Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers, Alexander Zvonkin (Moscow Center for Continuing Mathematical Education, 2007).
Well written . But your statement that the US can now afford to "waste a huge proportion of our talented population on humanities, arts, and other stuff that doesn’t involve you sitting in the school library until 3am" May indicate that you have a misunderstanding of the historical development of American scientific and engineering innovation. During America's most productive periods in science and technology, particularly in the early-to-mid 20th century, the educational and economic systems were far more decentralized and diverse compared to today.
Many of America's greatest scientific minds did not follow the rigid pathways we have now. For instance, figures like Thomas Edison or the Wright brothers were largely self taught, others like Richard Feynman or Nikola Tesla emerged from educational and research environments that were far more flexible and localized (Feynman once said he may not have even have made it in our new system, and he's far from the only on the old greats who said that). In those times, the system allowed for a much broader range of entry points into scientific and engineering fields, and many of our best old time innovators from its later stages who lived to see our new system said they might not have thrived in today's highly standardized, hyper-competitive academic environment.
Also, the U.S. was deeply committed to state-based education systems, which generated a diverse set of talents across different sectors. Today's more centralized, hierarchical system would probably exclude many of those earlier talents. The assumption that the U.S. is "wasting" talent by investing in the humanities and arts also ignores the role these fields have historically played in generating creativity and interdisciplinary thinking, although we may very well be doing so in the sense that the humanities themselves have become centralized, stale, and at best not performing their mission and at worst counterproductive to it.
Babies put inedible things in their mouths to tell what shape it really is. You could go pretty far with solid geometry with the right set of bead shapes.
Having taught electronics to 10 year olds, the best aid to practical trig / phasors / complex numbers is a coathanger wire helix of about two turns with one or two balls of wadded aluminum foil to slide down the wire. Side view is sine, top vew is cosine (or vice-versa), end / circle view is the Argand diagram, the moving ball traces out the sinusoid from the side and makes a changing angle with the fixed ball from the end view. Kirchoff's theorem that the voltage rises and drops around any closed loop must sum to zero really is just the same as saying that traversing any closed path one must end up where one started, or that any point on a landscape must have only one altitude, (It's close to the essence of gauge theories in general, too). I used plumbing analogies a lot, even for transistors, which the boys understood, but may be too complicated for mathematicians.
Cutting out shapes from paper and weighing them was the easiest way to do integrals. Line inegrals could be done with wire.
Anything is simple so long as you don't let mathematicians get ahold of it. You can learn everything you need to know about the Cliffird algebra of 3D Euclidean space in a couple of intuitive pages, as in most of the physics papers on Geometric Algebra, maybe 3 or 4 more pages for 4D Minkowski space, 5 or 10 more for 5D conformal represenatons of 3D, and not much more than double that to know all there is to know about arbirary signatures up to 8D, a couple pages more and all finite dimensional algebras. Go oveer it a few times and it will be second nature and Bott periodicity will seem as natural as the roots of unity. But read one Bourbaki-infected Wikipedia page of obscurantist pseudo-rigor on the topic and you may be brain-damaged for life.
I will read your piece on the "Needham question" with interest. Here is something that many miss. I took Mandarin for a year in college and found it daunting. Speaking and reading it was difficult enough, but even basic arithmetic was impossible for me. Almost as bad as Roman numerals I think that the adoption of Arabic (Indian?) numerals was a necessary condition for the West's scientific success. But I am no mathematician. Do others agree?
One thing that's often under-appreciated is how *hard* it is to do non-basic counting in other languages. It's like the first thing you learn in a new language (my six-year-old can count to twenty in Spanish and just one other word), but after years of learning Bulgarian and being able to follow and hold simple conversations, my mind shuts down on hearing the number "187" spoken in Bulgarian.
I too have noticed horseshoe theory applies to semitic tropes. So glad to see someone else say it aloud.
That aside, this is a fascinating read. I have a son who is good at math and somewhat bored in school, but I have hesitated to sign him up for "math enrichment" as it seems to be just more worksheets and pushing him further ahead of his peers, causing more boredom in school. A math circle would be a great alternative. I am, alas, not the source of the math genes in the family, but I can probably stay ahead of a group of 8yos....
the nice thing about circles, as described here, is that you don’t have to stay ahead! if you get confused and then puzzle out things in front of the group, thats expected and (sort of) the point
Zvonkin’s dedication is inspiring and also testifies to how well he understands what Math is. It could be argued that the whole business of Mathematics is basically defining two things differently and finally showing that they are, in fact, equal (through the Platonic tunnel that you mentioned).
