Wow, all I can think is how exceptionally lucky these kids are to be taught these concepts/exposed to these problems at such a young age by a willing and enthusiastic teacher. Most people professionally employed as K-12 math teachers wouldn’t even be able to teach this kind of curriculum well. I looked at the author’s blog history to find out if they were doing this for some huge amount of money and found a blog that suggests not only were they doing it for free (skimmed it, but didn’t see any reference to pay) but they had to try to convince other parents to let their kids participate! https://buttondown.email/j2kun/archive/a-foray-into-math-cir...
In my opinion, there is a relatively unknown (to those outside mathematics) huge “privilege” gap in mathematics education that makes it so those that only follow a cookie standard or accelerated curriculum are relatively unprepared for careers in mathematics compared to those tutored (or taught in special magnet programs, or by their mathematician parents) in these kinds of non-standard-curriculum concepts from a young age. Mostly, the problem is that the standard curriculum is almost purely rote-computational until you become a college ~Sophomore and it abruptly changes to being open ended and proof-based (which is the world most pro mathematicians live in) requiring skills in creatively applying logic. So students with this kind of exposure from a young age have a much easier transition to that while also scooping up all the math-career-builders like early papers and contest wins on the way.
Those other parents probably don’t know this but OP is providing an immensely valuable service that is hard to find in some areas and which some parents would pay a huge amount of money for.
+1, I didn't think of my upbringing as "underprivileged" in any way until I got to college and later did a math-adjacent PhD and was increasingly surrounded by people who had been doing math circle-type enrichment throughout their childhoods. I was OK at math but not especially precocious. I represented my middle school at a couple of local math competitions and didn't do very well, but looking back, it's kind of weird that I didn't have any help preparing at all.
Thinking about this more in the last year or two has led me to shift a lot of my charitable giving to math circle-type programs, even though I know they're less verifiable than a lot of the (pre-longtermist) effective altruism causes -- I think that kind of mathematical thinking is a very valuable tool that is not so easy to come by without these kinds of programs.
> those that only follow a cookie standard or accelerated curriculum are relatively unprepared for careers in mathematics
Culture-dependent? I recall the story from France of the second-grader who, asked what 2x3 equals, replied "3x2", knowing only that multiplication was commutative.
> the story from France of the second-grader who, asked what 2x3 equals, replied "3x2", knowing only that multiplication was commutative.
This is a classic joke making fun of the issues with French education based on the Bourbaki [1] school of mathematics, see [2] for more discussion. Different issues than the USA, but also bad in my opinion.
For those who are interested in getting involved with online math circles (as parents or potential instructors), check out https://theglobalmathcircle.org (Jeremy, the author of this post graduated from our training program)
> The Function Machine game (guess a function given the ability to query it as a black-box)
My kid has loved this one since I read about it on HN when she was 6. As she learned more advanced mathematical operations, we added them to the toolkit. It's great! I can tell she's mastered a concept when we can swap roles and she can accurately answer my queries.
Different age group, but I had success in engaging with high-school students with "Bitcoin mining". This was a ~15 minute exercise after a lecture on blockchains, during a cryptography seminar.
The original Bitcoin proof-of-work algorithm is to tweak the middle input of a hash so that the result starts with many binary zeroes (find x such that `sha256(sha256(a || x || b)) < H`). We simplified it down to `x^2 % N < 10^H` (calculators and computers allowed). You can freely tweak N and H.
The students had a blast, and I believe it was a lucky combination:
- It's more topical than ancient puzzles.
- The students were racing against each other.
- Rewards were semi-random (faster/smarter groups still had an advantage).
- The rewards were "physical bitcoins" (chocolate coins).
- Winning was more or less guaranteed by brute force, but there were plenty of shortcuts to find.
I'm disappointed to hear that ruler and compass constructions didn't engage them. I'm currently working on digitising a Hebrew translation of Euclid's Elements, and I was hoping that the very physical aspect (almost like art class) would be more engaging than theoretical stuff, and the fact that it's independent of any numbers would make it less scary than usual maths class. I still think there's educational value in Euclid's Elements beyond the historical importance, but it seems that I might be wrong on it being more engaging in an educational context for most kids.
> Trying to figure out who is better at penalty kicks based on counts of scores/misses.
If you have kids who care about a particular sport, this is a great way to teach linear algebra. There's a book 'Who's #1? - The Science of Rating and Ranking' that goes through different methods of ratings/rankings in great detail (it was one of the required readings for my MSc in Data Science).
In case this is of interest, and unknown, there's an aesthetically pleasant digitization of Euclid's elements[0]. But I'm not sure whether it's best for studying: usually, actively engaging with the material is key (e.g. performing the constructions, without a model), thought the effective rules for kids might be different.
I'm actually happy to see them interested in propositional logic, given how foundational it can be to coherent thinking. I would have guessed, as the author, that manual activities would have been preferred.
Well from the way the linked article describes it, there is no 'story' to maths like Euclidian geometry.
After all, what were the original motivations for compass and straightedge constructions? Ideas about perfection, symmetry, and constructability are all very abstract.
I found https://www.euclidea.xyz/ to be fun, but then I find drawing Girih patterns fun, and I'm also not 8 years old :)
I only gained an appreciation for ruler and compass constructions early in secondary school, at age 7 I'm not sure I could fully understand the beauty of deriving a huge system from axioms.
