I respect you point of view but... it won't hold for high level education. Let me explain why:

* I just wish people could forget the idea that Math is fun. It is not. It's interesting, but it's not fun and can be totally counter intuitive, thus there is no "fun" way to demonstrate it. You are mentioning about Geometry but the geometry you are talking about is a very very very tiny subset of what geometry is at a higher level. It's not an topic from Geometry but for example how do you demonstrate that the ensemble of invertible matrix is dense in the set of R(n,n) matrix in a visual manner? These concepts are so abstract that they become too hard to demonstrate. Achilles tendon of most american students I have known and I have worked with is that they are extremely dependent of a representation of what they are dealing with and they struggle with very abstract concept while in the Russian/ French/ Chinese education teachers teach very early to student to manipulate abstract objects (4-D vectors, advanced algebra etc... are often seen in high school).

* The "get less point if you hand back the report late" is actually a good thing for your students. Read one of the Dan Ariely book that showed that this pace helps students study and in the end they end up having better marks. That's the sad human psychology.

* For the theorem you need to learn to be a programmer I agree, you don't need them to be a decent coder. However... there's a say among lawyers: "A good lawyer knows the law, an excellent lawyer knows the judge". If you want to be a great programmer sometimes tools at your disposal fall short and... it's your turn to create. This is the moment where theorems become really important and a great programmer will know how to leverage them whereas a decent programmer will fall short. "A good programmer knows the tools, an excellent programmer knows the theorems" !

(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.

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).

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 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.

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.

>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.

I couldn't disagree more with your disagreement. Eigenvectors and eigenvalues are abstraction of things observed in many places and studied in one place so you can apply it to everywhere. Eigenface is simply an instance of linear space dimension reduction, whose implementation, by the way, is based on eigenvectors' connection to SVD and covariance matrix. Eigenface won't help with Taylor series definition of matrix exponential, nor solution of linear ODEs, nor Jordan canonical forms. Can anyone say those are not important topics of eigenvalues and eigenvectors?

If you truly want to understand something, there is no shortcut. You have to dig deep, look at and learn related topics, think hard about how and why scientists developed the subject this way. It takes time and concentration, a lot of them. Then you will gain something, and you need to keep at it to master it. No one said knowledge is easy, especially deep knowledge.

>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.

This demonstrates the pitfall of facile visualization, because the suggestion is wrong. There are dense sets with no boundary. The simplest example I can think of is rational numbers on the real line. It is dense on the real line, yet between every two rational numbers there is an irrational number, and between every two irrational numbers there is a rational number.

>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).

This is easier said than done. At best, it is impractical; at worst, fantasy. Who is going to spend a week of lectures to explain one application in a possibly obscure engineering field. What about the fact no everyone is from an engineering department.

Sometimes theorems are useful for proving other theorems and then for proving other theorems. Gershgorin circle theorem is mainly useful for proving bounds about eigenvalues. That is it. I couldn't motivate more than that. It has application in numerical linear algebra, but it amounts to another proof and you need to go pretty deep in matrix analysis to appreciate it.

>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).

This I strenuously disagree. Math, like every human endeavor, requires practice and lots of it. If you cannot recall a pertinent theorem at will, then you will not be able to use it to prove it. You don't have to remember every theorem for all time, but when you need it you better. And practice does develop insights and understandings, which I can personally attest to. Advanced math especially requires a familiarity with basics, for no computer will prove for you a countra-positive.

I am opposed to rote learning. Who isn't? Specifically, I am opposed to Chinese teachers' mind-numbing deluge-of-exercises approach. All the proofs are nothing more than bags of tricks and they seem to take special delight to confuse students by not explaining things fully. I so detest that mindset. The U.S teachers are much better. There are good teachers and bad teachers, of course, and I suspect college professors can be a lot better if they actually put the necessary time in. American textbooks are leagues ahead. But one thing I have learned since is that in the end you have to remember the theorems and tricks because: THEY ARE MATH.

>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.

