Interesting perspective on problem solving essentiating represntation change but I feel that that need not always be the case. I think the playing around leads to creation of the representation and intuition which the experts leverage to get to the answer (expertise i feel is more of developing representation where the problems are easy to solve rather than the act of shifting representation itself). This representation creation requires considerable time and effort. But this represntation can later on become a shackle as well since experts are not able to think beyond that which is why new insights/representation solve problems where expertise couldn't.
Could be that we are working with different definitions of expertise. There is an influential paper in my field with the title, "Adaptive Skill as the Conditio Sine Qua Non of Expertise" which would put adaptivity at the center of expertise. But the type of expertise we generally study is slightly different than, say, the ability to play the piano very well. Perhaps the latter doesn't require re-representation in the same way that a dynamic and changing housefire does.
Would need to read this paper before i respond properly but to define further of what I mean here is an example. We all are experts in language if you think about it (its just that its so common we forget its actually a pretty big feat what we can communicate). The idea is language representation has been earned and created over time and effort and its something that did not exist prior. Expert fire extinguishers have created representation for fire patterns that they can immediately figure out what needs to be done. In both of the cases a new representation was created with time and constant interaction with the environment rather than a shift in already present representations.
«It is, in the end, an empirical question whether this approach to math would solve the innumeracy issue. An empirical problem that cannot be resolved through IQ tests or parents teaching their own children, but only in classrooms with a diverse array of students.»
Not necessarily, I suspect.
One point is that (in my understanding) that parents (or relatives, or friends, or tutors, or the like) do overwhelmingly more mathematics teaching than classroom teachers, and that this has been true at least since the 1980s. (At first glance this might sound implausible but compare it to the situation outlined in Why Johnny Can’t Read—and schools place far more emphasis on reading education than math education!)
But a deeper point is that driving, walking, etc. are practical activities dedicated to the goals of our pre-intellectual nature. Even chess satisfies our (very strong!) primate-level desires for competition and social status. Mathematics does none of that: it satisfies our desire for beauty. How many people have much of a desire for beauty, e.g. in music?
And I’d say further (maybe I’m disagreeing with Bessis here) in saying that learning mathematics is more like learning a foreign language, or rather like learning a foreign Lebenswelt, the Lebenswelt der reinen Schönheit as we might call it, that can only be described in a foreign language! Not many people have a taste for that, just as not many people have a taste for learning languages or for excelling in the foreign world of music or of the visual arts. (And so you get people saying «I can’t draw», «I can’t sing», or «I can’t speak foreign languages» quite frequently—even though, clearly, almost any motivated person can do all three. And among the minority of people who can sing, how many of them take the trouble e.g. to sing four-part harmonies?)
Of course more people excel in music and drawing than in mathematics. But that has to do somewhat with social status and even more with the absence of people who want to tutor children in mathematics the way someone would tutor children in drawing or in playing the violin.
(By the way, if you’ve read this far, a word of advice: such tutoring is eudæmonic like almost no other activity (both for oneself and for one’s students) and it’s not even difficult so long as one knows one’s mathematics! (Although optimizing it, like optimizing anything else, is indeed difficult and time-consuming…))
Legend has it that the CIA used to teach its spies a foreign language in only three weeks by dropping them into an immersive environment where only that language was spoken. Hmm, if that approach worked then language schools that used it would be making a fortune. Apart from the fact that a commercial language school would find it very difficult to create that immersive environment, and most learners would cry that it's too inconvenient. Students might also complain if those who broke the rules were taken outside and shot.
My point is that from birth children are placed in just such an immersive environment where they are exposed to speech hundreds of times a day. I'm trying to imagine how a mathematical immersive environment could be created that would expose the infant child to the use of math more than once or twice a day, simple counting excluded.
The LDS missionary training center does language immersion. It probably takes about 2 weeks to get to the equivalent of what they call Spanish 2 in school. After 9 weeks it's maybe the equivalent of Spanish 4. Though it's not as immersive as that rumored CIA training, and there are obviously other objectives going on.
