Math acceleration for a 6.5 yo

[rachsr]

My younger DS who is 6 and a half years old is a very bright and happy child. he loves school and does not complain of it being boring. My older Ds was bored at schol so we hired a math tutor to work with him at home as enrichment. That worked out great for him. The tutor noticing the younger one hanging around offered to work with him for 10 mins after DS1's class just so that DS2 wouldnt feel left out. The 10 mins started streching to 20 and then nearly an hour within a couple of weeks. The tutor even bought him work books all on his own dime until finally I had to insist on paying him for the time he was spending on DS2. This was around 5 months ago. DS2 is currently doing Singapore math 3A (and Challenging math 2 )and breezing thru it. His tutor who is a retired math professor I must add thinks DS2 will be ready to take the AOPS Pre algebra 1 online course with DS1 in spring. I think its a huge step and am not sure if he can handle it .His tutor will work with him on the AOPS pre Algebra book too and assures me DS2 is capable of handling it. I feel we are going too fast but decided there is no harm in trying it out to see how it turns out. This child by the way happily does 2 digit addition in his "advanced" first grade math class.

Anyone else have a child this age who took that course ? How did it go ?

[go_go_gadget]

I've never heard that SM3 is sufficient preparation for AoPS pre-alg, but if it is I'll certainly try my seven year-old on it, since he's in SM4. My understanding, though, is that students should do SM through SM5 before starting AoPS.

[Crimson Wife]

How long is he going to be content doing the regular math work when it's so far behind what he is doing after school? If I were you, I'd work with the school to set up an independent study (maybe through EPGY since schools tend to be more familiar with that program) in lieu of the regular class work.

[wapiti]

My understanding, though, is that students should do SM through SM5 before starting AoPS.

 

This is correct.

 

DS2 is currently doing Singapore math 3A (and Challenging math 2 )and breezing thru it. His tutor who is a retired math professor I must add thinks DS2 will be ready to take the AOPS Pre algebra 1 online course with DS1 in spring. I think its a huge step and am not sure if he can handle it .His tutor will work with him on the AOPS pre Algebra book too and assures me DS2 is capable of handling it. I feel we are going too fast but decided there is no harm in trying it out to see how it turns out.

 

For the time being, perhaps consider having him complete the more challenging problems in Beast Academy alongside SM.

 

I would wait until he completes 5th grade math (operations with fractions and decimals, measurement, etc.), before having him take the pre-test for the Prealgebra book (with the understanding that the pre-test is easier than the difficulty level of the book). Only then have the tutor start working through the book with him.

 

Even though there's the opportunity for a refund before the third class, I would *not* lay out the money for the class without completing at least the first 2 or 3 chapters of the book first. IMO, it is quite likely that the book, class-assigned alcumus and class problem set will be too much all at once for such a very young kiddo.

[crazyforlatin]

I've never heard that SM3 is sufficient preparation for AoPS pre-alg, but if it is I'll certainly try my seven year-old on it, since he's in SM4. My understanding, though, is that students should do SM through SM5 before starting AoPS.

 

SM3 is not sufficient, but maybe by next spring, OP's child will be working on SM5. Is that what the math tutor is predicting? AOPS Pre-A is difficult and I think a child will have a better time with it if he has some level 5 preparation. DD is 7 and is using AOPS, but it's a lot of thinking if done every day, so we're still using grade 5/6 SM/MM for those days when we need a break to let those AOPS concepts sink in.

 

We're not taking the class since it would move too fast for DD. She's okay in math, but not really on that kind of level.

 

 

[rachsr]

Thank you for your responses.

Yes we expect him to be done with at least SM 4 by Spring next year. He is almost at the end of 3A now. DS1 is on SM 5 right now so he is signed up to take the spring class of PreAlgebra 1. The suggestion from the tutor was to let him sit in for the classes with his brother and then work with the tutor on the concepts taught that week. There might be going back and forth which would be a good thing. I am hoping it slows him down quite a bit but I am not sure if he is ready to think his way thru a problem which AoPS claims to do. I mean he seems to lack the maturity for that.

We do Life of Fred as bed time stories and recently started on Beast Academy. He loves to read life of Fred on his own too.

[rachsr]

How long is he going to be content doing the regular math work when it's so far behind what he is doing after school? If I were you, I'd work with the school to set up an independent study (maybe through EPGY since schools tend to be more familiar with that program) in lieu of the regular class work.

I am worried about this too especially since our school district has banned subject acceleration after adopting comman core standards. I will talk to the school soon and see what they have to say. But I expect to get all children are bright pushbacks from them. Our district does not start AG identification until the 3rd grade. My Ds2 seems so normal to me that it's hard for me to accept that he can easily work on math that is 2 grades ahead.So to have to be pushy in advocating for him is quite a task for me.

