I really like this pattern! The main challenge I see with applying it is to break down a concept enough for the target audience to follow along. For example, the article that was shared about Squirrel AI says they've broken down the math curriculum into 10,000 separate steps. That's pretty intimidating for the average teacher to try to reproduce.
My current project is to try to help my two-year-old son learn to count. He loves to read his counting books, and we've read some of them literally hundreds of times. But so far he still seems to be missing the point. He thinks that counting means saying the words, "1, 2, 3", but he hasn't figured out that he's supposed to match the numbers with the objects that he's counting. (He also stops at 3 - anything larger than 3 is "so many").
After reading this post and thinking some more, I think maybe the concept of one-to-one correspondence is actually a prerequisite for counting. I have some ideas to try on how to make that more concrete for him, but I'd love to hear if anyone else has suggestions.
Lastly, I read the examples of concept ladders that you shared. Unfortunately, it's not quite true that sqrt(x^2) = x. That works for positive numbers, and maybe you're assuming the students haven't learned negative numbers yet? But in general, sqrt(x^2) actually equals the absolute value of x.
The mathematician's explanation would be that you can only invert a function that's one-to-one, but just typing that sentence makes me start to feel like this teacher:
Re: roots and negatives: Ha! Yeah, I have this hazy remembrance from high school that in some situations, you're supposed to put the "±" (positive/negative) sign in front of the number... but in other situations, it's okay for you not to? I never got clear on when that was supposed to matter — it felt like one of those "conventions" things, that didn't reflect what's actually going on with the math. (Am I wrong about this? Or is that actually a separate thing?)
In any case, thanks for pointing that out. You're right in that there's definitely a trade-off between (A) absolute accuracy and (B) the ability to quickly convey an important idea. I tend to err closer to side (B) in general, and think this is wise, at least in most circumstances. (See a coming pattern, "π = 3"...)
Re: your two-year-old: I'm immediately reminded of Joe, the student from the third chapter of "Sideways Stories from Wayside School" by the inimitable Louis Sachar. If anyone HASN'T read that book, it's... beyond words.
I vaguely recall Egan writing about how to teach counting. One insight is that numbers above three or so isn't actually natural (see xkcd.com/764/ , but it's a real thing, too). As parents and teachers, we're in a similar situation to Rowan Atkinson's character: https://www.youtube.com/watch?v=u99LjJ32qOo
So just learning that there ARE numbers above three is an important step. Egan recommended relying on the human propensities for rhythm and rhyme, and learning poems like "One, Two, Buckle My Shoe..." (see the Wikipedia page for the full poem, which go WAY beyond what I learned: https://en.wikipedia.org/wiki/One,_Two,_Buckle_My_Shoe ) to prepare the way.
For roots, yeah, the idea is that we usually want the square root to be a function, which means it needs a single output for each input. By convention we choose the positive output.
But then if you have an equation like x^2 = 4, you're tempted to just take the square root of both sides and get x = 2. In doing that, we lost the solution x = -2. To make sure we have all the solutions, we have to put ± on one side whenever you take the square root of both sides of an equation.
For counting, you're right, nursery rhymes are probably the most fun way to begin. And I'll have to look up "Sideways Stories from Wayside School." Thanks!
Separate comment to address a general "how could this fail?" point that applies to a lot of things related to math. There's a general principle that students should learn not just the "how" but the "why", that they should understand rather than just learn a set of boring old rules. This is at least 95% true, but it has to be done in a way that avoids a trap.
Imagine you teach someone that 1 + 2 is the same as 2 + 1, and so on, so you can do a sum in any order you like (they're probably too young to need to know the difference between "associative" and "commutative"). Maybe later on they learn that 1 * 2 is the same as 2 * 1, and there's a general pattern here. The trap to avoid is that halfway through a longer problem, they encounter 1 - 2 and remember "you can swap things around in arithmetic" and turn it into 2 - 1.