After 6 months in the country, most missionaries are very comfortable with the language. At about a year in, I felt fluent and it was easier for me to speak Spanish than english, though my vocab was limited and my accent was never perfect.
I'm not sure what the equivalent would be for mathematics. But I would be interested in seeing it
Let us consider the possibility that learning a first language and learning math are very, very different. One requires no stick and carrot incentives. The other appears to go better if the math student is taught not math, but the pleasure of finding things out. Math, reading and experiment/play are tools employed in the activity of finding things out.
On the one hand the psycholinguists tell us that children know the rules of grammar at a very early age, and then we see a first reader that on page one says see john see john run run john run. Obviously the average psycholinguist has never changed a diaper or worked with a real live child. Gonna need a bigger shovel. Kids' verbal skills generally develop with age, the online world permitting. They learn more skillful comprehension and formulation. But at no stage does the child need intensive training to understand the rules of grammar. That's a paradox. I wonder if the years of learning about noun phrases and verb phrases and adverbs and adjectives and prepositions and subjunctive clauses have any benefit commensurate with the effort of learning. Using language involves the rule book very little. Sorry Noam, it looks like your entire life's effort was futile.
Math works very, very differently. Math doesn't work at all without operators, and a huge library of functions that have to be learned. There is nothing, absolutely nothing, that an mathematician enjoys more than finding an error in another mathematician's work. This after fifteen, twenty, thirty years of highly intensive study and a deep fundamental understanding of the objects being manipulated. Knowing the rules may not make a mathematician any more creative and original. But the rules of math are strict.
«Math works very, very differently. Math doesn't work at all without operators, and a huge library of functions that have to be learned. There is nothing, absolutely nothing, that an mathematician enjoys more than finding an error in another mathematician's work. This after fifteen, twenty, thirty years of highly intensive study…»
Are you yourself a mathematician (i.e., a discoverer of new mathematics)? Because none of the mathematicians I know have said anything of the sort, and the mathematicians who write about mathematics (e.g. Hardy, Thurston, Tao, …) speak pretty vehemently to the contrary.
As for the supposéd difficulty and fruitlessness of grammar, I don't know what to tell you—I personally used the «rule book» constantly and don't know where my philological skills would be without it!
To your point, though, the rules of mathematical grammar are certainly stricter in some ways than the rules of linguistic grammar (though those also are usually quite strict—*vide* e.g. a Greek or Arabic, or for that matter a Spanish or French, conjugation table), and the kind of «deep fundamental understanding» that you note is required for mathematics is very different from that which is required for languages. And I agree with you that that might well prevent an «immersion» program for working for mathematics the way it works for languages.
If you visit David Bessis's substack you will read of the common practice, to which he too has been subjected, of subjecting the papers of others to rigorous scrutiny. Google AI Overview says yes, this does happen, but not maliciously. Ha ha. If so this must be the only academic discipline immune to it.
I'm pleased to see that there are some still trying to uphold language standards.
Lockhart's Lament comes to mind here as another good essay from this perspective.
Also, though, I'm reminded of HollyMathNerd's point that practice to the point of reflex, when structured correctly, helps make the connection to the underlying structure obvious.
Lockhart's Lament comes to mind here as another good essay from this perspective.
Also, though, I'm reminded of HollyMathNerd's point that practice to the point of reflex, when structured correctly, helps make the connection to the underlying structure obvious.
About a year ago I got the chance to ask Paul Zeitz (founder of Proof School and mathematician at USF) how he would teach math if he didn't care at all about state standards or the conventional curriculum. Specifically, in the case of my 10 year old who was completing elementary school math a year early. His advice was to teach Geometry early, and if I can put words in his mouth, I suspect it was because of the modal argument you (and Bessis) make.
The best resource I've yet found for doing this is an old textbook called Zome Geometry, paired with Zome Tools. It still requires some memorization and "computation" to figure out different ways of understanding. But it always starts with a physical model, which my kids and I manipulate, play with, and try to understand.