[rachsr]

 

 

Is that what the math tutor is predicting? AOPS Pre-A is difficult and I think a child will have a better time with it if he has some level 5 preparation. DD is 7 and is using AOPS, but it's a lot of thinking if done every day, so we're still using grade 5/6 SM/MM for those days when we need a break to let those AOPS concepts sink in.

 

We're not taking the class since it would move too fast for DD. She's okay in math, but not really on that kind of level.

Thank you for writing about your DD's experience I expect Ds2 to go thru something similar . He will definitely take the pretest before March. I agree with you that the class would move too fast which is why I haven't signed him up for the class. The class is from 7:30 thru 8:45 which is too close to bed time. I am glad to hear that your Dd is able to work on the AoPS book though. I think that is the route we will end up on.

[rachsr]

 

For the time being, perhaps consider having him complete the more challenging problems in Beast Academy alongside SM.

 

I would wait until he completes 5th grade math (operations with fractions and decimals, measurement, etc.), before having him take the pre-test for the Prealgebra book (with the understanding that the pre-test is easier than the difficulty level of the book). Only then have the tutor start working through the book with him.

 

Even though there's the opportunity for a refund before the third class, I would *not* lay out the money for the class without completing at least the first 2 or 3 chapters of the book first. IMO, it is quite likely that the book, class-assigned alcumus and class problem set will be too much all at once for such a very young kiddo.

Thank you for the suggestions they are much appreciated. I especially liked the suggestion of trying the first three chapters of the book.

We should be getting the text book soon once I have a look at that I should be able to decide if he is even ready to start AoPS this spring.

He is doing BA 3A right now but not on a regular basis we should probably get it in his schedule. Yes, I will make sure he takes the pretest before he attempts to start PreAlgebra 1. I agree the course would be too much with all his other activities. After I started this thread Dh and I discussed it and decided to let him just try to sit thru the class with his brother who is already signed up. If all goes well we will sign him up for the next one. But I expect it will be tough on him to even to sit thru a 90 minutes class especially since the timing is not good for him. The best bet seems to be to let him work thru the book with his tutor whom he adores.

Thank you ladies for helping me think thru this issue. I am so glad to have such a support system. Happy Thanksgiving!

[Crimson Wife]

I am worried about this too especially since our school district has banned subject acceleration after adopting comman core standards. I will talk to the school soon and see what they have to say. But I expect to get all children are bright pushbacks from them. Our district does not start AG identification until the 3rd grade. My Ds2 seems so normal to me that it's hard for me to accept that he can easily work on math that is 2 grades ahead.So to have to be pushy in advocating for him is quite a task for me.

 

 

In this kind of a situation, I would strongly encourage you to get formal IQ and individual achievement testing. Going in with objective numbers placing your child in the 99.9____ th percentile will strengthen your hand in pushing for an independent study in lieu of the regular class work. If your child qualifies for the Davidson Young Scholars program, then their advocates may be able to help you as well.

 

Good luck!

[rachsr]

In this kind of a situation, I would strongly encourage you to get formal IQ and individual achievement testing. Going in with objective numbers placing your child in the 99.9____ th percentile will strengthen your hand in pushing for an independent study in lieu of the regular class work. If your child qualifies for the Davidson Young Scholars program, then their advocates may be able to help you as well.

 

Good luck!

Thank you Crimson Wife. We have been thinking about it as well. It costs such a lot here to get IQ testing done so we have to save up for it. Hopefully by summer we can get it done.

[serendipitous journey]

... if the tutor thinks it may work, and the child works fairly diligently for an hour of study, it may be a good fit. Button is performing at a similar level of math, but he's super wiggly; tempermental & intense; and I don't think the pressure of keeping pace with an online course would suit him at all. We looked at the sample of AoPS pre-A and DH didn't think it would be fun for Button (but I'm not sure and this thread has me revisiting the idea) -- the suggestion about doing the first three chapters seems excellent to me.

[mathnerd]

Thank you Crimson Wife. We have been thinking about it as well. It costs such a lot here to get IQ testing done so we have to save up for it. Hopefully by summer we can get it done.

 

One option for a "cheaper" IQ test in our area is to apply to our top rated, super exclusive private schools for admission. Part of their admission process is for the candidates to go through a psychiatrist administered IQ test and the full report is issued to you. This is how we got our son IQ tested for a reasonable price. You may check out this option if it is available in your area (though it is best to let them know that you are seriously considering sending your child to their school while enquiring :) ).