For making Euclid interesting to children, I remember really enjoying a game called Euclidea: https://www.euclidea.xyz/
I remember learning about hyperbolic and spherical geometry in middle school, and that was cool. Not really because of the axiomatic aspect, but more of the "how many lines can you draw through two points? Up to you!" sort of questioning of mathematical assumptions and the funny diagrams. The Fano projective plane model was interesting, the Poincaré disk model was interesting. I remember some animations and some interactive software you could play with. But yeah, after about a week I got bored of it. I would say one 2-hour session devoted to different geometries and constructions would probably be about right.
The Euclid's Elements approach of axiomatic geometry is interesting, and suitable for maybe a high school course. Before students learn algebra they don't really have an appreciation of deriving equations or proofs from a small starting point. And coordinate geometry is much more practical (some things are simply unconstructible with ruler and compass).
I don't know the setup used for the ruler and compass constructions, but there's a game called Euclidea [1] which may be of interest. I found the level progression, scoring system, and solving proofs to unlock new "tools" was done very well.
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In my opinion, there is a relatively unknown (to those outside mathematics) huge “privilege” gap in mathematics education that makes it so those that only follow a cookie standard or accelerated curriculum are relatively unprepared for careers in mathematics compared to those tutored (or taught in special magnet programs, or by their mathematician parents) in these kinds of non-standard-curriculum concepts from a young age. Mostly, the problem is that the standard curriculum is almost purely rote-computational until you become a college ~Sophomore and it abruptly changes to being open ended and proof-based (which is the world most pro mathematicians live in) requiring skills in creatively applying logic. So students with this kind of exposure from a young age have a much easier transition to that while also scooping up all the math-career-builders like early papers and contest wins on the way.
Those other parents probably don’t know this but OP is providing an immensely valuable service that is hard to find in some areas and which some parents would pay a huge amount of money for.
Thinking about this more in the last year or two has led me to shift a lot of my charitable giving to math circle-type programs, even though I know they're less verifiable than a lot of the (pre-longtermist) effective altruism causes -- I think that kind of mathematical thinking is a very valuable tool that is not so easy to come by without these kinds of programs.
Culture-dependent? I recall the story from France of the second-grader who, asked what 2x3 equals, replied "3x2", knowing only that multiplication was commutative.
This is a classic joke making fun of the issues with French education based on the Bourbaki [1] school of mathematics, see [2] for more discussion. Different issues than the USA, but also bad in my opinion.
[1] https://en.m.wikipedia.org/wiki/Nicolas_Bourbaki
[2] https://web.archive.org/web/20230315185224/https://www.uni-m...
My kid has loved this one since I read about it on HN when she was 6. As she learned more advanced mathematical operations, we added them to the toolkit. It's great! I can tell she's mastered a concept when we can swap roles and she can accurately answer my queries.
- (San Francisco) afterschool math team for 7-11yo kids: https://www.meetup.com/chess-games-inner-fire/events/2956372...
- (San Francisco) afterschool math circles at a couple of SFUSD schools: https://www.sfmathcircle.org/jose-ortega https://www.sfmathcircle.org/starr-king
- (Remote) 501(c)(3) nonprofit running weekly online competition math sessions for elementary and middle school students: https://mrmathonline.com/
If you're looking for something in between classes and circles, then perhaps:
- Engaging Math Circles (https://emc.school)
- Russian School of Math (RSM)
I suspect you can find an online pdf if you search for the title.
https://bookstore.ams.org/mcl-5
https://www.penguinrandomhouse.com/books/704500/mammoth-math...
The original Bitcoin proof-of-work algorithm is to tweak the middle input of a hash so that the result starts with many binary zeroes (find x such that `sha256(sha256(a || x || b)) < H`). We simplified it down to `x^2 % N < 10^H` (calculators and computers allowed). You can freely tweak N and H.
The students had a blast, and I believe it was a lucky combination:
- It's more topical than ancient puzzles.
- The students were racing against each other.
- Rewards were semi-random (faster/smarter groups still had an advantage).
- The rewards were "physical bitcoins" (chocolate coins).
- Winning was more or less guaranteed by brute force, but there were plenty of shortcuts to find.
Deleted Comment
> Trying to figure out who is better at penalty kicks based on counts of scores/misses.
If you have kids who care about a particular sport, this is a great way to teach linear algebra. There's a book 'Who's #1? - The Science of Rating and Ranking' that goes through different methods of ratings/rankings in great detail (it was one of the required readings for my MSc in Data Science).
I'm actually happy to see them interested in propositional logic, given how foundational it can be to coherent thinking. I would have guessed, as the author, that manual activities would have been preferred.
[0]: https://www.c82.net/euclid/#books
After all, what were the original motivations for compass and straightedge constructions? Ideas about perfection, symmetry, and constructability are all very abstract.
I found https://www.euclidea.xyz/ to be fun, but then I find drawing Girih patterns fun, and I'm also not 8 years old :)
For making Euclid interesting to children, I remember really enjoying a game called Euclidea: https://www.euclidea.xyz/
The Euclid's Elements approach of axiomatic geometry is interesting, and suitable for maybe a high school course. Before students learn algebra they don't really have an appreciation of deriving equations or proofs from a small starting point. And coordinate geometry is much more practical (some things are simply unconstructible with ruler and compass).
[1] https://play.google.com/store/apps/details?id=com.hil_hk.euc...
Deleted Comment