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Going BROADER in Math at home


Gil
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Disclaimer: I know that a good part of this is just 'What ifs' and stuff that maybe/sorta/could happen, but its really got me thinking right now and I can't sleep anyway so I decided I'd ask here, where someone might have been in a similar situation or know someone who is. After all, I've seen some pretty unusual things from the families on this board...This is mostly just fanciful thinking and not to be taken too seriously.

 

The Point:

It was always my intent to plan and provide 'exposure' to various fields of mathematics for my boys. I had hoped that by the time they were finished with highschool they will have a solid mastery of precalculus mathematics and a few rounds of exposure & some experience in branches of mathematics such as:

 

*Proof writing

*Statistics

*Differential Equations (ordinary)

*Linear Algebra

*Discrete/Finite Mathematics

*Calculus

 

Has anyone on these boards done these topics at home? I can't find any threads on this, but my search-fu is admittedly weak.

For what its worth I am not in the least bit concerned with high school credits or anything at this time. I don't even know if we will still be homeschooling for highschool, let alone in the states or what, so I am not worried about putting these 'extra' topics on a transcript. I am interested in getting these 'extra' topics to my kids though.


::::Back ground or "why I am even thinking about this"::::

I have some math major friends who are gearing up to take the GRE Math Subject test. They and I have been doing some major math revision this semester and the boys get a real kick out of watching us (and sometimes just them) mathing it up at the white board with all the different books and notes. Pal is now showing more and more interest in learning math that is typically taught at the undergraduate level and he asked me to think about teaching him 'serious math' once he is done with 'basic math'. Basic math = school math up to preCalc.

 

It was always my intention that they have exposure to other branches of math but I wasn't sure what path to take to get there. Right now, they are polishing off their 6th grade math books. Well, we are poised to finish the basic algebra sequence including PreCalculus by the time that the kids are 10-12 years old. Thats with time off + supplements.

 

The boys, inspired by my math sessions with my GRE-prep friends, have started reading some of my various math supplements from when I was tutoring math pretty seriously at the highschool and college level. ( [Math] for Dummies, Complete Idiots Guide to [Math], Humongous book of [Math], Problem Solvers Solution Guide to [Math] etc) They can follow some of it, because they know some algebra (almost done with the Keys to Alg. series and MM6 no problems) and some of it we have covered informally already. They have been asking me a ton of questions, some of which I explain, others of which I delay.

 

The boys seem to be re-entering a point where they are very receptive to new ideas and information in mathematics again. I am like this, when I get the 'itch' for something, I can learn it all very quickly and easily and am very sensitive to that topic in spurts.The supplements that we have range from Basic Math/PreAlgebra, Algebra 1, Algebra 2, PreCalculus, Trigonometry, Differential Equations, Geometry and Calculus I, II, and III. They have been at the supplements for several days now and so we spent sometime last night, putting the books 'in order' so that they can read through them in correct order. They have already polished off most of the Basic Math supplements thanks to MM and the very few questions in the books. Now they are perusing the Algebra books and working on some of those problems. I showed them which text books to reference if they get stuck/need an explanation.

 

I am not in anyway seriously expecting them to actually learn and master any of those books. Not in a long shot, but I do want to enable them to do as much as they want to with the supplements and we are getting ready to start algebra soon anyway. Again, this isn't about credit (highschool or otherwise) but about allowing the boys exposure and experience with other types of math in the course of their K-12 years. Many of these topics are on a level of difficulty comparable to Calculus, but math is so demonized in the USA that most people wouldn't believe it. I remember being pleasantly surprised by how fun and unbelievably easy Differential Equations were--it had been talked up to sound like purgatory or something--it was even easier than some parts of Calculus.

 

Pal says he wants to get a degree in math now. I don't know if he really means it, or he just thinks that he does, or if this will be passed up by his dream to do something else (like drive an ice cream truck!) but its something that I've gotten a little excited about thinking about it.

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My ds has taken classes in all of those topics except stats ( which he has self- studied to a certain degree ). Fwiw, I wouldn't assume that they will keep the same pace once they move beyond elementary math topics, It is not unusual for kids that master the elementary basics to slow down to a somewhat more normal pace with more advanced topics.

 

Of your list, several can be accomplished via AoPS.

*Proof writing

*Discrete/Finite Mathematics

*Calculus

 

I am not strong in math, so I can't teach these things at home, but AP stats is availBle through PA Homeschoolers online. DiffEQ and linear alg are both available through Stanford Online High School. Some people like Kathy in Richmond just teach the classes at home. We go via dual enrollment at the local university.

 

Math camps (if you go into the AoPS camp wiki you'll find a list) are a great resource for kids. Epsilon is the camp for little kids.

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Well, here's a kind of funny idea--but I really like Moses Richardson's Fundamentals of Mathematics book (my copy is from the 1940s, but there are more recent editions). The coverage is very broad, and gets into plenty of nifty things--it's a very crunchy math book for non-math majors, basically.

 

Maybe some of the New Math from the '60s might be of interest, too?  I've been having fun lately looking through the "Unified Modern Mathematics" courses by Howard Fehr and co.; the first three years (i.e. grades 7, 8, and 9) are in the public domain and are online at ERIC. (Why won't the site let me link this morning???)  Anyway, go to eric.ed.gov, and search for "unified modern mathematics" and check the box that says "full text available"; you'll see courses 1, 2, and 3 (each in two parts), plus the teacher commentary for each. They are wild! Sets, groups, fields...all kinds of very cool math there.

 

There is also the SMSG stuff available online, about which both stripe and wapiti have posted at various times, I think.

 

Hope you find something fun!

 

 

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My ds has taken classes in all of those topics except stats ( which he has self- studied to a certain degree ). Fwiw, I wouldn't assume that they will keep the same pace once they move beyond elementary math topics, It is not unusual for kids that master the elementary basics to slow down to a somewhat more normal pace with more advanced topics.

 

Of your list, several can be accomplished via AoPS.

