NOTE: This question will soon be duplicated, as I didn't make clear that I was a high school sophmore in the beginning. At first I thought it didn't matter, and somewhat arrogant to mention, but in the comments of HFBrowning's answer, we see:
@JohnClever KCd's observation is spot on. The fact that you are currently a student changes the advice everyone would be giving you; I am certain that all the answers you have received were geared around you being older. I would either heavily modify your question, or consider starting a different one, since I'm not sure this one (with all the answers given) can be salvaged in its current state. – HFBrowning 2 hours ago
I'm planning on creating a 'math club' that is, in truth, a semester's course covering set theory, logic, and abstract and linear algebra.
The bulk of the course is the algebra. The set theory and logic act as preliminaries and are covered in the first pair of months, where abstract and linear algebra are covered in the next two pairs of months respectively. That being said, although I can see myself replacing the algebra with combinatorics, analysis, or number theory, I see the foundations as essential.
We'll do 4-6 problems a day, usually about a third to a half of each section's problems, and working around 2-5 hours each day is expected (a little under an hour for each problem).
This gives about 6 sections a week, assuming you work on Saturdays. This is about a 1.5 chapters a week, and because there are 4.2 weeks in a month, that's 6.3 chapters a month, usually enough to move on to another book. I know this is rather fast, so I half the time to be 6.3 chapters every two months, or one book every two months. This allows us to do 2.75 books by the end of the school year, and assuming that we leave out some stuff, it seems perfectly reasonable.
School starts back up in January, and ends in June, leaving 5.5 months. Since the logic and set theory is rather short compared to either the linear or abstract algebra, we won't need 6 months.
Miscellaneous Things About the Course:
- I have no teaching experience, being a high school sophomore myself.
- The course will be taught from Mendelson, (for logic and set theory) Lang (for abstract Algebra), and Axler (for linear Algebra), the books I, myself, used to learn these subjects.
- It is assumed that anyone who joins the club has a great interest in math, and is sufficiently hardworking.
- Although the previous condition is obvious insofar as it is necessary, I have tried as much as possible to make it true insofar as it is sufficient.
- Thus, I have tried to make the curriculum require as few prerequisites as possible, and may even end up replacing linear algebra with something else to that end, considering the amount of calculus-based examples it often uses.
Has anyone else already done this? I would presume yes, but finding a specific example seems difficult.
How do I recruit students? I suppose I could leverage the concept of a 'super-hard math class covering concepts you've never heard of,' but it doesn't really seem true.
Does the curriculum make sense? In my experience, enumerative combinatorics would be the standard 'higher-math' example, but combinatorics is also the bane of my existence, and I hate it with every fiber of my being. If I had to replace the algebra, I could probably have abstract algebra be some point-set topology and a bit of measure theory, which I could use later on to teach fractal geometry, so that they finally understand what's on the posters to every math class in America.
How do I keep the students interested? I would assume it would be easier to do than a required curriculum course, especially considering how much more interesting I would suppose pure math would be than the unmotivated drudgery that is, for example, pre-calculus, but I also think that the sort of people who love math in high school would be the kind of people that sign up, and they might be expecting something more computational compared to the abstract stuff we're doing.
Clarifications: I now realize that people thought I was referring to Lang's graduate textbook, which I am not. I'm "ambitious", but not evil. I am instead referring to his undergraduate textbook.
At our school, I know that that basic computational linear algebra is taught in precalculus, which is why I thought Axler was reasonable, perhaps supplemented by some 3blue1brown for good measure.
Mendelson is required reading for many philosophy majors as an introduction to formal logic, so I hardly think it too harsh for a math enthusiast.