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Everybody who is in graduate mathematics had a moment where they realized that mathematics was "their thing", and they decided to dedicate their academic career to it. I don't know of many people who decided this before their fourth semester or so, but I do know a few who have had a life dream of being a mathematician since high school (they went to some charter school or something). For me personally, I entered undergrad as a physics major, and when I took my first abstract algebra course (as a just-for-fun elective) in my fourth semester, I loved it, I felt this feeling of "How have I not seen this [abstract mathematics] before?" I had had 13 years of mathematics education - from first grade up elementary school through three semesters of calculus and differential equations in my first two years of college - and never once had I even heard of these abstract mathematical structures - groups (-> universal algebra, free term algebras, commutative algebra), topological spaces (-> algebraic topology -> category theory), analysis (-> hardcore set theory), combinatorics (-> finite set theory)... etc. etc. etc.

Why? Why had I never seen these before? Was it because I wouldn't have understood them before? I certainly don't think this is true. I think as early as middle school, but definitely by 9th grade, if I had been introduced (in a suitable "fun" way, perhaps) to basic set theory, groups (set with an operation), topology (set with some sub-structure), combinatorics (playing with finite sets)... I would have understood these, I think. The standard two-semester undergraduate series in these topics, maybe with a slight slow-down to accommodate the lack of "mathematical maturity", could be understood by an (advanced) high schooler.

I have heard of examples where a student's prodigous interest in math in middle school will get recognized by a teacher and they will get put on some magic track to math success - they will design a cross-curriculum with the local university which allows them to spend the mornings doing their rote high school graduation requirements - biology, history, etc. - but in the afternoons they will go to a class or two at college - usually number theory or something basic like that. In the extreme (e.g. Gatton Academy) the students are effectively attending college for the last two years of their high school career.

But this system is obviously completely ad-hoc cannot reach everyone. Back to me personally, I never met a single "real mathematician" until I came to college. I was a "smart kid" - I got a 5/5 on the AP Calc test, 35/36 on my math ACT, 800/800 math SAT, etc. I did all the things "smart kids" are supposed to do - in particular I was a member of the math club at my high school. But, what happens in a math club? Who runs the math club? Who selects the topics? I'll spare you the sob story: it was essentially an engineering club. All of the fun activities were fundamentally engineering - launching rockets, designing bridges, etc. etc. I did not learn any math in the math club.

I entered college and decided to be a physics major. Here for my first two years I took calculus 1-3, diff eq, matrix algebra. Even in these classes, it felt very rote and "computational" (possibly because this university's engineering department is so big that it dictates the curriculum of the calculus courses since 75% of the students taking them are in the college of engineering...? just a crackpot theory). The professors would drop subversive hints sometimes - "They don't really want me to teach you guys this, but I'm going to spend today talking about the epsilon-delta definition continuity. This will not be on the exam." - but even then, I only got the feeling that there was some secret world behind calculus in particular - still no inkling that there existed a grand structure of abstract mathematics underlying everything.

Anyway, all this is a vague rant to suggest that somehow there should be opportunity for smart, interested students to access "deeper depths" of mathematics - to know that mathematics is a field where people still do research today, and that, bluntly, this shit goes mad deep. Ideally at a very early level - the earlier you start, the further head-start you have on your finite lifespan. I am already 21 years old! (Galois already made substantial advancements in mathematics by my age, and even had enough free time remaining outside of mathematics to die!) If I had learned about this stuff in high school, I could have had a 2-4 year head start on where I am now! It makes me feel stupid by comparison to "what I could have been". I feel like a large part of my life was wasted (not the point of depression, of course, but it is a nagging thought I've been having recently).

[Of course, it could be argued that such a structure would be impossible to implement without forcing the "uninterested" students through so much math that "they will never need in everyday life". After all, only a very small percentage of the world's children grow up to be abstract mathematicians.]

[Hypothetically the argument could be made, "Seventh graders just aren't ready for abstract math - they still have to learn about adding and multiplying and fractions and slope of a line in the plane." I don't believe this for a second. All it takes is a sufficiently strong teacher-student connection (which can often be obtained by simply a good enough or passionate enough teacher).]

What is the education-theoretic position on this phenomenon? Are there any avenues in place to combat this sad occurrence?