*Proof writing

*Discrete/Finite Mathematics

*Calculus

 

I am not strong in math, so I can't teach these things at home, but AP stats is availBle through PA Homeschoolers online. DiffEQ and linear alg are both available through Stanford Online High School. Some people like Kathy in Richmond just teach the classes at home. We go via dual enrollment at the local university.

 

Math camps (if you go into the AoPS camp wiki you'll find a list) are a great resource for kids. Epsilon is the camp for little kids.

Yeah, I'm sure that as we get deeper into Algebra 2/PreCalc level material they may slow down a bit. But, so far they have taken every bit of Algebra in stride. They get the concepts crazy fast and they love the patterns of the formulas and how it all works out on paper.  Pal is probably even a little better at the abstraction than Buddy, but Buddy is definitely more precise and careful in his written work which is why he gets a better score 90% of the time. Even if they slow down, with school Algebra done at ~12, that gives them 4-6 years to do other stuff in math.

 

I'm going to have to do something about the size graphs/dimensions on graphing paper because drawing equations as lines have been giving Pal a fit--he gets it, but he has trouble doing it on paper because the graphs are too small and then he finds that his algebra and his picture don't agree. Its making him nuts.

 

Which AoPS book teaches proofs, or is it intersperced in their other texts? The way that AoPS teaches, isn't exactly the way that my guys learn at this time. Personally I think that Buddy could grow into AoPS (he might outdistance the topics first though), but I don't think that Pal ever will. We shall see.

 

Thanks for those recommendations, I'll look up Kathy in Richmonds posts, see what/how she did for those classes. I will have to look into math camps in a couple of years, I think that the boys might really like to attend one! I haven't been able to get them excited about the idea of a math team yet but they're young still.

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Well, here's a kind of funny idea--but I really like Moses Richardson's Fundamentals of Mathematics book (my copy is from the 1940s, but there are more recent editions). The coverage is very broad, and gets into plenty of nifty things--it's a very crunchy math book for non-math majors, basically.

Ooh, sounds exciting. What is the reading level on it, if you had to guess? Do you think it would be something a kid could read, or would I need to ready/study ahead of them?

 

Maybe some of the New Math from the '60s might be of interest, too?  I've been having fun lately looking through the "Unified Modern Mathematics" courses by Howard Fehr and co.; the first three years (i.e. grades 7, 8, and 9) are in the public domain and are online at ERIC. (Why won't the site let me link this morning???)  Anyway, go to eric.ed.gov, and search for "unified modern mathematics" and check the box that says "full text available"; you'll see courses 1, 2, and 3 (each in two parts), plus the teacher commentary for each. They are wild! Sets, groups, fields...all kinds of very cool math there.

This is fantastic! Thanks so much this is fantastic! I've got the first course and it looks really, really exciting. This might be just the perfect thing for us!

There is also the SMSG stuff available online, about which both stripe and wapiti have posted at various times, I think.

I will look into these, I've never heard of them before.

Hope you find something fun!

 

Wow, looks like the hard part will be deciding what combination of stuff to use and how! I think that the boys will have fun with some of this stuff!

 

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So...just interspersed through out all the books? Do they start in Intro. to Algebra or another one?

ETA1: Nevermind, answered up thread. I don't type as fast as others!

ETA2: Dang! I don't delete as fast as others type either :). thank you luckymama and 8FillTheHeart!

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So...just interspersed through out all the books? Do they start in Intro to Algebra or another one?

 

I am unfamiliar with the lower books.   My ds started with the alg 3 and counting & prob books.  From what I understand, the geo book is full blown proof based, but the other upper books have lots of proofs.

 

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I agree that AOPS, starting with pre-algebra, will include some of those extra topics, and, for the most part, that has been my DD's favorite parts. AOPS also starts including some proof in Pre-algebra, although it's not as formal as it was when I took high school geometry and advanced algebra. There's a real focus on understanding the "Why".

I'd also suggest subscribing to Dover Books mailing list. They have a lot of math books that are simply wonderful and are often dirt cheap (and some get QUITE complex). My daughter is working through one on math history and the evolution of math now.

 

 

 

 

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Beast Academy has some beginning discrete math in it and lays the foundation for all kinds of higher-level math. Borac competitive math is at least 1/2 discrete topics. Looks like they most recently published book 10, in level 3 (considered preparation for AMC-8 and AMC-10 and partial preparation for AIME, designed for students aged 11-13 who "can solve linear equations, are fluent with fractions, and can factor into primes,") but 18 books are planned, up through level 4 for students aged 12-15.

 

The Borac books recommend their free site http://www.mathinee.com/ for actually teaching the topics for which their books provide practice. Looks like they've got some trig, combinatorics, probability stuff in there.

Beast also requires written response to a lot of problems, which is the beginning of proofs, and is conceptually consistent so there is no unlearning for your student; I am loving the distributive property section that my son is in now, his lightbulb moment about the difference between addition and multiplication reminded me of similar moments I had in college-level Discrete. You should look for posts on this board from mathwonk. He's shared a lot of terrific materials that he uses with Epsilon camp kids.

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Which AoPS book teaches proofs, or is it intersperced in their other texts? The way that AoPS teaches, isn't exactly the way that my guys learn at this time. Personally I think that Buddy could grow into AoPS (he might outdistance the topics first though), but I don't think that Pal ever will. We shall see.

 

Could you elaborate? This learning style stuff always fascinates me, I think because my kids are a little different from each other and I'm still trying to dial in the best way to work with my DD.

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Could you elaborate? This learning style stuff always fascinates me, I think because my kids are a little different from each other and I'm still trying to dial in the best way to work with my DD.

Well, first a disclaimer: I have only ever seen the Intro. to Algebra book in person and I only had it for several hours. In reading through it, it was written to the student (great) but it was written in a kind of round-a-bout style (not great). I can imagine that it is like a transcript to a lecture or something. I am having trouble putting this in words and I'm loathed to get 'technical' with the jargon because I don't speak educational-theory lingo. Instead I will try my hand at an analogy (or is it a metaphor?)

