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Let's assume that we also get to redesign our high school mathematics teachers as well, in the sense that we can assume that they know and can teach whatever material we choose to cover.

This is similar to these MO questions, although the first is more elementary school level, and the other two undergraduate.

I'm inclined to agree with much of Lockhart's Lament, on the rote learning vs. creative mathematics, although I would still worry about missing out on basic skills. On another issue, I know of at least two people questioning calculus as being the "goal" of high school mathematics, and advocating other subjects (Arthur Benjamin and less directly, Igor Rivin). Are there competing opinions?

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closed as not constructive by Andy Putman, Franz Lemmermeyer, Loop Space, Charles Siegel, Pete L. Clark Sep 29 '10 at 15:47

As it currently stands, this question is not a good fit for our Q&A format. We expect answers to be supported by facts, references, or expertise, but this question will likely solicit debate, arguments, polling, or extended discussion. If you feel that this question can be improved and possibly reopened, visit the help center for guidance.If this question can be reworded to fit the rules in the help center, please edit the question.

Man, the answers below seem a bit divorced from the needs of high school students. Remember that (at least in 2005) only 27.7% of Americans go on to complete college, and an even smaller percentage study technical subjects. I can't imagine that it would useful for them to study eg group theory or formal logic. If it were up to me, high school students would spend a lot of time on things like statistics. – Andy Putman Sep 29 '10 at 5:49
In fact, I am tempted to vote to close this question. Curricular development is something to which many math educators devote their entire careers. Answers that do not make reference to this large body of knowledge and history are not expert answers to this question, and the question is not phrased in a way that would motivate an expert to answer it. (You might as well ask me, "What are your ideas about teaching graduate level number theory?" Please be more specific!). If someone thinks this is too harsh, please let me know (and let me know why, of course). – Pete L. Clark Sep 29 '10 at 6:08
Pete, I think a bigger problem here is that the purpose of redesigning the curriculum has not been formulated. Hence anything can be an answer! – Victor Protsak Sep 29 '10 at 6:19
@Pete: Perhaps then MO is the wrong group of experts to be asking this to? I am actually interested in what competing opinions there are out there. To be specific, I can't recall talking to any mathematician who was strongly of the opinion that calculus is the best subject to put as the goal of high school mathematics. Does anyone think this? – Henry Segerman Sep 29 '10 at 6:28
I second Pete for the following reason: the answers given so far make my blood boil. I happen to teach mathematics at a high school, and too many people confuse "what I would have loved to have been taught in high school now that I am a graduate student in mathematics" with "what can you teach kids who sit in front of a TV screen or a PC for three or four hours a day and couldn't care less about the concept of a derivative". – Franz Lemmermeyer Sep 29 '10 at 9:42

I was fortunate enough to go to a high school that had a course called Further Math (if any of you have heard of the IB curriculum, this class covered most of the topics on the IB Further Math exam) where we did basic propositional logic, Peano arithmetic, basic group theory (up to normal subgroups) and set theory up to Cantor-Schroeder-Bernstein, plus some statistics. I loved this class, although the only new thing I saw all year was CSB, and I think students might like this more than something such as calculus that they only really work with in physics class.

I would say that every high school should start students with a general logic class - introduce the mathematical formalisms (up to truth tables) and then work with real-world examples of disingenuous statements or the like. I think that students very much like learning about how adults lie to them. :P

Beyond that, as much as I hate to say it, no student should leave high school in an ideal world without knowing probability and statistics (up to the basic properties of the normal distribution). It has become such a powerful tool in our times, and yet it is so easy to lie to people with statistics: take Twain's quote, "There are three kinds of lies: lies, damned lies, and statistics."

I guess the idea here is that most high school students will be distracted enough by learning about how adults are lying to them not to notice that they're also being taught how to think critically and how to pursue truth - and that's what mathematics is really about, right?

Edit: just saw Andy's comment.

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Probably a lot could be fixed if more people would read "How to Tell the Liars from the Statisticians"; it says much on how not to be fooled by "data pushers" with the barest minimum of math. – J. M. Sep 29 '10 at 6:11

I don't know about many other regions, but around here, it is a fairly common occurrence for students to enter high school not knowing how to add two fractions, so if we're going to rebuild high school, we may as well rebuild get them prepared for the new material while we're at it!

I agree with many of the points in Lockhart's lament, however I believe there is still a large necessity of getting students comfortable with all the basic objects, which can only really be done with a combination of exploratory exercise, and the standard "worksheet of 50 exercises". On top of this, I believe basic algebra should be moved fully to the elementary school level (no middle school around here, so elementary goes to grade 7) freeing up more space in the high school curriculum.

