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Imagine a child aged 11. They have just finished primary education and now moving into secondary education. This child has shown a great mathematical talent/disposition since a very young age. By the end of primary school they have completed studies that a top tier 13 year old would be doing (in my experience (UK) primary schools are better than secondary schools for giving stretching pieces of work like this).

How should the school/parents deal with this situation? The problem with learning with children their own age is that they are unlikely to learn anything new in class for two years, may get bored with the subject, and lose the enthusiasm they had for it.

One option is that the parents (or a tutor) could teach them extra maths outside of school. However the child may resent having to do extra learning, and will still be bored in class. The other factor is, what is the end-game in all of this? To do undergraduate maths at an earlier age? I'm not sure that's a good idea, nor that you should be pointing an 11-year-old down such a narrow path.

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  • $\begingroup$ Is there any math competition the child can attend? $\endgroup$ – user2139 May 10 '16 at 7:42
  • $\begingroup$ There is this: ukmt.org.uk/individual-competitions/junior-challenge which is run via schools across the UK. It's a chance to show how talented the child is but doesn't really solve any of the issues I've raised. $\endgroup$ – WelshGandalf May 10 '16 at 9:47
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    $\begingroup$ If it's not "just maths", it might be possible for the student to skip a grade. $\endgroup$ – Stefan Schmiedl May 10 '16 at 13:04
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    $\begingroup$ Another option is to get them started in computer programming (that's about the age where I started and it seemed downright addicting at the time). Implement this as an applied exercise parallel to their school math. Can you implement the new procedure in code? Can you automate the homework exercises? $\endgroup$ – Daniel R. Collins May 11 '16 at 15:16
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    $\begingroup$ There are several points to this. In my opinion learning 'new' maths is not the answer. Deepening what you already know is often better. Interpretating what mathematical talent is should be done with caution. How would you define a stretching peice of work? $\endgroup$ – Karl May 11 '16 at 19:08
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Here in the U.S. there's been a rise in the last decade of "math circles"; extracurricular math clubs with students of the same age, with some amount of play/competitiveness to hone their interest.

Disadvantages: They may not solve the problem of being bored in regular class; they may be expensive (not available to the economically disadvantaged); and in some sense they take pressure off public schools to provide high-quality education (increasing inequity between rich and poor students).

The Atlantic: The Math Revolution

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    $\begingroup$ That link was a great read. I wish we had stuff like that over here in the UK. I'm almost tempted to start one myself. $\endgroup$ – WelshGandalf May 11 '16 at 11:45
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    $\begingroup$ @WelshGandalf: Perhaps you should! $\endgroup$ – Daniel R. Collins May 11 '16 at 13:12
  • $\begingroup$ @Daniel R.Collins Take the pressure off public schools is an unfortunate phrase. The pressure on schools is one of the reasons for the shortage in qualified teachers. I'd prefer teachers, parents and students working in partnership with collective responsibility. $\endgroup$ – Karl May 11 '16 at 19:16
  • $\begingroup$ @Karl: I notice you left out administrators who actually run the place. $\endgroup$ – Daniel R. Collins May 12 '16 at 3:59
  • $\begingroup$ @DanielR.Collins adminstrators, the government, any stakeholder. My point was pressure has an adverse effect on a student's education. $\endgroup$ – Karl May 12 '16 at 7:15
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I might disagree with several implicit hypotheses: that mathematics is only a school subject; that the there is a single linear course through it; that the main option is just the speed with which one goes through the standard curriculum; that contests ("competition") (invariably problem-solving with time constraints) are the main alternative; that some sort of traditional teacher-student relationship (with its challenge-response aspects, if not actually adversarial) is the way that adults can help kids.

How about the "adult" role being to find not-necessarily-textbook math books written by real mathematicians about real mathematics, to recommend to the student?

Although it might be nice to have a social aspect to a kid's mathematical life, all my observations indicate that this is difficult in many ways. For one thing, kids have been taught (implicitly or explicitly) that school is about competition in the first place, and mathematics perhaps especially so. But not everyone enjoys math-as-competition, and contest-math tends to be heavily caricatured math. Further, although it's nice to be quick, mathematics is not only about speed, etc.

And I'm not a fan of most textbooks, either, since they most often make their topics dreary and authority-bound, and more complicated than they really are. Or exaggerated worries and insistence about "rigor" in relatively trivial situations. (E.g., a rigorous proof of a boring thing is not nearly as interesting as a compelling heuristic for an interesting thing.)

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    $\begingroup$ I like the direction of this answer and would give +1 if there were some examples of math books that could serve this purpose. $\endgroup$ – Daniel R. Collins May 11 '16 at 2:03
  • $\begingroup$ Doesn't answer "what do they do in school when they are being taught things they already know" - but my feeling is from the answers thus far, that there is little that can be done in that regard. $\endgroup$ – WelshGandalf May 11 '16 at 11:44
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    $\begingroup$ @DanielR.Collins, by this year, I'm out of touch with what might be appropriate books. Decades ago, there was semi-expository writing by quite a few good mathematicians, often collected. Some current analogues might be Mircea Pitici (ed.)'s collections "The Best Writing on Mathematics". For that matter, looking at Courant's and others' writings from last century probably wouldn't hurt anything. Littlewood's "Miscellany"? Martin Gardner's collections? One can learn calculus from the 2 pages in the 1960 Compton's Enclyclopedia, and see immediately how amazing it is. :) $\endgroup$ – paul garrett May 11 '16 at 18:41
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Permit me to support paul garrett's response,

not-necessarily-textbook math books written by real mathematicians about real mathematics

with an unusual suggestion:

Skiena, Steven S. Calculated bets: Computers, gambling, and mathematical modeling to win. Cambridge University Press, 2001.

I read this book in one day, I found it so engaging. It uses quite a bit of mathematical modeling, but always in the pursuit of a practical goal (which he achieves): winning Jai-Alai bets against the odds.


          Carom
          (Calculated Bets, p.12)
A nice interaction between mathematics, statistics, gambling—and algorithms and practical programming.1 It could form a nice math-adult-supervised reading.
1 Skiena is the author of: The Algorithm Design Manual. Springer Science & Business Media, 1998.

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This happened to me at an earlier stage - in an attempt to keep me occupied before I was old enough to go to school, my mum did everything up to long division with me by the time I was 3. She stopped after that as I had other things to keep me busy, but I can still remember the crisis point I hit when we got to long division at school, because I wasn't used to there being something mathematical that I didn't already know.

There are lots of great specific answers here on things you can do, but the important thing is yes, you need to keep them learning and experiencing things they don't know, otherwise they will struggle when they are next out of their comfort zone.

Generally speaking, there are two areas you can explore - Applications and Theory. I know from my own education that school maths tends to be lacking in both of these areas, so looking at an area of practical maths (physics and programming are the obvious ones, but maths is everywhere!), or delving into the theory behind what they have already learned (axioms of arithmetic, real analysis) can provide an extremely useful grounding that makes learning more complex ideas a lot easier.

There's an awful lot of resources for both of these online - even good old Wikipedia tends to be incredibly accurate when it comes to maths - so it needn't be a task that takes up a lot of the mentor's time. Set research projects and ask them to give you a presentation/short essay on what they found - writing about maths and reporting is another area that isn't necessarily taught very well and is a very useful skill!

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