This is super helpful and insightful. Having an 8 year old who finds school tremendously boring (plus stressful, boring + stressful = who would ever wanna go there) has definitely opened my mind to the issues with current education system. Do you believe schools will do a better job at teaching if the main focus was fostering curiosity and creativity? If it was a requirement for teachers to prioritize connection with their students? If they added a boat load of funny? If there is a degree of novelty? And teachers are paid in six figures at minimum?
Sarah, you get the prize for “most helpful comment of the open thread”! Your question gets at the heart of something that I’ve been trying (and failing) to say about Egan. The (too) short answer is “ha, no, not quite any of those things!”, and the fuller answer requires more than a comment. Would it be okay if I cited your question in a future post (“no thanks” is okay), and answer it there?
I really love the topics you choose and how you teach about teaching :) I teach English and "accidentally" forwarded your email about secret messages to a student. I would love to read some research about how kids are using LLMs and where it might lead. thanks for the blog <3
Re: where LLMs might be going — we’re going to explore this HARD in the LiD summer intensive! My thoughts are that eg GPT is the most powerful tool ever for hardcore learning, IF one knows how to use them…. and that Egan’s paradigm suggests a way forward.
Yeah! I don’t have anything by way of research, but I’ll tell you I’ve been thinking a lot this week about how LLMs can transform the Learning in Depth Intensive that we’re planning for this summer.
I highly recommend this podcast in regards to LLMs and education. It's definitely on the "doomer" end of things. It succeeded in introducing to me some new ways of thinking about this new paradigm in education.
Humans tend to compensate for environmental changes. If you assign bicyclists helmets, they tend to take more risks when biking and overall bike accidents can remain constant or even increase. https://en.wikipedia.org/wiki/Risk_compensation
When we test interventions in schools or learning, we try to measure the outcome on grades or performance on tests. But what if students who get an X10 teacher just study less to learn what they think they need to know, and spend more time on other things? This X10 teacher wouldn't show up in any traditional measurements.
Can this be the case for Egan? It does improve learning, but in a way that students compensate for so it's hard to measure?
Huh! Way to point out ANOTHER wrinkle in how hard it is to do educational research well… I suspect things like this were part of why Egan was disdainful of that enterprise. I think that in his mind — and I’m DEFINITELY not at a full understanding of this — educational interventions that actually worked would work so well that you wouldn’t need nuanced studies to see ‘em.
Anyhow, I think that describes my model pretty well. If you want your kids to learn about, say, ancient history, and they go to a normal American elementary school, well, you can see for yourself if they learn it or not. (Usually, nowadays, not.) When I look at what my kids were learning in school before we started homeschooling, I’m just aghast — it was virtually nothing interesting, nothing thought-provoking, nothing that could hold the attention of our ADHD kids. When we ran an Egan-inspired microschool, the kids effing learned interesting, meaningful things! (The story of the Buddha, how to cook, the science of atoms and molecules and life…)
I apologize: I suspect this is a non-answer to your question! But maybe it’s helpful nonetheless for me to say this somewhere: the scope of what we tell teachers to teach kids now is so narrow, and one thing Egan education can absolutely do is widen it, and make things that are dull interesting.
Here's what he wishes every practicing researcher knew about statistics:
"Hey dude, remember that statistics are just a tool for distinguishing signal from noise. If we really knew what was going on, we wouldn’t need statistics. It’s fine to use them, just like it’s fine to use training wheels and water wingies. But we should aspire to shed them eventually. You’ll never really learn to bike or swim until you ditch the extra wheels and the inflatables, and you'll never really understand the universe if you're stuck studying it with statistics."
Or, a little pithier: "Things seem random until you understand 'em."
Will admit I follow your blog zealously and have discovered a newfound enjoyment and interest in educational approaches! I’m not a homeschooler so more of an interest in how to make things interesting for myself and others in my day to day life.
Do you plan on ever looking to incorporate these patterns into a broader curriculum or similar? I’m quite familiar with your existing work for Science is Weird, and I think you mentioned you would look to give courses on happiness and evil in the future (if I recall correctly!) but did you have any views beyond that for the short to medium term?