What is amazing here is that some of the kids who went nuts (at the association of different puzzles being the same) will remember that feeling and that whole process hopefully, instead of the actual problem itself. That process is what joyful discovery in Math feels like. It’s like your most outlandish conspiracy theory coming true when seemingly unrelated topics are actually linked. If they have a nice memory of that experience, it would be interesting to see if some of them actually pursued Math.
Such a famous way to take credit away from the Chinese and the Indians nowadays- in some circles.
Sure, China and India have a huge population, but so does rest of Asia- Pakistan, Bangladesh, Indonesia. "Middle East", "Africa", or "Latin America" as regions have very high populations, too. You don't hear about famous STEM talent coming out of them often!
Great article! This point can't be overstated: "...young children are much more concrete thinkers than adults, whether that’s for neurological reasons or because they haven’t built the cognitive tools for abstraction yet." Yes to both. Young children progress from concrete to conceptual learning of mathematics, and the concrete stage is pretty important. They need to "see" and "hold math in their hands" (i.e., using manipulatives) before they can see it in their minds. This is actually now pretty well-established among math educators and curricula developers -- but then you get the occasional parent or teacher who asks a 3-year-old to do math in their heads.
I've been running a math circle for my daughters' homeschool co-op with mixed success. At first I tried to work through problems from this book: https://bookstore.ams.org/mcl-21/ but it didn't work with the group of kids at all. I've been having a lot more success with problems from this one: https://naturalmath.com/brightbraveopenminds/
The nice thing is that MSRI/AMS have translated and published a wealth of resources about math circles, so the approach is slowly making its way to America.
Not formally for me, but at my prep school for gifted children I had mathy friends and we did similar things. At 13, we solved problems in Complex Analysis, for example. All girls, by the way.
Well written . But your statement that the US can now afford to "waste a huge proportion of our talented population on humanities, arts, and other stuff that doesn’t involve you sitting in the school library until 3am" May indicate that you have a misunderstanding of the historical development of American scientific and engineering innovation. During America's most productive periods in science and technology, particularly in the early-to-mid 20th century, the educational and economic systems were far more decentralized and diverse compared to today.
Many of America's greatest scientific minds did not follow the rigid pathways we have now. For instance, figures like Thomas Edison or the Wright brothers were largely self taught, others like Richard Feynman or Nikola Tesla emerged from educational and research environments that were far more flexible and localized (Feynman once said he may not have even have made it in our new system, and he's far from the only on the old greats who said that). In those times, the system allowed for a much broader range of entry points into scientific and engineering fields, and many of our best old time innovators from its later stages who lived to see our new system said they might not have thrived in today's highly standardized, hyper-competitive academic environment.
Also, the U.S. was deeply committed to state-based education systems, which generated a diverse set of talents across different sectors. Today's more centralized, hierarchical system would probably exclude many of those earlier talents. The assumption that the U.S. is "wasting" talent by investing in the humanities and arts also ignores the role these fields have historically played in generating creativity and interdisciplinary thinking, although we may very well be doing so in the sense that the humanities themselves have become centralized, stale, and at best not performing their mission and at worst counterproductive to it.
I had the same question! The answer turned out to be that people tend to choose mates who are similar to them along multiple dimensions, so it's not just tall people marrying tall people, it's tall people with long arms and close-set eyes people marrying other tall people with long arms and close-set eyes. And because those are polygenic traits, you'd expect all their children to all be somewhat taller, longer-armed, and close-set-eyed than average, but probably not all *as* unusually tall, long-armed, and close-set-eyed as their parents. With enough kids, though, you have a pretty good chance of one very tall one, one very long-armed one, one with eyes very close together, etc.
Well written . But your statement that the US can now afford to "waste a huge proportion of our talented population on humanities, arts, and other stuff that doesn’t involve you sitting in the school library until 3am" May indicate that you have a misunderstanding of the historical development of American scientific and engineering innovation. During America's most productive periods in science and technology, particularly in the early-to-mid 20th century, the educational and economic systems were far more decentralized and diverse compared to today.
Many of America's greatest scientific minds did not follow the rigid pathways we have now. For instance, figures like Thomas Edison or the Wright brothers were largely self taught, others like Richard Feynman or Nikola Tesla emerged from educational and research environments that were far more flexible and localized (Feynman once said he may not have even have made it in our new system, and he's far from the only on the old greats who said that). In those times, the system allowed for a much broader range of entry points into scientific and engineering fields, and many of our best old time innovators from its later stages who lived to see our new system said they might not have thrived in today's highly standardized, hyper-competitive academic environment.