This I agree in general. It is after you gain insights and understanding can you innovate and advance. It is remarkable how many Ph.D.'s never master their fields. But it is hard and time consuming. My ideal of math textbook is Richard Courant's "Introduction to calculus and analysis." It is a perfect blend of mathematical rigor and insightful intuition. It is sad that I didn't have it as my introductory calculus textbook. It used to be the standard intro textbook in the West.

I suspect, and I could be wrong, the reason you blame teachers so much is that they didn't teach deep enough and you are smart enough to realize there is more. They had to be "easy" because it is already hard enough for some students. And this may be the ultimate problem with public education: it needs to teach the everybody but it can only do so by dumbing down the curriculum. That and it is expensive to teach but we want to do it on the cheap.

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."

A high school student doesn't have enough knowledge nor experience to be writing essays on educational theories. This entire post reads as a "I didn't get taught the way I thought I should, so I'm mad about it" rant.

Get your diploma, get a degree and come back in 5 years when you've got some solid education and life experience. You might be able to figure out why the education system is setup the way it is after that (hint: it isn't setup to promote effective education).

The point isn't to demonstrate the value of the homework. As you just stated, the homework itself often has very little intrinsic value. The value is to get you used to dealing with deadlines. Do you really think that your boss is going to let you miss deadline after deadline without a credible reason? Even if you don't have "boss" in the traditional sense (e.g. if you're doing a startup) do you think your investors are going to let you miss multiple ship dates without consequence?

You might not like it, but that's the way the world works. People need to plan ahead, and they need your products, assignments, code, or whatever by a certain date, otherwise it affects their plans. In the case of your teacher, he or she isn't assigning the deadline to be arbitrary or capricious. He or she is making the assignment due by a particular date so that he or she can get it graded and back to you in a timely manner. Submitting assignments whenever might be okay if you're the only one in the class. But if you take your attitude and multiply it out by the twenty, thirty (or sometimes even forty) other students he or she has to teach, it's a recipe for chaos.

> large portion of their responsibility for their education to the school.

> In other words, if you aren’t learning much, it’s the school’s fault.

I'm a senior at a public high school, and I absolutely disagree. The onus to learn always falls on the individual. I refuse to believe that there is a student anywhere that really wants to learn but cannot. Libraries and the internet are always available, as are other people — all of these resources can be learned from.

The author seems unhappy with their high school experience. Public schooling may have its flaws, but to blame one's lack of an education on one's school would be misguided.

> For example, a good way of teaching Geometry is to have students write a graphics renderer.

No it wouldn't. Not everyone is interested in programming or computer games.

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 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.

You need to do this whatever you're doing. Learning at school? Learn more than you need for the test. Learning at university / college? Learn more than you get in lectures. Reading a newspaper? Learn more than they tell you. Reading reports of research? Learn more about the research used.

> but ithe project actively shows students, “Here’s a concrete application of what I’m teaching that you can benefit from.”

BAH! Knowledge for knowledge's sake is a good thing. Clever exploration of unknowns is an important driver for science. The most well known example of this is Feynman. If you haven't read his books I thoroughly recommend them. He sees something, and wonders why it happens, and then goes off to try to understand it. He was a genius, and so his observations and explorations were usually of interesting things that involved high level science, often unknown at that time. But it's a useful principle.

> Following directions has no intrinsic value: it’s a means to an end. For example, why do students lose points for late homework? In what way is homework less valuable the day after it was due, besides a rule that just says so?

Unfortunately a lot of your school colleagues are going to end up in the kind of jobs where "following directions" is all that's required. Here is an important lesson: Sometimes you will have an idiot for a boss. That boss will tell you to do something. S/he will tell you to do it in a certain way. You will know, and have facts and evidence to support you that you're being asked to do something stupid, or in a stupid way. You might even be able to put a monetary figure on it - "doing this will cost us $X every month!". It is very frustrating, but sometimes NO ONE CARES. All they care about is the fact that you did what you were asked.

Also, getting work done early is usually a good thing. Learning to plan and prioritise work is an important life lesson, so learning to hand it in on time (with gentle penalties for lateness) is important. Often late doesn't mean "minus one point" it means "tough shit you blew it".