Or you teach them how to cancel products in fractions (2*3/2*5 = 3/5) and next thing you know, they cancel (2+3)/(2+5) = 3/5 too. We've all been there. Lockhart's lament, which I unfortunately couldn't join the reading group for, is big on building mathematical intuition; there's some book-length arguments on similar topics by Jo Boaler that I'm familiar with too. The trap is that if student A is encouraged to use her intuition, and generally gets the right answer that way, that's good; if student B uses their intuition and ends up cancelling sums, that's not good. Similarly with Lockhart's "area of a triangle with height h and base b" example, if student B thinks about it and comes up with "h + b" as a result, that's wrong, even if they have a mathematical-ish justification for it: "well if you increase the height then the area gets bigger, and if you increase the base then the area also gets bigger, and both of those principles apply if you add the two". There's a danger that student B learns the meta-rule "they tell me all the time to THINK, but every time I do that I'm told I've got the wrong answer. Maybe it's just a lie they tell children." The basic risk to all attempts at building mathematical intuition - which is not just good, but necessary too - is that until it's fine-tuned enough in an individual student, it can produce random results and lower grades than just learning a formula. That doesn't mean one shouldn't do it, but that one actively has to watch out and intervene to help the "student B" types along the way.
I've privately tutored a few students in the past who were struggling with school math taught in a "learn to think" style, and sometimes they were so confused by the problems that they fell back on heuristics like "find all the numbers in the word problem and multiply them" because at least that looks like they're doing something to the teacher. They were a lot happier, and their grades improved, when for some categories of problem that made up a good part of their tests we went through explicit formulas or procedures that they could just reel off in the test without "thinking". Classic example: the math rule of 3 for proportions (2 apples cost $8, how much are 5 apples?). After seeing my students struggle with the "just think about it" approach, I taught them the old-fashioned method: make a 2x2 table, "scenarios" go in rows, label the columns ("amount", "price") then multiply the two things diagonally opposite and divide by the one in the corner. That completely hides why it works, but it got some of my students from a failing to a passing grade.
A related problem is that in math especially, I find it's often much easier to get an intuition for some rule in general, than it is to properly sort out the edge cases when it does or doesn't apply. This is as true for "higher" math like calculus (famous example: de l'Hopital's rule - a minefield of edge cases if there ever was one), and the example you presented here (the intuition looks like it should extend to 0^0, which I'm sure is covered in the full version of your notes), and for example it's easy to reason from 2x=2y giving x=y and 3x=3y giving x=y and so on that ax=ay always reduces to x=y. That doesn't mean you can't build this up as a concept ladder (in fact it's a great idea), as long as you make sure the student doesn't think they've understood before they get to the warning about if a could be 0.
I think my ideas here are best seen as a complement to, rather than a rebuttal of, concept ladders - which I really like as an idea by the way! Or perhaps being aware of this failure mode is a way to make much more effective concept ladders and intuition-building exercises.
YES! You've just put your finger on something really important — that a necessary element of Concept Ladders (really, of all math teaching, but ESPECIALLY of these) is that they systematically identify and correct common misapprehensions that students have. And you've just listed out a few.
Misapprehensions seem like some of the MOST important math to teach, even beyond the practical value of "let's help students not screw up". Because (as you said) they arise when students use their intuition, they show us where students' intuitions are going wrong.
Truly, every common misapprehension could be its own separate table, included in the packet.
In fact, these misapprehensions seem to parallel today's science pattern — Secrets and Revelations° (losttools.substack.com/p/secrets-and-revelations) — in that they're not just adding more information, but upturning people's assumptions about the world.
YEAH!
One thing this suggests to me is that, if and when we seriously begin to make these, we have someone keep track of the versions — because teachers and tutors will forever be noting new misapprehensions, and want to be making new tweaks.
If anyone is interested in starting to make these, let us know!
On the AI question, back while I was teaching in China, there was a lot of hubbub about Squirrel AI (you can find their website in English, but it is really not useful as it is quite poorly translated), maybe a better thing is this article:
Where each lesson is broken down into mini MCQs and depending on the kind of error you commit (if any) you are given a small lesson to learn why that particular thing was wrong. Allegedly, really big success, but again, all data coming out of the company or media is skewed, but it could be pointing to what the future can be like.
