Thoughts about what worked in math circles

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.

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)

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.

> 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).

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).