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Oh, lots of things.

 

Personally I would do things at home outside the normal core curriculum for math majors, specifically avoiding subjects such as a rigorous course in modern algebra/real analysis as the school is unlikely to grant advanced credit. Rather I'd look at courses which are generally electives, that will greatly enrich his mathematical breadth and depth. 

 

There's some advanced elective books listed here: http://www.artofproblemsolving.com/Wiki/index.php/Math_books -- you'll notice many books from the Anneli Lax New Mathematical Library of the MAA, and the books listed by Gelfand are excellent. The Anneli Lax has more than are listed -- for example, an elementary introduction to topology. 

 

Another option could be looking at introductory university texts from other parts of the world -- for example, I really like Alan Beardon's Algebra and Geometry. It links linear algebra and group theory in a text for first-year undergraduates in the UK. One of the issues I see with American undergraduates is that too often they complete a modern algebra class with little understanding as to what it's actually used for, having become lost in the details. This book fills in these gaps and would be a truly excellent prequel to a serious course in abstract algebra. 

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And a pedagogical question: AoPS is good for having a lot of problem solving. But what about other things? What about writing things out properly? What about (even) more theoretical approaches, understanding theorems and proofs, and knowing how to prove things properly? How about trying some research? When do you worry about these things?

 

I found that AoPS does a very good job at introducing proofs, even in subjects where traditional curricula do not contain proofs (with other curricula, proofs are basically limited to geometry, but AoPS proves pretty much everything).

 

As for writing things out properly: I have taught my kids how to document their math work, beginning from algebra. I require this, and after a while, it becomes completely engrained to have only one equation per line, to show all work, to start with a figure if applicable.

I would think with two math parents, this should not be a problem at all.

 

And the calculus text is rather theoretical; I found it definitely a notch up in terms of formality of approach, and it does include many proofs. I have found it to be more theoretical and rigorous than some popular college calculus texts I have seen.

 

I have no suggestion for what you should do after running out of AoPS other than: worry about this in a few years, not now. Your student may slow down, may want to take classes at the university for other reasons, may NOT want to attend any live classes, may have developed a different area of strong interest and wish to focus on that one.

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I have no suggestion for what you should do after running out of AoPS other than: worry about this in a few years, not now. Your student may slow down, may want to take classes at the university for other reasons, may NOT want to attend any live classes, may have developed a different area of strong interest and wish to focus on that one.

 

Exactly.  We had the same issue for DS12, and we've found plenty of diversions.  Kiselev's geometry books, symbolic logic, discrete math, and even elementary abstract algebra work well.

 

At 9, the mathematical maturity just isn't there yet.  Your child may do the problems well now, and need some review.  It's way too early to tell.

 

Take it year by year, and see how it goes.

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I was adding up years and was curious how you are getting through all of the AoPS in four years.

 

If you are currently in Algebra, then:

10: Geometry

11: Intermediate Algebra

12: PreCalculus

13: Calculus

 

Has your son already completed both the C&P books? Number Theory? The Basics & Beyond books (these contain a lot more analytic geometry and stats as I understand.)?

 

Maybe I am misunderstanding. My son is one year behind yours and we chose AoPS for the exact reason of not running out of curriculum. My son is not very math intensive, however, so perhaps yours has flown through more of the books.

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When you have a kid who is really not "standard" in their development, then just try and take math planning just one year (or one term) at a time. AoPS is different from many other curriculum in the math world so I wouldn't bank on any kid just making it through in any particular amount of time. But, if *your kid* is making it straight through AoPS books still by the time that he's about to hit their PreCalculus book (are you planning on doing both of the Counting and Probability books and Number Theory?) then I'd begin to look ahead into some other options. Not before.

 

For mine, I've decided to go a different route-we've finished elementary mathematics, we've dabbled in Algebra and it went fine but we've declined to use AoPS at this time for a number of reasons. When they are a little bit older, I'll re-consider AoPS but for now, its not a viable option for them to use it. I've decided to wait until the boys are 10ish to look into AoPS. And unless something changes drastically, then that gives us 3+ years until they are ready I've only really thought about 2015 since its looming ahead of us and I want to see how they do with some "alternative" math books, so this is what I'm thinking of doing for 2015 and how I have, very crudely, broken it up. We do 2 - 3 hours of math a day, 6 days a week when we can.