[FairProspects]

One option for a "cheaper" IQ test in our area is to apply to our top rated, super exclusive private schools for admission. Part of their admission process is for the candidates to go through a psychiatrist administered IQ test and the full report is issued to you. This is how we got our son IQ tested for a reasonable price. You may check out this option if it is available in your area (though it is best to let them know that you are seriously considering sending your child to their school while enquiring :) ).

 

 

Unfortunately, that doesn't work here. They hand you a list of area psychologists and ask you to pay for the testing and come back with scores if you are truly serious about admission.

[Dana]

Thank you Crimson Wife. We have been thinking about it as well. It costs such a lot here to get IQ testing done so we have to save up for it. Hopefully by summer we can get it done.

 

 

We got IQ testing done through the local university. A grad student did the testing under the supervision of a faculty member. Prices were much cheaper than private testing. May be worth checking out...

[RayDad]

If your son wants to do it, I think it would be advantages to let him. I do not see a need for an IQ test as you have a Math Professor who knows first hand what your son's math knowledge is. I think that if your son is enjoying learning then never stop or slow his learning. Right now you have a child who is above average and enjoying himself, sounds like great thing to me. I wish you and your son the best in whatever path you decide to take.

[La Texican]

They were talking about an IQ test because the school won't differentiate or subject accelerate the six year old until third grade, then what will he do at school all day while he studies pre-algebra with a tutor in his spare time at home? imho That ship has sailed. If you have a child who could rise to the occasion for the tutor to teach him that far ahead then he's already outside of the box. I always quote that WTM poster's siggy, "Never let schooling get in the way of your education. -Mark Twain" Classic.

 

If somebody wants to teach your kid now I'd let them try. Maybe your kid's ready or not for what you have in mind, for the pre-algebra course, but if professor can engage your kid with what he needs for now, if it was me I'd be glad to let them keep going. I don't think we can know if this will make your son bored with school, or even if he would have become bored in school otherwise, but maybe now it was appropriate to engage him and might keep him engaged with school better in the long run. idk

[lewelma]

I am copying a post that I wrote a few months back. It may not fit your situation exactly, but I thought it would be something worth thinking about. So here are my 2 cents worth from a BTDT perspective.

 

Ruth in NZ

 

x-post:

 

Let me give you some background before I give you advice. At age 6.5, my ds surprised me when he solved 2x -6 = 5x - 12. He told me you "just subtract the 2x and then move the 12 and the answer is 2." We had never done any algebra. He finished all the IP and CWP in SM5, and 1 month after his 9th birthday, started AoPS intro algebra *independently* (There was no preA back then). This was not a very smooth transition, but he was *very* persistent, but that is a different story. Intro Algebra took him 2.5 years, but he did it completely independently including all the challengers. Another thing that might interest you is that I did 6 years of mathematical/scientific research in population dynamics, so I have been in the real world. I tell you all this so that you can evaluate my advice. ;)

 

I would suggest that the "best" way to use AoPS is to use it as written. This means the discovery approach and with the student working independently and using the text as a teacher. I don't want to get into an argument because I know that there are a lot of people using it in other ways including direct teaching (which I am likely to do with my younger who is not as mathy). However, given that your student is an incredibly strong math student, he would benefit greatly by the struggle, confusion, and frustration that goes along with using AoPS as written. Real mathematical problems (I mean in the real world as a job) are never clearly written, and are not laid out so that the approach is obvious. People in math related jobs try many different approaches before finding one that works. In fact, I have spent more than 3 months going down the wrong mathematical path, and eventually had to give up and try a new approach. I have had to get 6 math textbooks out of the library, lay them out on the floor all turned to the same topic, and read and compare all the different descriptions to try to understand. I have struggled my way through nonlinear probabilistic chaos papers in Economics journals hoping that I could apply the ideas to ecological systems. I fought for every mathematical equation that I ever published. This is reality, and this is what using AoPS as written will teach your child to do.

 

My point: I would be careful of starting the AoPS sequence too early, because your ds will not have the verbal skills to work through the books independently. He can learn the material with your help and guidance, but then you lose half of what the program is teaching. It teaches math but also it teaching the true process of doing math.

 

Given his strong skills I would suggest you do all the hardest problems in SM5 IP and CWP, and then switch to PreA. Start by helping, but by the second half of the book see if he can work through the book independently. Slow him down by making him do ALL the challengers. If he cannot do a challenger, then tell him to think about it and come back to it tomorrow. Make him struggle and fight for every answer. It seems like a waste of time compared to just teaching him the material, but you are after the true process, not just the math. After struggling through the entire Intro Algebra book by himself over 2.5 years, my son is now so good at problem solving that he has quit doing the review problems at the end of the chapter and is only doing the challengers. And geometry is supposed to be the hardest of the intro books. The AoPS way works! Don't short change it by starting too early.

