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.
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.
I want to add to this list for something that I teach to kids of similar ages: counting to a thousand on your hands. This is natural for any computer programmer, but there's some key aspects to it that help with math education.
This is quite fun for them as your question of "Do you know how to count to a thousand on your hands?" appears like magic or a superpower to them. So I demonstrate the beginning counting to 20 or so (quickly moving through 4 and 6). Then I start to ask them to predict certain unseen configurations (i.e. zero shot generalization). Re-demonstrate when failure to predict. Once the pattern is successfully learned, then I present a quiz/puzzle, and ask how many fingers "this many" is (all fingers unfurled). Always stumped, I provide the hint "if I had an additional finger and that finger were open and all others were closed, how many would that be? Can you figure out the other number from here?" It takes time, but they almost always get it.
The beauty of this is that we have a low barrier to entry, as the kid just needs to know how to multiply by 2 and know the names for numbers up to 1024. It surprisingly has many avenues of thinking that can help a kid better generalize concepts of math while still being entertaining (similar to concepts in this article). First, we teach the kids that there are multiple representations of things, and that we need to formulate things to match our goals. That we can break away from the common and expected thinking that most people have to gain "super powers" (i.e. not count like most people). Another important aspect is the above puzzle, where we specifically teach them that there are often better ways to go about solving a problem if we can find patterns. Rather than brute-forcing your way through this (summing each finger) you can exploit the iteration pattern to know that hinted at position is only one away from the desired. Frame of reference is such a crucial concept to mathematics and is at the root of solutions to many famous problems. Obvious post hoc, but inconceivable a priori.
We can even go quite deep and talk about proofs and how to design algorithms! I'll explain the algorithm identical to how we would perform a proof by induction (this is not how I teach kids, at the beginning):
k0th step: starting palms facing user, and an initialized position where all fingers are closed (thumb is a finger and at left most and right most positions). Starting from the right, increment the right most finger (thumb)
knth step: start from the right most position. If finger is closed, then unfurl. If finger is unfurled, close it and attempt to increment the next right most finger recursively following this condition.
There's more that you can build off of this one concept and similarly that with the topics in the article. What I've found is that which ever "game" the kid likes best is the one you should focus on and formulate your basis around. When they have difficulties with one game you use a different game that they are successful at to teach the difficult one. After all, math is a language and so many things can simply be rephrased.
I find that one of the difficulties many have with math is that the internalize it as quite strict. That it is often taught "this is the way," with no other methods accepted and thus people gain quite low generalizability of the concepts. Something that "word problems" are intended to resolve, but this approach is quite brute forced and more akin to how one might teach a machine rather than a human. This is coupled with the fact that so many are at a young age taught by people void of passion for the subject. This dispassion only passes from teacher to student (I'm sure many people can remember the breath of fresh air if they were lucky enough to find a teacher who loved math and encouraged the creative side of it. Honestly, that's how I came the love the subject and prior to that Junior in High School class, I hated the subject despite being good at it and in advanced classes).
Huh, my daughter and I taught her first grade class SET last year (I did most of the pedagogy…). They seemed both able to learn it and engaged. It wasn’t in a math circle setting, but a lesson to the whole class. Would be cool to investigate the differences!
Yeah Set has been dear to me since I was about 5 years old, and I remember bringing it into class for "favorite board game day" in 1st grade (so when I was 6). Really surprised the kids didn't like it!
There's a LOT of math you can do with it too, starting with some modular arithmetic. and I guess just the idea of abstracting the attributes to 0, 1, and 2. Then you can do a little bit of group theory and how the game is Z3^4 (iirc this is the right notation? been a while), 4 copies of Z3. And there are similar card games that represent other games that you can print out cards for if they're getting excited about group theory, so you can talk about axioms of groups, and how the games represent the groups, and why it makes sense.
Also you can introduce a bit of programming too and explain like how would you teach a computer how to play this game using the invariant property that a definition of a set is that sum c_i = 0 for all i 0 -> 3. and you can also easily generalize the game to other lengths by adding additional traits (background color of the card to add one) or fewer (remove bg color of the symbol).
We played Set with my son when he was ~6.5yo, and he immediately loved it. After a couple of games, he was spotting sets almost as quickly as we were. But his performance and interest varied based on distraction, tiredness etc.
If you want a one-player Set game, there's a nice open source one here:
The 'AI' looks for solutions randomly. Every 2 seconds (in easy mode) it picks two of the visible cards at random, and sees if the card needed to make a set is visible.
If your kid is totally new to Set, you probably want to adjust the delay to 5 seconds or something.
I think the specific topics and activities are hit and miss based on the specific skill level, interests, and daily mood of the group. Presentation style is a common theme though.
[+] [-] opportune|2 years ago|reply
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.
[+] [-] mycologos|2 years ago|reply
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.
[+] [-] pro-kythera|2 years ago|reply
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.
[+] [-] avital|2 years ago|reply
[+] [-] gnicholas|2 years ago|reply
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.
[+] [-] rahimnathwani|2 years ago|reply
- (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)
[+] [-] dan-robertson|2 years ago|reply
[+] [-] msla|2 years ago|reply
[+] [-] rahimnathwani|2 years ago|reply
[+] [-] dools|2 years ago|reply
[+] [-] amai|2 years ago|reply
https://www.penguinrandomhouse.com/books/704500/mammoth-math...
