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> 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.

(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 think this extends some of your same ideas:

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).

I just wish people could forget the idea that Math is fun. It is not.

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.

TL;DR: There's a time and place for everything learned in someone's education.

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."

> 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.

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.

>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.

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.

I just wish people could forget the idea that Math is fun. It is not.

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'd say fun is a very individual thing and everyone does not see fun the same way as you do. Abstract doesn't mean less fun.

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.

Very true. I am a person whose intuition works in a very weird way in that I try to abstract away problem details and try to apply rigor too soon. For most of my high school and early undergrad, I spent a lot of time being frustrated by "intuitive" examples that people used to give me which didn't really make sense at a fundamental level and seemed to be attempts at handwaving away the problems I had. It's only when I started learning more abstract mathematics did I really start having fun and seeking out how to create my own intuitions for things.

I'd disagree. I may be a little bit of a hypocrite for saying this (I'm about to finish my masters in cs), but while I found undergrad valuable I keep thinking of my grad degree as a complete waste of time and energy. And why? For the same reason stated by the author. So much of my energy goes into producing artifacts and making sure they fit some particular professors view of correct. I've learned so much more outside of school than I had in grad school for exactly the reasons enumerated.

To be fair, the author only claims to talk about high school.