 

HTH,

 

Ruth in NZ

[rachsr]

Thanks for your very informative advice Ruth in NZ.

I have always wondered how I can teach my kids persistance. My older DS just gives up if he cant figure out a problem in the first few minutes. He is disheartened easily. So if I were to ask him to do the AoPS pre algebra book by himself he would probably just give up. He needs someone to guide him but I hope by the end of the book he is able to be more independent. He started alcumus recently and I can hear his groans and moans each time he gets something wrong. But he is getting better at it hopefully he builds stamina over time.

My younger one I am not sure yet if he would be able to presist thru it. Mainly because of his comprehension skills and the look of the book. We got the text book last week, DS2 took one look at it and said " that is one biiig book". I could see that he was very concerned that he was expected to work on this book in a few months. We assured him that he will start work on it only when he is ready for it . He happily works on Beast academy but this one does look like a high school book. It will be interesting to see how it progresses. I guess I have to look into improving his comprehension skills.

Talking about the PreAlgebra book how wonderful is it? I am an Engineer and I am so jealous I didnt have such a book growing up. I couldnt put the book down for hours after I got it :)

[rachsr]

They were talking about an IQ test because the school won't differentiate or subject accelerate the six year old until third grade, then what will he do at school all day while he studies pre-algebra with a tutor in his spare time at home? imho That ship has sailed. If you have a child who could rise to the occasion for the tutor to teach him that far ahead then he's already outside of the box. I always quote that WTM poster's siggy, "Never let schooling get in the way of your education. -Mark Twain" Classic.

 

If somebody wants to teach your kid now I'd let them try. Maybe your kid's ready or not for what you have in mind, for the pre-algebra course, but if professor can engage your kid with what he needs for now, if it was me I'd be glad to let them keep going. I don't think we can know if this will make your son bored with school, or even if he would have become bored in school otherwise, but maybe now it was appropriate to engage him and might keep him engaged with school better in the long run. idk

 

@bold - I need to keep reminding myself of this !!

I agree with what you say - there is no harm in giving it a whirl. If it works good else that's good too. I am happy as long as he is having fun.

[boscopup]

I have always wondered how I can teach my kids persistance. My older DS just gives up if he cant figure out a problem in the first few minutes. He is disheartened easily. So if I were to ask him to do the AoPS pre algebra book by himself he would probably just give up. He needs someone to guide him but I hope by the end of the book he is able to be more independent. He started alcumus recently and I can hear his groans and moans each time he gets something wrong. But he is getting better at it hopefully he builds stamina over time.

 

 

It does take time, but it's sooooo good for them to learn that they aren't always going to be right! I have already told my son (who isn't even working on the AoPS books yet) that AoPS is designed to be hard, and that the authors think if you can do all the problems in the book perfectly, the book is too easy. So he will know going in that this won't be like Singapore Math... He will get problems wrong regularly.

 

I just finished chapter 4 in Prealgebra myself (I work on it when I get around to it). I have an EE degree, and I'm good at math. While most of this book is review for me (and I needed it - I haven't done some of this stuff in at least 15 years), I have learned some things that I didn't learn growing up. Very cool. I still get problems wrong sometimes. Not as often as a true prealgebra student would, I'm sure. I can tell that sometimes I'm drawing on previous knowledge. But I do miss problems, and though it's rare, I sometimes even have to look up a solution in the solutions manual. So they weren't kidding when they said they didn't think kids should be able to get all the problems right.

[songsparrow]

Also, consider going deeper and richer in math, rather than just accelerating.

 

Here's an article you might find interesting on the topic: http://www.artofproblemsolving.com/Resources/articles.php?page=calculustrap

 

Some resources to consider:

* Mathematics, A Human Endeavor, by Harold R. Jacobs. A text originally designed for college non-math majors, it introduces lots of intersting concepts to explore without requiring the student to know the language of algebra. It also ties math into lots of other disciplines and daily-life situations.

* Penrose the Magical Cat, and other books by that author

[mathwonk]

Caveat: I have a tendency to make dogmatic sounding statements, as here, but these obviously are merely opinions.

 

Opinions begin here:

 

As a retired college math professor, I would be inclined to put a lot of faith in the opinion of the retired college math professor. Such people not only have taught hundreds and hundreds of students, and hence have a sense of who can do what when, but are mathematicians themselves, and hence identify with the thought process of such a math-gifted child.

