I really appreciate the explicit connection you make to student’s experiencing success and an increase in motivation. You’re also correct to call out how these strategies are underused at the high school level. My 12th grade students benefit from all of these strategies just as much as the middle school students I used to teach.
Side note: Your description of fluid intelligence made me think of how VO2 Max functions in athletics. Many elite athletes have a very high VO2 Max because it’s relevant for being able to run fast, run for longer etc. It’s also possible for VO2 Max to improve with training, but people start off with different baselines for VO2 Max that is influenced by genetics. At the same time, having a high VO2 Max doesn’t guarantee that someone will be a great athlete because there are many other factors to athletics (coordination, balance, strength etc). This might be a terrible analogy but it’s what it made me think of!
Yea I agree about VO2 Max. You could extend the metaphor -- not everyone could become a top-level marathoner, but the vast majority of people could train for a run a 10k. Running a 10k is what we should expect of students in K12 education, and not have delusions that every runner can be an elite marathoner but also put really deliberate scaffolds in place to try and get everyone across a basic bar.
Interesting aside. My cousin was a D1 distance runner in college. He told me his coach measured all of the distance runners' VO2 max at one point, with the fancy oxygen masks running on a treadmill, etc (not just the number a Garmin watch spits out). But he didn't tell his runners their VO2 max. He used it to adjust their training plans but didn't see any use in telling runners because it didn't always correlate with their performance and he didn't think it was helpful for runners to know.
It's true that VO2 Max can be improved by training -- up to a point. But VO2 Max cannot be increased without limit, and for everyone there is some upper bound on VO2 Max that they will never be able to cross no matter how much they train. Ultimately, this limit is genetic; as you observe, some people have higher VO2 Max baseline, and that also applies to their VO2 Max potential.
This is also true of general cognitive ability. There are varying levels of baseline cognitive ability, and although everyone can in principle make improvements if they work at it, there will always be a limit for each person, regardless of how hard they study.
I think I would take the metaphor a bit further. I agree with you about VO2 max, but the goal of exercise isn't to have a high VO2 max, it's to meet a given goal. Maybe it's playing soccer, or running a 10k, or something else. A low VO2 max makes that harder -- you might have to train more intentionally and with more structure -- but it's still possible to reach reasonable goals. Sure, you won't be a world-class marathoner, in the same way a student with less fluid intelligence isn't likely to become a quantum physicist. But with the right instruction, I think the goals we set in K12 education are reasonable for most students.
So the limit isn't set by cognitive ability, it's an interaction between cognitive ability and instruction. And my hypothesis is that, for the vast majority of students who feel like they've hit their ceiling in K12, it's actually instruction and not cognitive ability that is the key constraint.
This post really reflects the depth of thought and care that goes into understanding how children learn — not just what they learn.
As teachers, we often sense these differences in processing and working memory but rarely see them articulated this clearly or tied back so practically to classroom strategies.
Your framing of “fluid intelligence as the bottleneck between learning and long-term memory” makes so much sense — and more importantly, it offers a path for teachers to act on it rather than feel helpless about “ability.”
Thank you for such a grounded and insightful perspective — this kind of reflection truly elevates teaching into a science and an art.
I really resonate with what you've written here... reading this also led me to wonder what kinds of classroom routines/lesson structures/classroom set ups might best help students "expand" their working memory/speed up their processing speed.
For example - and speaking of what we can learn from elementary school teachers - should we have more posters/anchor charts in HS classrooms? Does having a structured note taking system (e.g. Cornell notes) help students (temporarily) expand their working memory because they know exactly where to look for the concepts they need to work with? (Or does it not make a difference?) I also wonder if whiteboard desks help/hinder either of these elements of fluid intelligence.
However, one thing that I'm struggling with is how to apply these strategies when you're working with students who are years below grade level and/or have other learning needs. (Not saying you need to have answers - just saying that these are the things I'm struggling with now!)
Those are great questions. A few miscellaneous thoughts:
Routines are really important. Anything where students don't think about what to do and can think about the math they're learning disproportionately helps students with less fluid intelligence.
I'm a bit skeptical of anchor charts -- for the most important stuff, sure, but the walls can end up feeling chaotic and crowded, which isn't good. Notes similarly can be good or bad. Students with a slower processing speed in particular have a really hard time taking notes and listening at the same time, so notes need to be structured in a way that separates instruction and notetaking. I think giving students resources and examples can work better than taking notes under some circumstances. I'm not sure about whiteboard desks, I've never used them.
In terms of students years below grade level -- there's no easy answer. All of these strategies are important, but they also take time, and time is at a premium if you have more math to teach. My answer is prioritizing: choosing the most important math and doing it really well, and spending less time on some standards that I see as less important.
Super interesting, love this! Cool distillation of research and experience into why certain teaching strategies are critical, and narrating some tradeoffs amongst them since time is not unlimited. Also the phonics heuristic (useful for all, harmful for none, needed for some!
I also wonder about recall and synthesis, distillation, and use/application of LT memory. Even if LT memory is unlimited, that’s another limiting factor that is implied but I felt you didn’t make quite explicit: it’s underlying and connected to working memory and processing speed, but not just the new info someone takes in (e.g., what you’re teaching them and guiding them with those closer rungs on the ladder) but also how the use the existing memory / knowledge / experience.
Thanks! And your point about LT memory is important. There's a version of teaching for automaticity that is pure repetition, and while that will help with automaticity students often struggle to apply the knowledge when the context is slightly different. So retrieval practice needs to have some variation, gradually helping students apply what they know in different contexts. I think you're right that it connects with the idea of rungs on the ladder, and it's another example where having more cognitive resources makes it easier for some students to make those connections, but others need teacher support in knowing where to apply what they have learned.
