Comments

yeah, but Grothendieck was inventing something new

inventing something new is incredibly difficult; it's scaling a 1000-yard wall

what we're talking about is having to scale a 1000-yard wall just to learn how to add & subtract with borrowing

I do think that's different from language-based subjects, and I know there are kids who are naturally faster at it than others--and I think these kids aren't just the kids who are all-around smart at everything...although I don't feel I know this, for a fact.

Hey!

Let's bring up spelling brains!

I have a spelling brain. Period.

I can see it, dealing with Christopher (or Ed, for that matter).

I can spell, and I can spell naturally. I just look at a word, and then I spell it. I'm a Spelling Automaton!

Andrew, who can't even talk, can spell. Perfectly.

-- CatherineJohnson - 29 Sep 2005

I think Steven Pinker brought up something similar - that our brains had twenty million years to perfect 'more' vs. 'less', but only the last five thousand or so understanding 'six times seven'. Formal mathematics could only have developed after written language; written language could only have developed after spoken language. I doubt there were very many selection pressures for mathematical ability; my wild unqualified speculation is that until very recently, mathematically talented individuals were distributed randomly across the population. It remains to be seen what effect associative mating patterns will hold.

I always knew that Trig was a sin against nature.

-- IndependentGeorge - 29 Sep 2005

Right!

(I haven't read the book, only little snippets on math brains...)

-- CatherineJohnson - 29 Sep 2005

Well, uh, Simon Baron-Cohen says associative mating is giving us AUTISM!!!!!

-- CatherineJohnson - 29 Sep 2005

SO IT MAY BE TIME FOR ME TO POST "Is integer arithmetic fundamental to mental processing?: the mind's secret arithmetic"

-- CatherineJohnson - 29 Sep 2005

our brains had twenty million years to perfect 'more' vs. 'less', but only the last five thousand or so understanding 'six times seven'

I LOVE IT!

-- CatherineJohnson - 29 Sep 2005

Well, uh, Simon Baron-Cohen says associative mating is giving us AUTISM!!!!!

I hadn't heard of this - that's a very, very scary thought.

I brought it up because I was wondering about what was happening with Asian engineers - I had assumed that associative mating patters would be much more concentrated in China or India than in the U.S. I suspect that one of the reasons that math receives such emphasis there is because, given the social structures, math skills are the often the only meritocratic means of advancement. If you don't have family connections, often the only way for you to succeed in China or Inda is to study engineering.

It reminds me of somthing that happened in Singapore a while back. The government noticed that fewer and fewer educated women were getting married. They weren't just delaying their marriages until later in life; they weren't getting married at all. It turned out, social conventions meant that Singaprorean men wanted to marry only women with lesser educations than they, and Singaporean women only wanted to marry men with greater educations. In more crass terms, men wanted to marry down, and women wanted to marry up, which left a lot of educated, professional women without a spouse.

In one of the weirder stories of globalization, there soon followed a substantial migration of older, wealthier British men looking for educated wives.

-- IndependentGeorge - 29 Sep 2005

Hang onto your hat, cause I'm gonna START POSTING!

-- CatherineJohnson - 29 Sep 2005

I suspect that one of the reasons that math receives such emphasis there is because, given the social structures, math skills are the often the only meritocratic means of advancement.

I believe that's true, but I don't know that it's true, and don't remember why it is I've developed this impression.

-- CatherineJohnson - 29 Sep 2005

In more crass terms, men wanted to marry down, and women wanted to marry up, which left a lot of educated, professional women without a spouse.

My guess would be that that's true everywhere. Women want to marry up, men want to marry 'down.'

There was lots of writing about this years ago; one of the findings was that the standard age gap between spouses was 3 years--which is the exact age gap between Ed & me.

The good thing about these kinds of probably built-in preferences is that, culturally speaking, you can define them in all kinds of different ways.

If it's 'good enough' just to have a 3-year age difference, you don't necessarily have to have a 3-year education difference, too.

(Maybe)

Is it true British guys are going to Singapore?

Or was that a joke?

-- CatherineJohnson - 29 Sep 2005

It is true. Lee Kuan Yew even mentioned it in his autobiography - I'll post an excerpt tonight, when I get home.

The convention is universal, but it's a lot more pronounced in Asian societies. Personally, I'd rather marry up. Better yet, I'd love to be a trophy husband :)

-- IndependentGeorge - 29 Sep 2005

Boy, no kidding. You and me both.

Back in Studio City, a divorcee moved into one of the nicest re-done houses in the neighborhood: upstairs, downstairs, pool, the works.