Still, I really like the idea of concept ladders, I think it would have been useful to have this in my toolbox even for Econ which I was teaching, as some things in IGCSE are best taught through DI, which of course is despised!
Yeah — somehow I hadn't thought of using this in economics! (Which, as a former economics teacher, is pretty embarrassing...) The pattern "Everyone Learns Economics" is coming up in a few months; I'll make a note to mention Concept Ladders there...
How does your idea of concept ladders compare to the Beast Academy / Art of Problem Solving approach? So far that's my gold standard for new/innovative math curricula (works well for my son and my friends' kids who use it-- small sample size, I know), and it seems at least philosophically sympathetic.
Man, Beast Academy IS the gold standard: they've Egan-ized the elementary school curriculum! (GOSH their stuff is good. Actually, it's so good that if anyone out there has any NEGATIVE experience with Beast Academy, I'd love to hear it!)
My memory of using B.A. with my kids is foggy (and we leaned heavily on the monsters-and-comic-books part, rather than the workbook), but I'm recalling the curriculum does a good job of building up ideas conceptually, step by step (accompanied by dialogue and story). "Concept Ladders" differ in that they strip away the dialogue and story, and ask kids to do nothing but (1) solve tiny problems and (2) note whether they understood it or not.
My memory of the middle-school-oriented Art of Problem Solving is that it's almost the opposite: it throws kids into ludicrously difficult problems, and doesn't give much help. This is, of course, off-putting for most kids, but it's designed for math-obsessed kids (and for them, it's often wondrous).
But I've had limited experience with both, and they came from a long time ago. Anyone care to share their perspective?
Liked the exponent lesson. Intuitive, the progression you thought of. And the funny stuff - blurp/blurpy etc.
Why is this a nuclear option? Seems like a good “overall” problem set - ie, why is Concept Ladder reserved? What’s the “typical” lesson (non nuclear) you’d use, such that you’re reserving Concept Ladder somehow?
Yeah — if this is so good, why not just do it all the time?
I suppose someone COULD do that. And if "learning math" only means "learning the content of math", well, they'd learn a lot of math quite quickly!
But I think we also want "learning math" to include some other things. Just off the top of my head:
* puzzling out really complicated problems over the course of minutes, hours, days, and years
* finding patterns in the world around you
* asking mathematical questions
* appreciating the beauty of a pattern, a problem, or a proof
For the first of those, Concept Ladders° are too quick. (Boss Questions° — losttools.substack.com/p/boss-questions-in-math — are a better approach.) For the others, we need other patterns.
Thank you. Yes, I know John Mighton's JUMP, and appreciate its elegance.
Very much looking forward to Microscaffolded Math. I agree that strong math students don't get enough puzzling out or appreciating complex patterns.
Most of my effort, though, is the struggler. The kid I finished working with 4 minutes ago, "J", takes LOTS and lots of repetition to grasp how to add fractions with unlike denominator. Usually, curric lacks precisely the micro scaffolding you describe! Essentially I'm forced to create it, on the fly, using the white board.
If I had truly well-thought-out micro scaffolding in the curriculum, I'd protect my (limited) cognitive load to react to "J's" utterances and thoughts, instead of trying to come up with an appropriate micro step.
I've had students like that — it really feels like the dominant curriculum approaches really aren't powerful enough for this sort of kid. And while I was focusing on avoiding working-memory overwhelm for the student, I think you're wise to call our attention to the very real limits on the working memory of the TEACHER.
Mike, if you have any interest in trying to throw together some drafts of Concept Ladders (aimed for the student you're talking about, or another), feel free to share it with us, or even with me! (If you want feedback, I'd be happy to give it.) I suspect that someone could make a healthy amount of money making excellent Concept Ladders, and selling them — first to tutors, then to schools...
> "More and more, though, I’m suspecting that [repetition] is the way humans have used for millennia to learn hard skills."
