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School still too easy for my 10yo dd...


LisaK in VA
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I honestly didn't mind that my oldest dd could finish her daily work in a 2 hours or so when she was in the 2nd grade.

 

She's now in the 5th grade. In approximately 2 hours she has completed all of her required work (except for an hour of reading & Latin...which is the one subject she really must work, she's working through Wheelock's with her older brother who is enrolled in Scholar's Online). That 2 hours includes a requirement of about one hour in math daily...

 

She's still finishing up last year's reading, grammar and spelling (behind, due to moving...), and she's been working on a one-year acceleration in those subjects (definitely not challenged, but something I haven't worried too much about, either). However, as she's approaching the middle- and high school years, I think she should start facing a bit more challenging work (not just more, for the sake of doing more)

 

I haven't yet purchased her new reading/spelling/grammar...I was planning on Abeka grade 6 for those, but now I'm wondering if I should up the challenge a bit and see how she does in Abeka 7 for those subjects.

 

She is hard working, diligent, and rises to a challenge. She is tackling the pacing of Wheelock's like a champ. She will be starting BJU Earth/Space Science (with DVD) in another month...I'm certain that will be a challenging program as well. She's flying through LoF Pre-Algebra, and will be starting AoPS pre-Algebra & TT Algebra 1 probably within the next 4-5 months.

 

I think she'd do fine with Abeka 7, it would still be a bit on the "easy" side, and I could bump her up to Voyage with her older brother, vs. working through Town with her younger brother.

 

What do you think? I know Abeka isn't the best...but it works for us for grammar/lit/spelling (at least through 8th grade, after which we switch to BJU for Lit).

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I know this book I recommend may be unsuitable for all but maybe one in ten thousand kids, but for that one it may be a mind opening life altering experience. Even for the average kid, I suggest that the first chapter, article 1, paragraphs 1-5 of the first two pages of text in the book, are very useful. There the great Euler explains what mathematics and numbers are, and how they are applied to the study of "quantities", which he also explains, in a way one will never find in any other book. Since this forum in general and this thread in particular are concerned with meeting the needs of children who outstrip ordinary sources, this seems a good place to mention it again.

 

This is the book I propose for profoundly gifted learners, and I suspect it will help many others among us as well. And it is free.

 

http://archive.org/stream/elementsalgebra00lagrgoog#page/n35/mode/2up

Edited by mathwonk
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The section referred to is where Euler takes the trouble to explain how mathematics enters the study of the world around us. He points out it is useful for measuring "quantities", i.,e. anything that can increase or decrease. Moreover it can only do this after a unit quantity has been chosen. At that point the quantity becomes a number which can be calculated with. This crucial aspect of how to apply mathematics to practical matters is usually skipped over in silence, and the student is then greatly puzzled when he encounters various units in physics.

 

This one concept can form a whole lesson for the child learning about math and the real world. E.g. distance measure does not become a number until a unit distance like a foot or a mile has been chosen, or maybe a meter. To help the child understand why a meter is a better choice than a foot, just pose the question of communicating over a telegraph to someone in Europe how long is a foot? Obviously you cannot ask them to hunt up an ancient king and measure his foot!

 

But you can ask him to compute a certain fraction of the circumference of the earth, if he has the technical ability. On the other hand, what if we want to communicate with someone on another planet how long our unit if distance is? Then we see the value of using a "light year" or the radius of an atom of sodium, as Maxwell suggested.

 

A very interesting example is to analyze how one would measure area of plane figures in units which are multiples of a given square. I.e. how does one decide how many copies of a certain square fit inside another plane figure? This is the topic of a proposition in Book I of Euclid where he shows how to start from a given polygon and construct an equivalent rectangle having one side equal to the given unit. Then the area is measured by the length of the other side.

 

 

Anyway this is a fascinating subject and helps the student understand many things I wish my students in calculus had encountered.

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I really like this - but wanted to stop and note that in the right hand margin, the original owner of the scanned document made short little pencil marks which have been retained.

 

I kept wondering as I read along, "What's that for, lining up the scanning of it?"

 

A little farther in I suddenly realized the owner of the book was pulling out definitions and terms for distinctions.

 

Pretty cool. :001_smile:

 

I can imagine a situation where this marked portion of the book was paused at for a discussion in a class, or transcribed into a set of notes. I mean really, who says: "Hitherto" anymore?

 

Took a little bit to get into the typography of the document, but it really is lovely.

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I am so happy you gave yourself the pleasure of reading the real thing by the master Euler, and hope my little summary here did not spoil any of the pleasure. I tried to wait to give you time to get it from the horses mouth first, but also wanted to chime in with what it inspired in me too.