And she was DIVORCED. She was living way better than me, and she didn't even have a marriage in one piece.

I spent some time after that going around saying I wanted to be a trophy divorcee.

-- CatherineJohnson - 29 Sep 2005

Maybe an advantage I had over Catherine is that I never supposed there wouldn't be walls.

I think this is ABSOLUTELY true.

I hope to heck this doesn't sound ridiculous (I'm afraid it will) but I've come to believe that kids like me--the kids for whom 'book learning' came easily--developed a profoundly wrong 'metacognitive concept' of learning.

When learning is always easy, and then you hit a subject that isn't easy, you conclude that you aren't any good at it.

You (or at least, I) assume that people who can do that subject do it the same way you yourself did.....spelling, let's say.

I first figured this out when I finally learned to draw.

All my life, I wanted to draw.

But I couldn't draw. Other people could draw, not me. I noticed, too, that some people could just naturally draw (this is true) and I concluded that all people who could draw could just naturally draw.

I had NO idea that drawing was an incredibly simple skill to learn if someone--good teacher or good book--showed you how.

Zero.

(I realize it's not easy to become an artist, an expert in drawing, etc. But that isn't what I wanted to do. I just wanted to be able to draw something semi-realistic looking, period. And I assumed I was genetically incapable of doing this.)

Then I read the first chapter of Mona Charen's book, where she talks about the fact that people universally think of drawing as almost a magical talent, something you have or don't have, and then says that's bunk.

As soon as I read that chapter I a) realized I could learn to draw and b) started to have the horrible suspicion that I'd made a huge number of wrong judgments in my life because I thought that if it wasn't easy for me to do something, that meant I couldn't do it.

-- CatherineJohnson - 29 Sep 2005

I agree completely with Doug that there are walls in all fields. But I think math is different because for the most part it is nothing but walls.

It's a large mistake to believe that there's a group of people over there who get it all. There isn't. Math is really like a set of mountain peaks, and I think this applies all the way down to grade school. You climb one--it takes a lot of work, but you do it--but that doesn't do anything for all the other peaks out there. They all have to be climbed one by one. Some people have climbed several, a few people have climbed a hundred, but nobody could possibly climb them all.

This is incredibly important, and it's something I simply did not know. (And, yes, I think math is different from other fields in this respect. Walls from the get-go, and successive walls.)

And it lets the more mathematically talented off the hook because they think that just because they have some mathematical talent they don't have to work anymore. I've seen a lot of those, and they were all lying by the wayside.

That's interesting.

Now that is something you actually wouldn't see in other fields.

As a writer, I think of myself as a peasant (this comes from Scott Fitzgerald, who had some great line about novelists being peasants while short story writers were....something else. I forget what.)

I think of what I do as work, work, work.

Ed thinks the same way. (Historians really work. I personally could not be paid enough to set foot inside an archive.)

With writing (and probably all the language-based disciplines) it's obvious that the more you do, the better you get.

And it's equally obvious that you just have to do vast, mind-bending, heroic amounts of work to pull off a big project.

In Conclusion: all writers, universally, see writing as hard work.

I'd be surprised to find a single writer, anywhere, who thinks he or she doesn't have to work because he/she has writing talent.

-- CatherineJohnson - 29 Sep 2005

Tangential to the "math brain" discussion, my husband has made a very interesting observation.

A smidge of background here: He has always been one who has no fear of questioning or correcting his instructors, something that many of his primary school teachers didn't much care for, as you can imagine. He has a double major in mathematics and computer science and he'll graduate with his B.S. this spring. (He is 31, finishing his degree after a 10 year hiatus.)

What he has noticed is that while his CompSci^? and gen ed instructors often resent being corrected, his mathematics instructors do not.

His theory is that people who do math are accustomed to being wrong. They make mistakes all the time, and it's easy to do when working a complex problem on a blackboard. He thinks that you pretty much can't do math all the time and still maintain an infallibility complex, or superior attitude towards students. Especially since math is a young person's game, and most math professors are already past their "peak" in math ability, and know it.

In addition, in "soft" liberal arts areas, or conversely, extremely complex areas like programming, mistakes may not be obvious, or may be open to some debate. In math, an instructor can't wiggle around a mistake. If he has added 6 to 7 and gotten 14, that's just wrong, end of story.

What I think I'm getting at here is that making math easy for students through "no one answer", etc. is not helpful because it delays an understanding that math is hard for everybody including people like my husband, and that the best mathematicians in the world make mistakes all the time. This understanding actually makes me feel a lot better about my own anxieties about math.