I think the basic concepts of most first and second year undergraduate courses should be introduced at different times. For starters, basic logic in an abstract setting in elementary school (students scoff at you if you ask them why one car being blue doesn't imply all cars are blue, yet when you replace the visual part of the example, with functions and variables, are utterly lost), and proofs at least in early high school (contradiction, contrapositive, why converse can not be used, etc..). Elementary number theory and simple counting arguments can be taught right away, with divisibility, congruences. From here, one could create two streams; One stream involving math for people who just want to be functional, and math for those who want to see math! We shouldn't force math onto students if they truly don't find it interesting, nor should we prevent the neat stuff from being taught just because it's not a topic everyone will enjoy. From here, with these basics complete, we could introduce groups (..and rings and fields) with plenty of examples available. From here, vectors and matrices over $\mathbb{R}$ could be introduced, to provide even more examples of groups, while also displaying many simplified examples of real world situations using linear algebra. The idea behind the construction of $\mathbb{R}$ from $\mathbb{Q}$ (though no need for rigour) and the algebraic closure of $\mathbb{R}$. Again, no great rigour needed, but see if they can find something "missing" in $\mathbb{C}$ that may perhaps lead to another space. It would also be nice if throughout all this we could engage the students in "small" computational experiments.

All the while, it would also be great to explain what's out there in math. For example, mention there is an object called an elliptic curve, which just so happens to be a group if you look at it right. These curves are being used in modern cryptography, proof of Fermat's last theorem, and have many puzzling features to be discovered. Any high school student could understand that sentence, and it gives an idea of why we are doing this! (During my undergrad, one of my old friends from high school was actually under the impression math had been solved, since we were never lead to believe otherwise!!)

Now, this is certainly a perfect world I'm describing. Obvious problems I see are -If we are to teach more advanced concepts (well) in high school, then high school teachers need to be comfortable with these concepts. Preparing all our teachers for this is no simple task. -Even if a curriculum with all this were installed, with teaching staff ready to go, the enrollment would surely not be very high. If you tell students they can get through life perfectly well with 'easy' stream, you would have a hard time convincing most students (apart from scholarship hopefuls and those already fond of math) it would be worth their time to learn all this extra material. With enrollment very low at smaller schools, the extra funding required would be very difficult to justify to government officials with no background or fondness in these concepts.

Edit: As I'm just seeing other comments now, I feel I should mention probability and statistics are covered were covered in my high school curriculum, however if they are not being covered, this would fall in both of my mentioned streams! People should definitely know how to read articles in the newspaper with study results.

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Your point about telling students there's more math out there than what they learn in high school is important - lord knows I got a lot of, "so what are you taking next - calculus 15?" questions. And then there's the story of the student who, upon learning this, asks, "Why? The math we already have is hard enough!" – dvitek Sep 29 '10 at 23:20

I think basic group theory is a great thing to add early. I believe it develops abstract thinking much better than current high-school math subjects.

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In a perfect world, students would be exposed to formal logic and elementary proof theory. How can one meaningfully critique any media if one cannot negate a proposition, argue by contrapositive, etc.?

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The kind of logic needed in understanding critiquing arguments on non-mathematical subjects is much wider and less clear-cut than the formal logic appropriate for mathematics. There are few things more painful to hear than a smart mathematician applying mathematical logic naïvely to a real-world problem, and heading straight off on a highway to absurdity; to non-mathematicians, it just reinforces the worst stereotypes of our pedantry and disconnection from reality. – Peter LeFanu Lumsdaine Sep 29 '10 at 14:59
Be that as it may, a first course in mathematical logic does much to untangle knotted intuitions about what it means to argue. The number of students who cannot perform simple logical operations is astounding. Present an average high school student with the following scenario: "I claim that any time it is cloudy, it will also be rainy. What kind of evidence would prove me wrong?". In my experience, many will respond "A day when it's neither cloudy nor rainy." A student who cannot handle a situation when the solution IS clear-cut is certainly not ready for a non-mathematical setting. – Austin Mohr Sep 29 '10 at 18:54

I believe the basic notions of abstract algebra like groups and rings can be given at the high school taking the Integers as prototype. And since the students at high learn about the equations of geometric objects like line, circle, ellipse, etc., I am sure with utmost care the seed of Algebraic Geometry can also be sown. Other than that, I think we need to put more emphasis on theory of functions and calculus (including some proofs,if possible) and geometric insights.

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