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    $\begingroup$ It would be better if you could distill the key points of your post and edit them to pose a more compact/specific/tractable question. For example, you could essentially remove the paragraph that begins: "Anyway, all this is a vague rant..." and, more generally, it would be best to remove parts that satisfy either "vague" or "rant-like." You may wish to read through a few other questions on the site (if you haven't already) to get an idea of how they are written-up. (I'll delete my comment if/when your post is edited!) $\endgroup$ – Benjamin Dickman Nov 21 '15 at 9:24
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    $\begingroup$ To start, we could at least require high school math teachers earn an undergraduate degree in mathematics. That would at least give a student like you hope to get some guidance. Unfortunately, that seems like an unlikely development in the US, hence, I am afraid the internet is the bright student's only hope. But, frankly, to the curious, You Tube and other online course materials will allow the curious to stupidly outpace those who force themselves to follow the antiquated path of a "liberal arts education". Only catch, you gotta get into graduate school at some point... $\endgroup$ – James S. Cook Nov 21 '15 at 19:01
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    $\begingroup$ "I could have had a 2-4 year head start on where I am now! It makes me feel stupid by comparison to "what I could have been". I feel like a large part of my life was wasted..." -- Ok, so this is a sentiment I often struggle with myself. But the truth is that 2-4 years is not a huge difference. At this point I've met some people (not math) who went to college at age 16 and they've done literally nothing with their life. So it's not a magic bullet. $\endgroup$ – Daniel R. Collins Nov 22 '15 at 0:19
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    $\begingroup$ @DanielR.Collins indeed, a good point, if the advanced study is not properly coupled to a likewise nonstandard undergraduate it often leads to malaise. I saw this in several governer's school students I knew in my undergraduate. The university saddled them with unnecessary prereqs. which squelched their ambition. Of course, the best of such students will find a way around or through... $\endgroup$ – James S. Cook Nov 22 '15 at 13:53
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    $\begingroup$ @JamesS.Cook i don't think that requiring high school math teachers to have an undergraduate math degree would help the situation. Most math teachers that I know, including myself, have backgrounds in computer science, engineering, physics, etc. A very small part of being a high school teacher is content knowledge and adding this requirement to the long list of things that teachers need to get certified would only dissuade potential teachers even more. The real problem is finding teachers who are passionate enough to spend their free time to learn beyond their curriculum $\endgroup$ – celeriko Nov 23 '15 at 15:20
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Math circles are an attempt to bring the beauty of math to young people. My book, Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is an attempt to bring the joy of math to those who don't yet see it.

Many of us work at this problem. We will not reach everyone. There will be people who miss out for too many years. Teaching basic math in ways that relate it more to the beauties of abstract mathematics is possible. But it takes math teachers who enjoy math. I see many of them online, but they are not a representative sample.

I saw the beauty of math when I was young through logic puzzles and a book titled Mathematics: A Human Endeavor. But my introduction to higher-level math (at the University of Michigan) was unpleasant (too hard, proofs too long and unconnected to anything I deeply understood), and made me think I didn't like math. I am very lucky that I fell back in love with math at a less elite school (where I got my masters' degree).

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    $\begingroup$ Great post, and amazing that you should mention Jacobs' Mathematics: A Human Endeavor. We've been using that for our liberal-arts course at CUNY/Kingsborough for a number of decades. Apparently just in the past few months it ceased publication and we need to switch to another book. (Top proposal: Burger/Starbird's The Heart of Mathematics). Btw, I'm also "Delta" on the AngryMath blog. $\endgroup$ – Daniel R. Collins Nov 22 '15 at 0:25
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    $\begingroup$ Hi Daniel! How many students take that course? Couldn't you just get enough used copies to make it work? I wonder if the rights reverted to the author, and if he'd let you make copies through lulu.com or something? $\endgroup$ – Sue VanHattum Nov 22 '15 at 0:40
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    $\begingroup$ Good question. Traditionally most of the students at the college have taken that course (everyone who's not a STEM major, basically), which is lots and lots of books, and the college doesn't have any general way of handling out-of-print books. Unfortunately it hasn't been updated since 1994 and there's no e-book option or digital support or testbanks, even (a few years ago I wrote to the publisher and offered to make the latter and they declined). From a quick search I don't see any way of contacting Jacobs to ask that question. $\endgroup$ – Daniel R. Collins Nov 22 '15 at 4:16
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A lot of baby set theory was introduced into math in the 50s, 60s (New Math) but it did not have a good outcome for students IN GENERAL and so we went back to the traditional methods of 1900 or so. Which actually themselves had a lot of hard earned pedagogical evolution behind them.

Practically speaking rather than pushing abstract math to all kids, I think the answer could be little books or side activities or the like. But that said, it is still for most people easier to have a base of traditional skills and appliable classes (calculus and trig for physics, etc.) before getting into proofs and more abstract, useless math topics. Math maturity and all that. And even then only a very small group will really like the more pure math.

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