Are there good free online resources for such kids to help them explore and find their preferred directions, from reputable institutions?
A couple I’ve found:
1. https://science.mit.edu/diversity-and-inclusion/outreach-programs/#K-12
2. https://mity.org/online-programs/
Any others?
https://artofproblemsolving.com/
For the most part it is not free, but there are some free resources
https://artofproblemsolving.com/resources
and the paid programs don't cost too much, something like $400 for a 12-week online class. Also, the times are generally chosen to be convenient in the US, which might be difficult for your friend depending on where they are.
If you genuinely want something free, there is a ton out there, much of it good, but you might need to give up on "from reputable institutions". One person might post something online as a labor of love, and often that person might be employed at a reputable university, but if you want something backed up by an organization as a whole -- especially something more than "here are some notes you can read, or videos you can watch" -- then that organization will usually be asking what they stand to gain.
Here are some free notes on a variety of topics from the American Math Society:
https://www.ams.org/open-math-notes?grad_level=5
Here is my own contribution -- 150 pages on combinatorics, probability, and modeling based on TV game shows:
https://www.ams.org/open-math-notes/omn-view-listing?listing...
Best of luck!
You know what would be a really interesting problem in a weird-way related to game-show math, would be crowd flow specifically through a Casino and and determining probability for people interacting with a particular game, based on crowd flow theory [0] and how you can channel the crowd, knowing their personal avoidance and personal-space behavors to aim the flow through various $->$$$ temptations. and see what gets them to stop, stay, spend.
Human Plinko. For fun and profit!
[1] https://mathacademy.com/ [2] https://gmays.com/how-im-relearning-math-as-an-adult/
British, run by some fantastic people from the University of Cambridge. Arguably there is no institution with a better reputation in this domain. I did some front end and back end work for them over a decade ago and there's definitely some good mathematics in there. Their mission is to enrich mathematical learning for all levels, but there's plenty of content designed for gifted teens.
I suggest looking outside the US for a curriculum. The nation that came up with the terms "precalculus" and "prealgebra" surely has a pessimistic view on the value of elementary mathematics. The ideas do not exist only to prop up calculus and algebra!
MathPages.com could be good for enrichment.
For a 14 year old I'd recommend US universities as the first year is at a lower level there than UK universities (don't know much about non-anglophone universities). E.g. look for classes called "Precalculus" -- that will not be too difficult for them. And also introductions to mathematical logic (and set theory) to get their math thinking off on the right foot.
The other advantage of this is that it will get them in the mindset that a university environment is where they're going to be in their 20s. Doing an undergrad degree and then probably a masters/PhD, which it sounds like is what this teenager should be starting to think (not yet, but when they're 16/17).
Thatchaphol Saranurak is widely considered as one of the greatest upcoming TCS talents, pumps out meaningful results like a monster. Expect him to leave Michigan for MIT/Berkely/Stanford tbh. I took this class and it was comparable to the UMich honors math sequence.
Ross provides significant need-based financial aid (which can cover all fees for the six-week program).
- Algebra book by Gelfand [ amazon print n demand I guess ]
- A Decade of the Berkeley Math Circle by Zvezdelina Stankova and Tom Rike
- The Art and Craft of Problem Solving by Paul Zeitz
Also have a look at https://mathcircles.org and any math competitions available locally
It is not a course but a collection of tasks and problems.
Basically the same approach but with a strong focus on implementing cryptography and learning something about cryptology in the process.
(not having much money is no obstacle to reading books on math, so long as you are aware of Library Genesis)
https://ocw.mit.edu/search/?d=Mathematics&s=department_cours...
To be honest, this might be very challenging. BUT it may be possible. Just try entry level Analysis or Linear algebra. They often start kind of from scratch. How to construct numbers and so on. You could go very slow but at least you would be ready for real math...