 

last summer I successfully taught the method of Cardano for solving cubic equations, from chapter XII part I, pages 262-269 to brilliant 10 year olds. That will knock the socks off your kids' fifth grade teacher, guaranteed.

 

I wrote the material up myself as a supplement, in a few pages with worked exercises which I would happily share, but this site has a very small limit for pdf files. maybe some of it is in a copiable form.

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I'm really enjoying it...(have I said that already? :) ) - anyway, I'm plodding along, reading it with my morning coffee - got about an hour in again today...good brain food first thing in the morning you know?

 

Got curious and wondered, "Man, I hope this is a really, really long book".

 

Uh yep.

 

It's 600 pages..lol - I should be done in a couple of years.

 

Who uses viz. as a term anymore in math? That was my favorite "had to look it up for the proper definition" word today from the book

 

http://en.wikipedia.org/wiki/Viz.

 

And I was *sure* I was mentally mispronouncing it, so I had to look that up as well (yes, I was, I'm all better now)

 

http://www.thefreedictionary.com/videlicet

 

So that was the scene this morning, Prince Charming comes out to the kitchen where I am drinking coffee with noise cancelling headphones on, messy hair, half-awake saying:

 

Videlicet ---- over and over and over...:lol:

 

Was just a little embarrassed. :blush:

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I confess to using viz. in math papers I have written. What does it mean. I forget? for example? or something similar? Ah yes, namely! I knew that when i used it , honest.

 

if you really want to learn something not everyone knows try this next one out. These are some of the notes I sent out to my 10 year olds to supplement reading Euler. They assume a good understanding of solving quadratics (which we reviewed as well), i.e. that solving quadratics is the same as finding two numbers when given their sum and product. (the solutions u,v of X^2 - BX + C = 0, satisfy u+v = B and uv = C. Every kid should learn this in basic algebra but perhaps some only learn mechanical solution techniques.)

 

Cubic formula

Suppose we have x = a+b. Then cube both sides, and we get x^3 = (a+b)^3

= a^3 + b^3 + 3ab^2 + 3a^2b = a^3 + b^3 + 3ab(a+b) = a^3 + b^3 + 3abx.

Thus x= a+b is always a solution of the special cubic equation

X^3 = 3abx + (a^3+b^3).

 

In fact, this equation isn’t so special.

I.e. if we are given an equation x^3 = fx + g, we can try to solve it by finding a,b, so that f = 3ab, and g = a^3 +b^3. I.e. then x = a+b will be a solution. We claim such a and b can always be found.

 

E.g. given x^3 = 3x + 2. We can look for a,b so that 3ab = 3, and a^3 + b^3 = 2.

In this case, a=b=1 works. Thus x = 1+1 = 2 solves the equation, i.e. 2^3 = 3(2) + 2.

 

Another easy one is x^ 3 = 9x + 28. We want a,b so that 3ab = 9, and a^3 + b^3 = 28.

But a=1, b=3, works. Thus x = 1+3 = 4 solves the equation.

 

Try x^3 = 12x + 16. If you don’t guess it, notice that we want 3ab = 12, so we want ab = 4. And we want a^3 + b^3 = 16. Now this implies that a^3b^3 = 4^3 = 64. So if we try first to find a^3 and b^3, we know their sum is 16 and their product is 64.

 

From our study of quadratic equations, this means that a^3 and b^3 are solutions of the quadratic equation t^2 – 16t + 64 = 0. But t = 8,8 works for this. So we have a^3 = 8, and thus a = 2, and b = 2, work. Thus x = 2+2 = 4 should solve our equation, and it does so, since 4^3 = 64 = 12(4) + 16 = 48+16.

 

In fact this solution procedure always works, i.e. every cubic of form x^3 = fx+g, can always be written as x^3 = 3abx+(a^3+b^3) for some a and b. We find a and b by solving a quadratic equation like the one above.

 

To see it, we just have to show that if we are given f and g, then we can always solve 3ab = f and a^3+b^3 = g, for a and b. Lets start by solving for a^3 and b^3. If f = 3ab, then f/3 = ab, so f^3/27 = a^3b^3, and thus we are given a^3b^3 = f^3/27, and a^3+b^3 = g, and we want to solve for a^3 and b^3. But we know from our study of quadratic equations, that if we are given the sum and the product of two numbers, we can find the numbers.

 

So we can always find a^3 and b^3 so that a^3b^3 = f^3/27, and a^3 + b^3 = g, by solving the quadratic equation t^2 – gt + f^3/27 = 0.