Thanks! And yes: I haven’t formally announced it yet, but this summer I’m [deep breath] writing a book on education for parents. The book’ll lay out how we might give kids an Egan education in the early years — preschool, kindergarten, and grades 1–4 — that sets the foundation for everything that might come later.
Part of the reason to write that first is to force myself to sketch a clearer curriculum (in the humanities, sciences, arts, math, and beyond) that we can then develop into an EVEN clearer curriculum for schools.
The plans for it are still evolving, but anyone’s free to ask questions!
This sounds very exciting. Since it is such a significant effort, perhaps you can use this Substack "community", or specific members (some of whom may volunteer) as a resource.
Like the idea! I actually want to lean so hard into getting a community’s help that I’m planning to launch a one-year mostly-private Substack for just this purpose.
talk is related to tale, so the L probably was pronounced at one time:
"c. 1200, talken, "speak, discourse, say something," probably a diminutive or frequentative form related to Middle English tale "story," and ultimately from the same source as tale (q.v.), with rare English formative -k (compare hark from hear, stalk from steal, smirk from smile) and replacing tale as a verb. East Frisian has talken "to talk, chatter, whisper."
the other half of the story is that L has a general tendency to get lost before certain other sounds, such as K, because of the difficulty/difference in moving between how the two letters are formed in the mouth.
In a comment on his review, Andrew asks about why “ea” can get the various sounds it can get. I’d love to imagine how that bit of content could be Eganized, but I don’t know the story behind that. If you or @Christopher B. (or anyone else who’s knowledgable about the stories behind spellings) would like to explore that in the other comments section, I’d be happy to give an alley-oop and imagine how it might be linked to a cognitive tool or two.
I took a stab at it, and of course added my 2 cents about spelling instruction... :-D
Vowel questions are sort of the worst, because the reasons are almost always "Great Vowel Shift" and "Printers trying to standardize English!". I like other spelling questions so much better. ;-)
I've seen too many bad ideas about how to teach math, so I've been thinking recently about how I would teach math to kids in elementary school. Here are some ideas I thought I would share. All of them are things that fascinated me when I was a kid.
1. For 1st or 2nd grade:
Build a table of numbers from 1 - 100, with ten numbers in each row. (Substack doesn't seem to allow images in comments, so you'll have to try this yourself.)
a. Color the multiples of 10 in green. Why are they all stacked in the same vertical line?
b. Now color the multiples of 9 in blue. Why are they all on the same diagonal line?
c. Now color the multiples of 11 in red. What pattern do you see, and why?
2. For 3rd or 4th grade:
Can you explain the pattern you see in the following sequence?
1 * 1 = 1
11 * 11 = 121
111 * 111 = 12321
1111 * 1111 = 1234321
...
(Hint: Try the usual multiplication process by hand. What do you notice? Why does the pattern stop working once you reach 1111111111 (i.e., ten ones)?)
3. For 5th grade:
A magic square is a grid of numbers in which every row, column, and diagonal adds up to the same value.
a. Build a magic square using the numbers from 1 - 9.
b. Build a magic square using the numbers from 2 - 10. Is there a relationship between your solutions here and the solutions to part a? (Mathematicians call this relationship an isomorphism, which is so fundamental that math almost couldn't exist without it.)
c. Find a set of 9 numbers that can't be arranged into a magic square. Why doesn't it work?
d. Returning to the numbers from 1 - 9, which numbers can be placed in the center to from a valid magic square? What's special about those numbers? (This is much harder - I think it requires algebra to explain well, so it may be too advanced for 5th graders.)
4. For 6th grade:
Can you explain this pattern?
9 * 9 = 81
99 * 99 = 9801
999 * 999 = 998001
9999 * 9999 = 99980001
...
(Hint: The easiest solution here also requires proficiency with algebra.)
5. For 6th grade:
(Zeno's paradox) Start with a page of paper and take away half. Then take away half of what's left. Repeat this again and again until the page is too small to divide any further.
a. If we could continue this process forever, how much paper would be taken away?
b. What if we took just 1/3 of the remaining page each time instead of 1/2?
c. How is it possible that the answers to a and b are the same?!