 

Unified Modern Mathematics Course 1, Part 1. --- January - June

Hard Math for Elementary + Workbook.  -----------  January - June

Comprehensive Mathematics Practice Series  --  January - June

Mathematics: A Human Endeavor by Jacobs.   --  January - June

 

  • Elementary  Challenge Math...........................January - Mid-March

  • Upper Elem Challenge Math............................Mid-March  - May

  • Challenge Math.................................................June - September

  • Becoming a Problem Solving Genius.............January - September

  • Real World Algebra..........................................October - December

Essential Mathematics books 1 and 2 ------------------ July - December

Unified Modern Mathematics Course 1, Part 2. ---- July - December

 

Mental Math --------------------------------------------------- January-December

Art and Craft of Problem Solving.  January to December

 

 

Certain materials I hope will remain in our math-rotation for a few years.

Unified Modern Mathematics Courses 1-3 are online for free and I've downloaded them all and will be having them printed and bound as we progress. I intend for the boys to use the whole series over the next 3 years as it is very broad in its scope and sequence and what we've used so far the boys have enjoyed greatly. Next year we'll be more consistent in using it and trying to finish the whole first year.

The Art and Craft of Problem Solving by Zeitz is a great book to learn and practice solving tougher problems. I'm planning on keeping it around for 3-4 years for my boys to use and study from. 

How to Solve It! by Polya

 

I am hoping that 2015 will be the year that I begin to transition the boys into thinking harder and longer about problems that are less and less straight forward, I intend to use Zaccaros' and Ellisons' books since they are aimed at having kids problem solve, but will also be keeping Zeitz a constant presence.

Also, I intend to keep their basic skills sharp--and expand them--by using the two series from the UK Comprehensive Mathematics Practice --6 workbook series, they can do one book a month starting in January and then use Essential Mathematics--a 2 book series, they can have 2.5-3 months per book, starting in July.

 

I'll start to worry and wonder about 2016 around November 2015, for now I've got to figure out something for Economics and Civics in 2015.

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I was adding up years and was curious how you are getting through all of the AoPS in four years.

 

If you are currently in Algebra, then:

10: Geometry

11: Intermediate Algebra

12: PreCalculus

13: Calculus

 

Has your son already completed both the C&P books? Number Theory? The Basics & Beyond books (these contain a lot more analytic geometry and stats as I understand.)?

 

Maybe I am misunderstanding. My son is one year behind yours and we chose AoPS for the exact reason of not running out of curriculum. My son is not very math intensive, however, so perhaps yours has flown through more of the books.

 

Some kids do more than one per year.  DS12 can do ~3-4 in a typical year.

 

Here's a more thorough list for the extreme learner:

 

AoPS Prealgebra

AoPS Intro to Algebra

AoPS Intro to Number Theory

AoPS Intro to Probability and Counting

Kiselev Geometry (Planimetry and Stereometry)

Gelfand Algebra (the whole series, including functions)

AoPS Geometry

Symbolic Logic

Trigonometry

AoPS Intermediate Algebra

AoPS Intermediate Probability and Counting

AoPS Precalculus

Discrete Mathematics

AoPS Calculus <-- actually, I still prefer Leithold, which is beautifully presented

Linear Algebra

Abstract Algebra

Differential Equations

Probability and Statistics

Statistical Inference

 

That'll keep the kiddo busy through middle school.  :)

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I would be concerned about the ability for the knowledge to really be applied as well as densely absorbed if my student was covering three to four of these texts a year. Much like early readers "word calling" instead of truly reading, many kids can perform operations early, but not grasp the significance of the work they are doing. As Gil's concerns above for his younger kids with starting AoPS early and their mathematical maturity.

 

Stacking on courses is not equivalent to the intricacies of mathmematical reasoning and formulating a cohesive picture of the ideosyncricies of the mathematical world.

 

Some kids become obsessive, and I completely understand those that can intuitively understand the conceptualization of mathematics, but again I would be concerned with being able to fully grasp the implications. This is most important if the student is wanting to go into mathematics as the OP was thinking for his son. The side interests and diversity of topical implications becomes even more problematic than for a student like my son who is definitely not STEM.