I actually think there's an upstream argument that Bessis makes and I think many educational researchers would disagree with. To put it in my own terms: expertise and mastery leave clues, and those clues can be transferred across domains. In other words, learning what expertise feels like in mathematics (for instance) can help someone learn how to develop expertise in another field should they choose. I don't really know what the limits of that are (is expertise in mathematics, a decidedly intellectual practice, transferrable as a concept to being an F1 driver or a pilot). But most educational researches believe problem solving does not exist independent of domain, and as such training the expertise intuition of mathematics would not be valuable for other pursuits.
I highly recommend Measurement by Paul Lockhart. It takes this idea of play and objects in the imagination to heart. It’s the best book I’ve ever read on learning math.
Jared, thanks for the nice review :) — yes, this really is about "what it's like to do math"
Two comments on your (legitimate) challenge that I should run an experiment:
- First, there's a matter of expertise. It took me 20 years to fully articulate weird feeling I always had that math had never been explained to me, and to find a way to explain it to a wide audience. I'm no longer an academic and the skillset of a pure-mathematician-turned-tech-founder-turned-writer is quite different from that of an experimental education reformer.
- Second (and this one applies to everyone, not just me): the issue is as much about ideology and beliefs as it is about teaching itself. Dualism, Platonism and Innatism are all obstacles to fixing mathematical education, leading students to enter the classroom with the wrong habits and the wrong expectations. In my experience, overcoming their inhibitions usually requires prolonged 1-1 interactive conversations.
A third aspect is this important nuance: while understanding math really is about turning "math problems [or rather math expressions] into mathematical objects", this understanding has to be recreated by the students themselves. There are instances where these objets can be described by waving hands, but this isn't the normal situation and most intuitions cannot be directly transferred. Mathematical formalism really is the ultimate mooring of mathematical intuition. Teaching isn't so much about transferring intuitions as it is about transferring (helped by the rare transferrable intuitions) the right thinking habits that will drive the continuing progress of one's intuition.
One last note about the Simone Weil ref: I didn't mention it for a very stupid reason, because I wasn't aware of it ;)
You make good points about the difficulty of an experiment. But until the experiment is done, my book recommendation will always come with a little asterisk that indicates some uncertainty about the utility of the ideas. I would love for you to team up with someone and develop out an actual curriculum that could be tested. Even if it were small scale and outside of the classroom, or as a supplement to the classroom.
I don’t know if Simone Weil is worth reading or not. But the basic argument is interesting and worth being aware of given your thesis. She extends it far beyond geometry and calculus, and even to concepts like human rights.
To end on a high note, I just want to iterate how much I loved this book. Most psychology books are rather trash, and so perhaps its a low bar, but I would put Mathematica in my top 10 psych books. Maybe top 5. Pretty good for a non-psychologist.
This is the best kind of book review: book as personal journey into the topic. Lots to think about here. You may appreciate Gemma Mason's recent piece over at Folded Papers, on the mysteries of teaching math (she also references Weil):
I do wonder if Gauss can truly be said to have applied "expertise" to solve that puzzle, in the NDM sense. He was just a little kid; it's not like he had built up refined mental models through accumulated experience. Maybe he just had a special aptitude for looking outside the box for these kind of math problems? That hardly must imply he was a genius or that it's all about IQ. Just that certain people may have a natural aptitude for discerning creative solutions for certain kinds of problems. That said, I like your 90%-30%-30% Twitter-average American-academic formula in footnote 4 - I think that could work as a general principle for most things in life!
I'm also wondering if there could be key differences between arithmetical thinking, statistical thinking, and the more abstract mathematical thinking required at higher levels. Bessis's book seems to be mostly about the latter, but in your examples you were collapsing all three. Most people's stereotypical image of math is arithmetical, but statistics seems quite different and then once you get to advanced math it feels like a whole other thing altogether. That's not to say intuition isn't still central to all of them, but it might work differently in each case.
What a great essay. I suddenly feel so inadequate.