I think that's definitely true, for two reasons. One, it works, and so you'd expect humans to stumble upon it eventually by trial and error. (scientific evidence: something something Willingham something something spaced repetition studies.) Big caveat here though that it only works if you're practicing the right thing, and you generally need a competent teacher (in the most general sense of the word) to help you with that. Two, if you look at any activities where some form of learning happens, that are plausible candidates for pre-industrial-revolution origins.
Exhibit 1: music - practicing scales etc. was a thing long before the modern period, but also there's a whole genre of études which are pieces designed to practice particular techniques while also sounding good and fun to play. Big list at https://imslp.org/wiki/List_of_Piano_Études, some of the Chopin ones are famous enough they can stand on their own as concert music such as Op. 10 (1833).
Exhibit 2: traditional Chinese martial arts. A lot of time is taken up practicing forms (taolu) where you practice movements over and over again.
Exhibit 3: the longbow. In the middle ages, Englishmen were required to practice archery, at times other sports like football were even banned because they interfered with this matter of national importance. Every Sunday was Church in the morning and archery practice in the afternoon, as your duty to God and your King.
I'm sure there's a lot more examples I can think of, as far as I can tell the first industrial-era schools simply borrowed "practice, practice, practice" as their teaching method because it seemed to work everywhere else.
> "as far as I can tell the first industrial-era schools simply borrowed "practice, practice, practice" as their teaching method because it seemed to work everywhere else."
Wow — I don't get to have big educational A-HA!'s so often anymore, but this was a major one! (And can I point out that the "big history" of education is something that has yet to be written? If anyone's interested in taking this on as a project, let us know — Egan has ideas, but as this suggests, the complete picture will require more than his insights; it'll require details like this.)
I really like this pattern! The main challenge I see with applying it is to break down a concept enough for the target audience to follow along. For example, the article that was shared about Squirrel AI says they've broken down the math curriculum into 10,000 separate steps. That's pretty intimidating for the average teacher to try to reproduce.
My current project is to try to help my two-year-old son learn to count. He loves to read his counting books, and we've read some of them literally hundreds of times. But so far he still seems to be missing the point. He thinks that counting means saying the words, "1, 2, 3", but he hasn't figured out that he's supposed to match the numbers with the objects that he's counting. (He also stops at 3 - anything larger than 3 is "so many").
After reading this post and thinking some more, I think maybe the concept of one-to-one correspondence is actually a prerequisite for counting. I have some ideas to try on how to make that more concrete for him, but I'd love to hear if anyone else has suggestions.
Lastly, I read the examples of concept ladders that you shared. Unfortunately, it's not quite true that sqrt(x^2) = x. That works for positive numbers, and maybe you're assuming the students haven't learned negative numbers yet? But in general, sqrt(x^2) actually equals the absolute value of x.
The mathematician's explanation would be that you can only invert a function that's one-to-one, but just typing that sentence makes me start to feel like this teacher:
https://www.smbc-comics.com/?id=3565#comic
Re: roots and negatives: Ha! Yeah, I have this hazy remembrance from high school that in some situations, you're supposed to put the "±" (positive/negative) sign in front of the number... but in other situations, it's okay for you not to? I never got clear on when that was supposed to matter — it felt like one of those "conventions" things, that didn't reflect what's actually going on with the math. (Am I wrong about this? Or is that actually a separate thing?)
In any case, thanks for pointing that out. You're right in that there's definitely a trade-off between (A) absolute accuracy and (B) the ability to quickly convey an important idea. I tend to err closer to side (B) in general, and think this is wise, at least in most circumstances. (See a coming pattern, "π = 3"...)
Re: your two-year-old: I'm immediately reminded of Joe, the student from the third chapter of "Sideways Stories from Wayside School" by the inimitable Louis Sachar. If anyone HASN'T read that book, it's... beyond words.
