Why I’m taking full responsibility for my education

You can't always learn with the expectation that everything has a perfectly applicable use case (i.e the point about learning Geometry to build a renderer). This may be fine if all you want to do is (re-)implement things that have been done before but if you want to develop new things and solve new problems, you won't know beforehand what tools and what theory will lend to the solution. I'm doing a PhD now and I will take classes and read papers outside of my focus to see if anything will lend itself to my research. Much of this doesn't go anywhere until one day I go, "this problems kinds of sounds like what they did in [that other field]".

Also, many complex topics build upon many, many smaller results. I've found this true for large portions of math. Many readily applicable theorems, for example with applications in signal processing and control systems, require many years of math background and maturity. It's very difficult on the onset to connect the basics to the complex theories. (On a personal note, I've found that I had a relatively extensive math background as an undergrad. You could argue that the extra math I did was not worthwhile as it isn't all applicable. However, in some of the classes I've taken in grad school, I've noticed the gap in math background and familiarity as other students try to catch up.)

That said, I do admire the OP for taking hold of his education. I did not find my high school education, targeted at the average or at most students, to be well-suited for myself. I would just say that there's a reason the topics and material chosen are what they are. Extending what's done in school is a great idea, but if you are going to deviate, make sure you have a very good idea of what you are deviating from (and why).

(Disclosure: I haven't read the OP, or the rest of your post really beyond this sentence. I skimmed.)

(Double disclosure: I too am in high school.)

An emphasis on "fun" is one of the most misleading concepts in trying to get people to pay attention. Some things can be gamified, others well...not so much. (See: Educational games: The bane of every elementary school kids existence.)

In a way it's almost profane. Trying to make things interesting by making them fun when the underlying concept is already interesting. It's just that you have to focus on it in a small picture, so it makes it hard to see the bigger concepts that it's a part of. (Objectively I know that if I read through mathematics texts long enough it gets non-trivial and interesting, but actually doing that is so painful as to be impossible. (Impossible in the "I see no way to do it." sense of the word.))

Objectively I also know that I'll be mediocre at any sort of CS effort until I take the plunge. Still not motivating. Other things are just too distracting. I almost feel I should just unplug my Ethernet cable and hide it until I emerge a hacker.

I find most math interesting and boring, but there's also some parts of math I find fun believe it or not. Even at a high level.

This is probably a terrible explanation, but I'm giving it my best shot.

Point #1: Math is fun while you are learning it and figuring it out. After that, it isn't so much fun as it becomes exactly what it was meant to be: a tool. It's just useful at that point. "I know how to do X and it's useful in this case." To the point that there isn't a fun way to demonstrate it, I disagree. One of the best examples I can think of was when I took a science class and we were learning about velocity, vectors, and acceleration. The teacher had us build model rockets and launch them, guessing how high they would go based on calculations. That same teacher went to a different school and did the same thing, except on a bigger scale. All the students worked on one rocket with a goal to break the sound barrier. They put a few electronics in the rocket to measure acceleration and g-forces (IIRC). When the students calculated the altitude the rocket would reach, they were pretty close. Not terribly advanced, but advanced enough with actual application of the concept being learned. It wasn't entirely out of reach for the students to achieve, but a teacher needed to guide them there. The teacher needs to figure out how to present the information in an engaging way to draw students in. But, at the same time, the student needs to be willing to fish for the answer and occasionally accept the I-don't-know answer.

Point #2: While I understand the student's sentiment, I agree with your point. Unless a student realizes they could be FIRED over the fact that something is late at a real job, it won't sink in. It's a good thing to have deadlines, otherwise someone would be spending all their time on reddit/HN. A better approach would be to give the student an assignment that's a little tight on the schedule. That forces the student to concentrate on the material if they want to get a decent grade. I think it would be better if a student was kicked out of the class if they repeatedly turned things in late, and in a sense, be fired from the class.

Point #3: You don't need to learn the theory right away. There's no point in drowning a student in theory when they have nothing concrete to anchor it to. If I started talking about Optimization without giving a student an example of a slow program and a fast program that do they exact same thing with the only difference being optimization, they wouldn't have much of a reference point. If you look at a big program that's had execution analytics run against it and realize that a slightly less optimal solution is slowing the program down by a fact of 5, you then have a solid reference point to learn algorithms/data structures and optimization. You then understand the why behind needing to know something, rather than someone standing in front of a whiteboard saying, "Hey, this is important, you need to know this."

