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.
Of course, the better the instruction the greater the likelihood that each student will perform to their potential. And that's great. Nevertheless, anyone with common sense can see that, even with all the brilliant instruction in the world, there are cognitive thresholds which some students just do not surpass.
In my last few years of math teaching, I’ve been thinking a lot about what factors affect “smartness”, ie. what factors affect processing speed and short term memory. I know I’m hardly the first person to say this, but as a teacher it’s been really helpful to remember that processing speed and short term memory are affected by stress, hunger, language competence, distraction, social pressure, etc
When we make decisions based on student ability, we often have a lack of understanding of the temporary factors affecting that ability.
This comment may fit better under your post about tracking. I just read it as well and am thinking about both posts.
Yea that's a great point, I totally agree. That idea can help us why we see (on average) differences in learning between different demographic groups. And those are things we can mitigate -- not totally solve, but we can do our best to address them directly (routines that help with distraction, assigned seating mitigates social issues, we can provide meals to help with hunger). And then we can teach in a way that reduces the impact of those short-term memory demands, etc.
I've met lots of teachers who take a fatalistic attitude. They see the impact of the weight of poverty on students and think they can't do anything. And sure, it's a tall mountain to climb, but it's not intractable.
Doing something to change the social effects of poverty is precisely why we have a public education system! And there’s lots of evidence that it works, even if it’s tough for individual teachers to see. “I’m one cog in a system” is a something I tell myself when I’ve put in a lot of effort to helping a particular student without much visible result. Hopefully I’ve set up the student to make progress with another teacher.
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.
"Students do not show substantial differences in their rate of learning."
For chemistry specifically (what I teach), the prior math knowledge makes a huge difference in their initial performance, so weaker math students need a lot more practice to catch up.
Yup! I actually wrote a long post on that topic. I wouldn’t quite claim that all students learn at the same speed when controlling for prior knowledge, I would say that a large chunk of learning speed differences are actually prior knowledge differences.
Recent college grad here - the description of gaps left to fester was extremely accurate. Most of my math classes felt like a desperate attempt to stay above water, because I was never given the opportunity to learn the fundamentals. I had to spend an extraordinary amount of effort to fill in my own gaps while learning the current material.
College was a blessing, but also profoundly disappointing in many ways. "Higher education" gave me access to deeper learning but I often felt like the curriculum was an afterthought, and frankly, the teaching I received in high school was far superior to what I received in university. Isn't that the least bit embarrassing for the university?
I think there's a kind of ethics of rugged individualism in college that made many professors, and perhaps even the institution itself, feel absolved of any strong sense of responsibility to scaffold the learning they provide. This felt especially true in STEM classes. They reason that the knowledge is there - it's the students' responsibility to learn it. This is true in only the most trivial sense. Teaching and learning is a two-way street.
I hope this comment doesn't come off as "entitled student ungrateful for the education they received", but as "grateful student who wishes the system improves for those who come after."
Yea I agree with what you're describing. Everyone hits the wall in a different places -- many students would agree with what you said, but describe high school as where they hit the wall after success in middle school.
The other phenomenon related to what you're describing is that some educators, especially in college, see their role as sorting rather than teaching. They want to provide less support because their goal is to weed out students.
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.
This reminds me of the phrase "if you don't have time, you don't have priorities"
Ditto on feeling that it is useful to reframe some things in my math teaching from "we don't have time for (skill/concept)" to "(skill/concept) is less of a priority right now, we are choosing to focus on (skill/concept)"
Yeah I agree. There are lots of things pushing teachers away from doing this -- pressure about standardized testing (students will have better test scores in the long run if we prioritize, but in the short term shallow coverage of everything often makes more sense), rigid pacing guides, lack of content knowledge. Prioritization of key skills is so important in math.
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.
Of course, the better the instruction the greater the likelihood that each student will perform to their potential. And that's great. Nevertheless, anyone with common sense can see that, even with all the brilliant instruction in the world, there are cognitive thresholds which some students just do not surpass.
In my last few years of math teaching, I’ve been thinking a lot about what factors affect “smartness”, ie. what factors affect processing speed and short term memory. I know I’m hardly the first person to say this, but as a teacher it’s been really helpful to remember that processing speed and short term memory are affected by stress, hunger, language competence, distraction, social pressure, etc
When we make decisions based on student ability, we often have a lack of understanding of the temporary factors affecting that ability.
This comment may fit better under your post about tracking. I just read it as well and am thinking about both posts.
Yea that's a great point, I totally agree. That idea can help us why we see (on average) differences in learning between different demographic groups. And those are things we can mitigate -- not totally solve, but we can do our best to address them directly (routines that help with distraction, assigned seating mitigates social issues, we can provide meals to help with hunger). And then we can teach in a way that reduces the impact of those short-term memory demands, etc.
I've met lots of teachers who take a fatalistic attitude. They see the impact of the weight of poverty on students and think they can't do anything. And sure, it's a tall mountain to climb, but it's not intractable.
Doing something to change the social effects of poverty is precisely why we have a public education system! And there’s lots of evidence that it works, even if it’s tough for individual teachers to see. “I’m one cog in a system” is a something I tell myself when I’ve put in a lot of effort to helping a particular student without much visible result. Hopefully I’ve set up the student to make progress with another teacher.
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!
Some nice evidence aligning with this idea is here:
https://www.pnas.org/doi/10.1073/pnas.2221311120
"Students do not show substantial differences in their rate of learning."
For chemistry specifically (what I teach), the prior math knowledge makes a huge difference in their initial performance, so weaker math students need a lot more practice to catch up.
Yup! I actually wrote a long post on that topic. I wouldn’t quite claim that all students learn at the same speed when controlling for prior knowledge, I would say that a large chunk of learning speed differences are actually prior knowledge differences.
https://fivetwelvethirteen.substack.com/p/do-some-students-learn-faster-than
Recent college grad here - the description of gaps left to fester was extremely accurate. Most of my math classes felt like a desperate attempt to stay above water, because I was never given the opportunity to learn the fundamentals. I had to spend an extraordinary amount of effort to fill in my own gaps while learning the current material.
College was a blessing, but also profoundly disappointing in many ways. "Higher education" gave me access to deeper learning but I often felt like the curriculum was an afterthought, and frankly, the teaching I received in high school was far superior to what I received in university. Isn't that the least bit embarrassing for the university?
I think there's a kind of ethics of rugged individualism in college that made many professors, and perhaps even the institution itself, feel absolved of any strong sense of responsibility to scaffold the learning they provide. This felt especially true in STEM classes. They reason that the knowledge is there - it's the students' responsibility to learn it. This is true in only the most trivial sense. Teaching and learning is a two-way street.
I hope this comment doesn't come off as "entitled student ungrateful for the education they received", but as "grateful student who wishes the system improves for those who come after."
Yea I agree with what you're describing. Everyone hits the wall in a different places -- many students would agree with what you said, but describe high school as where they hit the wall after success in middle school.
The other phenomenon related to what you're describing is that some educators, especially in college, see their role as sorting rather than teaching. They want to provide less support because their goal is to weed out students.
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.
This reminds me of the phrase "if you don't have time, you don't have priorities"
Ditto on feeling that it is useful to reframe some things in my math teaching from "we don't have time for (skill/concept)" to "(skill/concept) is less of a priority right now, we are choosing to focus on (skill/concept)"
Yeah I agree. There are lots of things pushing teachers away from doing this -- pressure about standardized testing (students will have better test scores in the long run if we prioritize, but in the short term shallow coverage of everything often makes more sense), rigid pacing guides, lack of content knowledge. Prioritization of key skills is so important in math.
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.