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Old 2005-08-22, 18:03   #12
PhilF
 
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It was not part of my HS algebra either (early 70's).

-Phil
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Old 2005-08-22, 18:20   #13
xilman
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Quote:
Originally Posted by PhilF
It was not part of my HS algebra either (early 70's).

-Phil
On the other hand, it was part of my secondary school maths in the early 70's. As you should be able to tell from my carefully chosen vocabulary, I was not educated in the US.

I suspect that there are strong regional and temporal variations as to where and when such things were taught.

Paul
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Old 2005-08-22, 19:34   #14
R.D. Silverman
 
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Quote:
Originally Posted by Wacky
Bob,

Did yours? I do know that it was not a part of my HS math. (in the late '50s)

But times have changed ... By the time that I took Algebra in Grad school, they were introducing Sets to Elementary School students.
As part of pre-calculus (after 2 years of algebra and a year of very proof
oriented geometry) the honors curriculum included: (and taught by someone
who was very good!) [this was early 1970's]

set-theory
basic boolean logic
theory of equations; proof of fundamental thm. of arithmetic; we touched
briefly on soln of cubic, quartic.
advanced trig including DeMoives thm, polar coords
vector algebra, beginnings of linear & matrix algebra, transformation of coordinate systems
induction (and one sees summation there!)
limits WITH epsilon-delta proofs
derivatives based on formal definition
some combinatorial stuff
basic probability
structure of the real number line; Cauchy sequences; Cantor etc.
elementary into to abstract algebra; some basic group theory
LOTS of proofs!

My high school calculus course was superb. We used Apostol
Vol I for the first year and Vol II, supplemented with quite a bit of
diffeq's the second year. [plus some elementary measure theory/real
analysis]. Actually, I, along with my H-S girlfriend, were the first students
to have a second-year course.

BTW, my calculus teacher (perhaps not surprisingly) consistently had
his classes finishing in the top -10 in the U.S. on the competitive H-S
calculus exam (we did not have one when I was in school). His classes
were first in the U.S. on two different occasions.

I was fortunate to have such very good teachers.
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Old 2005-08-23, 00:53   #15
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Set-theory was not any part of my high school math education. Things like summation were introduced in calculus classes, if you took them.
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Old 2005-08-23, 01:23   #16
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Whoa! I guess those who say high school is getting easier and easier may have a point...

I entered my first high school (American system) in 2000, studied for two years before leaving in 2002 to my second high school (International Baccaleaureate), before graduating in 2004.

I do remember being taught the definition of sets and elements in my first high school, but that was as far as the class went. No Boolean logic...

There was a brief mention of the Fundamental Theorem of Calculus in my first high school, but it seems to have been threre just as a footnote...In the textbook for my first high school, there was a brief mention of solving general polynomial equations of degree three and higher. It said that third and fourth degree equations could be solved (although it wouldn't give any details, other than to say that the solutions were very complicated); but that fifth degree equations and higher were unsolvable. Although very little on general equations of degree higher than 2 were covered, methods to solve some such equations were taught (like factoring "by inspection" and Rational Root Theorem), but it was obvious even then that those methods would work only on "rigged" problems. Later, in my second high school, the roots of higher degree polynomials weren't mentioned at all, although I did read a book from the school library that went through the solution of the general cubic in painstaking detail. I read about the general quartic online, after I had read about the cubic. It seems to me that the main steps in the Cubic and Quartic Formulas aren't all that complicated, they just use a lot of tedious algebra.

DeMoivre's Theorem was taught in my second high school, as were the Compound Angle Formulas and their corollaries (Double Angle and Half Angle). Using DeMoivre's Theorem, the class learned how to derive triple, quadruple angle formulas, etc. And I learned polar coordinates in class too.

Vectors were covered (it was the last topic before graduation): addition, dot product, cross product and 3D geometry; but that was all. The phrase "linear algebra" wasn't mentioned at any time in either high school, but there was quite a large section on matrices in both high schools. Most of the time was spent on methods to solve systems of simultaneous linear equations.

Proof by induction was taught in both as well, but when I first saw it, I seem to remember wondering why it hadn't been taught years earlier, since it's conceptually very simple and doesn't seem to require any prerequisites. I think I first saw the summation symbol in grade 10, and used it quite often from then on.

Calculus (including limits) wasn't taught at all in my first high school, but was covered in quite a bit of detail in my final year before graduation at my second high school. However, I don't seem to recall seeing epsilon-deltas there. I did see them in my first year as a university undergraduate, but was never required to write my own epsilon-delta proofs. The professor used them when introducing new theorems (he liked to prove the theorems he taught, whenever possible), but that was really the only time they were used.

