Is there any sort of math curriculum for people who learned a math subject 10+ years ago and need something like a refresher, but with more foreshadowing. For example, there are many dots I connected between linear algebra and calculus, but I worry there are some dots I never connected (even though I did fine in the class).
Does such a curriculum or class exist?
> I worry there are some dots I never connected
This will always be true for anyone learning math in any circumstances.
The beauty of the subject is that there is always more to unravel, always more connections to make. Don't worry about what you didn't learn -- although follow-up classes to pick up some of the loose ends are always recommended :)
I don't believe that there is any one royal road to mathematics. Different people learn in different ways, and come with different backgrounds. For example, some people like to learn all the foundations and formalism up front, and then build on it. Personally, I like to learn a bunch of examples first, so that when I see the formalism I can relate it to what I already know.
Personally, I definitely appreciate foreshadowing (as you do) and try to incorporate it in my own teaching.
You'll find a lot of good math materials out there at higher levels, but generally speaking they will cover one subject at a time, as this is what creators enjoy creating. For example one book I like is Underwood Dudley's Elementary Number Theory (can be bought cheaply online) -- and one reason is that the author gives elementary proofs which mimic the structure of the higher-level formalism (i.e. abstract algebra), so that this formalism feels familiar by the time you actually learn it. This somewhat aligns with your request for "foreshadowing".
One of my colleagues said that this book drives him crazy for the same reason -- he thinks you should learn the higher-level formalism up front, so you can put it to good use. No one can say that one approach is better or worse than the other, just that it worked better for them.
Dudley’s book looks great, I love this kind of reading.
I have completed the entire engineering math curriculum in my school days, but have not taken abstract algebra. What resource do you recommend for learning that subject?
For a more traditional, formal approach, Saracino's "Abstract Algebra: A First Course" is my favorite among this genre.
I have also heard good things about Pinter's "A Book of Abstract Algebra", and it is available quite cheaply. I have not read it though.
Good luck!
There's absolutely no actual math education exercises, it's purely an overview of things a non-specialist would want to know about the existence of.
It's been partially looked over by people with real math knowledge, but there could still be some incorrect things here, and there's still some stuff I'd like to add but never got around to.
https://github.com/EternityForest/AnyoneCanDoIt/blob/master/...
There's likely nothing of interest to someone who knows anything like calculus already, this is all stuff that is kind blowing and exciting to those of us with boot camp coder or average non-technical person level math knowledge.
But perhaps someone who actually knows math could do a similar article, but for nontrivial math?
It's hard to understand how well it works until you've tried it yourself for a couple of weeks. Also it assumes you'll study at least a few times per week.
It has a system that can move you forward quicker for concepts that you know. It goes pretty far through a college math curriculum.
It also uses some spaced repetition as well to cement the ideas.