This is anecdotal, but I find at least the math I have learned to be fun. I got a BS in math and took some masters-level classes because I enjoyed it. I still work through recreational mathematics and play with project Euler and fractals when I have time.

You mention higher levels and more abstract forms of math, and I cannot say I would still have found it fun if I pursued a PHD and tried to truly breach new frontiers, but I can say that my professors generally all told me they enjoyed it. They did it because they had fun with it.

Different people find different things fun, but I think its safe to say that some people find even advanced mathematics fun.

Maybe it's not for you, but don't generalize that to everybody. It requires more work at higher levels of education, but that doesn't prevent it from still being fun for some people.

I couldn't disagree more. As an example, linear algebra is something that has been bothering me for years, long after my college math courses. One of its central topics is eigenvectors and eigenvalues. Every textbook I can remember would demonstrate these with pictures of a sheared rectangle (or a sheared box in 3d). That's fine, but it isn't fun and its not motivating, so no surprise that the concept never clicked.

Then very recently, while browsing wikipedia I came across an excellent demonstration of the concept: "eigenfaces" used in facial recognition software. Its fun, useful, and it clicked.

> how do you demonstrate that the ensemble of invertible matrix is dense in the set of R(n,n) matrix in a visual manner?

You could draw a picture of a circle, emphasizing the line of the circle as the boundary of the set. The invertible matrices are the interior of the circle, and they complement the singular (non-invertible) matrices which are represented by the boundary of the circle.

But more importantly, the teacher should first explain why this might be useful to know in the most concrete way possible, whether its theory behind an applied technique in engineering or a lemma used for an important abstract theorem (no handwaving - tell what's important or significant about the abstract theorem).

It is the teacher's job to motivate the students, good ones do so and bad ones don't. Lazy teachers who don't think its their job to motivate students will get exactly what they complain about: unmotivated students.

Students (American or otherwise) who are willing to just put their heads down and drill rote symbol manipulation are doing themselves a disservice - it generally does not lead to much insight or understanding of what they are doing (though it may in the exceptional cases of very smart students), especially as the math gets more advanced. Moreover, being adept at calculating is not useful in the age of computers. You only need to do a calculation once (as an algorithm in a computer program).

More important is to gain insight and understanding into the nature and limitations of the subject matter. Then the student is more likely to recognize the cases where it can be applied after they're done with the plug-and-chug problem sets at the end of the chapter (which plague even advanced math textbooks in the form of boilerplate theorem proofs). Of course, simply showing how to plug-and-chug is easier for teachers so they praise obedient students who are easily motivates and ask easy questions.

But as for I refuse to believe that there is a student anywhere that really wants to learn but cannot. That is true, but only to a degree. There remain plenty of places, even in America, where internet access is limited and a young student in particular may not have the resources to get consistent and frequent internet access. Similarly, there are places where the libraries are not accessible in a practical way.

A truly determined person, such as Srīnivāsa Rāmānujan can rise above, but they are the exceptions and even Ramanujan got help from people to reach his peak.

There's only so much you can do when everyone around you wants to know if "this is going to be on the test?" rather than "is there a deeper meaning to what this character said?".

And probably a poor employee. In the real world they pay you not to learn or to innovate but to do something. Usually, that something is something you have done many times before, that way you can do it faster and thus more cost effectively for them. The closer you become to machine the more profitable you become. That is the biggest hurdle the smart students face in the real world.

Reading has played the most fundamental role in my education. I have always loved learning from well-written books. At times, it might have been a disadvantage to not have a mentor to guide my progress or to answer questions. I'm certainly not a "gifted" learner and I have often spent long hours trying to understand a page of a book. But it is always incredibly satisfying when I finally do.

I think the most rewarding thing about taking responsibility for your own learning is gaining both humility and confidence. I am humbled by the amount of things I still have yet to learn in the enormous field of computer science. But I have also gained an enormous confidence in my _ability_ to learn. I know, from fifteen years of experience, that I can learn anything I want to as long as I apply myself.

Another benefit is that books are far cheaper than classrooms, they're portable, they go at your pace (you can turn the pages as quickly or slowly as you want), and you can keep them on a shelf to refer to for the rest of your life. And e-books are even better!