Oh, and as for "math brains", my husband's major the first time around, before the 10 year break, was Philosophy.

-- LesleyStevens - 29 Sep 2005

Catherine wrote:

"kids for whom 'book learning' came easily--developed a profoundly wrong 'metacognitive concept' of learning"

Catherine: That's it EXACTLY! Everything was always easy for me, reading, spelling, and grammar were nearly intuitive. Basic math was, too. I never had to study, so long as I paid attention in class. I never did homework, which was I found boring, because I could get A's on all my tests, and pass all my classes with C's or better (depending on how the homework was weighted).

My senior year in high school, I took pre-Calculus and Honors Physics. I was doomed. I didn't know how to study, had no work ethic for rote learning, hadn't learned algebra well enough to manage the math in physics. I dropped both classes, and assumed that the kids doing well in Physics, taking AP Calculus while I couldn't hack pre-Calc (kids who were in all the other honors classes I was in) were just better math people than I was. Now I know that however much that may have been part of it, they probably just worked harder than I did.

-- LesleyStevens - 29 Sep 2005

I used to be able to spell...

I was an early reader and speller, also. I wonder if things coming easy becomes a sort of self-identity where you secretly expect a certain speed of learning and when that doesn't happen you shut down. Of course, I wish my shut down came a little later. It really took me until grad school to realize that a longer process to something was perfectly fine.

There must be some visual aspect to it because my one son (who has no learning problems) will spell a basic word three different ways on paper. What is that all about? He does it all the time. How can one do it right once or twice in one sitting and then not remember the last time?

"I had NO idea that drawing was an incredibly simple skill to learn if someone--good teacher or good book--showed you how."

Or if you do it obsessively. I have years worth of notes with doodles all over them, to the point where you can barely see the writing. I doodled body parts, landscapes, anything I could think of, daily and all day. When you're trapped at a desk bored (or as was the case later, confused) the umpteenth time you've drawn that eyeball might be the one you're finally happy with.

-- SusanS - 29 Sep 2005

I used to doodle all the time when I got bored. Now, I post comments here.

-- IndependentGeorge - 29 Sep 2005

"His theory is that people who do math are accustomed to being wrong."

YES!!!! That's right.

-- CarolynJohnston - 29 Sep 2005

Catherine: "This is incredibly important, and it's something I simply did not know. (And, yes, I think math is different from other fields in this respect. Walls from the get-go, and successive walls.)"

What you and Bernie said is exactly right. It's not the existence of the walls, it's that it's walls all the way down.

Catherine again: "I had NO idea that drawing was an incredibly simple skill to learn...."

I've had the same experience. But for me the big breakthrough came when I realized that there is really no effortless way to create a good drawing.

Oh, there might be for someone, but none of the professional artists I've met (and my wife runs SF and Fantasy art shows) seems to have one either. Some things just take more work than you think or even can imagine. This one insight would have made life so much easier in college.

Oh well, too soon old, too late smart.

Lesley Stevens: "He has always been one who has no fear of questioning or correcting his instructors....

"His theory is that people who do math are accustomed to being wrong. They make mistakes all the time, and it's easy to do when working a complex problem on a blackboard."

I do the same thing, and occasionally notice people getting annoyed for what seems to me to be no reason. I suspect that I fail to notice most of the people who get annoyed. 8-/

Still, an openness to correction is a useful thing for anyone with an editor, so it has its upside, too.

-- DougSundseth - 29 Sep 2005

Lesley

Basic math was, too. I never had to study, so long as I paid attention in class.

Exactly!

I've had trouble expressing this to people.

The other day, when I was talking to my-neighbor-the-statistician about wanting to take a college calculus course, she said at some point, 'Well math was always hard for you."

And I said, "It wasn't hard."

And it wasn't!

Math was as easy as anything else right up to the moment I went to Wellesley and signed up for calculus and BOMBED.

I had simply sailed through 13 years of undergraduate education, and I had no idea how a person studies, or grapples with impenetrable material, or anything else.

More importantly, I had no idea that calculus might be hard for the kids who didn't flunk the test.

None.

-- CatherineJohnson - 29 Sep 2005

My mom, btw, had a photographic memory when she was a kid.

I didn't have a photographic memory, but I think there must be a 'photographic memory spectrum.'

I did memorize all kinds of stuff for tests, but I could just read it through a couple of times and remember it.