I'm using expertise in a more developmental sense here. I don't merely mean someone with significant experience, but the natural development of human capacity. In that sense a "special aptitude" is just a type of expertise.
One of the people who shared this article (Sharon Chou) made a similar comment about there being subtypes of mathematicians. I guess that's also what I'm hinting at when I say that different forms of mathematics are like different types of games, and we probably all have preferences for what we enjoy.
The upshot is that Gauss probably did *something* smart as a child to his teacher, but all of the details of what problem it was, how he solved it, or why the teacher gave it are later inventions.
The fascinating part to me is the ending part (Doing It the Hard Way): if you actually try to add the numbers 1-100 like a machine, shortcuts will start screaming at you from the paper. Try it! I think a reasonably smart fifth-grader motivated to figure this out might figure it out after playing with the brute-force method for a while (this would be interesting to try with actual children, though). I take that as a reminder that when pondering an abstract question ("How could Gauss have solved this?") it's often useful to get on the ground and do something concrete.
Thanks for sharing this. I'm traveling this week for a conference, but will have to add an update about the anecdotal nature of this story.
I think I would have tried to add 1-100 and not noticed any patterns. I wouldn't have even known there were patterns to look for and so would have missed them.
I encourage you to actually take a pencil and paper and try it! It takes ten minutes and is illuminating. Try to add the numbers 1 to 100 -- start writing them in a column, run out of space, put them in groups of 10, keep summing the same stuff in the ones column... You will see some patterns, and you will probably not even have to write all the numbers down before you get to the result. This is *also* what it is like to do math -- sometimes it takes some grunt work before you find a way around it.
I enjoyed this book a lot, but I think it misses an obvious point which is that the ability to connect mental abstractions, symbols, and concrete examples is *what it means to be good at math*. I agree that just teaching algorithms is very misguided, but math is about abstraction, generalization, and formalism. It's good to start with the triangle pictures but that's like the kindergarten version, you then have to be able to generalize.
I'm working with my kids at middle school math. They definitely struggle more than I did and even though I work hard to give visual and practical examples and ground the symbols in those examples, they really struggle to care about the examples or connect it to the symbols. Maybe I'm just a bad teacher, but a lot of kids can just make these connections themselves without a teacher. I'd bet good money that the author of Mathematica was a kid like that. *Of course* you want to have concrete pictures in your head, but just telling people that doesn't mean it levels the playing field.
I hear you. Unless Bessis, or someone sympathetic to him, is able to come up with an actual curriculum, it's just words on a page. I don't think I have enough to develop that curriculum. And I would always be unsure whether my error is getting the student to care enough, or if the issue is something else.
Richard Feynman called it the pleasure of finding things out. Define that? You can't. It's ineffable, beyond the reach of words. You can only give examples. But once you've experienced it, it's a drug that keeps you coming back for more.
The most gifted teachers are able to provide that finding-out experience, and showing how the use of expertise can yield that pleasure. For the student it might only be a five minute epiphany that changes their life.
Chess ELO is a measure of spatial IQ. I utterly fail to see why this should correlate strongly with numerical IQ.
Children of all cultures learn the extraordinarily complicated rules of grammar in their first few years of life, irrespective of how difficult their home language is. Psychologists are probably afraid to investigate the similarities and differences between learning grammar and learning math because this would explode a lot of pet theories. They might actually learn something.
I refer you to your chart "Neural Efficiency in tournament chess players" which asks us to note the high IQ individuals with poor ELO. There are a HECK of a lot of outliers, and the standard deviation is somewhere out in Northern Siberia.
My argument is that my own measured spatial IQ differs by 20 points from my verbal IQ, which is 20 points lower than my numeric IQ. I regard any supposed correlation between one specific IQ and another as complete nonsense.
Interesting perspective on problem solving essentiating represntation change but I feel that that need not always be the case. I think the playing around leads to creation of the representation and intuition which the experts leverage to get to the answer (expertise i feel is more of developing representation where the problems are easy to solve rather than the act of shifting representation itself). This representation creation requires considerable time and effort. But this represntation can later on become a shackle as well since experts are not able to think beyond that which is why new insights/representation solve problems where expertise couldn't.