One of my favourite writings on education is a pamphlet called "The Rhythm of Education" by Alfred North Whitehead in 1923. The language is of its time - education is about turning boys into men etc. - but put that aside, along with the implications for Latin and Ancient Greek teaching, and there's a model in there that has made me a much better teacher. To the extent that it seems to be forgotten in teaching circles around here these days, I nominate it as a "lost tool of learning". It can be found for free online with your favourite search engine.
Whitehead's model has three stages to Egan's five, but I think there's a way to translate between them. The three stages are, in order: Romance, Precision, Generalisation. However, Whitehead says that his three-stage model can be applied to everything from planning a single lesson to designing a whole school system.
First, he says, the student needs an idea of the subject - it doesn't have to be formal at this point, but it does have to be interesting and exciting. Or in the original: "The subject-matter has the vividness of novelty ; it holds within itself unexplored connexions with possibilities half-disclosed by glimpses and half-concealed by the wealth of material." Formal study can come later, for now let the student play with the material. This is the stage of romance.
After this comes the stage of precision, where we dissect the material, analyse it formally and logically, subject it to rules etc. This should not start too early: "It is evident that a stage of precision is barren without a previous stage of romance : unless there are facts which have already been vaguely apprehended in their broad generality, the previous analysis is an analysis of nothing. It is simply a series of meaningless statements about bare facts, produced artificially and without any further relevance."
What strikes me here is that, despite being among other things a mathematician and logician himself, Whitehead is quite open about something that I miss in lots of modern schooling that I've seen - kids need time to play, explore, get familiar with the material before you jump in to rules and facts. Even if the subject-matter is mathematics! And how better to do this than - well, Whitehead talks about "romantic emotion", but after reading the ACX review of Egan, I could flesh that out with things that can provoke such emotion: stories? drama? gossip? metaphors? jokes? binary oppositions?
So Whitehead's precision is a good match for Egan's Philosophic stage, and Whitehead's romance seems to be more or less an umbrella over the somatic, mythic and romantic stages. (I trust that Egan's work, being much longer than Whitehead's 40 or so pages, also goes into more depth here.) Both Whitehead and Egan would agree, I think, that the idea is NOT simply romance in primary school, precision in college and irony perhaps in postgraduate studies - but that elements of all stages can overlap, even in a single lesson. Whitehead does, however, suggest that the emphasis of education overall should shift as the boy (in his language) grows older, from mainly-romance to mainly-precision to mainly-generalisation. Whitehead actually sketches this out in more detail, for example that "Towards the age of fifteen the age of precision in language and of romance in science draws to its close, to be succeeded by a period of generalization in language and of precision in science."
Speaking of generalisation: "It is a return to romanticism with added advantage of classified ideas and relevant technique." It's hard to overstate what impact that phrase had on me when I first read it. "It is the fruition which has been the goal of the precise training. It is the final success." A caricature of the progressive-vs-traditionalist trench war in this model is that progressives think traditionalists want all education to be precision only, and traditionalists think progressives keep the students stuck at the stage of romance forever. Whitehead (and, to the best of my understanding, Egan) shake their heads at both straw men: precision (or philosophic understanding) is not the end goal, useful though it may be for a career in engineering. Rather, it is a stage that the mind must progress through to get to the real goal, ironic / generalised understanding. You can't leave the precision out. But you also shouldn't confuse the means with the end.
(Also, both of them have good things to say about Montessori.)
Professionally, I occasionally have to work with the Association for Computing Machinery (ACM) materials on computing curricula, and they too have a three-stage model of education, albeit with the boring terms beginner, intermediate, and advanced. But the description of these feels like they're as close to Whitehead as they can get without actually having to cite him!
Someone else who has clearly read his Whitehead is Terence Tao, a candidate for the title of "best mathematician alive" - he has a Fields Medal, the math version of a Nobel Prize among other things. And he writes (https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/) that "One can roughly divide mathematical education into three stages ... pre-rigorous, rigorous and post-rigorous". The details of his argument might not make sense unless you've done university-level calculus, but the model in which rigour is a stage to go through until you arrive at "intuition solidly buttressed by rigorous theory" sounds very Whitehead-adjacent (if not Egan-adjacent) to me.