Also, the U.S. was deeply committed to state-based education systems, which generated a diverse set of talents across different sectors. Today's more centralized, hierarchical system would probably exclude many of those earlier talents. The assumption that the U.S. is "wasting" talent by investing in the humanities and arts also ignores the role these fields have historically played in generating creativity and interdisciplinary thinking, although we may very well be doing so in the sense that the humanities themselves have become centralized, stale, and at best not performing their mission and at worst counterproductive to it.
Misc. thoughts:
Babies put inedible things in their mouths to tell what shape it really is. You could go pretty far with solid geometry with the right set of bead shapes.
Having taught electronics to 10 year olds, the best aid to practical trig / phasors / complex numbers is a coathanger wire helix of about two turns with one or two balls of wadded aluminum foil to slide down the wire. Side view is sine, top vew is cosine (or vice-versa), end / circle view is the Argand diagram, the moving ball traces out the sinusoid from the side and makes a changing angle with the fixed ball from the end view. Kirchoff's theorem that the voltage rises and drops around any closed loop must sum to zero really is just the same as saying that traversing any closed path one must end up where one started, or that any point on a landscape must have only one altitude, (It's close to the essence of gauge theories in general, too). I used plumbing analogies a lot, even for transistors, which the boys understood, but may be too complicated for mathematicians.
Cutting out shapes from paper and weighing them was the easiest way to do integrals. Line inegrals could be done with wire.
Anything is simple so long as you don't let mathematicians get ahold of it. You can learn everything you need to know about the Cliffird algebra of 3D Euclidean space in a couple of intuitive pages, as in most of the physics papers on Geometric Algebra, maybe 3 or 4 more pages for 4D Minkowski space, 5 or 10 more for 5D conformal represenatons of 3D, and not much more than double that to know all there is to know about arbirary signatures up to 8D, a couple pages more and all finite dimensional algebras. Go oveer it a few times and it will be second nature and Bott periodicity will seem as natural as the roots of unity. But read one Bourbaki-infected Wikipedia page of obscurantist pseudo-rigor on the topic and you may be brain-damaged for life.
I like your post.
Mathematicians can use the h-parameter model of a transistor, i.e. the linear part of the Taylor expansion in two variables.
I will read your piece on the "Needham question" with interest. Here is something that many miss. I took Mandarin for a year in college and found it daunting. Speaking and reading it was difficult enough, but even basic arithmetic was impossible for me. Almost as bad as Roman numerals I think that the adoption of Arabic (Indian?) numerals was a necessary condition for the West's scientific success. But I am no mathematician. Do others agree?
One thing that's often under-appreciated is how *hard* it is to do non-basic counting in other languages. It's like the first thing you learn in a new language (my six-year-old can count to twenty in Spanish and just one other word), but after years of learning Bulgarian and being able to follow and hold simple conversations, my mind shuts down on hearing the number "187" spoken in Bulgarian.
It's well understood that good notations have an oversized impact on the difficulty of the mathematics we can do.
I too have noticed horseshoe theory applies to semitic tropes. So glad to see someone else say it aloud.
That aside, this is a fascinating read. I have a son who is good at math and somewhat bored in school, but I have hesitated to sign him up for "math enrichment" as it seems to be just more worksheets and pushing him further ahead of his peers, causing more boredom in school. A math circle would be a great alternative. I am, alas, not the source of the math genes in the family, but I can probably stay ahead of a group of 8yos....
Google "The Art of Problem Solving" for your son.
the nice thing about circles, as described here, is that you don’t have to stay ahead! if you get confused and then puzzle out things in front of the group, thats expected and (sort of) the point
You might also be interested in Leitner’s forgetting curve. And if you’re interested in that, you might also be interested in the book Make It Stick.
Zvonkin’s dedication is inspiring and also testifies to how well he understands what Math is. It could be argued that the whole business of Mathematics is basically defining two things differently and finally showing that they are, in fact, equal (through the Platonic tunnel that you mentioned).
What is amazing here is that some of the kids who went nuts (at the association of different puzzles being the same) will remember that feeling and that whole process hopefully, instead of the actual problem itself. That process is what joyful discovery in Math feels like. It’s like your most outlandish conspiracy theory coming true when seemingly unrelated topics are actually linked. If they have a nice memory of that experience, it would be interesting to see if some of them actually pursued Math.
\\
losing is how you know you’ve picked an opponent worthy of a man.
