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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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    $\begingroup$ 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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    $\begingroup$ Could you elaborate on what "top tier 13yo would be doing" is, like specific, concrete examples? Say, does the child knows what a derivative is (maybe not a precise definition, but at least can spot relations like the one between distance/position and speed)? $\endgroup$ – dtldarek May 16 '16 at 7:02
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    $\begingroup$ BTW, In my opinion you should be cautious with much more advanced material, see this question. Perhaps introducing a few new concepts (like derivatives) is necessary to get into more engaging stuff, but do not do university-style classes with them (as some try to do). $\endgroup$ – dtldarek May 16 '16 at 7:03
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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
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    $\begingroup$ @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
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    $\begingroup$ @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
  • $\begingroup$ Strongly agree. Contest Math is in my experience a complete waste of time unless you care about contest Math itself. Studying deeper into subjects that do interest the student is a much better alternative $\endgroup$ – David Jul 13 at 11:58
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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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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.

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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.

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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.

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    $\begingroup$ I disagree. Enrichment does not mean holding anyone behind. You might find some of the essays in the appendices of this report interesting. $\endgroup$ – A. Goodier Jul 8 at 10:15
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    $\begingroup$ If I use X amount of time on enrichment, I am not using that X time for advancement. Maybe the enrichment is better. Maybe the advancement is better. But they idea that life lacks trade-offs is very non-strategic. Think about your lecture time. It is a fixed bucket. If you spend it on one thing, that doesn't allow doing other things. Unfortunately, life is an optimization problem and the hours are fixed. $\endgroup$ – guest Jul 8 at 15:26
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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.

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