 

Imagine that you are standing against the wall of a very large room with little to no lighting and there is a mess of junk in the floor...

There are two ways to guide someone through that room:

-a- Stand on the opposite side of the room and talk to them so that they can follow your voice.

-b- Walk beside them so that when they stumble, you are there to help them up and guide them.

 

AoPS seems to go for an all out option -a- approach. AoPS will let the students follow the teachers 'voice' and by extension allow them to run in to stuff as often as they happen to. If the student runs into every.single.roadblock on the way, so be it. If the student gets stuck/falls/stumble, well, its up to the student to get up unaided, reorient themselves in the right direction and try again. But the whole time the teachers 'voice' is calling them.

 

*I* have always used an option -b-  approach with the boys when I am lecturing/teaching them.

 

Math Mammoth, I feel teaches in a way that is a good deal option -b- but with a good dose of option -a- strategically thrown in from time to time. I think, quite honestly, that the disconnect between option -a- and option -b- type learning is a matter of maturity mostly and some personality thrown in there. I think that AoPS would stress and frustrate my boys more than teach and inspire them because of the  maturity issue more than anything. Thats not a chance I am willing to take. At this point in time, it is more important to me that they:

  • want to do the work
  • that they are excited to try again in the event of the occasional failure
  • eager to take on a new-to-them challenge (even if it isn't the most challenging)

One of the many reasons that I pulled them from PS is because PS was crushing my kids will and drive to learn. They had always done academics with me and it is something we did as a no-strings attached, low pressure thing and even when it was a bit difficult it gave us all satisfaction. I wasn't going to surrender that to the PS and I won't surrender that to AoPS either.

 

Developing and preserving those three traits in my kids with respect to education and life in general is far more important than doing the Most Challenging, Most Thought Provoking, Most Well Thought Out and the Most Rigorous program for any subject--let alone one that my guys love as much as they love math. My boys, at 6 and 7 years old, simply aren't mature enough to handle the attitude that AoPS requires. And thats okay with me.

 

I am letting them start Algebra as soon as they wrap up their Grade 6 work because I feel that they can do the math but I don't feel that they can handle the AoPS presentation of the lessons. From what I gleaned, AoPS is not an easy curriculum, it is not a medium curriculum, it is a difficult curriculum. It was meant to be that way, it was conceived with the express purpose of being difficult and feeding the need of kids who really love math and who can not only 'take the heat' but who bask in it with pencil, paper and a glass of lemonade. From what I can tell, you don't have to be great at math to use AoPS, but you have to possess that heat tolerance for when the going gets tough.

 

AoPS Intro. to Algebra, does its job, in the way that it was designed to do it, and it does it well. However, I feel that AoPS will just burn my kids. Like a fair skinned person who doesn't tan in sunlight--they just get straight up sun burn. Ouch! I really feel think AoPS will cause unnecessary stress and frustration and possibly sour them on math.

 

The boys have enjoyed math since they were toddlers, I would hate to ruin that for them while they are still so young by trying to force something more appropriate to a 12 year olds emotional maturity on them sooner than they can handle it. I'm a 'path of least moderate resistance' sort of fellow myself. I am not opposed to the boys ever using AoPS materials. In fact, I look forward to their offerings in the math, especially their books on Number Theory and Counting + Probability. But not yet.

 

Buddy is patient and more eager to meet people half way. (Which is why I said that he could grow into AoPS.) He is a good sport at 21 Questions (we are big riddlers here,) he doesn't mind puzzling over a riddle for a few hours sometimes but only sometimes and only if its a really good one. Buddy is careful and more willing to slow down to  think and spare himself the trouble of having to redo a problem. He'd rather do it one time and be right than rush and do it again. That is his personality--he decides which challenges he wants and I try to respect his choice. Buddy is a "ready...aim...double check the aim...okay? Good...fire! sort of kid.

Pal is not patient and he only cares about pleasing his big brother and me (sometimes). Pal, like Buddy is good at math and he is

good at riddles, but he has no patience for double and triple meanings and if he doesn't get it with in a few minutes he'll say: I don't know--what is it?" If you don't tell he can get pretty upset. If he doesn't 'get' the answer to the riddle, he will also get upset. Pal rushes when he does his work and he can work (and rework) a problem 3x in the same time that it takes Buddy to do it once.

Pal is more fire, fire, fire...okay, okay fine! I'll aim!--readyaimfire! YES! I got it! sort of kid. That is his personality.

 

I don't know if Pal will ever be the sort of kid to bask in the proverbial sun of difficulty. (Good luck to him, since he wants to be a math major!) But Buddy, he could grow into the sort of kid who, with sunscreen, shades and good company might spend sometime out there.

 

Addendum: I just wanted to clarify that I am totally open to using AoPS at a later time.  I fully believe that even someone who has done well with a standard PreAlgebra-PreCalculus series can get good use from the books. I think that AoPS books are very good example of what math is and how even someone who has had exposure to advanced math can glean more, learn more and gain more insight into the fundamental parts of a science and grow as a mathematician/scientist.

 

Heck, I might even buy the AoPS books for myself in a few years when all the errors have been smoked out and the 1st and 2nd editions are available used for cheap.

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I love your analogies, Gil, very clear! I've never held the main AoPS books myself, but we've looked at a lot of excerpts online and when I started reading to my kids from the first chapter of Pre-Algebra it was love at first listen. They kept demanding to do more of the problems. :p I showed them the pages and asked if this kind of book looked like fun or if it looked overwhelming and they both agreed it looked super fun! It seems to be exactly appropriate for them, though they're such different kids. (DD5 definitely burns in the sun of difficulty, but still she keeps trying to go out now and again. DS7 got a good burn when he was 6, but it seems to have kick-started his base tan, and now he can stay out longer and longer and even seeks out the sunny places when it really would be just fine to hang out in the shade a bit. ;) )

 

It's taken a good amount of grooming, but DS now sees more value in perseverance than in getting things right immediately. DD still preens if she gets a workbook page all correct, and pouts if something doesn't go her way, but we're building up to giving her work she can sink her teeth into. "Mathematics isn't meant to be easy, it's meant to be interesting!"