 

So the student has covered a pile of texts, but at what loss? Does that math have true meaning in the world, in the life of the student? I was a math freak studying group theory at an elite university before 15. Any of our kids could get geeked out on group theory as it has virtually no necessary prerequisites, but the problem becomes that the awe inspiring quality of group theory is how relational it is. It was delicious and exciting for me to be stretched with such lectures and problem solving until a theology professor came. One of the most painful experiences of my life when the visiting professor spoke of group theory confirming his belief in God. The man was brilliant. He used the unifying and recurring matrices, the abstract patterns of algebra, polynomials and dozens of examples from nature to paint a completely different picture of mathematics than I had ever witnessed. He showed me that math breathes. That pure math is a study of life, not numbers. That every moment of our lives is surrounded by fractiles in patterns my brain cannot even comprehend because of the infantismal and at the same time prodigious levels they are reoccurring at. I was too young to really understand my place in the universe, to have wrestled with such philosophical questions, to have really had to evaluate who I was. I was too young for Group Theory. I would argue that the philosophical questions of what place do we have in the universe, who am I, what does God look like are the basis for mathematics. There is no text book that can teach such things. Only life can teach such things.

 

.99 repeating equalling one is not some statistic to just be memorized. It is more a question about the fabric of existence. There is absolutely no way my son is going to understand those implications for easily the next seven to ten years. It is a stat for him, a tidbit. He will spit it out, work equations with it, even be able to superficially discuss it, but he will not know it. As his teacher, as his parent, putting him in a place where mathematics becomes stacking book after book of tidbits and stats is quite unfortunate. It turns beauty into rote and education into a status race.

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One thing that you can do that is really helpful is keep math books around--not just textbooks, but books on mathematics, history of mathematics, mathematicians, etc. I have a bunch of books that are on upper level mathematics that are well above and beyond the kids current abilities, but they'll read some of them anyway if it looks interesting enough for them.

 

Having extra books around for reference and relaxation helps for those periods of lower intensity or when you are stuck on a concept and need to take a break. It helps to see the lives of mathematicians and see that they are people too, to learn about the things that motivated them also.

 

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I'm certainly not qualified to give you advice, given that you have been giving *me* advice.   :001_smile: All I can do it relate our experience and hope it helps you in some way.  I will start by saying that my ds has not moved through the AoPS books in a 1 a year sort of way:

 

Firstly, my ds took *three* year to finish the AoPS Intro Algebra book working about 2 hours a day (age 9 -11).  This was because he had to do every. single. challenger, without hints or help.  Some times he would just sit and cry, it was horrible to watch .  But he would *not* let me help; he had to do it on his own!  (and there was a time when I stepped in a hid the book until we came to an arrangement).  But this was the book that he cut his teeth on, where he learned hard core problem solving and persistence.  Now, perhaps it would have been easier if AoPS's preA book had been out and he had done it first, IDK; but in hindsight it was clearly this experience that was critical to developing his mathematical maturity.

 

Geometry took 1.5 years at age 12-13; once again with all the challengers and no hints or help.

 

It was only after those 2 books that we started to ramp up speed. 

 

7th grade: AoPS Intro C&P; AoPS Intro Number Theory; and AoPS Intermediate Number Theory (class).  And this was the year that we worked really hard to understand *how* to write proofs, what are the different kinds of proofs and how to develop that illusive insight into a problem.  We used the first third of Art and Craft of Problem solving for this effort.  And I will agree with Gil that this is a book that you could use for 4 years --  it is designed to be used in a recursive manner at ever higher levels.  He also learned to use Latex to typeset his math, and spent a month on camp selection problems.

 

8th grade: AoPS Intermediate Counting (class); AoPS Intermediate Algebra (class); UMKT geometry (used to prep for -->); AoPS Olympiad Geometry (class).  Also learned how to use Geogebra and Asymptote to draw his diagrams, and spent a month on camp selection problems.  This was also the year that he began to learn how to help others solve problems on the AoPS board, how to evaluate where another student's understanding is, and how to give just the tiny but well-targeted hint to nudge him/her along.  This has been a very positive experience, and this teaching role has definitely helped him with developing insight into his own problem solving and with identifying errors in his own thinking.

 

You know where we are now, heading into 9th grade in January - he will take AoPS precalc class while concurrently self-studying Calc 1.  Come July it will be Calc 2 at the Uni.

