What should the high school math curriculum consist of? "Life is open book."
With the advent of widely accessible, inexpensive (or even free) computational tools and Computer Algebra Systems (TI-89, Wolfram|Alpha, etc.), much of what traditionally comprises a high school math curriculum can now easily be done by almost everyone.  Factoring polynomials, solving inequalities, graphing linear equations, differentiation and integration -- these are the types of skills high school math students spend most of their time learning, and yet all of it can be done for free by anyone with a web browser.
What does this mean for the high school math curriculum?  On the one hand, we could leave it more-or-less the same, insisting that today's student learn what we learned decades ago, while banning or carefully regulating the use of these new tools.  On the other hand, we could embrace the tools and the opportunities they create to spend more math class time on different topics and skills, perhaps focusing more on analytic and synthetic problem solving and less on mechanical symbolic manipulation -- but at the risk of students never learning some basic foundations.
So how about it?  Binomial coefficients?  The angle-addition formulas for trig functions?  The conditions under which a function has an inverse?  Basic computer programming?  Keeping in mind that the vast majority of high school students do not go on to become professional mathematicians, what should the high school math curriculum consist of?
Btw, I post this question (inspired by this discussion) here because this is a community of thoughtful mathematicians.  I recognize this discussion may belong in a different forum, but I don't what/where that forum is.  Any suggestions are welcome.
 A: I strongly believe that even if something can be done by calculator, a high school student should understand some nuts and bolts of how it's being done. 
I spent some time doing volunteer tutoring at CRLS (Cambridge Rindge and Latin School). I still remember the girl at 8th or 9th grade who was studying inequalities, yet she couldn't compute $2-1$ without the help of calculator. The most frustrating case was the kid in 9th or 10th grade who didn't know multiplication tables, he couldn't compute, say, 3 times 7 in his head. I was supposed to help him with square roots. Well, how do you explain that $\sqrt{8} = 2\sqrt{2}$? First of, you have to realize that $8 = 2 \times 2 \times 2$, and that's exactly what he couldn't do. I ended up spending some extra time with him, afterward been told that he passed the test or exam and was very happy with my help - but I wasn't! I felt we were wasting lots of time and just because he didn't know multiplication tables, he couldn't break down numbers into factors, and calculator wasn't really help. 
The bottom line: I think in school students still should learn basic arithmetic before moving to more advanced stuff. I think it's a pure idiocy to introduce the kids to set theory while having them relying on their calculators to compute $1 + 1$. Honestly, the advanced stuff can wait till college. 
A: I would include more probability, such as how to conduct a simple probabilistic experiment using statistical tests. Once you understand that, you can do a lot by simply looking up various tests. Without understanding probability, people are easily manipulated using statistics or make poor choices.
I would include definitely include more use of technology, including Wolfram Alpha and basic mathematical programming. These tools allow people to achieve much more.
Some basic mathematical philosophy would be good as well. Game theory can be very thought provoking and is quite simple as well. We could look at other areas even though we wouldn't be able to examine them rigorously. I think students should understand that maths is based on axioms, that not all maths problems can be solved (assuming ZFC is consistent) and that many problems can't be solved efficiently (P vs NP).
I would focus more on exploring various mathematical puzzles or unexpected results: Card doubling paradox, 3 hats puzzle, Monty hall problem and other results like this.
A: I spent over a decade thinking that "mathematics" is about completing page after page of identical long division problems. Only after I left school did I discover that mathematics actually has more to it than that. (And boy, am I glad I did!)
My opinions on the matter:


*

*I think there will always be people who like mathematics, and people who don't. However, I do think the number of people who "like mathematics" could be drastically increased if it was taught better.

*I don't think it's super-critical exactly which particular mathematical topics you teach. I think it is super-critical that you teach people what mathematics actually is. (A surprising number of people don't understand this.)

*I do not think that modern technology somehow makes understanding how to do X or how to do Y by hand "obsolete". Nor do I subscribe to the idea of "banning" calculators, computers and so forth. By all means, use the technology. (But don't let us fall into the trap of just blindly pushing symbols around without comprehending their meaning.)

