Seeking straight A's, parents push for pills

In article . com,
wrote:

I made up my own mnemonic for learning the names of the months in
Croatian. I thought it quite clever, but no-one else did. (Since
Villians Often Teach Slavic Languages, Smart Kids Really Like Studying
Privately.)

How interesting -- they must be quite different from the Russian, which may be
transliterated:
Yanvar'
Fevral'
Mart
Avril'
Mai
Iyun'
Iyul'
Avgust
Sentyabr'
Oktyabr'
Noyabr'
Dekabr'

--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

"Parenthood is like the modern stone washing process for denim jeans. You may
start out crisp, neat and tough, but you end up pale, limp and wrinkled."
Kerry Cue

Chookie wrote:
In article . com,
wrote:

I made up my own mnemonic for learning the names of the months in
Croatian. I thought it quite clever, but no-one else did. (Since
Villians Often Teach Slavic Languages, Smart Kids Really Like Studying
Privately.)

How interesting -- they must be quite different from the Russian, which may be
transliterated:
Yanvar'
Fevral'
Mart
Avril'
Mai
Iyun'
Iyul'
Avgust
Sentyabr'
Oktyabr'
Noyabr'
Dekabr'


Oh ... they are, which is part of the problem! In Serbian they are the
standard more-of-less international versions, Januar, Februar, Mart,
Abril, etc.... But Croatian isn't so lucky. Even with the mnemonic I
can't remember all of them (but can at least recognize them), but,
August through December is: kolovoz, rujan, listopad, studini,
prosenac. (And just to make things even more fun, studeni is an
adjective rather than a noun, so is declined differently!) Apparently
in Croatia they DO tend to take pity on foreigners and, when talking to
us, will simply refer to the months as 'first month', 'second month,'
etc.

Naomi


--
Chookie -- Sydney, Australia
(Replace "foulspambegone" with "optushome" to reply)

"Parenthood is like the modern stone washing process for denim jeans. You may
start out crisp, neat and tough, but you end up pale, limp and wrinkled."
Kerry Cue

"Chookie" wrote in message
...
In article ,
(Herman Rubin) wrote:

Possibly timing and meter, but I doubt it helps much with
fractions (the general idea is not there) or counting (this
should come much earlier); I do not know what "grouping"
means here, and as far as cooperating, I do not see this as
important in education; it is the individual who learns.

If it IS that kind of grouping, you have forgotten that peer demonstration
is
also an excellent way to learn a skill.

Grouping, in the context of music, refers to groupings of notes (at least,
in a techincal sense). Groupings of individuals are referred to as
orchestration or voicing. Grouping in music has definite skills cross-over
applications, because, like order of operations in math, it's a very rigid
formulaic language in which the meanings of a particular symbol are mutable
depending on the other symbols around it.


As far as groupings of individuals go, the whole is greater than the sum of
the parts in music, each person has a very individual role to play, which is
exceedingly important, and only if all parts work together does the desired
result occur. Each individual must learn on their own-unlike most
cooperative learning situations, there's no room for leaning on someone else
because one strong person can't do the work for the group, and the result is
very obvious. This is why a good music program includes not only private
lessons and individual work, but ensemble experiences as well, because
without both, a student is not a well rounded, developed musician. Even
pianists, who are perhaps the most prone to work individually, tend to spend
a large percentage of their time accompanying groups once they get beyond
the student level.

In article ,
Chookie wrote:
In article ,
Rosalie B. wrote:

There are some things that just have to be memorized. Med students
use "On Old Olympus Towering Top A Finn and German Viewed Some Hops"
for the 12 Cranial Nerves - Olfactory, Optic, Oculomotor, Trochlear,
Trigeminal, Abducens, Facial, Glossopharyngeal nerve, Vagus nerve,
Spinal Accessory nerve, and Hypoglossal nerve.