Memorizing material--any material at all--was extremely easy.

hmm

Actually, as I think about it, I'm not so sure I 'memorized' things. It was more like I remembered them.

It's been a huge shock to me, losing that capacity. I'm still struggling with my da** MasterCard number, though I seem to have mastered our pediatrician's phone number at last.

-- CatherineJohnson - 29 Sep 2005

I used to doodle all the time when I got bored. Now, I post comments here.

This is the kind of thing that tells me we are succeeding in our mission.

-- CatherineJohnson - 29 Sep 2005

Doug

I've had the same experience. But for me the big breakthrough came when I realized that there is really no effortless way to create a good drawing.

Wow.

Yet another I HAD NO EFFING IDEA moment.

I thought drawing eventually became easy, once you got fluent in it.

I believe you, though, because writing a book never, ever, becomes easy.

'Writing' becomes easy; writing a Comment here, or a post, is (usually) easy.

But writing-a-book, or writing-an-article, is never easy, ever, and it wouldn't cross my mind to expect it to be so.

-- CatherineJohnson - 29 Sep 2005

This one insight would have made life so much easier in college.

How so?

I mean, more specifically?

-- CatherineJohnson - 29 Sep 2005

I'm still struggling with my da** MasterCard number

Tell me this is a joke.

-- CarolynJohnston - 29 Sep 2005

I had an experience similar to that which you and Lesley report. In high school I could read or hear something once and remember it, so I didn't have to study much for facts. I could write a three-page opinion piece in 1-1/2 hours in the base snack bar before school. Math and science was easy, requiring nearly no homework.

Then I hit college and majored in physics; Washington University, St. Louis, National Merit Scholarship. I didn't have good study habits, and now I needed them. The problem was that I didn't realize it; I thought that I just hadn't figured out the trick. After all, everything has a trick, doesn't it? The results are pretty predictable, though it took me three semesters to leave the school.

I know my experience isn't unique and suspect it isn't even especially unusual. (I have a friend who had a similar experience at MIT.) I suspect that I might have figured the problem out sooner if I hadn't been able to skate through so many of my classes (note that I didn't actually figure out the problem until much later). I wish that I'd had a professor that I respected (a subset of "professors that I had") tell me that sometimes there's just no easy way for anyone to do [whatever], though. Not that I'd necessarily have believed him, of course; young and dumb isn't a cliche at random.

-- DougSundseth - 29 Sep 2005

I'm still struggling with my da** MasterCard number

Tell me this is a joke.

I wish.

-- CatherineJohnson - 29 Sep 2005

Doug

Then I hit college and majored in physics; Washington University, St. Louis, National Merit Scholarship. I didn't have good study habits, and now I needed them. The problem was that I didn't realize it; I thought that I just hadn't figured out the trick.

That reminds me; I have to post Ed's experience freshman year at Princeton.

He'd studied his tale off in high school--four hours a night, every night.

Even so, he had no idea what he was supposed to be doing in college.

-- CatherineJohnson - 29 Sep 2005

The good thing about our Math Struggle around here is that Christopher has (let's hope) internalized the idea that doing well in school means hard work.

It's spotty, though. He's so good at 'social studies,' which is not even a subject as far as I can tell, that he thinks social studies and history should be easy.

So....at some point we're probably going to have to fill him in on the fact that real history isn't something you read once & then know everything there is to know.

-- CatherineJohnson - 29 Sep 2005

Catherine: Maybe you could give him a project involving local history? Something that would require looking at microfiche newspaper archives, old land deeds, birth certificates, stuff like that. Something where he sees a little of what it's like to dig into primary sources, and local history so you guys can find it at your county courthouse and local library.

I'm just brainstorming, but I imagine something where I might be able to go "That spot where the mall is used to be a farm, and the guy that owned it founded this town, and this is his death certificate and his will, here's his obituary from the newspaper in 1893, and then his son gambled too much, and got into debt, and the bank took the farm, and here's the record of the estate sale, and here's his wife's diary that the library has" etc, etc.

This might give him an introduction to a number of research skills, and he might find out some cool stories to impress his friends with. Especially if he can find something juicy like "Where that Burger King is used to be the town square for public hangings."

-- LesleyStevens - 29 Sep 2005

For teaching kids history, the best books I've seen are the Story of the World series of books. Book one is here.

In fact, while you're at it you might as well pick up The Well Trained Mind by the same author. It covers all the rest of the school subjects besides math (It recommends Saxon or singapore for math). It is to English instruction what Singapore math is to math instruction.

