How to deal with a talented 11-year-old pupil?

I see two options: acceleration or enrichment. Middle school is the time when acceleration becomes possible, at least in most U.S. schools. So, instead of Math 6 he would be taking Math 6/7, instead of Math 7 he would be taking Math 7/8, and instead of Math 8 he would be taking Algebra I or HS Math I in districts that have switched to integrated math.

• Acceleration serves two goals: it attempts to rid the student from boredom, and allows taking more challenging courses, or more of them. It is also possible to combine courses in one year, say Geometry and Algebra II in grade 9, then precalc in grade 10, then AP Calc BC in grade 11, and then something else in grade 12. AP classes can earn college credits, allowing to save time and money in college.

• Enrichment is best to be taken before high school, because there are fewer options for acceleration yet the schedule is not crazy. This is the time for math circles, for recreational math, for leisurely going into vectors and matrices, for presenting basics of derivative and integral, for studying mechanics and solving a bunch of word problems about a swimmer who swam up and down the river losing and then catching his hat and stuff like that. Also introduce basics of logic and proofs.

I think there is a benefit to having alternate, conflicting answers to soft questions like this. My take is to keep going with the kid. Just ask for him to go with the kids a couple grades higher, when doing math. If they don't have that because of the grade structure, than have him be independent study for a period (or see if high school is near the junior high and he can walk over for an hour).

Note that ADM Rickover in his famous book on American education made strong arguments AGAINST what you are hearing here (conventional wisdom to do "enrichment"). Basically his argument is that it is an inefficient use of the time of strong students. Provided they are mastering the material like a decent of age kid would, than their time is better used by advancing them. After all, you are not holding back the of age kids for a couple years to enrich them. And all the "wonders of enrichment" pale compared to the wonders of a whole new topic, like Feynman moving to calculus from trig.

Since you are in the UK, I have a few specific suggestions below. I agree with existing answers and comments that there is little benefit in rapid acceleration.

The Royal Institution hosts mathematics masterclasses in some areas. See https://www.rigb.org/education/masterclasses for more information. Sometimes the sessions are for students in specific age groups and sometimes local schools are asked to invite a small number of pupils, but it may be worth researching whether there are any masterclasses in your area.

The UKMT has lots of books suitable for extension material for able students of all ages. Some books are specifically aimed at improving in their competitions. See https://www.ukmt.org.uk/shop. All Junior Maths Challenge and Junior Maths Olympiad papers going back to 1999 and 2003 resepctively are freely available on their website at the moment. I think it is quite common for JMC/JMO questions to be used for extension material in schools.

I also highly recommend the Art of Problem Solving books. These are based on an American curriculum, but there is plenty of excellent extension material for students at all levels to delve more deeply into topics they have learnt. See https://artofproblemsolving.com/store.

I would avoid accelerating any further. Be aware though that even without acceleration a talented child will have taught themselves or be able to infer without being taught much of the syllabus. The problem withh acceleration is that while it exposes the child to new material it doesn't necessarily provide intellectual challenge. A very bright child wil thrive off creative problem solving - this doesn't have to require much knowledge but will need lateral thinking and imagination.

Definitely look into the maths challenges for their age groups and higher - if they're 11 now they can do the primary maths challenge and probably have a go at the junior ones too. I would encourage them also to learn some programming. Just a kahn academy course to learn the basics of a programming language (e.g. python) then they can try some Euler problems. There are places like the London SMP currently offering classes for maths and computing online which are outside of the UK curriculum.

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!

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.

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          ^{(Calculated Bets, p.12)}

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A nice interaction between mathematics, statistics, gambling—and algorithms and practical programming.¹ It could form a nice math-adult-supervised reading.

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¹ Skiena is the author of: The Algorithm Design Manual. Springer Science & Business Media, 1998.

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.)

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