 

Your school will likely never be able or willing to do much for your highly gifted child. E.g. it is unlikely even one teacher there is as qualified in math as your child's tutor. So what can they be expected to do? A personal tutor who is a math professor, and whom the child likes to work with, is the holy grail in your situation.

 

As a completely independent suggestion, since your child already seems to know arithmetic, the next natural thing might be the two books by harold jacobs, algebra, and geometry. both are excellent in content, fun to read, and with challenging, fun problems.

 

they should also last your child a few years, and save you spending money for sequenced "home schooling" math series, even good ones like AOPS.

 

eventually a child like yours would be the ideal candidate for reading really great books, like euclid's elements, and euler's elements of algebra, especially since you have a tutor to help him.

 

but the important thing is to remember he is only 6, and does not really need to do any specific thing at this point except stay engaged, and not bored.

 

i second the recommendation of harold jacobs' human endeavor book, and other enrichment books.

 

another outstanding book for a little later, when the child has learned algebra and basic geometry, is the terrific book What is Mathematics, by Courant and Robbins, aimed at bright high school educated adults.

 

Double caveat: Opinions get possibly even more questionable and self oriented here:

 

I know there are good series of math books out there for home schoolers, like Singapore and AOPS, but as a mathematician I yearn for the chance to see a really gifted kid like this read exceptional books aimed at him, i.e. he should really be in another category of learning than the usual sources.

 

I may be off base here, but I will risk suggesting that a future mathematician is someone with potential to do more than just excel at the usual math skills we all are somewhat familiar with. Being a good mathematician is not just being at the top of the usual scale of skills and knowledge. it means exploring in a much larger realm, it means not parroting but creating mathematics. Such students can quickly grow to appreciate math that is seldom even offered to most students even gifted ones, and they deserve to see that material, which is only available from the best authors.

 

 

Opinions with some counterpoint to previous ones begin here, (out of simple honesty, a mathematician's failing):

 

However I recall that participating in math contests was a source of pleasure and satisfaction to me as a child, so even if that does not correlate directly with doing real math, it can be an enjoyable way to keep a child engaged, and this is the focus of books like AOPS. Whatever the child likes and seems fit to you, I would suggest doing that, and not follow some path for creating a future mathematician that I who have not even met your child would design. Choice of his future will be ultimately made by the child.

 

For now, have faith in yourself to offer the best options, you are obviously dedicated to doing it well.

[FairProspects]

eventually a child like yours would be the ideal candidate for reading really great books, like euclid's elements, and euler's elements of algebra, especially since you have a tutor to help him.

 

but the important thing is to remember he is only 6, and does not really need to do any specific thing at this point except stay engaged, and not bored.

 

i second the recommendation of harold jacobs' human endeavor book, and other enrichment books.

 

another outstanding book for a little later, when the child has learned algebra and basic geometry, is the terrific book What is Mathematics, by Courant and Robbins, aimed at bright high school educated adults.

 

Double caveat: Opinions get possibly even more questionable and self oriented here:

 

I know there are good series of math books out there for home schoolers, like Singapore and AOPS, but as a mathematician I yearn for the chance to see a really gifted kid like this read exceptional books aimed at him, i.e. he should really be in another category of learning than the usual sources.

 

I may be off base here, but I will risk suggesting that a future mathematician is someone with potential to do more than just excel at the usual math skills we all are somewhat familiar with. Being a good mathematician is not just being at the top of the usual scale of skills and knowledge. it means exploring in a much larger realm, it means not parroting but creating mathematics. Such students can quickly grow to appreciate math that is seldom even offered to most students even gifted ones, and they deserve to see that material, which is only available from the best authors.

 

Any ideas what to do in the meantime? I feel like most days we just "play" math while I am waiting for the maturity of persistence to kick in and the frustration level to lower. The concepts are already there, but the maturity to deal with hard problems is not, so instead we find the exact center of 1000s of hexagons in hotel lobbies and such.

[mathwonk]

i am tempted to answer "lighten up?" (for your own sake) but that sounds flippant. for a 6 year old understanding hotel lobbies sounds fine. how old is your child? we can't really remake our child's personality, and least not in my experience. (just a grandparent talking now. but we can make them hate math, or maybe even us. i.e. been there, done that wrong myself.)

[FairProspects]

No he's 8, but a very intense personality who frustrates easily.

[jennynd]

No he's 8, but a very intense personality who frustrates easily.

 

 

My 8 yr old is like that. There are only 2 types of question, too easy or too hard. there is nothing in between.

[wapiti]

 

My 8 yr old is like that. There are only 2 types of question, too easy or too hard. there is nothing in between.