 

A funny thing about option A and option B...option A is how I parent. :) I didn't spoon feed them, they fed themselves starting at 6 months. I didn't help them climb playground equipment, they figured out how to navigate ladders and things before 18 months while I stayed several strides away and pretended to look the other way. When they play together, I don't intervene unless the noise is getting on my nerves. So maybe that's why my kids and I embrace an approach that has you feel your way slowly and awkwardly when things are new, in order to have greater confidence, balance, nerves, flexibility once you get a bit of practice under your belt?

 

Interesting stuff! :)

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I love your analogies, Gil, very clear!

Good, I'm glad that it was clear for at least one other person. I was worried I had revealed the severe lack in my vocabulary.

I've never held the main AoPS books myself, but we've looked at a lot of excerpts online and when I started reading to my kids from the first chapter of Pre-Algebra it was love at first listen. They kept demanding to do more of the problems. :p I showed them the pages and asked if this kind of book looked like fun or if it looked overwhelming and they both agreed it looked super fun! It seems to be exactly appropriate for them, though they're such different kids. (DD5 definitely burns in the sun of difficulty, but still she keeps trying to go out now and again. DS7 got a good burn when he was 6, but it seems to have kick-started his base tan, and now he can stay out longer and longer and even seeks out the sunny places when it really would be just fine to hang out in the shade a bit. ;) )

Wow, lucky! I always have to balance what the boys (think) they want to do/achieve against their boundaries while also respecting their right to break through any limits they have. I'm really getting more and more into the idea of filling out their next couple of year(s) with Algebra + 'broader math exposure. I'm going to try and take some time to seriously reflect on what they are doing, want to do and how to get them there. I have never been crazy about the Algebra 1 --> Geo -->Algebra 2 -->Trig/PreCalculus --> Calculus. Just feels grossly artificial to me.

 

It's taken a good amount of grooming, but DS now sees more value in perseverance than in getting things right immediately. DD still preens if she gets a workbook page all correct, and pouts if something doesn't go her way, but we're building up to giving her work she can sink her teeth into. "Mathematics isn't meant to be easy, it's meant to be interesting!"

See, this is where I need to get the boys to be. They know that 'math isn't meant to be easy, its meant to be interesting' but they don't know that 'math isn't meant to be easy its meant to be interesting.' Possibly, this is my fault because I should have challenged them more and I was busy trying to keep it 'fun' that I didn't let them struggle enough early on. So despite all my cute slogans and inspirational speeches, they are used to math being easy and novel and that ease and sense of novelty is now a big part of why they enjoy math all the time. What I wouldn't give for a do-over, *sigh*.

A funny thing about option A and option B...option A is how I parent. :) I didn't spoon feed them, they fed themselves starting at 6 months. I didn't help them climb playground equipment, they figured out how to navigate ladders and things before 18 months while I stayed several strides away and pretended to look the other way. When they play together, I don't intervene unless the noise is getting on my nerves. So maybe that's why my kids and I embrace an approach that has you feel your way slowly and awkwardly when things are new, in order to have greater confidence, balance, nerves, flexibility once you get a bit of practice under your belt?

Yeah, its funny, I feel that I parent more in option -a- mode myself! I find that because they were so young when I started teaching them that I am intentionally kinder, softer or more sensitive to their emotions during lessons/work. Outside of school related things, I have had more than one mom chide or gasp in horror at me for being harsh with the boys.

(ie I'd sit on the bench and watch as they plummet from the top of the climber and if they were still crying after a minute or two say "Alright, be quiet already!!! If you're all that hurt you should get over here so I can get a look at you!" Only to tell them "alright, you have made it across the park. You will live, now scram dammit!" :)) I :wub:  tough love.

Heck, come to think of it, I've had moms on here chide me for being too harsh with my standards about school work also. Hmm...interesting!

Interesting stuff! :)

 

 

I have a number of "going broad" resources listed in my siggy if you'd like more ideas. Good luck!

I think I'm in love! Those are amazing! Thank you! Thank you! Thank you!

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I have a number of "going broad" resources listed in my siggy if you'd like more ideas. Good luck!

I think I'm in love! Those are amazing! Thank you! Thank you! Thank you!

 

You are welcome! That post is from a few years ago and I just realized that the free Crypto Club workbook link was a broken link so I updated it. Do let me know if any other links are broken and I'll do my best to go back and update them.

 

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See, this is where I need to get the boys to be. They know that 'math isn't meant to be easy, its meant to be interesting' but they don't know that 'math isn't meant to be easy its meant to be interesting.' Possibly, this is my fault because I should have challenged them more and I was busy trying to keep it 'fun' that I didn't let them struggle enough early on. So despite all my cute slogans and inspirational speeches, they are used to math being easy and novel and that ease and sense of novelty is now a big part of why they enjoy math all the time. What I wouldn't give for a do-over, *sigh*.

 

I don't think there's any way of knowing if one way is better than another! We do what seems right with what we know and when we know different we do different. I expect my kids to crumble and lose motivation if asked to do lots of practice in basic computation, using more difficult tools but not using more complicated thought processes; you expect your kids to crumble and lose motivation if they're asked to do lots of heavy-lifting thought processes before they are even allowed to start multiplication; as parents we probably have some insight and we may well be right about our own kids! Time will tell, but chances are, all our kids will probably do okay regardless, since they have parents who want them to be their best selves. ;)

 

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Possibly, this is my fault because I should have challenged them more and I was busy trying to keep it 'fun' that I didn't let them struggle enough early on. So despite all my cute slogans and inspirational speeches, they are used to math being easy and novel and that ease and sense of novelty is now a big part of why they enjoy math all the time. What I wouldn't give for a do-over, *sigh*.