Derivatives based on the formal definition (but using the intuitive "definition" of the limit) were taught in my second high school; in fact, that was the first method we learned for computing derivatives. However, we didn't use the formal definition for long, it was soon replaced by the Power Rule, Chain Rule, etc. We went on to cover all the topics (except linear differential equations with constant coefficients, numerical definite integration and power series)that I would later see in single-variable calculus at university

Combinatorials (definition and basic uses) and Pascal's Triangle were taught in my first high school. In my first high school, combinations and permutations were used to solve basic probability problems. In my second high school, probability was covered in much more detail; in addition to reteaching the old material, the teacher also covered continuous probability distributions (using definite integrals) and statistics (mean, median, mode, standard deviation, stem-and-leaf plots, outliers, normal distribution).

But the next two items on Dr. Silverman's list are far ahead anything I've seen, even in my first year at university. I was taught (probably since before high school) that the real numbers are just the rational numbers and the irrational numbers, and I've known for a long time that irrational numbers can be seen as the limit of a sequence of increasingly good rational approximations. But I haven't seen Cauchy sequences at all, although I did read a bit about Cantor's theory in books (but it was never mentioned in the classroom). I'm not really sure what you mean by abstract algebra; but I have tried to read a bit on group theory (I didn't get very far); and none of these were mentioned in high school or my freshman year at university. Glancing at the courses available in the math department, I probably won't see these topics until my junior and senior years as an undergraduate.
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Old 2005-08-25, 12:02   #17
mfgoode
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[QUOTE=R.D. Silverman]My high school calculus course was superb. We used Apostol
Vol I for the first year and Vol II, supplemented with quite a bit of
diffeq's the second year. [plus some elementary measure theory/real
analysis]. Actually, I, along with my H-S girlfriend, were the first students
to have a second-year course.]

May I ask Bob did you eventually marry your H-S- girl as you are doubly good in Maths undoubtedly
Mally
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Old 2005-08-26, 08:00   #18
Orgasmic Troll
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Quote:
Originally Posted by jinydu
Whoa! I guess those who say high school is getting easier and easier may have a point...

I entered my first high school (American system) in 2000, studied for two years before leaving in 2002 to my second high school (International Baccaleaureate), before graduating in 2004.

I do remember being taught the definition of sets and elements in my first high school, but that was as far as the class went. No Boolean logic...

There was a brief mention of the Fundamental Theorem of Calculus in my first high school, but it seems to have been threre just as a footnote...In the textbook for my first high school, there was a brief mention of solving general polynomial equations of degree three and higher. It said that third and fourth degree equations could be solved (although it wouldn't give any details, other than to say that the solutions were very complicated); but that fifth degree equations and higher were unsolvable. Although very little on general equations of degree higher than 2 were covered, methods to solve some such equations were taught (like factoring "by inspection" and Rational Root Theorem), but it was obvious even then that those methods would work only on "rigged" problems. Later, in my second high school, the roots of higher degree polynomials weren't mentioned at all, although I did read a book from the school library that went through the solution of the general cubic in painstaking detail. I read about the general quartic online, after I had read about the cubic. It seems to me that the main steps in the Cubic and Quartic Formulas aren't all that complicated, they just use a lot of tedious algebra.

DeMoivre's Theorem was taught in my second high school, as were the Compound Angle Formulas and their corollaries (Double Angle and Half Angle). Using DeMoivre's Theorem, the class learned how to derive triple, quadruple angle formulas, etc. And I learned polar coordinates in class too.

Vectors were covered (it was the last topic before graduation): addition, dot product, cross product and 3D geometry; but that was all. The phrase "linear algebra" wasn't mentioned at any time in either high school, but there was quite a large section on matrices in both high schools. Most of the time was spent on methods to solve systems of simultaneous linear equations.

Proof by induction was taught in both as well, but when I first saw it, I seem to remember wondering why it hadn't been taught years earlier, since it's conceptually very simple and doesn't seem to require any prerequisites. I think I first saw the summation symbol in grade 10, and used it quite often from then on.

Calculus (including limits) wasn't taught at all in my first high school, but was covered in quite a bit of detail in my final year before graduation at my second high school. However, I don't seem to recall seeing epsilon-deltas there. I did see them in my first year as a university undergraduate, but was never required to write my own epsilon-delta proofs. The professor used them when introducing new theorems (he liked to prove the theorems he taught, whenever possible), but that was really the only time they were used.

Derivatives based on the formal definition (but using the intuitive "definition" of the limit) were taught in my second high school; in fact, that was the first method we learned for computing derivatives. However, we didn't use the formal definition for long, it was soon replaced by the Power Rule, Chain Rule, etc. We went on to cover all the topics (except linear differential equations with constant coefficients, numerical definite integration and power series)that I would later see in single-variable calculus at university

Combinatorials (definition and basic uses) and Pascal's Triangle were taught in my first high school. In my first high school, combinations and permutations were used to solve basic probability problems. In my second high school, probability was covered in much more detail; in addition to reteaching the old material, the teacher also covered continuous probability distributions (using definite integrals) and statistics (mean, median, mode, standard deviation, stem-and-leaf plots, outliers, normal distribution).