 

E.g. in the case of x^3 = 18x + 35, we have f = 18, g=35, so f/3 = 6, and f^3/27 = 6^3 = 216. So to find a^3 and b^3, we want to solve t^2 – 35t + 216 = 0. Since 216 = 6^3 = 2^3.3^3 = 8.27, we see that this equation factors as (t-8)(t-27) = 0, so a^3 = 8 is one solution, and we can take a = 2, and then b = 3. So x = 2+3 = 5 is a solution. Indeed then 5^3 = 125 = 18(5) + 35.

 

Try this one: x^3 = 24x + 72.

And this one: x^3 = 30x + 133.

And this one: x^3 = 18x + 217.

 

Of course a cubic should have three roots and we are only getting one root. The reason is that, at the step where we take a cube root, we are only taking the one real cube root. But every real number has three complex cube roots, two of which are imaginary. If we took also the other two imaginary cube roots we would get the other two roots of our cubic.

 

E.g. if we consider x^3 = 27x + 54, we have f/3 = 9, so f^3/27 = 9^3, and g = 54,

So we want to solve t^2 – 54t + 9^3 = 0 for t = a^3.

 

Now 9^3 = 3^3.3^3 = 27.27, and 54 = 27+27, so we get a solution of t = 27 = a^3 and a =3, b = 3, and x = 6.

But we also have two other complex cube roots of 27, namely 3 times the complex cube roots of 1, i.e. a = (3/2)(sqrt(3)i – 1), and a = (3/2)(-sqrt(3)i-1).

 

Then since 3ab = f = 27, we get b = 27/3a = 9/a = 6/( sqrt(3)i – 1), and

b = 6/(- sqrt(3)i – 1). Then x = a+b = (3/2)(sqrt(3)i-1) + 6/( sqrt(3)i – 1) ,

and (3/2)(-sqrt(3)i-1) + 6/( -sqrt(3)i – 1).

 

If we rationalize the fraction b = 6/( sqrt(3)i – 1) by multiplying top and bottom by

( sqrt(3)i + 1), it becomes 6( sqrt(3)i + 1)/(-3-1) = (-3/2) ( sqrt(3)i + 1). Thus the sum a +b = (3/2)(sqrt(3)i – 1) + (-3/2) ( sqrt(3)i + 1) = -(3/2) – (3/2) = -3. And indeed x = -3 also works as a solution.

 

Then last one should work too, since then a = (3/2)(-sqrt(3)i-1), and b =

6/(- sqrt(3)i – 1) = (-3/2)(- sqrt(3)i – 1), so

a+b = (3/2)(-sqrt(3)i-1) + (-3/2)(- sqrt(3)i + 1) = -3.

 

So x = -3 is a double root, as we can test further if we know calculus, by taking the “derivative” of the equation, getting 3x^2 = 27, and we see that x=-3 is also a root of this equation. Or without calculus, we can check it by multiplying out (x+3)^2.(x-6) = x^3-27x-54. The square on the factor (x+3) means that x = -3 is a double root.

 

This procedure thus gives all three roots of a cubic of form x^3 = fx+g, but only by using complex numbers, even when the solutions themselves are all real.

 

Finally, every cubic can be transformed into one of that form, i.e. without an x^2 term, just as a quadratic can be changed into a perfect square, i.e. one without an x term. The reason is that just as the coefficient u, of x^2 –ux+v, is the sum of the roots of the quadratic, so also the coefficient u of x^3 – ux^2_+vx-w, is the sum of the roots of the cubic. Thus if we subtract u/3 from each root, their new sum will be zero. I.e. let x –u/3 = y, or x = y+u/3, and multiply out x^3 - ux^2 + vx –w =

(y+u/3)^3 – u(y+u/3)^2 +v(y+u/3) – w, we should get a cubic in y with no y^2 term.

 

I.e. the y^2 terms come from the first two terms, the first one has y^2 term y^2.u, while the second term has y^2 term –y^2.u. So yep, they cancel.

Next we solve our equation of form y^3 – fy –g = 0, or y^3 = fy+g, by the method explained above, and then we take x = y+u/3 to solve the original equation.

 

Some hardy soul figured out how to extend these ideas also to solve quartic equations (see Euler), and then it took 300 more years of effort for someone to show that quintics cannot be solved by these methods.

In particular the solutions of X^5 = 80X – 2, cannot be written as algebraic combinations of roots of algebraic combinations of the coefficients. I.e. there is no formula for the roots involving only integers and the symbols +, -, . , /, and nth.root, for various n>1.

