My 6yo daughter has exceptional talent and interest for Math. She's already discovering some fundamental theorems for herself (number theory, algebra, geometry). Does anyone here know any resources to help her development?
Other thoughts:
* The "Exploding Dots" project came up with a creative way to present place value that leads into a lot of other concepts like number bases, algebraic structures, polynomials, and other things. (It's actually like a generalization of the abacus, with fun computer animation.) I'm not sure which presentation of this is best -- I think the original inventor has gone through like three different versions and there are several versions available online now.
* I'm a huge fan of the late math journalist Martin Gardner, who wrote a lot of really entertaining stuff about math as well as puzzles and games. His books are really brilliant and cover a very wide territory. On the other hand, they were mostly written for adult audiences (originally, mainly readers of Gardner's Scientific American column) and may contain cultural references that kids wouldn't get, not least because some of them go back to the 1960s. :-( Also in some cases there have been further discoveries in the decades since the columns were published, so learning about them only from Gardner's old work may not give a good sense of where things have gone since then (e.g., he famously originally introduced both the Game of Life and the RSA algorithm to mass audiences, which then learned a lot about both topics in the ensuing decades). But maybe have a peek because they do stress the "math is fun and you can do it for fun" and also "math is all kinds of stuff and not just arithmetic" notions.
* Gardner does have two math books for kids (the aha! series) that are very fun and introduce a lot of fascinating stuff about logic and paradoxes, among other topics.
* Two online games that involve proof and circuit synthesis (spoiler alert: these are actually often isomorphic to each other because of Curry-Howard and stuff), which I think I heard about on HN:
I totally loved both of these and I think they could conceivably be accessible for a very motivated and very talented child.
Another online logic game that I also learned about on HN and loved:
https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...
Unlike the other two, this one uses (only) text and symbols instead of blocks and arrows. It's meant for college undergraduates, but I envision highly mathematically-talented middle-schoolers possibly being able to finish it. The user interface can probably also be improved a lot to make it clearer what you're allowed to do in each context. Maybe keep this one in mind for the future? :-)
Contemporary pre-college math education is famously very weak on discrete math topics (like number theory, logic, and combinatorics). I don't know if this is due to a desire to train aerospace engineers for the Space Race, or because discrete math has radically increased its profile only in the past few decades with the rise of computer science, or as a kind of backlash against the New Math which tried (mostly unsuccessfully) to teach kids formal logical foundations through set theory. I wish I knew some more good discrete stuff for younger audiences.
I haven't ever done any of it myself, but I hear Khan Academy's explanations (especially in math) are great and very self-paced.
I find having a map of the "terrain" is always good when exploring.
1: https://www.amazon.com/Breaking-Numbers-into-Parts-Second/dp...