 

 

and some questions are both too easy and too hard :tongue_smilie:

[mathwonk]

easier said than done of course, but the traditional technique is patience and praise. I.e. wait until the child exhibits the desired trait of application of himself on a hard problem and then praise it.

 

this technique enabled that loony Harvard psychologist B.F. Skinner even to teach pigeons to distinguish different symbols on a chart.

 

that association makes me hesitate to recommend it, but used responsibly it can do some good. Fortunately you are young and strong,

 

and patience can be learned. I have almost learned a little after 70 years, but not much.

[shawthorne44]

...I know there are good series of math books out there for home schoolers, like Singapore and AOPS, but as a mathematician I yearn for the chance to see a really gifted kid like this read exceptional books aimed at him, i.e. he should really be in another category of learning than the usual sources.

 

I may be off base here, but I will risk suggesting that a future mathematician is someone with potential to do more than just excel at the usual math skills we all are somewhat familiar with. Being a good mathematician is not just being at the top of the usual scale of skills and knowledge. it means exploring in a much larger realm, it means not parroting but creating mathematics. Such students can quickly grow to appreciate math that is seldom even offered to most students even gifted ones, and they deserve to see that material, which is only available from the best authors....

 

Interesting idea. "Living Books" applied to math. I never would have thought of it.

[serendipitous journey]

easier said than done of course, but the traditional technique is patience and praise. I.e. wait until the child exhibits the desired trait of application of himself on a hard problem and then praise it.

 

this technique enabled that loony Harvard psychologist B.F. Skinner even to teach pigeons to distinguish different symbols on a chart.

 

that association makes me hesitate to recommend it, but used responsibly it can do some good. Fortunately you are young and strong,

 

and patience can be learned. I have almost learned a little after 70 years, but not much.

 

 

:lol: ... there are some things I really can't pass on, and Skinner training animals is one of them ... may I say that while the techniques of patience, praise, and hitting the motivational sweet spot are excellent, Skinner couldn't teach a pig or a dog to peck: pigeons are natural peckers, and naturally pay attention to the visual world. Though that's really consonant with your point of waiting until the child naturally shows an interest or behavior -- some things just can't be imposed from outside.

[serendipitous journey]

RE the OP:

 

Now that I've started using AoPS prealgebra with my 7yo, I must say I'd hesitate to do the couse with him. It is very thorough and rich, and I like being able to go through it at his pace; and I'm doing my best to follow Ruth's advice above, and have him use the program as written to himself. I believe in another thread she suggested using the Prealgebra book to transition from parent-led to textbook-led and that'w what I'm trying to do.

 

At the moment we work in the book 3 days/week, for 10 - 30 minutes or so at a time. Button reads most of it and we discuss as we go. We are just starting, so I am not sure how this will go in the long run, but for the moment it's fun.

 

mathwonk and I disagree somewhat about the charms of Euclid's work; but I don't feel too bad about that, there are other mathematicians who think Euclid would have driven them from the field. So perhaps try Euclid when he's ready but don't despair if it isn't a hit. :)

 

-- FWIW, the other 3 days we're doing MUS Algebra right now.

[Reya]

Your school will likely never be able or willing to do much for your highly gifted child. E.g. it is unlikely even one teacher there is as qualified in math as your child's tutor. So what can they be expected to do? A personal tutor who is a math professor, and whom the child likes to work with, is the holy grail in your situation.

 

 

My mother is a STEM education researcher. One of her friends, a math prof, joked, "What's the difference between elementary teachers teaching reading and teaching math? All elementary school teachers can read."

 

The follow up is, "And virtually none can do math."

 

As a completely independent suggestion, since your child already seems to know arithmetic, the next natural thing might be the two books by harold jacobs, algebra, and geometry. both are excellent in content, fun to read, and with challenging, fun problems.

 

 

This worked well for my son, who started with Algebra at 7 and will finish Geometry at just over 10. AoPS wasn't hard for him, but there are more words per page and other issues that made it just less user friendly for his level of maturity.

 

To succeed, the child needs not only a GRASP of the math but the ability to rapidly "see" multiplication facts and their inverse, division. A child should be able to give all factors of any 2 digit number without much difficulty. This is about fluency, not just understanding.

 

I know there are good series of math books out there for home schoolers, like Singapore and AOPS, but as a mathematician I yearn for the chance to see a really gifted kid like this read exceptional books aimed at him, i.e. he should really be in another category of learning than the usual sources.

 

I may be off base here, but I will risk suggesting that a future mathematician is someone with potential to do more than just excel at the usual math skills we all are somewhat familiar with. Being a good mathematician is not just being at the top of the usual scale of skills and knowledge. it means exploring in a much larger realm, it means not parroting but creating mathematics. Such students can quickly grow to appreciate math that is seldom even offered to most students even gifted ones, and they deserve to see that material, which is only available from the best authors.

