Everything you always wanted to know about mathematics (2013) [pdf]

I can completely understand the skepticism and agree that many online courses are paper thin. That's where we're different.

For example, our BC Calculus course comprises 302 topics, each containing 3-4 knowledge points, so ~1060 knowledge points in total. Students must master each knowledge point to move on to the next. Our spaced repetition algorithms ensure that students are repeatedly tested on the material (we have quizzes every 150 XP or so). If they fail a question on a quiz or topic review, the system requires that they retake the failed topic. Students _cannot_ complete a course without mastering the entire thing.

Each knowledge point is connected to key prerequisites in the same course and lower courses. If a student stumbles on a particular knowledge point, our system can determine the most likely point of confusion and refer them to the associated key prerequisite topic (which they must pass to continue making progress).

We also have a couple of dozen multistep questions, similar to those you'd find on the BC exam (although the BC exam has about 4-5 parts per question, ours have about 9-10).

Regarding results, we had an 11-year-old sit the BC exam recently, and it looks like they will get a 5, the top mark. (For those that are unaware, students usually sit the BC Calc exam at the end of high school in the US, so 18). I admit that's an extreme case, but it's not isolated. I could reel off many success stories of students achieving real results on real tests after self-studying using our curriculum. We also have an associated school district program in Pasadena, California, where dozens of 8th-graders have achieved 4s and 5s in the BC exam, mostly learning using our system.

In terms of the required effort - provided you have no issues with the necessary prerequisite knowledge, you can get through our entire BC Calculus course by committing 40-50 minutes per day, five days per week, for around 5-6 months. Of course, if there are gaps in the prerequisite knowledge, then it'd take a little longer - but thankfully, our algorithms can detect missing knowledge and fill the gaps. That’s one of the advantages of having an intelligent, interconnected system comprising over 3000 topics!

As for our higher-level courses - some of these are still in development. However, our linear algebra course is comparable to several high-quality books on the subject (I like Lay, Anthony & Harvey, and Axler, though we use others). It currently has 176 topics, but many foundations are laid out in our Integrated Math III / Precalculus courses (vectors, matrices, basic determinants, inverse matrices, linear transformations in the plane), so the real number is around 200.

https://mathacademy.com/courses/linear-algebra

(click on the "content" tab to get a complete list of topics).

Could one of our students ace the GRE? That's a great question. We still need content on several key areas required for the GRE (e.g., Abstract Algebra, Real Analysis, Complex Analysis, and Graph Theory). These courses are still in development - we already have a lot of this content behind the scenes. That said, I'm confident that our students have the necessary tools to succeed in the parts of the GRE we currently cover. We don't "teach to the test," not even with BC Calc, but equipping our students with the necessary knowledge and skills to go from 4th grade math right the way up to acing the GRE (just as we've done with BC Calc) is one of our medium to long-term goals.

Happy to answer any further questions about the curriculum you may have.

You would probably get more traction if you offered a free month up front because so many platforms before you have failed to deliver on the hype.

EDIT: to be clear, "code" only works if your set is finite, but the principle of "does this question even have an answer" remains "yes" for injective and "no" for surjective without the codomain being named explicitly.