagorean theorem for a triangle A,B,H by adding a square of side H to the hypotenuse, then adding three more triangles to form a square of side (A+B), then subtracting the total area of the fours triangles from the area of the larger square to show that H^2 = A^2 + B^2 ? Or, early on , using similar geometry to explain the distributive property? Or, he could cite a book he has written on the subject?

  4. Priya Talreja 3 months ago3 months ago

    Now I know, and after reading this, why my kid might have struggled in middle school math. Firstly, it was all taught online, secondly, the curriculum was going at a speed just so everything was completed, pretty much like a checklist of procedures, and even though the kid was learning, the vast amount of work was just too much to make any sense at all. Now, in high school, Math has become a strength, just … Read More

    Now I know, and after reading this, why my kid might have struggled in middle school math. Firstly, it was all taught online, secondly, the curriculum was going at a speed just so everything was completed, pretty much like a checklist of procedures, and even though the kid was learning, the vast amount of work was just too much to make any sense at all.

    Now, in high school, Math has become a strength, just due to the fact that it makes sense now, kids can enjoy it along with music they love! It’s been a game changer!! It’s just not about quantity any more, it’s about a more encouraging teacher.

  5. Lili Ibanez 3 months ago3 months ago

    I agree. My grade school-age grandson is high on the autistic spectrum. The only subject he enjoyed was math. But now he considers it work, the counting forwards and backwards and sideways and left and right and so on using the 120 chart. He is at a slight disadvantage, however, his teachers and aides never make him feel that way so he trudges on. But I hate that a curriculum he enjoyed only felt like … Read More

    I agree. My grade school-age grandson is high on the autistic spectrum. The only subject he enjoyed was math. But now he considers it work, the counting forwards and backwards and sideways and left and right and so on using the 120 chart.

    He is at a slight disadvantage, however, his teachers and aides never make him feel that way so he trudges on. But I hate that a curriculum he enjoyed only felt like curriculum after the 120 chart, which is supposedly meant to put him on level ground with his classmates yet has made him lose interest. It takes a lot of work to add two and three figure numbers now. This is not explorative education for him, it’s just counting which he can do in his head but is forced to use a chart for.

    I hardly think his is a special case nor is a specialized method of learning numbers needed here. But some in the pack don’t have quite the capacity for catching on as the rest, yet given the space can enjoy the journey of new ideas and discovery that they may otherwise miss out on. Instead I feel like he’s being left behind and made to do extra just to catch up and stay up. It almost feels like punishment.