 

 

As long as the child is kept engaged, I don't see any conflict between Jacobs or AoPS and your end goal. It may look like "the same thing but earlier," but in many ways, it isn't because it leaves time later for deeper explorations while building a solid foundation. With my son, I wanted to foster true independence, and Jacobs was the best for that, partly because the pages are so visually uncluttered and simple. (My son is 2E, but some things are age-related.) Some other books that are excellent don't work as well for him because the reading level is too high or the pages are just too busy.

 

AoPS has a game-like approach that is really appealing to a lot of kids, too. It just didn't work for us. It keeps the joy and discovery of numbers alive, however.

 

Mathwonk would probably love the EMACs series, which really is GREAT, but it just didn't work for my son because it's JUST PLAIN too wordy. When he's more mature (and, yes, has some "standard" credits under his belt), they will be easy for him and fun.

 

For us, Jacobs has been a huge blessing because it's allowed my son to take a huge stride into completely independent learning. He reads the book, understands it, and does meaningful work. That means a lot when he can take this to other programs later.

[mathwonk]

@Serendipitous Journey: Good point about Euclid being off putting to some. It may well require a tutor for benefit. I was myself put off by it years ago, from just opening it up and not getting past the first few completely incomprehensible definitions.

 

Unfortunately I did not realize at that time that one should ignore those and begin with the propositions. My brilliant students last summer were also not enamored of either Euclid or Euler when left entirely on their own, but with my help they enjoyed and benefited from it.

 

There is of course a difference between books you can just hand to the student, like Harold Jacobs books, and books that are potentially much more valuable, like Euclid and Euler, but which need a little hand holding to get the child started, plus probably progressive consultation.

 

Without such a helpful introduction I myself passed almost my whole life without returning to Euclid, and realizing that it really is the best book there is on the topic. I tried therefore to provide a better introduction for my charges last summer, and that is why I also try here to help future generations of mathematicians to have the exposure they really deserve. In college also we received introductions to great literature from our professors who gave us samples from them to whet our interest.

 

There may be some comparison to encountering Shakespeare or Herman Melville. A child may be put off by being handed a gigantic "great books" volume containing Shakespeare, but enchanted by a reading from a single passage, like the balcony scene of Romeo and Juliet, or the pre - battle speech by Henry V. Many people are bored by the long factual chapters from Moby Dick, but most would be charmed and entertained by the first chapter where Ishmael suddenly and startingly meets Queequeg.

 

I just advocate not being put off by the initial challenge of encountering great books. Indeed that is why I am here, and other tutors, to help enter the gateway. Books that are easier to enter are so for a reason - they may well contain less of substance. Another approach is to use the easier books as warmups to the harder ones.

 

I would also be inclined to ask mathematicians who do not appreciate Euclid (as I originally did not) to take another look; I would ask e.g., whether they have noticed for that Euclid anticipates in Prop. III.16, Newton's definition of a tangent line to a general curve by over a thousand years. Calculus would be so much easier to teach if students actually came to it with the sort of deep background provided in Euclid but not in his imitators.

 

But again there is no one size that fits all. Indeed my advice may not be appropriate for home schooling, since there we may prefer materials that we do not need to be expert in ourselves, and that are more self contained. But if a tutor is available, I would suggest shooting for the best materials, with the deepest content, always maintaining an eye for the child's degree of engagement.

 

As always you are the best judge, since you know your child best.

[mathwonk]

I had forgotten just how reluctant I myself was to read Euclid until recently, when I was "forced" to do so by a book from the famous contemporary geometer Robin Hartshorne. Hartshorne wrote a companion to Euclid, Geometry: Euclid and beyond, in which he does not quote Euclid, but requires the reader to have at hand a copy of Euclid to read along. This technique obliged me to get a copy of Euclid and read it in order to follow the discussion and exercises in Hartshorne. To my delight, after my brief initial resistance, I fell in love with Euclid. Someone just had to get me to give up the wrong notion I had long harbored, that Euclid is inaccessible; until then I was not willing to look at it.

 

The best version of Euclid for an introduction is the beautiful Green Lion edition, under $20 at Amazon, without the lengthy commentary of the translator Heath. A beautiful essay by Hartshorne on "Teaching Geometry according to Euclid" can easily be googled as well, and is a very eloquent and persuasive argument for doing just that.