[+] [-] BoppreH|2 years ago|reply
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.
[+] [-] unknown|2 years ago|reply
[deleted]
[+] [-] verbify|2 years ago|reply
> 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).
[+] [-] mbivert|2 years ago|reply
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
[+] [-] gilleain|2 years ago|reply
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 :)
[+] [-] noncovalence|2 years ago|reply
For making Euclid interesting to children, I remember really enjoying a game called Euclidea: https://www.euclidea.xyz/
[+] [-] Mathnerd314|2 years ago|reply
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).
[+] [-] elesiuta|2 years ago|reply
[1] https://play.google.com/store/apps/details?id=com.hil_hk.euc...
[+] [-] unknown|2 years ago|reply
[deleted]
[+] [-] godelski|2 years ago|reply
This is quite fun for them as your question of "Do you know how to count to a thousand on your hands?" appears like magic or a superpower to them. So I demonstrate the beginning counting to 20 or so (quickly moving through 4 and 6). Then I start to ask them to predict certain unseen configurations (i.e. zero shot generalization). Re-demonstrate when failure to predict. Once the pattern is successfully learned, then I present a quiz/puzzle, and ask how many fingers "this many" is (all fingers unfurled). Always stumped, I provide the hint "if I had an additional finger and that finger were open and all others were closed, how many would that be? Can you figure out the other number from here?" It takes time, but they almost always get it.
The beauty of this is that we have a low barrier to entry, as the kid just needs to know how to multiply by 2 and know the names for numbers up to 1024. It surprisingly has many avenues of thinking that can help a kid better generalize concepts of math while still being entertaining (similar to concepts in this article). First, we teach the kids that there are multiple representations of things, and that we need to formulate things to match our goals. That we can break away from the common and expected thinking that most people have to gain "super powers" (i.e. not count like most people). Another important aspect is the above puzzle, where we specifically teach them that there are often better ways to go about solving a problem if we can find patterns. Rather than brute-forcing your way through this (summing each finger) you can exploit the iteration pattern to know that hinted at position is only one away from the desired. Frame of reference is such a crucial concept to mathematics and is at the root of solutions to many famous problems. Obvious post hoc, but inconceivable a priori.
We can even go quite deep and talk about proofs and how to design algorithms! I'll explain the algorithm identical to how we would perform a proof by induction (this is not how I teach kids, at the beginning):
k0th step: starting palms facing user, and an initialized position where all fingers are closed (thumb is a finger and at left most and right most positions). Starting from the right, increment the right most finger (thumb)
knth step: start from the right most position. If finger is closed, then unfurl. If finger is unfurled, close it and attempt to increment the next right most finger recursively following this condition.
There's more that you can build off of this one concept and similarly that with the topics in the article. What I've found is that which ever "game" the kid likes best is the one you should focus on and formulate your basis around. When they have difficulties with one game you use a different game that they are successful at to teach the difficult one. After all, math is a language and so many things can simply be rephrased.
I find that one of the difficulties many have with math is that the internalize it as quite strict. That it is often taught "this is the way," with no other methods accepted and thus people gain quite low generalizability of the concepts. Something that "word problems" are intended to resolve, but this approach is quite brute forced and more akin to how one might teach a machine rather than a human. This is coupled with the fact that so many are at a young age taught by people void of passion for the subject. This dispassion only passes from teacher to student (I'm sure many people can remember the breath of fresh air if they were lucky enough to find a teacher who loved math and encouraged the creative side of it. Honestly, that's how I came the love the subject and prior to that Junior in High School class, I hated the subject despite being good at it and in advanced classes).
[+] [-] sieste|2 years ago|reply
Indeed, beware of playing this game with kids! Mine started shouting "645!" at each other.
[+] [-] thenobsta|2 years ago|reply
[+] [-] RheingoldRiver|2 years ago|reply
There's a LOT of math you can do with it too, starting with some modular arithmetic. and I guess just the idea of abstracting the attributes to 0, 1, and 2. Then you can do a little bit of group theory and how the game is Z3^4 (iirc this is the right notation? been a while), 4 copies of Z3. And there are similar card games that represent other games that you can print out cards for if they're getting excited about group theory, so you can talk about axioms of groups, and how the games represent the groups, and why it makes sense.
Also you can introduce a bit of programming too and explain like how would you teach a computer how to play this game using the invariant property that a definition of a set is that sum c_i = 0 for all i 0 -> 3. and you can also easily generalize the game to other lengths by adding additional traits (background color of the card to add one) or fewer (remove bg color of the symbol).
Anyway yeah this game is the best.
[+] [-] rahimnathwani|2 years ago|reply
If you want a one-player Set game, there's a nice open source one here:
https://playset.netlify.app/
The 'AI' looks for solutions randomly. Every 2 seconds (in easy mode) it picks two of the visible cards at random, and sees if the card needed to make a set is visible.
If your kid is totally new to Set, you probably want to adjust the delay to 5 seconds or something.
Set is more fun against human players of course.
[+] [-] hgsgm|2 years ago|reply
[+] [-] jacknobody|2 years ago|reply
[+] [-] akermany|2 years ago|reply
[+] [-] benlivengood|2 years ago|reply
[+] [-] unknown|2 years ago|reply
[deleted]