It's also why, in any debate on whether we should include learning facts or "drilling" multiplication tables in education, I'm emotionally on team YES without believing for a moment that facts are the only goal of education. Whitehead again on the end goal: "The really useful training yields a comprehension of a few general principles with a thorough grounding in the way they apply to a variety of concrete details ... The function of a University is to enable you to shed details in favour of principles. When I speak of principles I am hardly even thinking of verbal formulations. A principle which has thoroughly soaked into you is rather a mental habit than a formal statement."
I think most mathematicians would agree that playing with ideas is a core part of math. It's the non-mathematicians who took out all the actual interesting math from the math curriculum and made it boring. (The author of A Mathematician's Lament made a similar argument: https://maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf)
The other day I came across a book by someone who admitted to hating math but suggested that you could make it more useful by teaching kids about how to balance a checkbook. To me, that sounds almost as bad as having them read the manual to the vacuum cleaner as part of their English class.
Recently, I have been quoting Gall's Law, and somebody mentioned that John Gall had written books on parenting. I found a few of them on Amazon (self-published) and have been really enjoying them -- (warning: they are difficult to navigate because some of them consist almost entirely of case studies from his pediatric practice). Though it's classified as parenting versus education, I think his work may be interesting to some members of this community.
This is super helpful and insightful. Having an 8 year old who finds school tremendously boring (plus stressful, boring + stressful = who would ever wanna go there) has definitely opened my mind to the issues with current education system. Do you believe schools will do a better job at teaching if the main focus was fostering curiosity and creativity? If it was a requirement for teachers to prioritize connection with their students? If they added a boat load of funny? If there is a degree of novelty? And teachers are paid in six figures at minimum?
Am I thinking about this the right way?
So what should schools really teach?
Sarah, you get the prize for “most helpful comment of the open thread”! Your question gets at the heart of something that I’ve been trying (and failing) to say about Egan. The (too) short answer is “ha, no, not quite any of those things!”, and the fuller answer requires more than a comment. Would it be okay if I cited your question in a future post (“no thanks” is okay), and answer it there?
Sure. Please do
I really love the topics you choose and how you teach about teaching :) I teach English and "accidentally" forwarded your email about secret messages to a student. I would love to read some research about how kids are using LLMs and where it might lead. thanks for the blog <3
Re: where LLMs might be going — we’re going to explore this HARD in the LiD summer intensive! My thoughts are that eg GPT is the most powerful tool ever for hardcore learning, IF one knows how to use them…. and that Egan’s paradigm suggests a way forward.
Yeah! I don’t have anything by way of research, but I’ll tell you I’ve been thinking a lot this week about how LLMs can transform the Learning in Depth Intensive that we’re planning for this summer.
I highly recommend this podcast in regards to LLMs and education. It's definitely on the "doomer" end of things. It succeeded in introducing to me some new ways of thinking about this new paradigm in education.
https://podcasts.apple.com/gb/podcast/emerge-making-sense-of-whats-next/id1057220344?i=1000610403148
Humans tend to compensate for environmental changes. If you assign bicyclists helmets, they tend to take more risks when biking and overall bike accidents can remain constant or even increase. https://en.wikipedia.org/wiki/Risk_compensation
When we test interventions in schools or learning, we try to measure the outcome on grades or performance on tests. But what if students who get an X10 teacher just study less to learn what they think they need to know, and spend more time on other things? This X10 teacher wouldn't show up in any traditional measurements.
Can this be the case for Egan? It does improve learning, but in a way that students compensate for so it's hard to measure?
Huh! Way to point out ANOTHER wrinkle in how hard it is to do educational research well… I suspect things like this were part of why Egan was disdainful of that enterprise. I think that in his mind — and I’m DEFINITELY not at a full understanding of this — educational interventions that actually worked would work so well that you wouldn’t need nuanced studies to see ‘em.
