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This may not be a research level math question, but I believe it is still relevant to Math Overflow.

What general resources exist for students in highschool who are very interested in Mathematics? What advice would you give to a young student to encourage them, and nurture their interest in mathematics? If a young high school student came to you and said they were very interested in math, and wanted to know what to do to keep learning, what would you tell them?

Thank you for your help,

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    $\begingroup$ Have them learn to code algorithms. If they already know how to code, great. If they don't, it'll be great for them to learn. Analysis of algorithms demands attention to detail and mathematical thinking without an emphasis on proofs that might be off-putting to a high schooler. $\endgroup$
    – Steve Huntsman
    Commented Jul 26, 2011 at 18:41
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    $\begingroup$ @SteveHuntsman: it may not work for everybody: if a high school teacher had tried to make me learn to code algorithms, I think I would've never chosen to study maths at university level! $\endgroup$
    – Qfwfq
    Commented Jul 26, 2011 at 19:30
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    $\begingroup$ This question would be more acceptable on math.stackexchange. Alternatively, if the question was about more accessible areas of research or how much time research mathematicians could devote to mentoring such students, that would be a somewhat better fit for MathOverflow. Gerhard "Ask Me About System Design" Paseman, 2011.07.26 $\endgroup$
    – Gerhard Paseman
    Commented Jul 26, 2011 at 19:31
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    $\begingroup$ Gerhard, since the questioner is a regular math.SE user, I think his concious decision to ask this here rather than there should be accepeted. $\endgroup$
    – user9072
    Commented Jul 26, 2011 at 20:35
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    $\begingroup$ To follow up on Steve's comment, projecteuler.net is a fantastic resource for challenging but non-technical programming problems related to mathematics. Get them coding in a user-friendly language like python, and point them to that site. $\endgroup$
    – Simon Lyons
    Commented Jul 27, 2011 at 12:01

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This is an answer to your question,

What general resources exist for students in highschool who are very interested in Mathematics?

A patient teacher is the best resource for an interested high school student. Fundamentally what students lack is not access to mathematical content (cf. Wikipedia and countless books) but of mathematical thinkers. The high school curriculum is rigid and this steers high school math teachers into a pretty doctrinal way of thinking (bless those teachers who break out of the mold); consequently, the students follow along because they are not exposed to anything else.

The best thing you can do for a high school student is to show him or her how to think mathematically. Walk through problems with the student, explore until you find something you don't know and are curious about, ask the natural questions, and most importantly, try to set the student to up to ask the natural questions. Try to present things in a way so that you are not the lecturer who knows everything, but rather a fellow intellectual explorer.

I definitely agree with Steve Huntsman: any project should feature a computational component. Students today aren't afraid of computers, and strong programming skills are invaluable in both the science and business worlds. As Steve says, implementing an algorithm forces students to really think, and isn't entirely unlike mathematical reasoning. Plus, serious proofs tend to be out of reach of even most undergraduates, whereas numerical simulations can let any student get a hands-on interaction with a mathematical problem.

Let me demonstrate all of this with a timely anecdote.

I am teaching this summer at the annual Stony Brook summer math camp. After my introductory probability class today, a student came up to me saying she wanted to do research in mathematics. She had done some computer modeling of molecular dynamics in biochemistry, but I have no training in that area and even if I did, I imagine the mathematical models would be quite complicated.

Since any such model must contain some random dynamics (namely, diffusion), I decided to introduce random walks to her. I set up some definitions, went through some examples, and we played around with a simple, lattice-based model of molecular dynamics. After chatting for an hour or so, she asked the natural question: if we have a random walk on the 2-dimensional lattice, is it guaranteed to return?

The answer is given by Pólya's theorem: a simple random walk is recurrent in 1 and 2 dimensions, but is transient in 3 dimensions and above. But a theorem statement is not a satisfactory answer, and I knew that I couldn't explain the proof of the theorem at the level of an advanced high school student. Thus her research project was born:

  • understand the ideas behind an elementary proof of Pólya's theorem, and
  • use a computer to estimate $p_d(n)$, the probability that a $d$-dimensional random walk returns to the origin in exactly $n$ steps.
Voilà, a tailor-made project motivated by the student's interests, featuring both a mathematical and a computational component.