I like looking at mnemonics, but I've never been able to make them work for
me. Not only do you have to remember the mnemonic, you have to remember what
it's associated with, *and* the words you had trouble remembering in the first
place! The only aide-memoire I use is "Thirty days hath September..."

The particular mnemonic listed here would not help me
at all. About the only non-trivial one that I have
seen which could be used is the one for the order of
the classes of stars, as each class is designated by
only one letter.

There are others of limited use, such as ones giving
the digits of pi by counting the numbers of letters
of the words. But I do not think that Roy G. Biv
as the "rainbow man" is of that much use.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

"Herman Rubin" wrote in message
...
In article ,
Chookie wrote:
In article ,
Rosalie B. wrote:

There are some things that just have to be memorized. Med students
use "On Old Olympus Towering Top A Finn and German Viewed Some Hops"
for the 12 Cranial Nerves - Olfactory, Optic, Oculomotor, Trochlear,
Trigeminal, Abducens, Facial, Glossopharyngeal nerve, Vagus nerve,
Spinal Accessory nerve, and Hypoglossal nerve.

I like looking at mnemonics, but I've never been able to make them work
for
me. Not only do you have to remember the mnemonic, you have to remember
what
it's associated with, *and* the words you had trouble remembering in the
first
place! The only aide-memoire I use is "Thirty days hath September..."

The particular mnemonic listed here would not help me
at all. About the only non-trivial one that I have
seen which could be used is the one for the order of
the classes of stars, as each class is designated by
only one letter.

G I know the feeling...there is another one--make a fist, and "count" your
months on the exposed knecked on the back of your hands, with January on the
knuckle, February in the crease, March on the knuckle, etc...August starts
the next round, on top of a knuckle...the ones up high have 31 days, the
ones down low do not.......a visual representation G

--
Buny

" Nobody realizes that some people expend tremendous energy merely to be
normal."
~ Albert Camus

Herman Rubin wrote:
In article . com,
laraine wrote:
Herman Rubin wrote:

I also question the utility of English literature, unless
you are going to discuss that literature.


Well, discussion is the core of liberal arts, IMO.
(And, as far as rigor, literary critism might be going
in the direction of philosophy in its logical sense, I
believe, or is trying to do so --I just saw a reference
to a book on the philosophy of feminism.)

But, here's another example of the utility of English
literatu Dr. Rubin writes a book, non-fiction or
fiction, describing a math classroom as he would
like it to be taught. Perhaps he could just outline
ideas, or perhaps he could write a story where
an intelligent student was not able to make a
difference in the world because she was not
properly taught the concepts, or something
like that.

You are assuming that mathematics should be taught
in a classroom run as the typical present elementary
school classrooms are run.

No, I haven't assumed that--I don't yet feel
knowledgeable enough to make a judgment
about it.


Consider that when those who understood the concepts
taught it, the original "new math" was hardly a
failure, or it would never have been adopted.

I think that I was taught 'new math' in elementary
school in the 70's. In any case, I personally found
the textbooks excellent, and I don't recall too many
people having trouble with them. My recollection is
that people did reasonably well until they got to
algebra. The 'new math' was not as complex as
what Tom Lehrer makes fun of, though, when
I studied it.

What I didn't like was the year and a half
I had in 6th and part of 7th grade when
there were no math lectures, and we
essentially worked independently. It would
have been better, I think, if a computer could
have given us each the right kinds of
problems geared to our individual abilities,
but we were essentially on our own (with
a teacher who would privately answer
questions).

Also,
we have evidence that strong logic can be taught in
elementary school,

I recall learning a lot of elementary set theory.
That seems similar.

and that a good proportion of
high school students could handle "Euclid" geometry.
In my time, in Chicago the high school enrollment
was quite high.

We also know that 60 years ago (and earlier) the
standard high school curriculum for going to college
required a good proof-oriented geometry course.


It's funny--I have met people who have told me
that they found geometry much easier than
algebra. Now, I found algebra much easier
than geometry. I don't know the reason for that.