Anyhow, I think that describes my model pretty well. If you want your kids to learn about, say, ancient history, and they go to a normal American elementary school, well, you can see for yourself if they learn it or not. (Usually, nowadays, not.) When I look at what my kids were learning in school before we started homeschooling, I’m just aghast — it was virtually nothing interesting, nothing thought-provoking, nothing that could hold the attention of our ADHD kids. When we ran an Egan-inspired microschool, the kids effing learned interesting, meaningful things! (The story of the Buddha, how to cook, the science of atoms and molecules and life…)
I apologize: I suspect this is a non-answer to your question! But maybe it’s helpful nonetheless for me to say this somewhere: the scope of what we tell teachers to teach kids now is so narrow, and one thing Egan education can absolutely do is widen it, and make things that are dull interesting.
I appreciated this post along the same lines: https://www.experimental-history.com/p/there-are-no-statistics-in-the-kingdom.
Here's what he wishes every practicing researcher knew about statistics:
"Hey dude, remember that statistics are just a tool for distinguishing signal from noise. If we really knew what was going on, we wouldn’t need statistics. It’s fine to use them, just like it’s fine to use training wheels and water wingies. But we should aspire to shed them eventually. You’ll never really learn to bike or swim until you ditch the extra wheels and the inflatables, and you'll never really understand the universe if you're stuck studying it with statistics."
Or, a little pithier: "Things seem random until you understand 'em."
Wait, what's that about bike helmets?
https://en.wikipedia.org/wiki/Risk_compensation
For risk this phenomena has been studied, but I assume it exists for all kinds of behaviors e.g. learning in schools.
Will admit I follow your blog zealously and have discovered a newfound enjoyment and interest in educational approaches! I’m not a homeschooler so more of an interest in how to make things interesting for myself and others in my day to day life.
Do you plan on ever looking to incorporate these patterns into a broader curriculum or similar? I’m quite familiar with your existing work for Science is Weird, and I think you mentioned you would look to give courses on happiness and evil in the future (if I recall correctly!) but did you have any views beyond that for the short to medium term?
Thanks! And yes: I haven’t formally announced it yet, but this summer I’m [deep breath] writing a book on education for parents. The book’ll lay out how we might give kids an Egan education in the early years — preschool, kindergarten, and grades 1–4 — that sets the foundation for everything that might come later.
Part of the reason to write that first is to force myself to sketch a clearer curriculum (in the humanities, sciences, arts, math, and beyond) that we can then develop into an EVEN clearer curriculum for schools.
The plans for it are still evolving, but anyone’s free to ask questions!
This sounds very exciting. Since it is such a significant effort, perhaps you can use this Substack "community", or specific members (some of whom may volunteer) as a resource.
Like the idea! I actually want to lean so hard into getting a community’s help that I’m planning to launch a one-year mostly-private Substack for just this purpose.
talk is related to tale, so the L probably was pronounced at one time:
"c. 1200, talken, "speak, discourse, say something," probably a diminutive or frequentative form related to Middle English tale "story," and ultimately from the same source as tale (q.v.), with rare English formative -k (compare hark from hear, stalk from steal, smirk from smile) and replacing tale as a verb. East Frisian has talken "to talk, chatter, whisper."
(From https://www.etymonline.com/)
the other half of the story is that L has a general tendency to get lost before certain other sounds, such as K, because of the difficulty/difference in moving between how the two letters are formed in the mouth.
YES! Thank you, Kirsten!
In a comment on his review, Andrew asks about why “ea” can get the various sounds it can get. I’d love to imagine how that bit of content could be Eganized, but I don’t know the story behind that. If you or @Christopher B. (or anyone else who’s knowledgable about the stories behind spellings) would like to explore that in the other comments section, I’d be happy to give an alley-oop and imagine how it might be linked to a cognitive tool or two.
His comment: https://open.substack.com/pub/aslowcircle/p/review-an-imaginative-approach-to?r=f8lzw&utm_campaign=comment-list-share-cta&utm_medium=web&comments=true&commentId=52405436
I took a stab at it, and of course added my 2 cents about spelling instruction... :-D
Vowel questions are sort of the worst, because the reasons are almost always "Great Vowel Shift" and "Printers trying to standardize English!". I like other spelling questions so much better. ;-)
I've been looking forward to one of these!
