This post arises due to Paul Lockhart's work -- A Mathematician's Lament

The current math education
I have lived and experience the mathematical education system in three countries. China, Australia and US. Australia is quite the same as US.

While each system have it's own curriculum and it's own strong point, I found non of them really lead students to their full potential.

I only know China's middle school math, but I can imagine the high school level math too.
Chinese math focus on math itself. It rarely find real applications. These years, the Chinese government are changing the textbooks(yes, textbooks are dominated by a few publishers, and under great influence by the Ministry of Education) to make it more applied and filled with very lame cartoons.
The textbooks become colorful and more applied. But trust me, teachers don't teach those applied stuff, at least never in my class. They teach the pure mathematics. They talk in mathematical terms. Applied math problems show up on tests once in a while, but it's worded so easily, you have to be retarded to not set up the equations.

The teachers does go into depth into topics, but all the topics are created for the most important test for Chinese high school students--National College Entrance Examination(NCEE). How difficult are those tests? VERY. You have to know almost every possible method of mathematical object manipulation(vectors is a big part of the test)

The teachers basically doesn't teach ANYTHING ELSE except the ones important for the NCEE. Let's say, functions. Teachers will teach you 9 different techniques to solve values of a function. I can't remember all of them, because Chinese math class is 50% stuffing information into the brain. The condensed notes can take up a 60 page notebook. I suck at remembering, but all those questions requires a great handle of the techniques to solve them.

In China, not just math, the entire High school years is preparing for a test. The eventual application of math is for the test and the test only.

They are not allowed to use calculators for tests. How interesting that I can use a graphing calculator for my AP calc test in US.

Some NCEE problems are more difficult than AIME problems.

From what I see, the 100 getting Chinese students can basically solve any problem with-in the NCEE skill level, but I don't know about their strength outside NCEE material. For example, elementary number theory. But they are equipped with so much tools(they are the math cheat sheet and almost human calculators), it's not difficult for them to apply their technique on something new.

I believe the NCEE math overall skill requirement is not any where lower than AIME.

In US and Australia, the math education concerns about it's application more than China. They are always questions in an chapter ask students to solve a real world problem. But, still, it's very little compare to math classes for engineers.

US and Australia, math is basically for yearly final test and college related tests like the SAT andHSC
I don't have a lot unique things to say about the US math system. It's basically featured in A Mathematician's Lament

Why I don't agree with Lockhart and what math education should be like
First of all, I have strong disagreement with Lockhart's idea.
I don't believe math is an art.
Math is NOT an art. Math require creativity, inspiration. That doesn't make it art. Everything requires creativity and inspiration, should everything be art?

Notations are important.
Notation are used to explain ideas.
Natural language can explain ideas too.
But only students who can express idea with English should be ones good with math?
Why is notation ever invented? its used to make sharing ideas easier. People agree upon on certain notations, idea sharing can move beyond natural language.

Please express that in Chinese natural language.
Or start proving everything with English, lets see how long the proof of Fermat's Last Theorem can ever be generated.
In the old days, there isn't anything like algebra. Why isn't there? because there isn't a notation. Let's from now on, stop using notations. or everyone use their own notation... I will use "1231sda123asd" for the first positive integer. Someone else use "1aaasss"
Numbers are notations.
English is also a notation. Let's not learn English all together and use something more natural. Everything written is a notation. Let's abandon writings and find a new way to store notion.

Basic techniques should be learned.
Grammar are best learned while reading.
How about alphabets...?
How can one learn reading without alphabets.
Did anyone read English out loud by first reading novels then understand how to pronounce each alphabet?
Basic concepts have to be learned. There isn't learning unless there is something to learn. Lockhart is not suggest learning, he is suggesting researching and experimenting.

Rigors proofs can be helpful
Like me, I have problem understand words. I found rigors proofs are step by step, leading me to the final solution. That's what I like. I like ideas, but I like ideas I can understand.

Lockhart basically makes math education a intellectual conversation.
That's NOT education. That should be something out side education. That should be some small after school activity.

Math education should like the current one. Except teachers are more a observer than an instructor. A teacher gives student problems, students try to find solutions. If no one have an idea, teacher can give hints until someone find a solution. The process of finding the solution is done, so students actually practice their skills of exploring new areas or make use of what they learned before. Teachers can then expand the thinking process by adding informations on each step of the solution student demonstrated.
Everything valued currently should still be valuable. Include those definitions Lockhart believe should not be given directly to the student but make them find out themselves.
No, student don't find definitions. Definitions should be taught. Maybe with some additional info on why did mathematicians decide on a definition. There is no way people can communicate when definitions are too different.

That's what I believe math class should be.
Feeding students are definitely not what math is about. A class created just for tests is even worse. But make math an art and let all students create w/e they want is also what I don't want to see.

Someone should set up a trial. Let Lockhart(he is a high school teacher) design his perfect class and start feeding students in there and see what become of the students.