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I am a high school student really into algebra and algebraic geometry.

I want to expose this sort of math to other high school students that have the motivation and ability.

For a long time, I had no idea how to learn more math after learning calculus, and I feel my middle school and early high school years were wasted as a result. One day, I coincidentally found a post on r/math that had a link to a bunch of Springer PDFs that were being given out for free. This experience was incredibly eye-opening, and I was exposed to all sorts of math that I had never seen before.

I want to find similar students to me that never got the chance to find an online copy of Lang, or their father's calculus textbook. Furthermore, I want to give a mathematical experience better than picking up a random book and struggling through it while you're supposed to be doing homework.

In short, I want to teach advanced mathematics (leaning towards abstract algebra/foundations) to high school (or even middle school) students, with no prerequisites other than demonstrated commitment to learning and calculus.

In order to accomplish this goal, I have a few questions.

(1) When I first asked this question, I thought that we were going to go through parts I and II of Lang's Algebra, all of Mendelson's logic book, and Axler's linear algebra book. I based the curriculum and pacing heavily on Math 55 at Harvard. But then the answers made me realize that the majority of students are not willing to neglect their grades and social life to study math for 60 hours a week. Should I postpone the project to the summer looking for the students who can, or lower my standards?

(2) How do I find and recruit students? Nobody at my school is interested, and even if they were, I doubt most of them would put in the effort.

(3) What general methods for teaching should I know of? How do I write a lesson plan, schedule a curriculum, and grade proofs?

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    $\begingroup$ I think the percentage of people in your target age who can do what you want is extremely small (probably only a few dozen in the U.S.), and it's likely that each of them is as individualistic and goal-oriented as you, but in directions sufficiently different from you that this would not work. My recommendation is to devote your energies toward your learning, and not worry about other people's progress and about trying to "reform the system". One compromise would be to write detailed solutions and/or notes, and share them with like-minded people you meet on MSE or other places. $\endgroup$ Oct 12 '21 at 17:31
  • $\begingroup$ A close personal friend of Lang's: "We all knew his books were terrible, but we loved him anyway." $\endgroup$ Oct 14 '21 at 1:43
  • $\begingroup$ Damn it, I meant to do a line break and it sent the message. --- This happens to me a lot, probably as a result of so many years on usenet (mostly sci.math, but other places as well). I click on "edit", then copy what I've written, then "cancel", then I delete the comment, then I start a new comment by pasting in what I'd copied, and continue writing, trying to remember not to use what used to be the carriage return key on electric typewriters (now the "Enter" key, at least on U.S. keyboards). $\endgroup$ Oct 15 '21 at 16:22
  • $\begingroup$ @DaveLRenfro I must echo your sentiments. The OP should focus on his own learning and not worry about teaching others... except perhaps on the MSE etc. Locally, almost certainly not. This actually hits close to home for me, I must admit, I have allowed too much of my personal study of math be guided by my teaching. I'm a hostage to what kids are interested in in some sense. On the one hand, it's self-preserving since I only have so much time, but on the other hand, with scarcity of students it's a bad habit I need to break. $\endgroup$ Oct 17 '21 at 1:38
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For (2), I'd suggest creating the lessons first as blog posts or youtube videos and then sharing them. I think the math reddit might be a good place, but I'm not active there. These can then in turn serve as advertisements for any active curricula you'd like to implement.

For (3):

  1. Start with an overall structure of the course. Make a list of course-level goals.
  2. Create assessments (the equivalent of a midterm/final) based on these course-level goals. Even if you don't wind up using these assessments, it's useful to have them to structure the course.
  3. Expand your list of goals to a list of goals for each week/module/lesson. If your target audience is really no prerequisites, pay close attention to everything someone needs to know to be able to answer all of the questions you created in step 2.
  4. Create assessments based on each week's goals.
  5. Create lessons in preparation for those assessments.
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Re (1), you should lower your standards. As previously discussed, the population for Math55 in high school is tiny. (Add onto that the difficulty of finding them and it just makes little sense to try to develop this.) In addition, you really lack the math knowledge OR the practical pedagogical experience to develop Math55 for high school (even the time, given you are in high school).

But I think it would be better to abandon this project and concentrate instead on your own learning. If you need interaction (natural human need, you talked before about wanting a club), I would look into math teams, Olympiads, etc. Yes, I know the math team at your school didn't satisfy you, but typically if you do well there are playoffs, recognition, etc. and in that case you'll be drawn into a circle of bright mathy youngsters at other schools.

Re (2), this is a problem people all over the world struggle with this, in various topics. In some ways, it's really more of a practical marketing question (but non-trivial). Honestly rather than trying to build something, you're better off just concentrating on your own development. If you want, you could blog about it. But don't be surprised how hard it is to get traffic. And again, you really lack the chops or the time to develop even an interesting blog on the topic. But if you treat it more as journaling than maybe OK.

Re (3), you need to develop, try, and read about several practical approaches. This is a huge topic.

Overall, lots of interesting things here that you could learn about (Fermi problem estimation of a population, how to build interest across the Internet, and pedagogy/engagement in high end math). But I would just learn about them in the course of trying to create this thing. Which you won't do successfully. It's like starting a mail order computer kit company. You're unlikely to become Dell, but you'll learn some interesting things along the way. (And if you do become Dell, go you. But still, you need to learn by doing.) Really, I would just get through high school and do Olympiads. And learn Japanese.

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