I've seen too many bad ideas about how to teach math, so I've been thinking recently about how I would teach math to kids in elementary school. Here are some ideas I thought I would share. All of them are things that fascinated me when I was a kid.
1. For 1st or 2nd grade:
Build a table of numbers from 1 - 100, with ten numbers in each row. (Substack doesn't seem to allow images in comments, so you'll have to try this yourself.)
a. Color the multiples of 10 in green. Why are they all stacked in the same vertical line?
b. Now color the multiples of 9 in blue. Why are they all on the same diagonal line?
c. Now color the multiples of 11 in red. What pattern do you see, and why?
2. For 3rd or 4th grade:
Can you explain the pattern you see in the following sequence?
1 * 1 = 1
11 * 11 = 121
111 * 111 = 12321
1111 * 1111 = 1234321
...
(Hint: Try the usual multiplication process by hand. What do you notice? Why does the pattern stop working once you reach 1111111111 (i.e., ten ones)?)
3. For 5th grade:
A magic square is a grid of numbers in which every row, column, and diagonal adds up to the same value.
a. Build a magic square using the numbers from 1 - 9.
b. Build a magic square using the numbers from 2 - 10. Is there a relationship between your solutions here and the solutions to part a? (Mathematicians call this relationship an isomorphism, which is so fundamental that math almost couldn't exist without it.)
c. Find a set of 9 numbers that can't be arranged into a magic square. Why doesn't it work?
d. Returning to the numbers from 1 - 9, which numbers can be placed in the center to from a valid magic square? What's special about those numbers? (This is much harder - I think it requires algebra to explain well, so it may be too advanced for 5th graders.)
4. For 6th grade:
Can you explain this pattern?
9 * 9 = 81
99 * 99 = 9801
999 * 999 = 998001
9999 * 9999 = 99980001
...
(Hint: The easiest solution here also requires proficiency with algebra.)
5. For 6th grade:
(Zeno's paradox) Start with a page of paper and take away half. Then take away half of what's left. Repeat this again and again until the page is too small to divide any further.
a. If we could continue this process forever, how much paper would be taken away?
b. What if we took just 1/3 of the remaining page each time instead of 1/2?
c. How is it possible that the answers to a and b are the same?!
One of my favourite writings on education is a pamphlet called "The Rhythm of Education" by Alfred North Whitehead in 1923. The language is of its time - education is about turning boys into men etc. - but put that aside, along with the implications for Latin and Ancient Greek teaching, and there's a model in there that has made me a much better teacher. To the extent that it seems to be forgotten in teaching circles around here these days, I nominate it as a "lost tool of learning". It can be found for free online with your favourite search engine.
Whitehead's model has three stages to Egan's five, but I think there's a way to translate between them. The three stages are, in order: Romance, Precision, Generalisation. However, Whitehead says that his three-stage model can be applied to everything from planning a single lesson to designing a whole school system.
First, he says, the student needs an idea of the subject - it doesn't have to be formal at this point, but it does have to be interesting and exciting. Or in the original: "The subject-matter has the vividness of novelty ; it holds within itself unexplored connexions with possibilities half-disclosed by glimpses and half-concealed by the wealth of material." Formal study can come later, for now let the student play with the material. This is the stage of romance.
After this comes the stage of precision, where we dissect the material, analyse it formally and logically, subject it to rules etc. This should not start too early: "It is evident that a stage of precision is barren without a previous stage of romance : unless there are facts which have already been vaguely apprehended in their broad generality, the previous analysis is an analysis of nothing. It is simply a series of meaningless statements about bare facts, produced artificially and without any further relevance."
What strikes me here is that, despite being among other things a mathematician and logician himself, Whitehead is quite open about something that I miss in lots of modern schooling that I've seen - kids need time to play, explore, get familiar with the material before you jump in to rules and facts. Even if the subject-matter is mathematics! And how better to do this than - well, Whitehead talks about "romantic emotion", but after reading the ACX review of Egan, I could flesh that out with things that can provoke such emotion: stories? drama? gossip? metaphors? jokes? binary oppositions?
