Very good. It's strange that we often separate concepts from doing things.
My current favourite explanation of an idea is simply as a tools for thinking - and just as protractors are made out of plastic, ideas are made out of words (or numbers etc). Thus ‘doing things’ is essential for possessing concepts.
well i think it demystifies because a) it avoids any talk of 'inner' goings on, which always lead one down rabbit holes, and b) I don't think it's a metaphor really. Ideas are literally tools/devices/instruments made of words. If you think of an example like a counting-out rhyme (eeny meeny miny moe...) then that's easier to see. Normal counting works pretty similarly. And all ideas/concepts are like this: patterns of words that we can use to do stuff.
Great piece! I love your definition. How wild that in so many cases we ask teachers to teach toward some nebulous vision of deepening conceptual understanding without having an actual working definition that informs their pedagogy. And appreciate your point that discovery/inquiry-based learning is not the only way to get at conceptual understanding as these generalizations don’t *only* need to be invented or discovered by students. That said, this type of discussion was what I loved best about teaching/leading when using CGI — the moments when you could guide students toward a generalization or conjecture based upon what came up in their own strategies for solving a problem, asking, “will that always work?” or “what makes that true?”
I haven't taught K-2, but there is something special about those early addition/subtraction ideas. To a real extent they are more discoverable -- so closely related, they're almost inevitable. I've noticed this with my own children.
This is so great from start to finish. "We should teach a generally true principle...so that they can apply that to any problem." Another way of describing this is that conceptual understanding leads to the transfer of knowledge to novel problems; your example of the function + derivative graphs is a perfect illustration.
Great article. Somehow EdReports has determined that alignment is a measure of understanding. IMO conceptual and numerical variation are the building blocks for a strong curriculum. Supported by teacher analysis of student understanding at the moment of teaching.
I do wonder if we'll look back at 2025 (and the adoption of Illustrative Math in NYC) as the peak of EdReports' influence. I feel like it can only go down from here, though maybe I'll be proven very wrong!
Well said! I'll take a shot at defining "conceptual understanding" in context. I think my version reinforces your ideas and calls out Ed Reviews for how they are "measuring" this.
I define "conceptual understanding" as students asking "why" and "how does this connect?" rather than just "how do I solve it?" This definition is much more about how students approach learning mathematics than about what kind of problems they are solving (though both are influential).
Assessing discrete math skills is straightforward, so textbooks and adaptive supplement systems often focus on them. However, conceptual understanding is found much less in individual problems and much more in connecting these "zoomed-in" skills to see the "zoomed-out" big picture.
This means how we teach—by fostering a culture of productive struggle and explicitly highlighting connections—has a more direct impact on conceptual understanding than what specific curriculum we use. What makes this hard is that any given classroom has a range of students with varying "zones of proximal development". Meeting students' needs requires enough opportunity to struggle, but also enough support to ensure the struggle is productive, not harmful.
"Focus on conceptual understanding" isn't a feature of the student-facing materials alone. It's a feature of the teacher-facing supports that help them create opportunities for "why" and "how" questions, and then provide the right scaffold to the right student at the right time. Teachers, like students, can apply procedures without understanding the concepts -- so this really comes down to good teaching.
Very good. It's strange that we often separate concepts from doing things.
My current favourite explanation of an idea is simply as a tools for thinking - and just as protractors are made out of plastic, ideas are made out of words (or numbers etc). Thus ‘doing things’ is essential for possessing concepts.
https://open.substack.com/pub/bernardandrews/p/what-is-an-idea?utm_source=share&utm_medium=android&r=35t6uv
Ideas as "mental tools" is definitely a cool metaphor. I'm not sure if it has the demystification effect on me. Still thinking about this...
well i think it demystifies because a) it avoids any talk of 'inner' goings on, which always lead one down rabbit holes, and b) I don't think it's a metaphor really. Ideas are literally tools/devices/instruments made of words. If you think of an example like a counting-out rhyme (eeny meeny miny moe...) then that's easier to see. Normal counting works pretty similarly. And all ideas/concepts are like this: patterns of words that we can use to do stuff.
Great piece! I love your definition. How wild that in so many cases we ask teachers to teach toward some nebulous vision of deepening conceptual understanding without having an actual working definition that informs their pedagogy. And appreciate your point that discovery/inquiry-based learning is not the only way to get at conceptual understanding as these generalizations don’t *only* need to be invented or discovered by students. That said, this type of discussion was what I loved best about teaching/leading when using CGI — the moments when you could guide students toward a generalization or conjecture based upon what came up in their own strategies for solving a problem, asking, “will that always work?” or “what makes that true?”
I haven't taught K-2, but there is something special about those early addition/subtraction ideas. To a real extent they are more discoverable -- so closely related, they're almost inevitable. I've noticed this with my own children.
This is so great from start to finish. "We should teach a generally true principle...so that they can apply that to any problem." Another way of describing this is that conceptual understanding leads to the transfer of knowledge to novel problems; your example of the function + derivative graphs is a perfect illustration.
I am the Eggman, coo-coo-ca-chooooo...
Yeah transfer is certainly lurking around the corners of this discussion.
Excellent!
Very much enjoyed reading this!
Thanks for reading!
Great article. Somehow EdReports has determined that alignment is a measure of understanding. IMO conceptual and numerical variation are the building blocks for a strong curriculum. Supported by teacher analysis of student understanding at the moment of teaching.
I do wonder if we'll look back at 2025 (and the adoption of Illustrative Math in NYC) as the peak of EdReports' influence. I feel like it can only go down from here, though maybe I'll be proven very wrong!
Well said! I'll take a shot at defining "conceptual understanding" in context. I think my version reinforces your ideas and calls out Ed Reviews for how they are "measuring" this.
I define "conceptual understanding" as students asking "why" and "how does this connect?" rather than just "how do I solve it?" This definition is much more about how students approach learning mathematics than about what kind of problems they are solving (though both are influential).
Assessing discrete math skills is straightforward, so textbooks and adaptive supplement systems often focus on them. However, conceptual understanding is found much less in individual problems and much more in connecting these "zoomed-in" skills to see the "zoomed-out" big picture.
This means how we teach—by fostering a culture of productive struggle and explicitly highlighting connections—has a more direct impact on conceptual understanding than what specific curriculum we use. What makes this hard is that any given classroom has a range of students with varying "zones of proximal development". Meeting students' needs requires enough opportunity to struggle, but also enough support to ensure the struggle is productive, not harmful.
"Focus on conceptual understanding" isn't a feature of the student-facing materials alone. It's a feature of the teacher-facing supports that help them create opportunities for "why" and "how" questions, and then provide the right scaffold to the right student at the right time. Teachers, like students, can apply procedures without understanding the concepts -- so this really comes down to good teaching.
I think most of the rest is how well the curriculum supports "explicit attention to concepts" as outlined nicely in this article: https://pubs.nctm.org/view/journals/mte/11/2/article-p93.xml