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    $\begingroup$ +1, but I disagree with the decree that any project should feature a computational component. Also, while it is perhaps technically true that students these days don't lack access to mathematical content, there is still a relative dearth of good books on real mathematics for high/middle/elementary school students. So another role of the teacher should be to identify and recommend the good books on actual mathematics. Good books can play a role similar or complementary to the role of a a good teacher. $\endgroup$
    – Kevin H. Lin
    Commented Jul 27, 2011 at 21:34
  • $\begingroup$ Nice answer! Thanks for giving such a great example of teaching as well. I'm wondering whether you could please update the link for Polya's theorem; the "an elementary proof" link no longer works. Thanks! $\endgroup$
    – Linus Rastegar
    Commented Oct 19, 2017 at 22:14
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There are a number of excellent summer math programs for advanced highschool students. I have my own opinions of which are better or worse, but a quasi-objective list of the programs you'd want to consider can be found at the MIT list of "other selective math programs.

In addition to the usual AMC math competitions, there's USAMTS and ARML. USAMTS is nice because it's not done in an exam setting. ARML is in person and so lets you meet lots of other math people. There may be more good competitions that have cropped up since I was in highschool.

Some internet resources include AoPS and math.SE. I don't know a good list of all the good internet resources.

Any other advice is going to be highly location specific. For example, in Boston there's the new PRIMES program. One bit of generic location specific advice is to see if the nearest good college has any programs for advanced highschoolers, or even just ask around the nearest math dept.

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  • $\begingroup$ I think, if I had to give three words of advice to such a high schooler, they would be "Go to Ross." $\endgroup$
    – Vivek Shende
    Commented Mar 12, 2013 at 5:18
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There are two outstanding resources in English: 1. R. Courant and Robbins, What's Mathematics?, and 2. Hilbert, D. and Cohn-Vossen, S. Geometry and the imagination.

Many of my friends (including myself) read these books in their high school years, and decided to become mathematicians.

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  • $\begingroup$ I agree completely. I would suggest also Euclid's Elements, and Euler's Elements of Algebra. These both start essentially from nothing and go past what most college math majors know well. The clarity of Euler's explanation of Cardan's formula is unequaled by any college algebra text I know of, and that's just part 1 of Euler's 2 part work. $\endgroup$
    – roy smith
    Commented Mar 12, 2013 at 19:32
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Mathematicians cannot be properly classified into middle school students, high school students, undergraduates, postdocs, tenure track job holders, etc. The learning advice depends on the person's knowledge, experience, and level of understanding of the subject, not on his formal position and career stage. (It also depends on what he finds interesting and aesthetically appealing, of course.) The mathematical level of high school students varies so much that no general advice based on the "high schooler" status is possible.

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    $\begingroup$ I agree with Leonid. The answer depends very much on the student. It isn't completely individual, as Leonid points out: there are different natural classes of students, and different things would appeal to (and be appropriate for) different classes. I am suspicious of universal answers. Except in algebraic geometry. $\endgroup$
    – Ravi Vakil
    Commented Jul 26, 2011 at 20:09
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    $\begingroup$ I appreciate the response, but this type of non constructive answer, while completely correct and reasonable, is not exactly what I am looking for. It is not too difficult to post on every advice oriented soft question and say that it is to broad to answer perfectly. Indeed, the best advice will always be specific to the individual inquiring, but this does not mean my above question is unanswerable. I have edited to add more specifications, and to (hopefully) put things more concretely. $\endgroup$
    – Eric Naslund
    Commented Jul 26, 2011 at 20:32
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For some such students (including myself, at that stage) the main thing is to have a good supply of interesting problems to think about. I never did competitions, but I worked through lots of Cambridge entrance papers. The Cambridge entrance system has changed since then, but there is still interesting stuff here: http://www.admissionstests.cambridgeassessment.org.uk/adt/step/Test+Preparation

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Any of the books in the Elementary section of Chris Jeris's undergraduate bibliography will be fun for a talented high schooler to look at.

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This only answers a small part of the question, but for secondary-school students interested in math I can't help but recommend C. Stanley Ogilvy's book Excursions in Geometry.

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Problems of http://www.itym.org

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