Good luck though, you sound motivated and committed.

Naively, I like to think that nobody should look back on high school, university, or anything until many years AFTER it has completed. Only then have you progressed far enough away from the experience to make an objective view of it.

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?".

In fact, everybody should do this.

It is called being an adult.

However, I think he was on the right track when he wrote "High school isn’t educational because it incentivizes a credential only meaningful to universities instead of educating students." This isn't entirely true; high school students will learn things that will be useful even if they don't attend college. But much of the material of a high school education (and the way it is taught) supports the college-preparatory model.

Many students would do well to enter a trade school at an earlier age. Perhaps the US high school education system could be streamlined if students decide which track they were on earlier. Students who were set on college would continue with the traditional HS model, while trade school students could begin vocational training at age 16 or 17.

The real root of this issue is that there is a demand for workers (in the US economy) with skills above that of a HS diploma, but below that of a bachelors degree. These certifications are specialized trade skills. These jobs are relatively well-paying and continue an American economic tradition of entry-level jobs which enable workers to work their way into the middle class.

However, trade schools are looked down upon. Many people go to college and end up working in jobs which they are (supposedly) over qualified for. We need to refocus our entire education system on what matters, and the perception of these non-college degrees will change.

I love learning, and that’s why I’ve never been a great student. In middle school and high school, I spent my free time learning about things that interested me—especially programming. In that time I learned about everything I could because school wasn’t enough. Dynamics and kinematics, geometry, optics, digital signal processing, splines, vector and matrix math, programming language design, lambda calculus, type systems—things no student my age was expected to know or care about.

I got high grades whenever I bothered to put in the busywork, but I had little reason to. Points and grades aren’t worth anything.

Pretending that grades have intrinsic value is toxic to learning and innovative thinking. You get what the teacher explains straight away, and want to move on? Not allowed: you have to go at everyone else’s pace. You want to know more about a related topic? Too bad: you have to follow the curriculum. You want to do projects with real-world value? As if! You have to do homework and memorise information—sadly, the kind of information that might have deep meaning to you if it were framed in another way.

I do think losing points for late assignments is fair, though. It inconveniences the person doing the grading, and in the real world, lateness isn’t tolerable. Besides, if you don’t follow the (meaningless) directions, you should expect (meaningless) punishment.

But here’s the thing: would I have bothered to learn on my own if I hadn’t had the insufficient school system to piss me off? Somehow, I think not.

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.

I'm going to get a Certificate of Profiency (I'm currently waiting for the results). In California, it's legally required to be seen as the equivalent of a high school diploma. It's essentially a GED for people under 18.

I plan on going to a high-end university in 1-2 years.

Strong will and individual drive to learn define great university students - hopefully this kid adheres to that path and capitalizes on the opportunities University will open up for him. But he's currently dropping out of High School and trying an alternate route because it doesn't meet his rather ill informed expectations - if this occurs again at a higher educational institute, he will be making a mistake.

One thing I've noticed about recent high school grads: they all seem to think the world should meet their expectations and if not, there is something wrong with everyone else. University is a great place to have that attitude beaten out of you. If you don't come out of University humbled, you didn't go to a good one.

Message to the author: university is going to kick your ass - start preparing for it. If you do so, it sounds like you have the attitude and drive to do very well for yourself. But make sure you learn what you don't know before you start to solidify your opinions on things.

A lot of what you learn has benefits beyond being obviously useful and insightful. It teaches you how to think.

I hated maths and couldn't understand the practical purposes of it. However it helped me develop a way to think about problem solving much later in life. I hated ancient history, What possible benefit does understanding the battle of Troy have to do with me getting a job? It helped teach me critical thinking, which is incredibly important.

While I agree there are problems with modern western education, you've just taken a few of personal anecdotes and written off thousands of useful programs.

"A self-taught man usually has a poor teacher and a worse student." - Henny Youngman

"The difference between the university graduate and the autodidact lies not so much in the extent of knowledge as in the extent of vitality and self-confidence." - Milan Kundera (The Unbearable lightness of being)

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!