-- KDeRosa - 30 Sep 2005

Lesley,

They say his son gambled too much, but that's only because everybody got the story from his wife's diary, which she conveniently had placed in the town library. I think she was really his downfall. That golddigger took him for all he was worth, then hooked up with the guy who made a fortune with the town's first Burger King, which happens to be where the public hangings used to happen.

-- DanK - 30 Sep 2005

I'm just brainstorming, but I imagine something where I might be able to go "That spot where the mall is used to be a farm, and the guy that owned it founded this town, and this is his death certificate and his will, here's his obituary from the newspaper in 1893, and then his son gambled too much, and got into debt, and the bank took the farm, and here's the record of the estate sale, and here's his wife's diary that the library has" etc, etc.

You know....he'd have fun with that.

One of the mysteries around here is: Who owns our road?

We live on a private, one-lane road that's so private no one knows who owns it. The city refuses to maintain it (which means no snowploughing in the winter), because they don't own it. Meanwhile we don't know who owns it, and whoever it is does nothing.

It would be cool for Ed & Christopher to go dig up the title.

There wouldn't be a lot of analysis involved, but We'd Have A Name.

-- CatherineJohnson - 30 Sep 2005

We like History of US, and we'll probably try to get to the ancient history series.....whose name I forgot. I think a couple of different friends of ours wrote at least two volumes.....

I'll get the titles.

-- CatherineJohnson - 30 Sep 2005

I have both the Story of the World series and the History of Us.

If you have grade schoolers, the Story of the World is perfect to read to them. Older grade schoolers and middle schoolers can read it themselves. It reads as well as any book I've ever read to my kids. I can't recommend it enough. It takes some of the most complicated times in history and brings them down to a child's level. The books are also incredibly throrough.

I have recently been reading The History of Us to the kids and I really appreciate the "readability" of these books, as well. I only wish she had a world history series.

The Well-Trained Mind changed my life a few years ago. Along with The Educated Child (William Bennett), I finally felt like I could figure out what my school was doing well, and what I was going to have to supplement.

The Well-Trained Mind is written for classical homeschoolers so it goes through all stages of childhood with lessons and resources written out and explained. I cannot state enough what a gem of a book it is. I still go back into to it every few months. It mounts a sprited defense of traditional teaching methods for anyone who still needs more convincing.

-- SusanS - 30 Sep 2005

wow

I'll have to try to read that again.

I've picked it up in bookstores, but always have trouble with....the tone or the style or something.

-- CatherineJohnson - 30 Sep 2005

Maybe there's a cheap used one somewhere. I'll bet there is. You can always borrow mine the next time you swing by.

The author (co-author, really)was a teacher in the 70's, I think, when she decided to pull her child out and homeschool. She got a lot of grief for it even though she was a trained teacher. The "child" grew up to have an obnoxiously high SAT score and is the author of the Story of the World series, among other books.

I was sold when she said in her book that grade schoolers do not need to have everything relate back to them. She said that this was the time for the Big Stories. So I got a great book (Black Ships of Troy) and started reading it to my grade schoolers. The look of shock on their faces when they heard the Greek myths for the first time was unbelievable and I knew then that she was right. They had this look of, "Mommy, are you sure you're supposed to be reading such things to us?" Quite a bit different than the usual fare they read.

-- SusanS - 30 Sep 2005

Whoa! I've never heard of the Well-Trained Mind OR the Educated Child!

To echo Catherine: I know nothing.

And if we BOTH know nothing, that means this whole venture is up to you guys. Good luck.

-- CarolynJohnston - 30 Sep 2005

You know, I just need to order & read The Well-Trained Mind.

There's obviously A REASON why people love that book.

Is Bennett worth wading into?

I have trouble with him, too, for some reason. (I'm finally getting a copy of Hirsch's What Your 6th Grader Should Know, assuming I ever complete my Amazon order, that is.)

-- CatherineJohnson - 30 Sep 2005

I was sold when she said in her book that grade schoolers do not need to have everything relate back to them. She said that this was the time for the Big Stories.

Yup, I need to read this book.

It's obviously going to crystallize a lot of stuff I semi-know intuitively.

-- CatherineJohnson - 30 Sep 2005

The mother's name is Jessie Wise and knows some very old-school ways of taching reading.

If you are going to teach your kid how to read, and I highly recommend that you do not leave this important task up to the school, I highly recommend using The Ordinary Parent's Guide to Teaching Reading.

It is scripted, so it makes the parent's job very easy. It uses a teaching method that happens to be in accord with both the cognitive science and the reading research. It expects mastery of the basics and memorization [shudder] of rules.