So Whitehead's precision is a good match for Egan's Philosophic stage, and Whitehead's romance seems to be more or less an umbrella over the somatic, mythic and romantic stages. (I trust that Egan's work, being much longer than Whitehead's 40 or so pages, also goes into more depth here.) Both Whitehead and Egan would agree, I think, that the idea is NOT simply romance in primary school, precision in college and irony perhaps in postgraduate studies - but that elements of all stages can overlap, even in a single lesson. Whitehead does, however, suggest that the emphasis of education overall should shift as the boy (in his language) grows older, from mainly-romance to mainly-precision to mainly-generalisation. Whitehead actually sketches this out in more detail, for example that "Towards the age of fifteen the age of precision in language and of romance in science draws to its close, to be succeeded by a period of generalization in language and of precision in science."
Speaking of generalisation: "It is a return to romanticism with added advantage of classified ideas and relevant technique." It's hard to overstate what impact that phrase had on me when I first read it. "It is the fruition which has been the goal of the precise training. It is the final success." A caricature of the progressive-vs-traditionalist trench war in this model is that progressives think traditionalists want all education to be precision only, and traditionalists think progressives keep the students stuck at the stage of romance forever. Whitehead (and, to the best of my understanding, Egan) shake their heads at both straw men: precision (or philosophic understanding) is not the end goal, useful though it may be for a career in engineering. Rather, it is a stage that the mind must progress through to get to the real goal, ironic / generalised understanding. You can't leave the precision out. But you also shouldn't confuse the means with the end.
(Also, both of them have good things to say about Montessori.)
Professionally, I occasionally have to work with the Association for Computing Machinery (ACM) materials on computing curricula, and they too have a three-stage model of education, albeit with the boring terms beginner, intermediate, and advanced. But the description of these feels like they're as close to Whitehead as they can get without actually having to cite him!
Someone else who has clearly read his Whitehead is Terence Tao, a candidate for the title of "best mathematician alive" - he has a Fields Medal, the math version of a Nobel Prize among other things. And he writes (https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/) that "One can roughly divide mathematical education into three stages ... pre-rigorous, rigorous and post-rigorous". The details of his argument might not make sense unless you've done university-level calculus, but the model in which rigour is a stage to go through until you arrive at "intuition solidly buttressed by rigorous theory" sounds very Whitehead-adjacent (if not Egan-adjacent) to me.
It's also why, in any debate on whether we should include learning facts or "drilling" multiplication tables in education, I'm emotionally on team YES without believing for a moment that facts are the only goal of education. Whitehead again on the end goal: "The really useful training yields a comprehension of a few general principles with a thorough grounding in the way they apply to a variety of concrete details ... The function of a University is to enable you to shed details in favour of principles. When I speak of principles I am hardly even thinking of verbal formulations. A principle which has thoroughly soaked into you is rather a mental habit than a formal statement."
I think most mathematicians would agree that playing with ideas is a core part of math. It's the non-mathematicians who took out all the actual interesting math from the math curriculum and made it boring. (The author of A Mathematician's Lament made a similar argument: https://maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf)
The other day I came across a book by someone who admitted to hating math but suggested that you could make it more useful by teaching kids about how to balance a checkbook. To me, that sounds almost as bad as having them read the manual to the vacuum cleaner as part of their English class.
Thank you for sharing this. Having looked it up and read it, I agree with your assessment.
https://notability.com/n/FMUuJgr85zAHHUyKwvUOO
Boss Question Idea For High/Middle School
Recently, I have been quoting Gall's Law, and somebody mentioned that John Gall had written books on parenting. I found a few of them on Amazon (self-published) and have been really enjoying them -- (warning: they are difficult to navigate because some of them consist almost entirely of case studies from his pediatric practice). Though it's classified as parenting versus education, I think his work may be interesting to some members of this community.
¹ I think we do. Or half do. The way the French half pronounce the second 'n' in 'non' or the' r in "Sartre". Or at least I do?