 

The proof of this took me a whole quarter last time I taught it, using Galois theory, but maybe if we read Abel’s proof it will look easier. (Euler did not know this, it seems.)

Edited by mathwonk
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Viz.

 

It is permitted to know - it means a stepping back, like a breath in the precise pattern of a solution, I think I ran across the same idea yesterday which is commonly thrown around as "dual N back" memory tracing.

 

It's like waltzing the previous step(s) at a certain count; or as Lawrence Welk would have it, " a one ah, a two ah, a three ah" You step back, pick it up as a whole and hit "four ah." Like an echo, make sense?

 

Just ate pie (lol, get it..pi?) anyway, will re-read prior post with great interest several times, and when I strike it rich and settle, print it.

 

Gem! :D

Edited by one*mom
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Speak of the debbil...."those errant knaves" - just dying to know if your research on Tartaglia lines up with this bio line from the Catholic Encyclopedia:

 

http://en.wikisource.org/wiki/Catholic_Encyclopedia_(1913)/Nicol%C3%B2_Tartaglia

 

(clip, bolding/color mine)

 

In 1521, he was teaching mathematics in Verona and in 1534 he went to Venice. B 1541, he had achieved the remarkable triumph of solving the cubic equation. In a mathematical contest with Antonio del Fiore, held in 1535, he had shown the superiority of his methods to the method previously obtained by Scipione del Ferro (d. 1526) and known at that time to del Fiore alone. The glory of giving these results to the world was not for Tartaglia, as Cardan (q. v.) having in 1539 obtained a knowledge of them under the most solemn pledges of secrecy, inserted them, with some additions and with some mention of indebtedness, in his "Ars Magna", published in 1545. A long and bitter controversy ensued in which Cardan was supported by his pupil Ferrari.

 

Did Tartaglia really sandbag on this whole thing for *that* reason?

 

Any history reading you've done that leads you to a conclusion one way or the other?

 

ps: Try some Wendell Berry, you'd dig him.

Edited by one*mom
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My understanding is just what you've said, the result was due to tartaglia, and communicated in secret to cardano, then published with credit by him, but bitterly resented by Tartaglia. This is with some justification, since publication is the surest form of credit, and for some 500 years since, the method has been known unjustly as Cardano's method.

 

But the error was tartaglia's since he should have known this rule, and should not have refrained from publishing. Many of us have experienced the same.

 

my earlier explanation of this stuff is in the first 29 pages of the notes:

 

http://www.math.uga.edu/%7Eroy/844-2.pdf

 

especially page 24-29. You will easily notice however that this stuff is impenetrable, and hence that I did not understand it myself. hence I could not even explain it to grad students at that time. After reading Euler however, I actually understood it and could then explain it to 10 year olds.

 

 

by the way, until your post i did not realize that "arrant" is a variant of "errant". thank you again!

 

http://www.thefreedictionary.com/arrant

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Your version is much better. More smack to it.

 

Scanning through, just enjoying the ideas moving by, I noticed on 25 or so LaGrange (*which is NOT related to the ZZ Top musical issue of note*) floated by.

 

It's been bugging me since I sort of separated one from the other, and I kept wondering at what junction (or even if) nano-tech came in. It was LaGrange...neat.

 

I grew up very close to this place: http://www.mmi.org/ - spent a lot of time in the yards & gardens nearby.

 

Conversational proximity I guess you'd call it. So that general idea has sort of been set and rooted for me now. :) Fun!

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It was once a small town, I can remember when it was about 15K. The MMI is about amazing in the people they pull through there. The library is on the other side of the MMI, it's a pretty profound place...just across the lawn there are gardens, and the architectural palaces, and archives. But these places all radiate their own things, but were run by beautiful people who always had time for children.

 

There is an arts community there, large theater...never had to knock, just go in. The people that came in and out, the conversations of science; people from all over the world, I was so lucky....so we'd sit on the lawns, talk, watch hands fly around, or pointed to the gardens for real work, side by side.

 

Artists of stage, brain..all sorts and kinds, it was a place where at night, in the library, they'd meld, speak, talk, trade ideas, make meetings, make friends..

 

I saw in LaGrange were he spent time try to figure out the sides of the moon, very nice.

 

I heard the video where they wanted the kids to learn a machine this last week, and bypass the understanding..

 

And then I think of them, those people back there, and I miss them. But they are still out there. I found one here:

 

 

If you watch, you'll see/hear where she played with dishes to try to understand the one side of the moon, just like LaGrange did.

 

Pretty neat filming. It's a keeper.