I really appreciate the explicit connection you make to student’s experiencing success and an increase in motivation. You’re also correct to call out how these strategies are underused at the high school level. My 12th grade students benefit from all of these strategies just as much as the middle school students I used to teach.
Side note: Your description of fluid intelligence made me think of how VO2 Max functions in athletics. Many elite athletes have a very high VO2 Max because it’s relevant for being able to run fast, run for longer etc. It’s also possible for VO2 Max to improve with training, but people start off with different baselines for VO2 Max that is influenced by genetics. At the same time, having a high VO2 Max doesn’t guarantee that someone will be a great athlete because there are many other factors to athletics (coordination, balance, strength etc). This might be a terrible analogy but it’s what it made me think of!
Yea I agree about VO2 Max. You could extend the metaphor -- not everyone could become a top-level marathoner, but the vast majority of people could train for a run a 10k. Running a 10k is what we should expect of students in K12 education, and not have delusions that every runner can be an elite marathoner but also put really deliberate scaffolds in place to try and get everyone across a basic bar.
Interesting aside. My cousin was a D1 distance runner in college. He told me his coach measured all of the distance runners' VO2 max at one point, with the fancy oxygen masks running on a treadmill, etc (not just the number a Garmin watch spits out). But he didn't tell his runners their VO2 max. He used it to adjust their training plans but didn't see any use in telling runners because it didn't always correlate with their performance and he didn't think it was helpful for runners to know.
It's true that VO2 Max can be improved by training -- up to a point. But VO2 Max cannot be increased without limit, and for everyone there is some upper bound on VO2 Max that they will never be able to cross no matter how much they train. Ultimately, this limit is genetic; as you observe, some people have higher VO2 Max baseline, and that also applies to their VO2 Max potential.
This is also true of general cognitive ability. There are varying levels of baseline cognitive ability, and although everyone can in principle make improvements if they work at it, there will always be a limit for each person, regardless of how hard they study.
I think I would take the metaphor a bit further. I agree with you about VO2 max, but the goal of exercise isn't to have a high VO2 max, it's to meet a given goal. Maybe it's playing soccer, or running a 10k, or something else. A low VO2 max makes that harder -- you might have to train more intentionally and with more structure -- but it's still possible to reach reasonable goals. Sure, you won't be a world-class marathoner, in the same way a student with less fluid intelligence isn't likely to become a quantum physicist. But with the right instruction, I think the goals we set in K12 education are reasonable for most students.
So the limit isn't set by cognitive ability, it's an interaction between cognitive ability and instruction. And my hypothesis is that, for the vast majority of students who feel like they've hit their ceiling in K12, it's actually instruction and not cognitive ability that is the key constraint.
This post really reflects the depth of thought and care that goes into understanding how children learn — not just what they learn.
As teachers, we often sense these differences in processing and working memory but rarely see them articulated this clearly or tied back so practically to classroom strategies.
Your framing of “fluid intelligence as the bottleneck between learning and long-term memory” makes so much sense — and more importantly, it offers a path for teachers to act on it rather than feel helpless about “ability.”
Thank you for such a grounded and insightful perspective — this kind of reflection truly elevates teaching into a science and an art.
Thanks Padma!
I really resonate with what you've written here... reading this also led me to wonder what kinds of classroom routines/lesson structures/classroom set ups might best help students "expand" their working memory/speed up their processing speed.
For example - and speaking of what we can learn from elementary school teachers - should we have more posters/anchor charts in HS classrooms? Does having a structured note taking system (e.g. Cornell notes) help students (temporarily) expand their working memory because they know exactly where to look for the concepts they need to work with? (Or does it not make a difference?) I also wonder if whiteboard desks help/hinder either of these elements of fluid intelligence.
However, one thing that I'm struggling with is how to apply these strategies when you're working with students who are years below grade level and/or have other learning needs. (Not saying you need to have answers - just saying that these are the things I'm struggling with now!)
Those are great questions. A few miscellaneous thoughts:
Routines are really important. Anything where students don't think about what to do and can think about the math they're learning disproportionately helps students with less fluid intelligence.
I'm a bit skeptical of anchor charts -- for the most important stuff, sure, but the walls can end up feeling chaotic and crowded, which isn't good. Notes similarly can be good or bad. Students with a slower processing speed in particular have a really hard time taking notes and listening at the same time, so notes need to be structured in a way that separates instruction and notetaking. I think giving students resources and examples can work better than taking notes under some circumstances. I'm not sure about whiteboard desks, I've never used them.
In terms of students years below grade level -- there's no easy answer. All of these strategies are important, but they also take time, and time is at a premium if you have more math to teach. My answer is prioritizing: choosing the most important math and doing it really well, and spending less time on some standards that I see as less important.
Super interesting, love this! Cool distillation of research and experience into why certain teaching strategies are critical, and narrating some tradeoffs amongst them since time is not unlimited. Also the phonics heuristic (useful for all, harmful for none, needed for some!
I also wonder about recall and synthesis, distillation, and use/application of LT memory. Even if LT memory is unlimited, that’s another limiting factor that is implied but I felt you didn’t make quite explicit: it’s underlying and connected to working memory and processing speed, but not just the new info someone takes in (e.g., what you’re teaching them and guiding them with those closer rungs on the ladder) but also how the use the existing memory / knowledge / experience.
Thanks! And your point about LT memory is important. There's a version of teaching for automaticity that is pure repetition, and while that will help with automaticity students often struggle to apply the knowledge when the context is slightly different. So retrieval practice needs to have some variation, gradually helping students apply what they know in different contexts. I think you're right that it connects with the idea of rungs on the ladder, and it's another example where having more cognitive resources makes it easier for some students to make those connections, but others need teacher support in knowing where to apply what they have learned.