 

What Kathy in Richmond suggested at this point would be to take 3 years of the AoPS WOOT for the hard core problem solving while concurrently working his way through the 200 level classes at the university which may or may not be challenging, hard to tell.

 

As for writing out proofs, the AoPS graders are excellent and give very detailed and very targeted suggestions on both the content and style of the proof.  Well worth the money to us as I cannot do this for him.  So two typeset formal proofs each week in addition to all the hand-written somewhat-less-complete  proofs that he does.  He has now had one full year of this feedback (8th grade) and his proof writing style is excellent.

 

Hope that helps! 

 

Ruth in NZ

 

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I would be concerned about the ability for the knowledge to really be applied as well as densely absorbed if my student was covering three to four of these texts a year. Much like early readers "word calling" instead of truly reading, many kids can perform operations early, but not grasp the significance of the work they are doing. As Gil's concerns above for his younger kids with starting AoPS early and their mathematical maturity.

 

Stacking on courses is not equivalent to the intricacies of mathmematical reasoning and formulating a cohesive picture of the ideosyncricies of the mathematical world.

 

Some kids become obsessive, and I completely understand those that can intuitively understand the conceptualization of mathematics, but again I would be concerned with being able to fully grasp the implications. This is most important if the student is wanting to go into mathematics as the OP was thinking for his son. The side interests and diversity of topical implications becomes even more problematic than for a student like my son who is definitely not STEM.

 

So the student has covered a pile of texts, but at what loss? Does that math have true meaning in the world, in the life of the student?

 

I get your points and completely agree.  However, if the OP's house is anything like ours (both parents mathematicians), then those kids may already have the misfortune/blessing of bathing in the world you describe, on a daily basis.  We explore wild theories at the breakfast table, just for fun.

 

Much depends on the child, too.  DS8 is on a normal pace, but is starting out with higher math than other kids his age.  DS12 is a wildly asynchronous learner who devours theory at an astonishing pace.  He only likes the artistry, though, and doesn't fully grasp application until used in the further development of theory, and then he hits mastery on a single pass.  It's freaky, even to DW and I, who were both fast learners ourselves.  His 3-4 courses per year always include hard theory / problem solving as well as rote application.  It's why we love AoPS, Gelfand, and Kiselev.

 

I do not espouse reckless racing through math.  I think such an approach to be a travesty.  But, the question posed is legit.  Some kids consume math at a freakish pace, and this is the forum to find them!

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I have a bunch of stuff linked in my siggy for anyone interested in what we did. We have primarily taken a delight-driven route while using a much more qualified to teach tutor for the more rigorous work. AoPS is only a supplement here (I've always wished to make it our core but DS's tastes changed from book to book, e.g. he loved the NT, INT class and CP but didn't love the algebra and geometry and now loves the precalc and beyond books).

 

Apart from what is linked in my siggy, we also explored a portion of Geometry Revisited by Coxeter during his geometry-crazed days (both his geometry courses, first Jurgensen and then Coxeter, were very proof heavy), and some advanced math logic through eIMACS. To further understand and apply the math he is learning, we started on high school physics when he was 9 and he will take engineering physics at the CC in spring.

 

ETA: just realized that my siggy doesn't list all of the rigorous math post elementary school level:

8yo: algebra 1 with Dolciani, AoPS intro to CP class, some AoPS intro to NT

9yo: more AoPS intro to NT, AoPS intro to alg review and challenge problems (half of book), Jurgensen geometry

10yo: Jurgensen geometry and Coxeter geometry revisited plus assorted AoPS books for challenge problems, eIMACS logic

11yo: algebra 2 and trig with Dolciani, AoPS intermediate number theory class, abstract algebra and group theory fundamentals

12yo: MIT OCW single variable calc (a portion only -- summer work), calculus 1 at CC; to be followed by calculus 2 and possibly AP calc BC if he wants it, AoPS precalc at home

13yo: considering uni courses...TBD

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I'm no math wiz, but having a math loving DD, I've been surprised at the twists and turns her math journey has taken. I could not have predicted 2 years ago that she would like Gelfand's Algebra over AOPS intro to algebra. That she would prefer vintage math over singapore math.

 

I second the suggestion of enrichment, through reading about mathematicians and by mathematicians. We have collected books written by Theono Pappas and Martin Gardner, and they're great to curl up with every once in a while.