*I think it might be useful to teach a wider range of mathematical subjects. Not in exhaustive detail, obviously. In many cases, that requires a whole heap of complicated ground-work. But introduce the interesting ideas and explore the main properties.
Personally, I don't see why you couldn't do something like group theory with school-age children. (Obviously, we're talking older children here.) "You know how to add and subtract, right? OK, so let's just throw the rule book out the window and invent our own brand-new system of adding and subtracting objects. Hey, look, we invented this from scratch, but now all these interesting properties keep appearing..."
My other pet idea is to take some of the computer software out there which does stuff with user-defined functions. Let the kids play with it, and see what neat stuff they can come up with. If you just type random gibberish, nothing interesting happens. But if you understand coordinate systems and functions and have an intuition for how simple mathematical operators affect numbers, you can make the computer draw trippy stuff.
Perhaps the biggest problem with mathematics is that most people don't understand it. Many people think that mathematics "is about numbers". (That's like saying that science "is about test-tubes"!) Many think that mathematics "is one subject". (Again, that's like claiming that science is only "one subject".)
The way that I do mathematics is that it's this fascinating journey of exploration, experimentation and discovery. But the way the textbooks do it is "Here is a type of problem. Here is the exact procedure for solving this type of problem. Here are 300 identical problems of this type. Go solve them. Your teacher will then compare your answers to the published answer book, and give you some ticks." Wow, how thrilling. :-P
Having said all that, we do seem to currently live in an age where being stupid is something to be proud of. This doesn't just affect mathematics. How to solve that larger meta-problem, I have no idea....
A: In high school in the 1970s I got tremendous benefit out of a one-term elective course called Mathematical Logic, which was nothing fancy, just basic prepositional stuff.  Understanding the basic structure of logical arguments empowers one to learn anything more efficiently. If college-bound students could ditch half of their AP Calculus for a course in Logical Reasoning that includes basic material from Daniel Velleman's book How to Prove It and some basic facts about statistics, they would be in a far better position to pursue many subjects, not just math.
A: I recall the Asimov short story about humanity that has completely lost the knowledge of how to do simple arithmetic without use of a computer, called "The Feeling of Power". The point is, despite computers being able to do these things for us, it is still important and valuable to know how to do simple operations. At times, we are amazingly close to that point in this very day, certainly in the current crop of kids growing up now.
Basic mathematical skills are tremendously valuable in a variety of places. Just because you CAN factor a polynomial by computer does not mean that at least understanding the tools to do so are no longer necessary for 99% of the people in the world.
A: Students should learn math. You know, the stuff that most of us do. When we use the word "math" we certainly don't have in our minds adding, multiplying, factoring polynomials, or even calculus. So why do we teach children that this is what math is?
This view has been in several published places like Devlin's The Math Gene. He makes a good argument that the concept of a group isn't any harder than other stuff done in grade school. The problem is that it is introduced later.
The best thing I've read about math education is A Mathematician's Lament by Paul Lockhart.
A: Whatever you think the high school mathematics curriculum "should" be, in the United States a curriculum has been put in place, known as the Common Core Standards (CCS), which will significantly - I believe - change American mathematics education. (The CCS extend to all K-12 mathematics.)
http://www.corestandards.org/the-standards/mathematics
I don't think these changes are for the better. A short summary of my personal views are available here:
http://www.education.umd.edu/MathEd/conference/vbook/public-perceptions_Malkevitch.pdf
A: I am surprised people advocate the white box approach to learning math. In computing, it is imperative from programmers of lower layers(operating system, compilers) to hide their code internals to higher layers programmers. I don't see anything wrong with hiding low-level math from kids through the use of CAS(computer algebra system) and mathematical programming languages. In fact it need be such low level "algorithms" can be taught using such CAS eg "Write an algorithm that performs long division". Math assimilation has been regressive because of old methods of teaching which were clearly paper-based before the advent of computers. Einstein had to hire an "assistant" who acted basically as a computer to grind the numbers. My argument is we have computers now and let us incorporate them in learning "high" math which matters. In fact, low-level math has no applications in the real world which makes sense to relegate it to the computer while kids are taught how to use math as professionals do. 
My ideal syllabus for high school, assuming they have been introduced to computer earlier on, especially programming lessons. 


*

*discrete mathematics (binary, Boolean algebra, logic)

*recursion theory (theory of functions, classifications of functions, graphing using computer)

*history of mathematics 

*calculus (allowed to use computer) 

*mathematical modeling. 

*computer programming. 

*linear algebra (matrices)