Looking forward to your next article ☺️
Could be that we are working with different definitions of expertise. There is an influential paper in my field with the title, "Adaptive Skill as the Conditio Sine Qua Non of Expertise" which would put adaptivity at the center of expertise. But the type of expertise we generally study is slightly different than, say, the ability to play the piano very well. Perhaps the latter doesn't require re-representation in the same way that a dynamic and changing housefire does.
Would need to read this paper before i respond properly but to define further of what I mean here is an example. We all are experts in language if you think about it (its just that its so common we forget its actually a pretty big feat what we can communicate). The idea is language representation has been earned and created over time and effort and its something that did not exist prior. Expert fire extinguishers have created representation for fire patterns that they can immediately figure out what needs to be done. In both of the cases a new representation was created with time and constant interaction with the environment rather than a shift in already present representations.
«It is, in the end, an empirical question whether this approach to math would solve the innumeracy issue. An empirical problem that cannot be resolved through IQ tests or parents teaching their own children, but only in classrooms with a diverse array of students.»
Not necessarily, I suspect.
One point is that (in my understanding) that parents (or relatives, or friends, or tutors, or the like) do overwhelmingly more mathematics teaching than classroom teachers, and that this has been true at least since the 1980s. (At first glance this might sound implausible but compare it to the situation outlined in Why Johnny Can’t Read—and schools place far more emphasis on reading education than math education!)
But a deeper point is that driving, walking, etc. are practical activities dedicated to the goals of our pre-intellectual nature. Even chess satisfies our (very strong!) primate-level desires for competition and social status. Mathematics does none of that: it satisfies our desire for beauty. How many people have much of a desire for beauty, e.g. in music?
And I’d say further (maybe I’m disagreeing with Bessis here) in saying that learning mathematics is more like learning a foreign language, or rather like learning a foreign Lebenswelt, the Lebenswelt der reinen Schönheit as we might call it, that can only be described in a foreign language! Not many people have a taste for that, just as not many people have a taste for learning languages or for excelling in the foreign world of music or of the visual arts. (And so you get people saying «I can’t draw», «I can’t sing», or «I can’t speak foreign languages» quite frequently—even though, clearly, almost any motivated person can do all three. And among the minority of people who can sing, how many of them take the trouble e.g. to sing four-part harmonies?)
Of course more people excel in music and drawing than in mathematics. But that has to do somewhat with social status and even more with the absence of people who want to tutor children in mathematics the way someone would tutor children in drawing or in playing the violin.
(By the way, if you’ve read this far, a word of advice: such tutoring is eudæmonic like almost no other activity (both for oneself and for one’s students) and it’s not even difficult so long as one knows one’s mathematics! (Although optimizing it, like optimizing anything else, is indeed difficult and time-consuming…))
Legend has it that the CIA used to teach its spies a foreign language in only three weeks by dropping them into an immersive environment where only that language was spoken. Hmm, if that approach worked then language schools that used it would be making a fortune. Apart from the fact that a commercial language school would find it very difficult to create that immersive environment, and most learners would cry that it's too inconvenient. Students might also complain if those who broke the rules were taken outside and shot.
My point is that from birth children are placed in just such an immersive environment where they are exposed to speech hundreds of times a day. I'm trying to imagine how a mathematical immersive environment could be created that would expose the infant child to the use of math more than once or twice a day, simple counting excluded.
The LDS missionary training center does language immersion. It probably takes about 2 weeks to get to the equivalent of what they call Spanish 2 in school. After 9 weeks it's maybe the equivalent of Spanish 4. Though it's not as immersive as that rumored CIA training, and there are obviously other objectives going on.
After 6 months in the country, most missionaries are very comfortable with the language. At about a year in, I felt fluent and it was easier for me to speak Spanish than english, though my vocab was limited and my accent was never perfect.