\\
Some of us research mathematicians are female.
> But China and India have large populations
Such a famous way to take credit away from the Chinese and the Indians nowadays- in some circles.
Sure, China and India have a huge population, but so does rest of Asia- Pakistan, Bangladesh, Indonesia. "Middle East", "Africa", or "Latin America" as regions have very high populations, too. You don't hear about famous STEM talent coming out of them often!
Great article! This point can't be overstated: "...young children are much more concrete thinkers than adults, whether that’s for neurological reasons or because they haven’t built the cognitive tools for abstraction yet." Yes to both. Young children progress from concrete to conceptual learning of mathematics, and the concrete stage is pretty important. They need to "see" and "hold math in their hands" (i.e., using manipulatives) before they can see it in their minds. This is actually now pretty well-established among math educators and curricula developers -- but then you get the occasional parent or teacher who asks a 3-year-old to do math in their heads.
Beautiful. Beautiful writing.
I've been running a math circle for my daughters' homeschool co-op with mixed success. At first I tried to work through problems from this book: https://bookstore.ams.org/mcl-21/ but it didn't work with the group of kids at all. I've been having a lot more success with problems from this one: https://naturalmath.com/brightbraveopenminds/
The nice thing is that MSRI/AMS have translated and published a wealth of resources about math circles, so the approach is slowly making its way to America.
Well, you do now.
On the other hand, his FRHS calculus teacher showed him the calculus of variations, which changed his life.
As to math circles:
Not formally for me, but at my prep school for gifted children I had mathy friends and we did similar things. At 13, we solved problems in Complex Analysis, for example. All girls, by the way.
Well written . But your statement that the US can now afford to "waste a huge proportion of our talented population on humanities, arts, and other stuff that doesn’t involve you sitting in the school library until 3am" May indicate that you have a misunderstanding of the historical development of American scientific and engineering innovation. During America's most productive periods in science and technology, particularly in the early-to-mid 20th century, the educational and economic systems were far more decentralized and diverse compared to today.
Many of America's greatest scientific minds did not follow the rigid pathways we have now. For instance, figures like Thomas Edison or the Wright brothers were largely self taught, others like Richard Feynman or Nikola Tesla emerged from educational and research environments that were far more flexible and localized (Feynman once said he may not have even have made it in our new system, and he's far from the only on the old greats who said that). In those times, the system allowed for a much broader range of entry points into scientific and engineering fields, and many of our best old time innovators from its later stages who lived to see our new system said they might not have thrived in today's highly standardized, hyper-competitive academic environment.
Also, the U.S. was deeply committed to state-based education systems, which generated a diverse set of talents across different sectors. Today's more centralized, hierarchical system would probably exclude many of those earlier talents. The assumption that the U.S. is "wasting" talent by investing in the humanities and arts also ignores the role these fields have historically played in generating creativity and interdisciplinary thinking, although we may very well be doing so in the sense that the humanities themselves have become centralized, stale, and at best not performing their mission and at worst counterproductive to it.
Not to mention that a good humanities education teaches you to write well.
So many papers in math and physics are poorly written.
Also, most of the students working in the university library at 3AM are in fact humanities majors.
My thesis advisor--a mathematical physicist of some renown--had an undergraduate degree in Theater and a math PHD from NYU.
Feynman attended Far Rockaway High School--a public school that produced three science Nobel Laureates, the he attended MIT and Princeton.
And Feynman, among others, said that those places had become very different than they were when he had attended them
In fact, I heard him give a lecture on how much he hated the rigidity and competition at Far Rockaway High School when he attended it.
That school has been closed for decades.
I was referring to the colleges he attended, I dont what his opinion of that HS was
On the other hand, his FRHS teacher showed him the calculus of variations
which was the key to his adult research.
He was otherwise bored stiff.
Why should assortative mating increase the variance among *your* offspring? Of course it will on a population level
I had the same question! The answer turned out to be that people tend to choose mates who are similar to them along multiple dimensions, so it's not just tall people marrying tall people, it's tall people with long arms and close-set eyes people marrying other tall people with long arms and close-set eyes. And because those are polygenic traits, you'd expect all their children to all be somewhat taller, longer-armed, and close-set-eyed than average, but probably not all *as* unusually tall, long-armed, and close-set-eyed as their parents. With enough kids, though, you have a pretty good chance of one very tall one, one very long-armed one, one with eyes very close together, etc.
Ahhh thank you!
Reading back through a bunch of your math posts now. Perhaps this will be the day I finally work out what a "group" is...