 

It *is* still early on, IIRC your kids' ages.  Besides, it's not as though there's zero depth in MM - maybe it was a just-right amount of challenge for what they've done so far.

 

I have found it tricky to find the right level of challenge for my boys.  Right now, they're at different places in AoPS Intro to Alg; sometimes it's more challenge than feels comfortable for them but every time we turn to another, more direct text for instruction on a topic, they are so uninterested that they only want to take in the procedure without really thinking (in which case they forget quickly).  So, we're using AoPS for teaching topics and sometimes adding in other stuff for practicing topics.  FWIW, I like Intro to Alg but I like Prealgebra better - more contest problems and overall more fun, IMO, perhaps simply due to the style of topics covered.

 

As you know, there's more than one path to problem solving, with or without AoPS resources - keep in mind the supplemental AoPS options for later (Prob Solving Vols 1 and 2 and the Intro to C&P and NT texts).  From what you have written, if you were to use Prealgebra for the actual teaching while they're on the young side, I would suggest you working together with them socratic-style at the white board - the sequence of the individual problems within a lesson works perfectly for that.

 

For problem solving generally, you might ease into problem solving at a level that's below their current math level.  You might try to move from working with you to stewing by themselves over a hard problem for a tiny bit longer at a time, until they're working more on their own; this can be a long, gradual process.  I just skimmed what's mentioned above for resources - you might look at the MOEMs books, particularly Volume 2, since it separates the elementary contests from the middle school ones, so you can choose which level might be appropriate (or try both).  Have you looked at Beast Academy?  There are some interesting topics covered there in a more age-appropriate way than the deeper version of those topics in AoPS Prealgebra.

 

For algebra, if you are looking for a text, I would recommend Jacobs, which seems well-suited for younger students and is perfect for following MM6.  The concepts are further developed gradually within the set of exercises that follows a lesson, but in a very gentle way.  There may be some logistical difficulty with any algebra text in that it isn't in a workbook, write-on-the-page style.  I'd use white boards.  For my handwriting-challenged boys, I often make up worksheets with problems from a text, though you probably don't have that sort of time on your hands.

 

Just my two cents.  I vote for continuing to make it fun at this age.  Fun and challenge need not be mutually exclusive (even if my boys might say so in the case of alg 1 :tongue_smilie:).

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One more thing, reading your post above more closely, IMO the point of AoPS is to learn mainly by working through the lesson problems, not by reading a transcript or lecture (i.e., the lesson problem solutions).  The solutions are there for help and clarity and to point out important things; I generally do that while they're doing the lesson problems and we spend very little time skimming through the solutions together.  On the one hand, you're absolutely correct that AoPS was written to be difficult; the learning philosophy is one of learning by doing hard problems.  On the other hand, depending on how you use AoPS as tool (as in, how much you work together through the lesson problems), you may be overstating the extent and inevitability of student stumbling, which IMO is adjustable - even moment-to-moment - based on amount of independent discovery vs socratic vs direct instruction.  In our house, we usually take a socratic approach, even though we have gone to the other extremes of independent or direct instruction at various times, especially in the Prealgebra text.  (I find that when I veer too much to direct instruction, my kids do not pay attention, basically.)

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Thanks for all the support everyone! This is fast turning into the most useful board I've ever started!

I was being a bit dramatic or facetious, but with the whole "what I wouldn't do for a do over" bit, but the support that you guys offer is amazing.

 

We are probably going to have to take a breather when they finish their 6th grade math books while I get their next set of materials and their new and improved Math Binders ready. I really want them to do their 'Arithmetic Reference' books soon and will probably continue it through the summer and they will probably continue on using the supplementary Algebra manuals and working on Math Mammoth algebra worksheets while I get their new books in order, but I'm still having them start in Chapter 1 of the Algebra text. It looks like we will be going through beginning algebra and some of Unified Modern Math C1P1.

 

I'm really loving those Unified Math Books, so thanks so much for telling me about them, Emerald Stroker. The boys are finally telling time on face clocks with easy now, so the first section on clock arithmetic must be kismet or something! The boys will get a kick out of UMM, I think.

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I can imagine that it is like a transcript to a lecture or something.

 

Imagine that you are standing against the wall of a very large room with little to no lighting and there is a mess of junk in the floor...

There are two ways to guide someone through that room:

-a- Stand on the opposite side of the room and talk to them so that they can follow your voice.

-b- Walk beside them so that when they stumble, you are there to help them up and guide them.

 

AoPS seems to go for an all out option -a- approach. AoPS will let the students follow the teachers 'voice' and by extension allow them to run in to stuff as often as they happen to. If the student runs into every.single.roadblock on the way, so be it. If the student gets stuck/falls/stumble, well, its up to the student to get up unaided, reorient themselves in the right direction and try again. But the whole time the teachers 'voice' is calling them.

 

*I* have always used an option -b-  approach with the boys when I am lecturing/teaching them.

 

The whole approach of AoPS is that there is no teacher lecturing, but that the student learns through problem solving. And yes, that means stumbling into obstacles and learning from that. The explanations come in after the student has worked on the problems, not before.

So, there is no "teacher's voice calling" - the student is faced with carefully constructed puzzles and led through them in a Socratic way (the way the problems and the steps in each problem build on each other is genius).

The explanations are exactly what a math teacher would need to explain for understanding of the concept; if the text is perceived as "wordy", it is because it is meant to be used without a teacher, and thus every word a good teacher would say to explain needs to be in the text.

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So, the boys and I have been talking about this and I'm trying to get them refocus and slow down a little bit. They had been speeding through an Algebra textbook but I talked to them about their current Math Mission (Finish all of Math Mammoth Materials--aka "8th grade") and we had a talk about finishing what you start, not giving up and not being distracted etc...

 

Together, we have come up with some tentative goals for carrying us through to the end of the summer.

0) Finish their current math stuff (MM6, ARME6, SSM, KtA)

1) make their own Math Guide*

2) Get as many Math Supplements/Challenge books through the library as possible (Penrose, Zacarro, Borac, etc), read and work through (some of) them.