I'm not sure what the equivalent would be for mathematics. But I would be interested in seeing it
Let us consider the possibility that learning a first language and learning math are very, very different. One requires no stick and carrot incentives. The other appears to go better if the math student is taught not math, but the pleasure of finding things out. Math, reading and experiment/play are tools employed in the activity of finding things out.
On the one hand the psycholinguists tell us that children know the rules of grammar at a very early age, and then we see a first reader that on page one says see john see john run run john run. Obviously the average psycholinguist has never changed a diaper or worked with a real live child. Gonna need a bigger shovel. Kids' verbal skills generally develop with age, the online world permitting. They learn more skillful comprehension and formulation. But at no stage does the child need intensive training to understand the rules of grammar. That's a paradox. I wonder if the years of learning about noun phrases and verb phrases and adverbs and adjectives and prepositions and subjunctive clauses have any benefit commensurate with the effort of learning. Using language involves the rule book very little. Sorry Noam, it looks like your entire life's effort was futile.
Math works very, very differently. Math doesn't work at all without operators, and a huge library of functions that have to be learned. There is nothing, absolutely nothing, that an mathematician enjoys more than finding an error in another mathematician's work. This after fifteen, twenty, thirty years of highly intensive study and a deep fundamental understanding of the objects being manipulated. Knowing the rules may not make a mathematician any more creative and original. But the rules of math are strict.
«Math works very, very differently. Math doesn't work at all without operators, and a huge library of functions that have to be learned. There is nothing, absolutely nothing, that an mathematician enjoys more than finding an error in another mathematician's work. This after fifteen, twenty, thirty years of highly intensive study…»
Are you yourself a mathematician (i.e., a discoverer of new mathematics)? Because none of the mathematicians I know have said anything of the sort, and the mathematicians who write about mathematics (e.g. Hardy, Thurston, Tao, …) speak pretty vehemently to the contrary.
As for the supposéd difficulty and fruitlessness of grammar, I don't know what to tell you—I personally used the «rule book» constantly and don't know where my philological skills would be without it!
To your point, though, the rules of mathematical grammar are certainly stricter in some ways than the rules of linguistic grammar (though those also are usually quite strict—*vide* e.g. a Greek or Arabic, or for that matter a Spanish or French, conjugation table), and the kind of «deep fundamental understanding» that you note is required for mathematics is very different from that which is required for languages. And I agree with you that that might well prevent an «immersion» program for working for mathematics the way it works for languages.
If you visit David Bessis's substack you will read of the common practice, to which he too has been subjected, of subjecting the papers of others to rigorous scrutiny. Google AI Overview says yes, this does happen, but not maliciously. Ha ha. If so this must be the only academic discipline immune to it.
I'm pleased to see that there are some still trying to uphold language standards.
Lockhart's Lament comes to mind here as another good essay from this perspective.
Also, though, I'm reminded of HollyMathNerd's point that practice to the point of reflex, when structured correctly, helps make the connection to the underlying structure obvious.
Lockhart's Lament comes to mind here as another good essay from this perspective.
Also, though, I'm reminded of HollyMathNerd's point that practice to the point of reflex, when structured correctly, helps make the connection to the underlying structure obvious.
About a year ago I got the chance to ask Paul Zeitz (founder of Proof School and mathematician at USF) how he would teach math if he didn't care at all about state standards or the conventional curriculum. Specifically, in the case of my 10 year old who was completing elementary school math a year early. His advice was to teach Geometry early, and if I can put words in his mouth, I suspect it was because of the modal argument you (and Bessis) make.
The best resource I've yet found for doing this is an old textbook called Zome Geometry, paired with Zome Tools. It still requires some memorization and "computation" to figure out different ways of understanding. But it always starts with a physical model, which my kids and I manipulate, play with, and try to understand.