That would be much easier and possibly more
sobering to read than the many letters on this
newsgroup, which, while they are quite
enlightening and entertaining at times (and
a sort of literature in themselves), are also
repetitive, disorganized, and don't seem to
lead to many clear suggestions of solutions.
The existence of such books might also
encourage discussion from educators
or at least mathematicians everywhere.

The books will almost entirely have to be written.

So perhaps you are proposing that a new
textbook needs to be written specifically
for exceptionally gifted children in math.


The books that my son studied to learn logic are
available, but I doubt that many primary school
children can handle them. The one he learned
algebra from might be a possibility for someone
who has already learned to read.

The book which I recommend for "adults" to learn
the ordinal approach to the integers was definitely
written for those who were adept at using algebra
and had some idea of a proof, and who were willing
to see that the details could be filled in. It is
not appropriate for beginning teaching, and in fact
it is even inadequate in its treatment of the integers,
as it does not go into positional notation. But
making sense of positional notation requires ordinals.


I recall learning about different bases in elementary
school--it didn't seem like a big issue. Wasn't that
part of the 'new math?'

Those who produced the new math materials did not start
with a finished product. We can to some extent go back
to the old structured materials we had before the idea
that children should always be with their age groups
was imposed, and the idea of "relevance" instead of
learning for the distant future, and doing things
in a manner removing repetition, were the rule.
--

So, you mean to start more from scratch than
what those who designed 'new math' started
with?

C.

In article om,
laraine wrote:
Herman Rubin wrote:
In article . com,
laraine wrote:
Herman Rubin wrote:

I also question the utility of English literature, unless
you are going to discuss that literature.


Well, discussion is the core of liberal arts, IMO.
(And, as far as rigor, literary critism might be going
in the direction of philosophy in its logical sense, I
believe, or is trying to do so --I just saw a reference
to a book on the philosophy of feminism.)

But, here's another example of the utility of English
literatu Dr. Rubin writes a book, non-fiction or
fiction, describing a math classroom as he would
like it to be taught. Perhaps he could just outline
ideas, or perhaps he could write a story where
an intelligent student was not able to make a
difference in the world because she was not
properly taught the concepts, or something
like that.

You are assuming that mathematics should be taught
in a classroom run as the typical present elementary
school classrooms are run.

No, I haven't assumed that--I don't yet feel
knowledgeable enough to make a judgment
about it.


Consider that when those who understood the concepts
taught it, the original "new math" was hardly a
failure, or it would never have been adopted.

I think that I was taught 'new math' in elementary
school in the 70's. In any case, I personally found
the textbooks excellent, and I don't recall too many
people having trouble with them. My recollection is
that people did reasonably well until they got to
algebra. The 'new math' was not as complex as
what Tom Lehrer makes fun of, though, when
I studied it.

This was not that common; the big problem was the
teachers.

Many new math books had considerable use of
variables. As I keep pointing out, it is the
linguistic use of these which is the important
one, and do not try to skimp on how many are
used. Most algebra books discuss solution of
one equation in one unknown, and then give many
problems which should not even be tried with
one unknown. Use as many as you want, and use
the rule of equality to eliminate.

What I didn't like was the year and a half
I had in 6th and part of 7th grade when
there were no math lectures, and we
essentially worked independently. It would
have been better, I think, if a computer could
have given us each the right kinds of
problems geared to our individual abilities,
but we were essentially on our own (with
a teacher who would privately answer
questions).

Also,
we have evidence that strong logic can be taught in
elementary school,

I recall learning a lot of elementary set theory.
That seems similar.

No, this is merely sentential logic. Very little
can be done with it; what is needed is predicate
logic, with quantification only over individuals.
This is definitely more complicated, but not that
much, in its rules, and when it comes to using it,
it is much harder. It is adequate for all of
mathematics, and any question in sentential
calculus can be automatically answered.

and that a good proportion of
high school students could handle "Euclid" geometry.
In my time, in Chicago the high school enrollment
was quite high.