But the next two items on Dr. Silverman's list are far ahead anything I've seen, even in my first year at university. I was taught (probably since before high school) that the real numbers are just the rational numbers and the irrational numbers, and I've known for a long time that irrational numbers can be seen as the limit of a sequence of increasingly good rational approximations. But I haven't seen Cauchy sequences at all, although I did read a bit about Cantor's theory in books (but it was never mentioned in the classroom). I'm not really sure what you mean by abstract algebra; but I have tried to read a bit on group theory (I didn't get very far); and none of these were mentioned in high school or my freshman year at university. Glancing at the courses available in the math department, I probably won't see these topics until my junior and senior years as an undergraduate.
I would say my high school math experience was very similar
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Old 2005-08-26, 15:34   #19
mfgoode
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[QUOTE=jasong]I've been wanting to study number theory on my own, but have hit a similar snag in all the books I bought from Amazon. I have encountered two symbols, both approximately shaped like a capital letter E. I am going to describe them and hope that someone can give me a link that can adequately explain one or both of them.[unquote]
This website should help
http://members.aol.com/jeff570/mathsym.html
Mally
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Old 2005-08-26, 15:56   #20
mfgoode
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Quote:
Originally Posted by TravisT
I would say my high school math experience was very similar

When 42 years have passed, like mine, after your formal training, the question is not what you have studied in High school or university but how much you have retained in your memory. Memory is like a muscle-if you dont use it-you lose it! It just atrophies. Also if you have kept in touch or left off.
In my case I have kept abreast with the subject in spite of my globe trotting.
And remember the more you learn the more convoluted the brain becomes and so can retain more and more.
The herb 'bacoma monnieri' is widely available in India and is known as Brahmi
and is said to improve memory with no side effects at all. The normal dose is
one capsule (250 mg ) taken twice a day. There is also Brahmi hair oil. I use both and can vouch for its effectiveness.
Does the word Brahmi ring a bell? The Indain numerals were first known as Brahmi numerals but thats for another post.
Mally
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Old 2005-08-26, 16:08   #21
mfgoode
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[QUOTE=jinydu]Whoa! I guess those who say high school is getting easier and easier may have a point...

I entered my first high school (American system) in 2000, studied for two years before leaving in 2002 to my second high school (International Baccaleaureate), before graduating in 2004.[unquote]
Could anyone kindly explain to me the various steps in the American system of education in detail right up to the PhD level? and the length of time it takes for each step? And the various names like IB, freshman etc.
It seems to me that the word graduate is used quite cheaply. In the British system which I am familiar with, a degree is quite difficult to attain without sweat and toil !
Thank you,
Mally
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Old 2005-08-26, 18:19   #22
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Quote:
Originally Posted by mfgoode
Could anyone kindly explain to me the various steps in the American system of education in detail right up to the PhD level? and the length of time it takes for each step?
I think that it can perhaps be better explained in a somewhat historical perspective. In recent decades, there have been various variations which I will mention.

Public Education is mandadated by law and funded by the tax payers. Generally it is considered to cover 12 years (from age 6 to 18) commonly called Grade 1 through 12. In practice "Kindergarten" has now been added as a partial day of learning/child care for 5 year olds. Grades 1-6 are taught in "Elementary" schools my teachers who teach all subjects (except perhaps "Music" and a few other special subjects that each receive perhaps one or two hours per week of instruction by a specialist) The students in Elementary school remain in one classroom and with the same class throughout the year. Between years, there may be some reshuffling in the larger schools, but you generally spend the entire time with one set of classmates. The Primary Education continues through the 8th grade. Secondary Education is Grades 9-12. In larger school districts, they have the opportunity to make a finer division of children by age and may split grades 6-8 into a separate "Middle" school. In my day, Grades 7-9 were in a "Junior High". As you reach this age grouping, the classes become less monolithic and you begin having individually scheduled periods of instruction where you mix with the entire student body at that grade level.
Whether in a "Junior High", or in the "High School", from the 9th grade, you collect "credits" for courses completed. 9th grade students as called Freshmen in High School, (but the term usually gets applied when they are attending the same facility as the Sophomores, Juniors, and Seniors).
When I attended High School, the only REQUIREMENT for graduation was the accumulation of the required credits, including both a core requirement (eg. English and Math) and a total. More recently, they have set exit exam requirements at various points. However, these requirements are typically 2 years below the customary work at that level. I have seen gifted 9th grade students successfully complete the exit exam for High School. I feel that one factor is that there was a backlash to "social promotion" which sends individuals on to the next grade on the theory that the social damage of failure was worse than the lack of mastery of the subject. As a result they were granting High School Diplomas to some who did not know arithmetic (I don't mean the nuances that Dr. Silverman would discuss. I mean 16+7). (To be Continued)
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