I actually think there's an upstream argument that Bessis makes and I think many educational researchers would disagree with. To put it in my own terms: expertise and mastery leave clues, and those clues can be transferred across domains. In other words, learning what expertise feels like in mathematics (for instance) can help someone learn how to develop expertise in another field should they choose. I don't really know what the limits of that are (is expertise in mathematics, a decidedly intellectual practice, transferrable as a concept to being an F1 driver or a pilot). But most educational researches believe problem solving does not exist independent of domain, and as such training the expertise intuition of mathematics would not be valuable for other pursuits.
I highly recommend Measurement by Paul Lockhart. It takes this idea of play and objects in the imagination to heart. It’s the best book I’ve ever read on learning math.
Jared, thanks for the nice review :) — yes, this really is about "what it's like to do math"
Two comments on your (legitimate) challenge that I should run an experiment:
- First, there's a matter of expertise. It took me 20 years to fully articulate weird feeling I always had that math had never been explained to me, and to find a way to explain it to a wide audience. I'm no longer an academic and the skillset of a pure-mathematician-turned-tech-founder-turned-writer is quite different from that of an experimental education reformer.
- Second (and this one applies to everyone, not just me): the issue is as much about ideology and beliefs as it is about teaching itself. Dualism, Platonism and Innatism are all obstacles to fixing mathematical education, leading students to enter the classroom with the wrong habits and the wrong expectations. In my experience, overcoming their inhibitions usually requires prolonged 1-1 interactive conversations.
A third aspect is this important nuance: while understanding math really is about turning "math problems [or rather math expressions] into mathematical objects", this understanding has to be recreated by the students themselves. There are instances where these objets can be described by waving hands, but this isn't the normal situation and most intuitions cannot be directly transferred. Mathematical formalism really is the ultimate mooring of mathematical intuition. Teaching isn't so much about transferring intuitions as it is about transferring (helped by the rare transferrable intuitions) the right thinking habits that will drive the continuing progress of one's intuition.
One last note about the Simone Weil ref: I didn't mention it for a very stupid reason, because I wasn't aware of it ;)
Thanks again for the great piece!
Thanks for the response.
You make good points about the difficulty of an experiment. But until the experiment is done, my book recommendation will always come with a little asterisk that indicates some uncertainty about the utility of the ideas. I would love for you to team up with someone and develop out an actual curriculum that could be tested. Even if it were small scale and outside of the classroom, or as a supplement to the classroom.
I don’t know if Simone Weil is worth reading or not. But the basic argument is interesting and worth being aware of given your thesis. She extends it far beyond geometry and calculus, and even to concepts like human rights.
To end on a high note, I just want to iterate how much I loved this book. Most psychology books are rather trash, and so perhaps its a low bar, but I would put Mathematica in my top 10 psych books. Maybe top 5. Pretty good for a non-psychologist.
I highly enjoyed this piece, as usual. I look forward to that article about system 3
This is the best kind of book review: book as personal journey into the topic. Lots to think about here. You may appreciate Gemma Mason's recent piece over at Folded Papers, on the mysteries of teaching math (she also references Weil):
https://foldedpapers.substack.com/p/a-little-pure-gold
I do wonder if Gauss can truly be said to have applied "expertise" to solve that puzzle, in the NDM sense. He was just a little kid; it's not like he had built up refined mental models through accumulated experience. Maybe he just had a special aptitude for looking outside the box for these kind of math problems? That hardly must imply he was a genius or that it's all about IQ. Just that certain people may have a natural aptitude for discerning creative solutions for certain kinds of problems. That said, I like your 90%-30%-30% Twitter-average American-academic formula in footnote 4 - I think that could work as a general principle for most things in life!
I'm also wondering if there could be key differences between arithmetical thinking, statistical thinking, and the more abstract mathematical thinking required at higher levels. Bessis's book seems to be mostly about the latter, but in your examples you were collapsing all three. Most people's stereotypical image of math is arithmetical, but statistics seems quite different and then once you get to advanced math it feels like a whole other thing altogether. That's not to say intuition isn't still central to all of them, but it might work differently in each case.
What a great essay. I suddenly feel so inadequate.