3) Focus on Mental Math and do daily practice.

4) Math Mammoth 7ab + Algebra1ab worksheets with guidance from me + some math books to help them when needed.

5) Unified Modern Mathematics Course 1

 

I'm going to try and steer them into more problem solving type work, so I've been on the library site requesting/reserving books from Zacarro and others.

I really want them to turn their attention towards using what they know to work more interesting problems, not just dashing ahead...so I'm thinking that a rehash of elementary math using out-of-the-box resources will be beneficial to them.

 

MY BIG PROJECT: is to create a sort of 'workbook' out of UMMC1 for them to use when the time comes. I had thought about just jumping in, but it will be too much for them right now. They can't handle the dense textbook format + separate book for writing out answers just yet. Plus going through the books with a fine toothed comb will help me prepare to guide/assist them more readily. (Yay! Maybe this can be one way for me to "help" my kids--adapting materials to their abilities!)

 

Whenever we get through items 0-4 on that list, we'll begin doing UMM and a formal course in Algebra 1, the boys picked a textbook that they like so I will probably be writing a workbook for that also--cutting down on the problems, condensing as needed and adding in challenging problems from other books and of course making sure that they will have enough room...Plus, I can build in the spiral review that I want my kids to have for Algebra 1. They seem to get it already, but I want to be sure that we are really and truly rock solid on the fundamentals of Algebra!

 

We won't be starting for several more months, but Pal insists that he wants to do Geometry so does anyone know if Saxon Algebra 1 and 2 would be a good idea for him? Don't those books have Geometry written into the Scope and Sequence of Algebra?

 

*Math Guide: A book of notes, examples, explanations, summaries, and definitions that is written and assembled by the student that is meant to serve as a reference to lower level math/arithmetic and the making of which provides a comprehensive review of lower level math. The idea is that this will take weeks (if not months) to do and will allow us to check for any cracks in their foundation while we slowly add in more.

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5) Unified Modern Mathematics Course 1

 

<...>

MY BIG PROJECT: is to create a sort of 'workbook' out of UMMC1 for them to use when the time comes. I had thought about just jumping in, but it will be too much for them right now. They can't handle the dense textbook format + separate book for writing out answers just yet. Plus going through the books with a fine toothed comb will help me prepare to guide/assist them more readily. (Yay! Maybe this can be one way for me to "help" my kids--adapting materials to their abilities!)

 

<...>

 

We won't be starting for several more months, but Pal insists that he wants to do Geometry so does anyone know if Saxon Algebra 1 and 2 would be a good idea for him? Don't those books have Geometry written into the Scope and Sequence of Algebra?

 

I like both UMM and the new math curriculums in general. The first chapter on various finite number systems reminds me alot of the "Elements of Mathematics" modular arithmetic material. Since Burt Kaufmann was on the UMM board this isn't surprising. The stuff on mappings, groups, and fields is a gentler version of Allendoerfer and Oakley. It is good stuff. However, I'm not convince it is the best option for *very* young accelerated kids. If feel like there is more formalisms and jargon than you really need. Many of these topics are included in AoPS without the new math jargon. Sure introducing abstract algebra early has some *huge* benefits but there is also a bunch of cruft that comes along with that approach. If you really want to pursue that approach you should try searching from some of myrtle and charon's old posts. Unfortunately, most of those were on the old boards which are gone. If you use archive.org's wayback machine you can still access the blog posts referenced on some of the WTM posts.

 

For geometry(or any other topic), I wouldn't use saxon. Jacobs' Geometry is highly regarded and the 1st ed is available on Amazon for <$20 w/ shipping. I have a copy of this and plan to use it with DS7 sometime in the next year or so.  Jacobs' books all do a good job of balancing approachability and rigor. If your kid loves that... mathwonk has great advice on moving on to  Euclid.

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Hi Gil. I am finding your story and problems so very interesting.  You have definitely identified math talent earlier than I did with my son, and this has led you to a fascinatingly different path than I have had with older ds.  My goals were to challenge him at all times, but *he* needed to do it without me.  So he was challenged not by the math, but by *his* requirement to do it without any instruction.  I'm not talking teacher instruction, I'm talking any instruction at all - text, internet, personal interaction, anything. His driving desire for this approach started at about age 7. It was as if he *needed* to be wandering in the dark at all times, stumbling into obstacles until he finally found the light switch.  His genius was forged in the fire of frustration. 

 

Now I would not suggest that your 6 and 7 year olds should do this kind of trial by fire, but given their talent I think that you should start moving away from instruction type 'b' and towards instruction type 'a' (or actually what Regentrude describes in post 29).  Is it less efficient? YES for understanding the math techniques, but NO for problem solving.  The kids at the IMO prep camp talked about having the inspiration or insight into a problem and *how* to actually get it.  It is this inspiration that the wandering-in-the-dark approach teaches. Yes, it does take longer.  I think my older could be done with calculus by age 12 if we had used direct instruction. But what he has found is that there are kids in his AoPS classes that are young and brilliant, but they just cannot do the problem solving.  My guess is that they learned through direct instruction which is why they accelerated so quickly (as in 5th grade and AoPS algebra 3 class).  But if you think of math as 2 streams - the techniques and the problem solving, their technique skill is light years ahead of their problem solving skill.  In contrast, my son is advancing in both streams more evenly because of the AoPS approach.

 

As you know, there is no hurry.  And as you know, you simply cannot predict the future with kids like ours.  My older took almost 3 years to do the AoPS Intro Algebra book (equivalent to USA algebra 1 and 2), but that was because he did ALL the challengers in the entire book with NO help of any sort.  He fought, hard, to do the work.  It seemed so slow to *me* but that was because I could not actually see what he was learning.  And apparently it was a LOT. The following year he did AoPS intro geometry, intro counting, intro number theory, intermediate number theory, first 1/3 of a university text (Art and Craft of Problem Solving), the IMO training camp take home exam, and the BMO (British Math Olympiad).  Wow!  What a jump.  This year, he will advance to the intermediate level in geometry, algebra, counting, and pre calc AoPS material, and hopes to get on the 12 member squad for NZ.  At that point he will have just started 9th grade.  If he had finished calculus at 12 with a focus on the techniques rather than the problem solving, he would not have an eye on the IMO. 