 

These children are still so young and their likes/dislikes are still developing, iykwim. If I had a child who flew through abstract math theory very fast, I would just take her/his lead and only plan a few weeks at a time.

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One problem: I can almost never get DS to write anything down. I've tried to explain various reasons for writing things down, but he (misguidedly) thinks it's a sign of weakness, and he likes to do algebra in his head. This is a block we need to break through. It's partly a developmental thing, I suppose. But it's also to do with our inability as parents to get our children to do what they're supposed to (e.g. getting DD5 to go to bed before midnight).

 

This leads to the question of when to be concerned with being more theoretical and rigorous. Personally I like the idea of being more intuitive and instinctive (assuming your intuitition and instinct are usually pretty good) and just worrying about rigor when you really need it. I suppose I'm thinking it would not be fun for a young person to be weighed down by too much formality, when they may not need it yet.

 

 

Alcumus will help with the first.  DS8 tries to do everything in his head (and usually succeeds).  When we got to chapter 10 of PreAlgebra, we had him go back to the start with Alcumus for practice, and he had to get out pen and paper (yes, pen -- we don't use pencil for math practice because it can be erased, and we're working on the value of retaining abortive attempts which may need to be revisited).  To be honest, I'm not sure how much to worry about it.  Writing down your thoughts is a must, so that your brain is free to be creative.  However, at this age, stretching one's working memory is not at all a bad thing.  I'm inclined to encourage the latter for the time being, as the former can be corrected at any point.

 

As for rigor, I certainly would not be concerned about it until the need becomes obvious.  Exploration is far more valuable.  Last year (before AoPS Pre-A), DS8 was playing with Theory of Arithmetic, and also with the development of axiomatic mathematics.  It was just fun, without pressure.  But, it has given him a unique perspective on his math work.  Hopefully, that will carry forward until he is ready for real rigor.  We'll eventually do symbolic logic, and that is where I expect rigor will start setting in.  Plus, that is SUCH a fun course!

 

I have no experience with Epsilon camp -- sorry...

 

 

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One problem: I can almost never get DS to write anything down. I've tried to explain various reasons for writing things down, but he (misguidedly) thinks it's a sign of weakness, and he likes to do algebra in his head. This is a block we need to break through. It's partly a developmental thing, I suppose. But it's also to do with our inability as parents to get our children to do what they're supposed to (e.g. getting DD5 to go to bed before midnight).

I have another question. What do people think of Epsilon camp (apart from being really expensive)?

 

I had the same issue myself with writing stuff down. If forced I would write it down but I would do so grudgingly and with a bad attitude, and I'd solve the problem in my head before I wrote it down. I actually just found my old report cards from the early 1990s where my teacher was discussing this. This caused issues when I finally got to a course I couldn't do in my head.

 

I'd consider it probably more useful to work on it by getting to problems that are not feasible to do in one's head, and while I don't have a copy of AOPS algebra I would be very surprised if he didn't run into it by the end of the book.

 

As far Epsilon camp, I have no experience but if you have any questions I would pm mathwonk.

 

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Yes, with AoPS your son will soon find problems he can't do in his head. Do the Alcumus as Mike suggested and think about signing him up for one of their online classes eventually. Then he'll have to regularly submit written solutions to the graders, who will critique his written style & clarity in addition to his getting the right answer. Later on, he might add in the USAMTS, which involves sets of proof problems done over a month or so at home & sent in for grading & comments.

As for Epsilon, I'm no longer involved, but I was the original academic director & was involved in organizing the camp back in 2011, which was the year mathwonk taught for us. Feel free to PM me if you want.

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DS often looks at various math websites (for kids, like mathplayground.com or mathisfun.com) and learns a lot of math that way, often before he covers it in his "proper" math study. Has anyone experienced any advantages or disadvantages in this?

 

I can mostly only think of advantages for DS as long as he was made aware of healthy boundaries like keeping to meal and sleep times. Only disadvantages were for me:

1. I was often stumped about how he learned something that I was reserving for him to learn at some later time

2. I often had to face and find solutions for his boredom when I tried teaching him certain things because he had already learned them from some website

3. I had to reschedule then slowly let go of some subjects so he could have more time to chase these fascinating math sites (you might also like to add the nrich maths site to his mix)

 

Basically, I just missed being the one to lead him to these fascinating topics...but perhaps it was for the better because I learned along too.

 

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