I vaguely recall Egan writing about how to teach counting. One insight is that numbers above three or so isn't actually natural (see xkcd.com/764/ , but it's a real thing, too). As parents and teachers, we're in a similar situation to Rowan Atkinson's character: https://www.youtube.com/watch?v=u99LjJ32qOo
So just learning that there ARE numbers above three is an important step. Egan recommended relying on the human propensities for rhythm and rhyme, and learning poems like "One, Two, Buckle My Shoe..." (see the Wikipedia page for the full poem, which go WAY beyond what I learned: https://en.wikipedia.org/wiki/One,_Two,_Buckle_My_Shoe ) to prepare the way.
For roots, yeah, the idea is that we usually want the square root to be a function, which means it needs a single output for each input. By convention we choose the positive output.
But then if you have an equation like x^2 = 4, you're tempted to just take the square root of both sides and get x = 2. In doing that, we lost the solution x = -2. To make sure we have all the solutions, we have to put ± on one side whenever you take the square root of both sides of an equation.
There's a similar catch in this proof that 1 = 2, but I won't give it away yet until people have a chance to try it: https://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html
For counting, you're right, nursery rhymes are probably the most fun way to begin. And I'll have to look up "Sideways Stories from Wayside School." Thanks!
Separate comment to address a general "how could this fail?" point that applies to a lot of things related to math. There's a general principle that students should learn not just the "how" but the "why", that they should understand rather than just learn a set of boring old rules. This is at least 95% true, but it has to be done in a way that avoids a trap.
Imagine you teach someone that 1 + 2 is the same as 2 + 1, and so on, so you can do a sum in any order you like (they're probably too young to need to know the difference between "associative" and "commutative"). Maybe later on they learn that 1 * 2 is the same as 2 * 1, and there's a general pattern here. The trap to avoid is that halfway through a longer problem, they encounter 1 - 2 and remember "you can swap things around in arithmetic" and turn it into 2 - 1.
Or you teach them how to cancel products in fractions (2*3/2*5 = 3/5) and next thing you know, they cancel (2+3)/(2+5) = 3/5 too. We've all been there. Lockhart's lament, which I unfortunately couldn't join the reading group for, is big on building mathematical intuition; there's some book-length arguments on similar topics by Jo Boaler that I'm familiar with too. The trap is that if student A is encouraged to use her intuition, and generally gets the right answer that way, that's good; if student B uses their intuition and ends up cancelling sums, that's not good. Similarly with Lockhart's "area of a triangle with height h and base b" example, if student B thinks about it and comes up with "h + b" as a result, that's wrong, even if they have a mathematical-ish justification for it: "well if you increase the height then the area gets bigger, and if you increase the base then the area also gets bigger, and both of those principles apply if you add the two". There's a danger that student B learns the meta-rule "they tell me all the time to THINK, but every time I do that I'm told I've got the wrong answer. Maybe it's just a lie they tell children." The basic risk to all attempts at building mathematical intuition - which is not just good, but necessary too - is that until it's fine-tuned enough in an individual student, it can produce random results and lower grades than just learning a formula. That doesn't mean one shouldn't do it, but that one actively has to watch out and intervene to help the "student B" types along the way.
I've privately tutored a few students in the past who were struggling with school math taught in a "learn to think" style, and sometimes they were so confused by the problems that they fell back on heuristics like "find all the numbers in the word problem and multiply them" because at least that looks like they're doing something to the teacher. They were a lot happier, and their grades improved, when for some categories of problem that made up a good part of their tests we went through explicit formulas or procedures that they could just reel off in the test without "thinking". Classic example: the math rule of 3 for proportions (2 apples cost $8, how much are 5 apples?). After seeing my students struggle with the "just think about it" approach, I taught them the old-fashioned method: make a 2x2 table, "scenarios" go in rows, label the columns ("amount", "price") then multiply the two things diagonally opposite and divide by the one in the corner. That completely hides why it works, but it got some of my students from a failing to a passing grade.
A related problem is that in math especially, I find it's often much easier to get an intuition for some rule in general, than it is to properly sort out the edge cases when it does or doesn't apply. This is as true for "higher" math like calculus (famous example: de l'Hopital's rule - a minefield of edge cases if there ever was one), and the example you presented here (the intuition looks like it should extend to 0^0, which I'm sure is covered in the full version of your notes), and for example it's easy to reason from 2x=2y giving x=y and 3x=3y giving x=y and so on that ax=ay always reduces to x=y. That doesn't mean you can't build this up as a concept ladder (in fact it's a great idea), as long as you make sure the student doesn't think they've understood before they get to the warning about if a could be 0.