 

I don't even know what I am trying to tell you here, just that perhaps you cannot predict what will happen, or perhaps that you want to carefully and gently move away from direct instruction with such gifted math learners.  I don't know.  But I will say it again.  It was the frustration, the tears, the raging that forged my boy's math talent and achievement.  He was a bit older than yours and it was the approach that he desired, but the lack of teaching was the key to his success.

 

Ruth in NZ

 

 

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The trick, I find, comes of taking a topic in which the student has had direct instruction, and somehow making it fresh enough to be worth problem-solving with. This is where I like competitive math. The problems come out of left field with some of this stuff and no amount of computational expertise will make you see the patterns that need seeing in some of the puzzles. Then you can throw in some stuff that encourages you to see the same ol' computations in more elegant ways...computing the sums of the first 100 integers or whatever.

 

I'm watching your threads more closely than ever, because while I know my hooligans don't have the passion or the genius of yours, my DD5 went to bed last night begging for learning...she worked through the first few chapters of LoF Apples yesterday on her own, writing out her answers on a separate piece of lined paper, and at first her suggestion for learning time was Fred but then she sighed, "Fred is too EASY. My other books [singapore 1A] are easy TOO." I said, "I've noticed that addition and subtraction facts are coming a lot easier to you lately. It might be time to start skipping around a bit in your Singapore book." She sat up in bed with wild grin and said, "Oh, THANK YOU MOMMY!" and hugged me hard around the neck.

 

Um. Still holding tightly to the last fraying thread of my firmly-held theory that kindergarten (which begins in fall for her) needs NO academics even for the most accelerated kids?

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Um. Still holding tightly to the last fraying thread of my firmly-held theory that kindergarten (which begins in fall for her) needs NO academics even for the most accelerated kids?

 

My older did NO formal academics until first grade, and at 6.5 I bought him a singapore math workbook.  I did not add in any other formal academics until 2nd grade.  We just unschooled.  We went to museums, concerts, the grocery store.  We drew, rode bikes, went to the pool.  I talked to him about science and we did a big mushroom project in 1st grade by wandering around in the woods identifying and counting mushrooms. His dad read to him about history and played with maps and globes. He played the violin and read books.

 

Formal work is not necessary, and really depends on what your kid wants.  And delaying certainly did not hold back my older, even in math.  You just have to be willing to compact, a lot.

 

Ruth in NZ

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Formal work is not necessary, and really depends on what your kid wants.  And delaying certainly did not hold by my older, even in math.  You just have to be willing to compact, a lot.

 

 

I dont think it is necessary and with my kids I have never compacted anything.   Whatever happens simply just happens.   :)

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I should put that a little differently. I still don't think kindergarten *needs* academics. But because of my views about kindergarten being primarily for fun, I hadn't been fretting about the complete lack of academic challenge in public school kindergarten. So now I'm seeing that my daughter is different. If she's already getting frustrated with work that's too easy, then maybe a kindergarten classroom for kids who are just learning to count past 10 is not the right place for her after all.My point in posting this here was that I empathized but didn't really understand the folks with very small kids who insist on progressing their computation skills before their problem solving skills catch up. Now I'm getting a taste of it and it's kind of upsetting my apple cart.

 

Still, I'm pretty happy to have a little guy who'll mess around with his "theories" for "solving subtraction problems with a blank" or who wants to invent "one of those symbol things like a plus or multiplication" that will take the inputs and create new numbers or digit sums for the outputs, who is willing to tackle a rate problem with only rudimentary multiplication skills. I would like it a lot if his sister would also be willing to think about stuff from a few angles before leaving it behind, and I'm currently brainstorming ways to make that happen. A rousing game of "There are Zero" from LoF was a great start today.

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You are welcome! That post is from a few years ago and I just realized that the free Crypto Club workbook link was a broken link so I updated it. Do let me know if any other links are broken and I'll do my best to go back and update them.

 

What? Free Crypto Club workbook? Headed over now. Thank you!

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Thank you lewelma and everyone else who replied. I'm still wrestling with this a bit. A good bit. A lot. A whole, whole, whole lot. One minute I have a plan in mind, the next minute I'm in turmoil again. I don't know if I feel up to this, I really don't. I understand why parents send their kids to school--I really, really do. There are already a million and one ways for a parent to screw up their kids and I feel like I've just had my potential for failure doubled twice over now....

 

I do know this much: As we move beyond elementary arithmetic/mathematics, I'll be cutting back quantity and going for quality. While before we have done every problem, every time, that approach isn't going to be as feasible or very logical for Algebra where the problems are easily 4-8 steps and many of them are precision exercises. Besides, our reason for doing so in MM was different, there was more to learn by doing all the problems in elementary math and not all of them were math related. But those reasons won't take as high of priority in upper level math. I'm going to have switch the focus to problem solving for '2nd grade'. I realized that I feel uncomfortable with what I have seen of AoPS - Intro. to Algebra so I'm going to try and take another, more in-depth look at the program and see what it will take 1) put my mind at ease 2) to better understand the method that can be used to guide/coach precocious children and 3) to get them ready for that type of instruction. Not the book, per se, but definitely that type -a- instruction/guidance we were discussing up above...

 

I'm rethinking my original view on the Singapore CWP series (see here) and am thinking of tailoring their math syllabus toward the 'challenge' problems (especially for Algebra) in general. The way that they are talking now, I'm going to have let them drift apart for math sooner or later. Pal really, really wants to do Geometry. He has always enjoyed the geometry chapters in MM and his favorite math-friend is a big fan of Geometry also so theres that. I'm digging out all the resources that I have on hand and looking through the free stuff linked in the forums also. I had really hoped to go with Singapore upper level math programs, but it looks like none of them are complete anymore and I haven't been able to find any sets online 2nd hand, but that's a whole 'nother box of blooms.