We also know that 60 years ago (and earlier) the
standard high school curriculum for going to college
required a good proof-oriented geometry course.


It's funny--I have met people who have told me
that they found geometry much easier than
algebra. Now, I found algebra much easier
than geometry. I don't know the reason for that.

Which geometry? The "Euclid" course is rarely
given now, except as an honors course. But
Euclidean geometry is a totally different subject
than algebra; high school level algebra essentially
deals with real numbers, while geometry deals with
idealized points, lines, circles, etc. It does
contain some algebra, in that it uses special cases
of the rule of equality.

That would be much easier and possibly more
sobering to read than the many letters on this
newsgroup, which, while they are quite
enlightening and entertaining at times (and
a sort of literature in themselves), are also
repetitive, disorganized, and don't seem to
lead to many clear suggestions of solutions.
The existence of such books might also
encourage discussion from educators
or at least mathematicians everywhere.

The books will almost entirely have to be written.

So perhaps you are proposing that a new
textbook needs to be written specifically
for exceptionally gifted children in math.

An exceptionally gifted child can use the
existing books. My son did, with not that
much help. In fact, for logic he used two
books, with somewhat different notations,
around age 6.

It would not be difficult to write a logic
text which could be used by average elementary
school children with a good teacher; with some
modifications, my late wife's book can almost
do this, although it was written for juniors
in college. If someone is interested in making
the modifications, I would be glad to cooperate,
including introductory material on variables
which could be used with beginning reading.
I am NOT a writer, and I know how much work
is involved in writing a book. In this case,
there are good templates to work from, so it
will be easier.

The books that my son studied to learn logic are
available, but I doubt that many primary school
children can handle them. The one he learned
algebra from might be a possibility for someone
who has already learned to read.

The book which I recommend for "adults" to learn
the ordinal approach to the integers was definitely
written for those who were adept at using algebra
and had some idea of a proof, and who were willing
to see that the details could be filled in. It is
not appropriate for beginning teaching, and in fact
it is even inadequate in its treatment of the integers,
as it does not go into positional notation. But
making sense of positional notation requires ordinals.

I recall learning about different bases in elementary
school--it didn't seem like a big issue. Wasn't that
part of the 'new math?'

It is a minor part. The important aspect of this is
that there are lots of representations for numbers,
and it is mainly a matter of convenience which to use.
I am not that familiar with the Mayan representation,
but most of the representations did not use the same
characters in different positions, which requires the
"zero" character. It does not make that much difference,
until one gets into numbers with large numbers of digits,
which is relatively modern.

Those who produced the new math materials did not start
with a finished product. We can to some extent go back
to the old structured materials we had before the idea
that children should always be with their age groups
was imposed, and the idea of "relevance" instead of
learning for the distant future, and doing things
in a manner removing repetition, were the rule.


So, you mean to start more from scratch than
what those who designed 'new math' started
with?

In my opinion, and it was my opinion more than
50 years ago, the new math made the mistake of
concentrating on the cardinal representation of
numbers. It is deceptively easy, but there are
major difficulties, including what is a finite
number. The only fully adequate "definition"
of this is one which can be counted and the
counting terminates. This brings in the ordinal
counting process.

The ordinal approach is self-contained, and
the definitions are clearly definitions. One
should do both; the cardinal (how many) and
the ordinal (count in order) concepts describe
the same finite objects, but give different ways
of looking at them.


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

Herman Rubin wrote:
In article om,
laraine wrote:
Herman Rubin wrote:
In article . com,
laraine wrote:
Herman Rubin wrote:

I also question the utility of English literature, unless
you are going to discuss that literature.


Well, discussion is the core of liberal arts, IMO.
(And, as far as rigor, literary critism might be going
in the direction of philosophy in its logical sense, I
believe, or is trying to do so --I just saw a reference
to a book on the philosophy of feminism.)