I think my ideas here are best seen as a complement to, rather than a rebuttal of, concept ladders - which I really like as an idea by the way! Or perhaps being aware of this failure mode is a way to make much more effective concept ladders and intuition-building exercises.
YES! You've just put your finger on something really important — that a necessary element of Concept Ladders (really, of all math teaching, but ESPECIALLY of these) is that they systematically identify and correct common misapprehensions that students have. And you've just listed out a few.
Misapprehensions seem like some of the MOST important math to teach, even beyond the practical value of "let's help students not screw up". Because (as you said) they arise when students use their intuition, they show us where students' intuitions are going wrong.
Truly, every common misapprehension could be its own separate table, included in the packet.
In fact, these misapprehensions seem to parallel today's science pattern — Secrets and Revelations° (losttools.substack.com/p/secrets-and-revelations) — in that they're not just adding more information, but upturning people's assumptions about the world.
YEAH!
One thing this suggests to me is that, if and when we seriously begin to make these, we have someone keep track of the versions — because teachers and tutors will forever be noting new misapprehensions, and want to be making new tweaks.
If anyone is interested in starting to make these, let us know!
On the AI question, back while I was teaching in China, there was a lot of hubbub about Squirrel AI (you can find their website in English, but it is really not useful as it is quite poorly translated), maybe a better thing is this article:
https://www.technologyreview.com/2019/08/02/131198/china-squirrel-has-started-a-grand-experiment-in-ai-education-it-could-reshape-how-the/
Where each lesson is broken down into mini MCQs and depending on the kind of error you commit (if any) you are given a small lesson to learn why that particular thing was wrong. Allegedly, really big success, but again, all data coming out of the company or media is skewed, but it could be pointing to what the future can be like.
Still, I really like the idea of concept ladders, I think it would have been useful to have this in my toolbox even for Econ which I was teaching, as some things in IGCSE are best taught through DI, which of course is despised!
Yeah — somehow I hadn't thought of using this in economics! (Which, as a former economics teacher, is pretty embarrassing...) The pattern "Everyone Learns Economics" is coming up in a few months; I'll make a note to mention Concept Ladders there...
How does your idea of concept ladders compare to the Beast Academy / Art of Problem Solving approach? So far that's my gold standard for new/innovative math curricula (works well for my son and my friends' kids who use it-- small sample size, I know), and it seems at least philosophically sympathetic.
Man, Beast Academy IS the gold standard: they've Egan-ized the elementary school curriculum! (GOSH their stuff is good. Actually, it's so good that if anyone out there has any NEGATIVE experience with Beast Academy, I'd love to hear it!)
My memory of using B.A. with my kids is foggy (and we leaned heavily on the monsters-and-comic-books part, rather than the workbook), but I'm recalling the curriculum does a good job of building up ideas conceptually, step by step (accompanied by dialogue and story). "Concept Ladders" differ in that they strip away the dialogue and story, and ask kids to do nothing but (1) solve tiny problems and (2) note whether they understood it or not.
My memory of the middle-school-oriented Art of Problem Solving is that it's almost the opposite: it throws kids into ludicrously difficult problems, and doesn't give much help. This is, of course, off-putting for most kids, but it's designed for math-obsessed kids (and for them, it's often wondrous).
But I've had limited experience with both, and they came from a long time ago. Anyone care to share their perspective?
Great blog!
Liked the exponent lesson. Intuitive, the progression you thought of. And the funny stuff - blurp/blurpy etc.
Why is this a nuclear option? Seems like a good “overall” problem set - ie, why is Concept Ladder reserved? What’s the “typical” lesson (non nuclear) you’d use, such that you’re reserving Concept Ladder somehow?
Yeah — if this is so good, why not just do it all the time?