 

Still, there is a part of me that begins to feel more and more conflicted with encouraging or even just allowing this pursuit in my too-young-sons. Especially Pal. I love seeing him excited and engaged about something, I love seeing him thrive in 'our' subject, but a part of me--the selfish part--feels robbed and cheated of the chance to teach and guide my sons the way a normal teacher or parent would want to. Another part of me, the selfless, fretful paternal part is just afraid because I don't want him to burn out, I don't want him to crash into a glass ceiling at break-neck speed and I don't want him to come to hate math the way I have seen some of my friends grow to hate what was once a passion of theirs that their parents or coaches ruined for them. (ex: I have a friend who plays music exceptionally well, especially strings and yet, she hasn't played a note in 5 years after having been soured on the whole experience right after starting college)

 

I guess that 2nd grade will be a year of big changes--I'll have to let go of course, if I want to allow them to reach their potential (in general, not just in math) but I don't want to 'drop them' so to speak and just let them crash and burn either. I feel ridiculous for even getting this worked-up over math for 6yos but...anyway, I'll stop there. After I made my last post in this thread, I decided to give myself some room and time from our math issues. I need to figure out the rest of our homeschool because despite what the boys say I think that elementary should include more than just math.

 

Thank you all so much for all that you have contributed, both to this thread and to the forum in the past. I have been sifting through the archives a lot and I continue to unearth a trove of information, insight and inspiration on these boards.

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I think one thing that really causes profound interests to be *ruined* for young kids is when the kid wants to back off and go at normal speed for a while/pursue other interests and the parent/teacher/coach keeps pushing.

 

Be alert for this tendency in yourself, and also watch your children. Make sure that they know (and you probably are doing this already, but I'm just reminding) that it's okay to be interested in other things too, and that although you love doing math with them, you'd love to support their interests in other areas too.

 

You sound like you're doing everything right so far.

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I love seeing him thrive in 'our' subject, but a part of me--the selfish part--feels robbed and cheated of the chance to teach and guide my sons the way a normal teacher or parent would want to.

Gil, I am definitely fully involved in my ds's math. He *needs* me to care enough to listen to his discoveries. Instead of spending an hour explaining something *to* him, he spends an hour explaining something to *me*. When I am too tired to listen or really not in the mood, he is hurt, really hurt, so I must make a conscious effort to be there for him for at least an hour a day if not more. He wants me to proof read his proofs, to bounce ideas off of me, to have me learn with him. I am NOT separate from his learning and I am really sorry I did not make that clear. He really needs me even through it is his strongest subject. The camaraderie is incredibly important.

 

Another part of me, the selfless, fretful paternal part is just afraid because I don't want him to burn out, I don't want him to crash into a glass ceiling at break-neck speed and I don't want him to come to hate math the way I have seen some of my friends grow to hate what was once a passion of theirs that their parents or coaches ruined for them.

We all worry in this way. It is very hard to know when encouragement turns to badgering, and I have learned to be *very* observant of his emotional cues. He got very burned out right before the BMO last year, and I backed way off. Right now, he is in a good space and is doing about 4 to 6 hours a day. It varies, a lot.

 

It appears to me that you are doing everything right. So take a deep breath, and enjoy your little people!

 

Ruth in NZ

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Gil, here is an x-post from last year. You are clearly more skilled and knowledgeable in math than I am, but I thought that this description might help you understand how my boy and I worked together last year:

 

x-post

 

Yes, that is the book. My son is 12. We are currently able to understand only 30% of it. It is a University math majors textbook. The author states that you should read each chapter until you don't understand, then move to the next chapter. When you finish the first pass of the book, you start again. This approach allows you to work at your personal level in each topic, and allows your ability in all topics and problem solving to be increased concurrently.

 

There is NO way that my son could work with this book independently. We work on each problem together and then read through the proofs together. If the problem is easier, we each separately investigate it and write up a formal proof and then compare. My goal is to find ideas in each problem that will be generalizable to other problems. We keep a list, and I quiz him every day about the different generalizable skills we have learned. For example, what kinds of problems are would likely be helped by the extreme principal? or what kind of problems suggest a proof by induction? How can you recognize parity in geometry problems? These types of questions are not directly answered in the text -- they are more of a way for us to really internalize what we are reading and categorize all the ideas. Plus, it helps us review esoteric ideas by recalling specific problems that reflect them. We've decided that if there are 20 different tactics that are possible, and we can recognize that 4 are good candidates for a certain problem, we can try those four. If one works, great, if none work, then at least we have gotten our hands dirty and have a much better understanding of the problem and can go from there.

 

To help in proof writing, I drill him on specific phrases like "This specific case is generalizable because the only special feature of 11 that we used is that it is odd." (yes, I am memorizing all this too, so that just popped out of my brain). This drill has really helped him not only with the language of math, but also helped him realize different approaches he could use to prove a conjecture. For example, the above case showed us that you can use an example as your proof in many cases of parity. This is very important to know, because most proofs do not allow this. Our overall goal is to get as many tools in our tool box as we can, and then remember what tools we have in there!

 

All this is really working. I cannot believe how far we have come in 2 months.

 

I told someone last week that I could only go through this process once because what I am giving my son is not a knowledgeable tutor, but rather a skilled learner who is at his exact level in math. If I ever go through this material again with a student, I would be much much more knowledgeable and I would loose the confusion that has been so critical in helping him battle through this material. What I am finding is that because I don't know the answers and I cannot teach him how to do it, I am instead teaching him how to learn problem solving -- what questions to ask, what answer to hunt for, how to compare problems, how to really interact with this material. No tutor who knows the material well could do this as well as I can, because once you have the knowledge, it would be virtually impossible to relive the confusion.

 

But then I realized that because my memory is so shaky, I could probably do it one more time. :001_smile:

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