 

If one is willing to buy Hartshorne's book mentioned above as well, that is the ideal way to read Euclid, guided by a master. Hartshorne's also contains vastly more material that would last a young student for years and years. Just the first chapter of Hartshorne is sufficient guide for reading much of Euclid. (The plane geometry is mostly finished in the first 4 chapters or so of Euclid.)

 

Here is another argument in favor of using Euclid, perhaps not quite as eloquent for me as Hartshorne's marvelous essay, but one I can link directly to.:

 

http://www.kysu.edu/...Perfections.pdf

[serendipitous journey]

mathwonk, thanks for the guide to Euclid! will revisit him. :)

[melmichigan]

My DS is on a similar track. I do not plan to let him start AoPS after SM PM at this point in time. The current plan is to keep moving through the upper SM levels with his tutor. He will get to AoPS when his handwriting and patience catch up to his math ability, probably with the online classes because of the pacing and TA support. I'm thinking we'll consider it when he's around 9 years old. Older DD did AoPS Prealgebra with RR last year and it was a great experience, but not one DS is ready for at this point in time, not to be done the way I want to see it done at least. YMMV.

 

I also think you will have your hands full in the near future if DS is sitting through grade level math. That will get old very fast in my experience.

[mathwonk]

Here is my own little introduction to Euclid, excerpted from the epsilon camp notes located near the bottom of my web page:

 

http://www.math.uga.edu/~roy/

 

 

Euclid’s Elements: Introduction to “Proofsâ€

 

Euclid is famous for giving proofs, or logical arguments, for his geometric statements. We want to study his arguments to see how correct they are, or are not.

 

First of all, what is a “proof� We may have heard that in mathematics, statements are proved to be either true or false, beyond any shadow of a doubt. In my opinion, this is not quite right. Rather we play a game of logical deduction, beginning from a set of assumptions that we pretend are true. Then assuming or pretending those things are true, we ask what other things would be true as well. We do not discuss whether the things we are pretending to be true really are true, but if we run across a world where they are really true, then we may be sure that anything else we deduce logically from them will also be true in that world. So there are statements we take for granted, called axioms or postulates or assumptions, and then there are statements called theorems that we deduce or “prove†from our assumptions. So we don’t know that our theorems are really true, but in any world where the assumptions are true, then the theorems are also true.

 

In Euclidean geometry we describe a special world, a Euclidean plane. It does not really exist in the real world we live in, but we pretend it does, and we try to learn more about that perfect world. So when we “prove†a statement in Euclidean geometry, the statement is only proved to be true in a perfect or “ideal†Euclidean plane, but not on the paper we are drawing on, or the world we are living in.

 

It’s a game like Monopoly, or dungeons and dragons, where we have a certain goal we want to achieve, but there are rules we agree to play by. We only win if we follow the rules. Sometimes the game gets boring, or too hard, and then we may just change the rules to make it easier or more fun.

 

The rules Euclid tried to play by are stated in his 5 postulates, and his “common notionsâ€. One fun thing about reading Euclid is trying to catch him using a rule he forgot to state. There are several places where this happens. It is also fun to see how clever he is at getting by without some of the rules we usually give ourselves to make the proofs easier. E.g. he makes his constructions not with ruler and compass, but with “straightedge†and compass. We are so used to saying “ruler†that I am going to do this sometimes, but his straightedge does not have marks on it like our ruler.

 

So Euclid’s geometry has a different set of assumptions from the ones in most schoolbooks today, because he does not assume as much as we often do now. That makes some of his proof harder than ones in schoolbooks, because he does not give himself as much to go on. His geometry is also different from that of professional mathematicians because he forgot to state some postulates that he actually uses, and mathematicians have had fun suggesting postulates he might have or should have stated. Different people have different opinions about what the best postulates should be, so there is more than one way to do Euclidean geometry. We will look at Euclid’s own version and make some choices of our own to fill in any gaps we notice.

 

At first we are going to try to use only postulates 1-4, as Euclid did, as well as his common notions. Those postulates are roughly as follows:

1) We can draw a finite line segment between any two different points.

2) We can extend a finite line segment as far as we want in a line.

3) We can draw a circle if we are given a center and a point on the circumference.

4) All right angles (half of a “straight angleâ€) are equal.

 

Euclid also compares the size of different figures, the size of a collection of segments is like the sum of their lengths, and the size of a plane figure is something like its area. But Euclid does not use numbers to measure either length or area, so he needs some rules to tell when two figures have the same size, or smaller or larger size. His “common notions†are really postulates for the concept of equal size. They mostly say that “equality†of (size of) figures behaves as we expect E.g. if you add figures of the same size you get new figures of the same size, and a figure that fits inside another with room to spare is not equal to it in size.