I'm using expertise in a more developmental sense here. I don't merely mean someone with significant experience, but the natural development of human capacity. In that sense a "special aptitude" is just a type of expertise.
One of the people who shared this article (Sharon Chou) made a similar comment about there being subtypes of mathematicians. I guess that's also what I'm hinting at when I say that different forms of mathematics are like different types of games, and we probably all have preferences for what we enjoy.
That Gauss story is most likely apocryphal. I recently found this article, which you might find fascinating, if a bit party-pooper-y: https://www.americanscientist.org/article/gausss-day-of-reckoning
The upshot is that Gauss probably did *something* smart as a child to his teacher, but all of the details of what problem it was, how he solved it, or why the teacher gave it are later inventions.
The fascinating part to me is the ending part (Doing It the Hard Way): if you actually try to add the numbers 1-100 like a machine, shortcuts will start screaming at you from the paper. Try it! I think a reasonably smart fifth-grader motivated to figure this out might figure it out after playing with the brute-force method for a while (this would be interesting to try with actual children, though). I take that as a reminder that when pondering an abstract question ("How could Gauss have solved this?") it's often useful to get on the ground and do something concrete.
Thanks for sharing this. I'm traveling this week for a conference, but will have to add an update about the anecdotal nature of this story.
I think I would have tried to add 1-100 and not noticed any patterns. I wouldn't have even known there were patterns to look for and so would have missed them.
I encourage you to actually take a pencil and paper and try it! It takes ten minutes and is illuminating. Try to add the numbers 1 to 100 -- start writing them in a column, run out of space, put them in groups of 10, keep summing the same stuff in the ones column... You will see some patterns, and you will probably not even have to write all the numbers down before you get to the result. This is *also* what it is like to do math -- sometimes it takes some grunt work before you find a way around it.
I enjoyed this book a lot, but I think it misses an obvious point which is that the ability to connect mental abstractions, symbols, and concrete examples is *what it means to be good at math*. I agree that just teaching algorithms is very misguided, but math is about abstraction, generalization, and formalism. It's good to start with the triangle pictures but that's like the kindergarten version, you then have to be able to generalize.
I'm working with my kids at middle school math. They definitely struggle more than I did and even though I work hard to give visual and practical examples and ground the symbols in those examples, they really struggle to care about the examples or connect it to the symbols. Maybe I'm just a bad teacher, but a lot of kids can just make these connections themselves without a teacher. I'd bet good money that the author of Mathematica was a kid like that. *Of course* you want to have concrete pictures in your head, but just telling people that doesn't mean it levels the playing field.
I hear you. Unless Bessis, or someone sympathetic to him, is able to come up with an actual curriculum, it's just words on a page. I don't think I have enough to develop that curriculum. And I would always be unsure whether my error is getting the student to care enough, or if the issue is something else.
Richard Feynman called it the pleasure of finding things out. Define that? You can't. It's ineffable, beyond the reach of words. You can only give examples. But once you've experienced it, it's a drug that keeps you coming back for more.
The most gifted teachers are able to provide that finding-out experience, and showing how the use of expertise can yield that pleasure. For the student it might only be a five minute epiphany that changes their life.
Chess ELO is a measure of spatial IQ. I utterly fail to see why this should correlate strongly with numerical IQ.
Children of all cultures learn the extraordinarily complicated rules of grammar in their first few years of life, irrespective of how difficult their home language is. Psychologists are probably afraid to investigate the similarities and differences between learning grammar and learning math because this would explode a lot of pet theories. They might actually learn something.
I think the correlation is .4, which is pretty meaningful. But I'm not sure what your argument is
I refer you to your chart "Neural Efficiency in tournament chess players" which asks us to note the high IQ individuals with poor ELO. There are a HECK of a lot of outliers, and the standard deviation is somewhere out in Northern Siberia.
My argument is that my own measured spatial IQ differs by 20 points from my verbal IQ, which is 20 points lower than my numeric IQ. I regard any supposed correlation between one specific IQ and another as complete nonsense.