To clarify my below comment, I use your running analogy:

  - while two runners may run at the same speed, they may not start from the same position. I do not care who "wins" (is the fist) I care only how far they get at the end
  - unlike running, it is not immediately obvious who runs faster
  - running speed is not constant, but grows slowly. For some students it starts growing later but stronger and lets them close the gap
  - some students are behind because of an injury -- you cure them and they will be as fast as the others
  - unlike running, you are not alone

Of course there are guys that will never be up to the challenge. It is just quite complex to figure out which ones. Have you been to "class reunion" (is this the correct English term?) events? Which are the students who lived up to everyones expectations? Who were the ones that surprised you?

I personally think that assessing people is very hard, and most of us think that we are good judge of character. I keep a mental list on people I misjudged and that constantly reminds me how hard is to judge others.

There is a misunderstanding here. My original comment was a reaction to the comment that "most people don't need the math". Following that same logic you end up realizing that most people do not use any (or most) of the stuff their learned -- so we should not teach it.

Of course I do not agree with this conclusion at all.

Why don't we?

You see, lack of smartness is just one cause of struggling, but there are others. There are many vectors for a student and you will see them in very different mixtures:

SmartnessDumbness
ConfidenceLack of confidence
BraveryCowardice, passiveness
Hard workingLaziness
InterestedUninterested, skeptic
Sense of safetyWorries, depression
etc...

I am a teacher, not a fucking judge. Who am I to decide which students deserve hours of work and who don't? I rather leave it to life and I do instead my best to do whatever to teach them. And I am no idealist, I know that there are children as stupid as a rock, still, I give a chance at least.

Primary sources -- what makes them more reliable?

The fallacy you make is that "closing the gap" means "slowing down".

"Democracy does not work if people are not well educated."

Exactly. Also, the best way to learn is not necessarily to learn always the things most relevant to your current work. There is a reason why most of us like games, puzzles, riddles, even though they are not directly applicable to the real world.

"It even harms the kids who are good at math and want to do it because the teacher has to slow down for the kids who have no talent for math, aren't going to go into a math related profession and shouldn't be forced to learn about the square roots of negative numbers or quadratic equations."

Returning for this sentence for a moment: my experience is that the best teachers were those who were actually able to close the gap between the students with different abilities.

I agree. My personal experience is that courses, systems do not really matter. What matters are good teachers. If a school system is good, then it encourages the talented to become teachers and allows them to do their best.

"Most people don't need the math."

Let's be honest. Strictly speaking most people do not need ANY of what they learn in school except reading and writing and basic algebra.

The problem of history, economics and political science is that many of the sources are actually the work of "manipulative talking heads".

"Languages?"

I give you that one. You cannot have enough of languages (applies to programming languages, too ;) ).

"practical skills like woodworking or metalworking?"

Um, they actually teach those things in my country -- in primary schools definitely, and then in higher schools were the children not interested in math and such go.

"Grammar and history are probably more useful than imaginary numbers whatever else you could say about them."

Imaginary numbers are not taught under university (at least in my country), history is not that relevant for most students at all -- they do not make the mental leap from "boring past events" to the current world around them. Exactly the same problem that they have with mathematics.

"Which is fantastic if you're ever likely to go into a math related profession. For everyone else, far less useful."

That is true but I also had to learn a lot of stuff that I never used since -- it was a tradeoff for me, too. But as time progresses, students specialize more and more, so I do not see this as a huge problem. Also, it is quite common that young people do not realize what they really want to do at about their 20th birthday.

"To everyone else it's a waste of time which could be spent far better learning things which might ever be useful to them."

Exactly what? Grammar, history, geography, physics, basketball? Which one of these is important or useful?

In mathematics the basics are not about being directly important. They prepare your mind for the harder stuff. One of the basic things to learn is exactly that there are things that are NOT easily translated into direct day-to-day practice, but this doesn't mean they are useless. Mathematics is all about abstraction and manipulation of symbols.

On the other hand I agree with you that basic math courses need a major overhaul. Probability theory is a must, I do not even understand why they havent included it in the first place.

There is one problem, though. Achieving a really useful level of math needs about 15 years. Now trim the math from basic education and you are harming those who actually want to use it professionally later. It is like piano -- you have to start learning very early to be able to reach the top. While I understand that this increases the pressure on those students who will never use it, but I think that is an acceptable tradeoff.

I'm not saying that everyone in the optimization business is a fool! I do not even know, where did you get this idea that I think Linux kernel developers are Type 2 fools.

My observation basically is that there are just too many programmers who _think_ they are good at optimizing, but in the end they just mess up everything.

And yes, I am a Java programmer _and_ a C programmer. In fact I am working now on a real-time project and correcting the several mistakes the devs made thinking it improves the speed.

Speed is not about less assembly instructions. And I am quite sure that the Linux kernel devs know this.