I suppose someone COULD do that. And if "learning math" only means "learning the content of math", well, they'd learn a lot of math quite quickly!
But I think we also want "learning math" to include some other things. Just off the top of my head:
* puzzling out really complicated problems over the course of minutes, hours, days, and years
* finding patterns in the world around you
* asking mathematical questions
* appreciating the beauty of a pattern, a problem, or a proof
For the first of those, Concept Ladders° are too quick. (Boss Questions° — losttools.substack.com/p/boss-questions-in-math — are a better approach.) For the others, we need other patterns.
Look forward, though, to Microscaffolded Math°, which will integrate the elements of Concept Ladders° into the actual math-teaching lesson. (If you'd like a preview, take a look at JUMP Math; the best intro is by David Leonhardt at the Times: https://opinionator.blogs.nytimes.com/2011/04/18/a-better-way-to-teach-math/?_r=0 and https://opinionator.blogs.nytimes.com/2011/04/21/teaching-math-advanced-discussion/ )
Thank you. Yes, I know John Mighton's JUMP, and appreciate its elegance.
Very much looking forward to Microscaffolded Math. I agree that strong math students don't get enough puzzling out or appreciating complex patterns.
Most of my effort, though, is the struggler. The kid I finished working with 4 minutes ago, "J", takes LOTS and lots of repetition to grasp how to add fractions with unlike denominator. Usually, curric lacks precisely the micro scaffolding you describe! Essentially I'm forced to create it, on the fly, using the white board.
If I had truly well-thought-out micro scaffolding in the curriculum, I'd protect my (limited) cognitive load to react to "J's" utterances and thoughts, instead of trying to come up with an appropriate micro step.
I've had students like that — it really feels like the dominant curriculum approaches really aren't powerful enough for this sort of kid. And while I was focusing on avoiding working-memory overwhelm for the student, I think you're wise to call our attention to the very real limits on the working memory of the TEACHER.
Mike, if you have any interest in trying to throw together some drafts of Concept Ladders (aimed for the student you're talking about, or another), feel free to share it with us, or even with me! (If you want feedback, I'd be happy to give it.) I suspect that someone could make a healthy amount of money making excellent Concept Ladders, and selling them — first to tutors, then to schools...
> "More and more, though, I’m suspecting that [repetition] is the way humans have used for millennia to learn hard skills."
I think that's definitely true, for two reasons. One, it works, and so you'd expect humans to stumble upon it eventually by trial and error. (scientific evidence: something something Willingham something something spaced repetition studies.) Big caveat here though that it only works if you're practicing the right thing, and you generally need a competent teacher (in the most general sense of the word) to help you with that. Two, if you look at any activities where some form of learning happens, that are plausible candidates for pre-industrial-revolution origins.
Exhibit 1: music - practicing scales etc. was a thing long before the modern period, but also there's a whole genre of études which are pieces designed to practice particular techniques while also sounding good and fun to play. Big list at https://imslp.org/wiki/List_of_Piano_Études, some of the Chopin ones are famous enough they can stand on their own as concert music such as Op. 10 (1833).
Exhibit 2: traditional Chinese martial arts. A lot of time is taken up practicing forms (taolu) where you practice movements over and over again.
Exhibit 3: the longbow. In the middle ages, Englishmen were required to practice archery, at times other sports like football were even banned because they interfered with this matter of national importance. Every Sunday was Church in the morning and archery practice in the afternoon, as your duty to God and your King.
I'm sure there's a lot more examples I can think of, as far as I can tell the first industrial-era schools simply borrowed "practice, practice, practice" as their teaching method because it seemed to work everywhere else.
> "as far as I can tell the first industrial-era schools simply borrowed "practice, practice, practice" as their teaching method because it seemed to work everywhere else."
Wow — I don't get to have big educational A-HA!'s so often anymore, but this was a major one! (And can I point out that the "big history" of education is something that has yet to be written? If anyone's interested in taking this on as a project, let us know — Egan has ideas, but as this suggests, the complete picture will require more than his insights; it'll require details like this.)