For example, the first 26 lessons force the child to incrementally learn and repeat a song that teaches each letter and its associated sound. This makes the memorization aspect much more palatable to kids.

The reason for the rigor at this tender age is because in the classical curiculum the child will be expected to be reading and, more importantly, understanding the classics by high school.

-- KDeRosa - 30 Sep 2005

The best part about the well Trained Mind is that it is a coherent curriculum.

The curriculum breaks down human civilization into four periods: Classical, Medieval, Renaissance to 1850, and 1850 to Present. You run through the sequence the first time in grades 1-4, again in grades 5-8, and the third time in grades 9-12. Clearly, this is setting the framework for history instruction. But, the English language arts are also tied to this structure, so all the literature used also comes from the appropriate period. (The Story of the World series follows the progression). In grades 1-4, you read the stories to your child from the Story of the World and other simplified versions of literature (As Susan wrote, Black Ships before Troy is a simplified version of the Iliad). In grades 5-8 the progression is repeated but this time the child is reading the simplified stories. In high school, after being exposed to the stories twice already, the child is expected to be reading the classics himself, largely self-study.

It is very much a basics skills curriculum: reading, spelling, writing, grammar, geography, logic, and latin (mostly to help with grammar and vocabulary) are all taught directly and explicitly. The child’s toolbox is systematically filled with the tools she will need to read, understand, and analyze, and write about difficult literature in high school.

-- KDeRosa - 30 Sep 2005

-- DanK - 30 Sep 2005

I have no experience with The Well Trained Mind. I did browse through Susan Wise Bauer's The Well Educated Mind, though. It's aimed at adults like me who missed a lot of a classical education. You kind of have to browse it, because it tells what classic novels to read. So, I set it aside and began reading Don Quixote. 8 months and about $4 in library fines later, I finished it. I actually had to check it out three or four times. I enjoyed it a lot. My problem is that Susan Wise Bauer says that I should read it twice, being more analytical on the second pass. Sheesh. I'm sure she's right, but life may be too short. I do need to check her book again, though, to see what I should read next.

-- DanK - 30 Sep 2005

Dan, I have that book. A girlfriend and I decided we'd read Don Quixote (and of course, all the rest, in order.) I think we got around halfway through the book before we gave it up. I decided I'd have to rely on that memory of doing Man of La Mancha many moons ago. And the idea of reading it two or three times had us bursting into laughter.

I do really like her list of genres and books, though. Now, I just jump around and pick one when I have some time.

-- SusanS - 30 Sep 2005

Susan,

The first half is the better part. The second part was written years later. In the meantime, someone else in Spain had published an unauthorized sequel to the first part. Cervantes spends a lot of time in the second half disparaging the fraudulent sequel.

I read the translation by Burton Raffel. It looked tons more modern and readable than the other version on my library's shelf. I think that helped me a lot.

-- DanK - 30 Sep 2005

Only slightly less brutal and three times as boring are 90% of the legal decisions written before the mid 20th century which every law student has to read at a clip of a few hundred pages a night, brief, and be prepared to field any question the professor may ask when called upon randomly the next day in class. Still beats engineering school though.

-- KDeRosa - 30 Sep 2005

Classical education is (gasp) developmentally appropriate!! In the correct way. The curricula is repeated 3 times because it fits the way kids are at those times. The first cycle is aimed at the big picture and filling little minds with knowledge since that is what young children crave. This stage is called the "grammar" stage because kids are learning the language of everything. The second cycle begins around age 10-12. This cycle focuses on logic, and this is when children begin to formally study logic as well as examining the hows and whys of history and science and literature. This matches the child's intense interest in arguing, thus this stage is called the "logic" stage. The third cycle focuses on putting everything together in a logical and well written way and is called the "rhetoric" stage. It begins in 9th grade or so. This is when students read original texts and sources and learn to write elegantly and logically.

-- LoneRanger - 30 Sep 2005

Wow, Ken--what a fantastic recommendation on reading.

I was looking for things back when Christopher was....hmm.

Now I'm thinking I may have been worrying about Andrew.

It's all a blur.

In any case, I'm very happy to have a good reading curriculum for parents.

-- CatherineJohnson - 30 Sep 2005

Still beats engineering school though

LOL

I took a communications law course in grad school, and the court decisions were great.

Very droll.

-- CatherineJohnson - 30 Sep 2005

OK, I've got the reading book in, yes, MATH REFS.....

-- CatherineJohnson - 30 Sep 2005