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I'm kind of ok with him only spending 2ish hours on school work because he also does other things on his own that are challenging (computer programming, electronics, etc.). He also willingly reads a lot on his own. So aside from upping the level of the books, maybe you could get her involved with an interest/hobby where she could use her talents and motivation.

 

Oh, she has plenty of hobbies/interests...drawing, knitting, AHG badgework (she finished all of the GS Junior badges last year), reading...she reads roughly 1000 pages a week outside of school (takes after her momma). I'm not interested in having her in school for the same amount of time as a high schooler, just making sure the work she has is more appropriate. BJU 8th grade science is going to be difficult (according to BJU, that course is *really* written at a HS level), her Latin course is difficult (high school course), math is at an appropriate level, but LA is really too easy right now. I'm not trying to punish her diligence and worth ethic, but she does need a bit more.

 

She should at least spend 4 hours in school a day, and that would still have her finished with school by lunch, giving her 2 hours to swim, and plenty of time to draw, paint, knit, work on badges, and read to her hearts content.

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School is also too easy and simultaneously too hard for my gd. I have an 11 year old gd who tests as a 19 year old in problem solving and as an 8 yr old in addition skills. This seems to cause a problem for the cretins who teach at her school. helloooo.... one of those skills is more suited to a calculator and the other to a human being, guess which? advice? boarding school is being discussed. what?????

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School is also too easy and simultaneously too hard for my gd. I have an 11 year old gd who tests as a 19 year old in problem solving and as an 8 yr old in addition skills. This seems to cause a problem for the cretins who teach at her school. helloooo.... one of those skills is more suited to a calculator and the other to a human being, guess which? advice? boarding school is being discussed. what?????

 

:DI haven't heard that word used in a long time:lol:

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welcome to the world of "impatient man".

 

 

by the way I haven't heard this quote for 40 years:

 

"This world is a comedy to those that think, a tragedy to those that feel."

Horace Walpole

 

At that time I "flattered" myself that to me the world was a comedy. Perhaps i just prefer to spend my time laughing rather than crying.

 

As to "cretins" this is a word I recall hearing last in 1981 from a friend of mine at Harvard, a Fields medalist, in reference to the math faculty of a certain liberal arts college in the northeast, also containing friends of mine from high school, that did not see fit to rehire a brilliant mathematician who was apparently too intelligent to teach their students.

 

Did you realize that being too intelligent is a job liability even as a faculty member of a college? I have known several cases of this phenomenon, e.g. a full professor at Johns Hopkins who could get hired at a state school in the mountain west. One should never try to "drop down in class" too far. The appreciation level does not go up, it goes down, if you are overqualified, since then the people you try to appeal to cannot understand what you have to offer.

 

if you can't dumb it down enough, you can't survive in the USA, not even in academia.

 

It was extremely embarrassing to me as an academic to realize that although one cannot be too strong to be hired as a meat lugger (a job I held proudly for a couple of years), one can be too intelligent to be hired as a professor.

Edited by mathwonk
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School is also too easy and simultaneously too hard for my gd. I have an 11 year old gd who tests as a 19 year old in problem solving and as an 8 yr old in addition skills. This seems to cause a problem for the cretins who teach at her school. helloooo.... one of those skills is more suited to a calculator and the other to a human being, guess which? advice? boarding school is being discussed. what?????

 

When I am empress of the world, we will separate arithmetic skills (which I really do think are necessary in the long run) from conceptual skills in the same way we separate penmanship from reading.

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I must admit as well that one year I was among those who paid no federal income taxes, as a grad student with a wife and two children, teaching two courses, writing a PhD thesis, running 4 miles a day, sleeping 5-6 hours a night, eating vegetarian food partly because we could not afford meat, and earning only $5,000 a year for the same teaching load for which a professor earned $50,000+. But i have tried to make up for it since then.

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What about an on-line class of her interest?

:iagree:

Dd9 is finding her Landry English class challenging & fun. Grammar Town, CE and Para Town, taught by a pro, is such a blessing. Dd loves it. This content is meant for a class setting, imho. Mrs. Hathaway adds a lot of color commentary and interest. Athena's geography class keeps dd busy also -- but it is not 'busy' work. Athena's Greek Myths starts soon and she'll be over the moon to have friends to chat with about mythology.

 

Interacting with the teacher and other students online is a thrill for her. Her online math classes are stimulating also -- on many levels -- as well as her music, mandarin, art & tech classes.

 

When work is done she reads, reads and reads. I never hear, "I'm bored," or "I'm done with everything and I don't have anything to do." Thankfully I don't hear that from any of my dc. Phew. :)

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