But, here's another example of the utility of English
literatu Dr. Rubin writes a book, non-fiction or
fiction, describing a math classroom as he would
like it to be taught. Perhaps he could just outline
ideas, or perhaps he could write a story where
an intelligent student was not able to make a
difference in the world because she was not
properly taught the concepts, or something
like that.

You are assuming that mathematics should be taught
in a classroom run as the typical present elementary
school classrooms are run.

No, I haven't assumed that--I don't yet feel
knowledgeable enough to make a judgment
about it.


Consider that when those who understood the concepts
taught it, the original "new math" was hardly a
failure, or it would never have been adopted.

I think that I was taught 'new math' in elementary
school in the 70's. In any case, I personally found
the textbooks excellent, and I don't recall too many
people having trouble with them. My recollection is
that people did reasonably well until they got to
algebra. The 'new math' was not as complex as
what Tom Lehrer makes fun of, though, when
I studied it.

This was not that common; the big problem was the
teachers.

Many new math books had considerable use of
variables. As I keep pointing out, it is the
linguistic use of these which is the important
one, and do not try to skimp on how many are
used. Most algebra books discuss solution of
one equation in one unknown, and then give many
problems which should not even be tried with
one unknown. Use as many as you want, and use
the rule of equality to eliminate.

What I didn't like was the year and a half
I had in 6th and part of 7th grade when
there were no math lectures, and we
essentially worked independently. It would
have been better, I think, if a computer could
have given us each the right kinds of
problems geared to our individual abilities,
but we were essentially on our own (with
a teacher who would privately answer
questions).

Also,
we have evidence that strong logic can be taught in
elementary school,

I recall learning a lot of elementary set theory.
That seems similar.

No, this is merely sentential logic. Very little
can be done with it; what is needed is predicate
logic, with quantification only over individuals.
This is definitely more complicated, but not that
much, in its rules, and when it comes to using it,
it is much harder. It is adequate for all of
mathematics, and any question in sentential
calculus can be automatically answered.

and that a good proportion of
high school students could handle "Euclid" geometry.
In my time, in Chicago the high school enrollment
was quite high.

We also know that 60 years ago (and earlier) the
standard high school curriculum for going to college
required a good proof-oriented geometry course.


It's funny--I have met people who have told me
that they found geometry much easier than
algebra. Now, I found algebra much easier
than geometry. I don't know the reason for that.

Which geometry? The "Euclid" course is rarely
given now, except as an honors course. But
Euclidean geometry is a totally different subject
than algebra; high school level algebra essentially
deals with real numbers, while geometry deals with
idealized points, lines, circles, etc. It does
contain some algebra, in that it uses special cases
of the rule of equality.


Interesting... I have never thought of algebra as
dealing with real numbers--I thought that was
more of a calculus issue. I always thought that
the goal of algebra was to set up the problem,
then solve for the unknown variable (or variables),
whether they represented integers, reals, or
something else. Sorry if I am woefully ignorant
about the philosophical structure of mathematics.

I did take a proof-oriented geometry course in
high school in the late 70's. Looking at Euclid's
"Elements" on the web, though, I'd have to say
that my course was taught at a more basic level.

Perhaps the issue with such courses now is
that the homework can take quite a bit of time,
and homework seems to be a no-no these days.

That would be much easier and possibly more
sobering to read than the many letters on this
newsgroup, which, while they are quite
enlightening and entertaining at times (and
a sort of literature in themselves), are also
repetitive, disorganized, and don't seem to
lead to many clear suggestions of solutions.
The existence of such books might also
encourage discussion from educators
or at least mathematicians everywhere.

The books will almost entirely have to be written.

So perhaps you are proposing that a new
textbook needs to be written specifically
for exceptionally gifted children in math.

An exceptionally gifted child can use the
existing books. My son did, with not that
much help. In fact, for logic he used two
books, with somewhat different notations,
around age 6.

It would not be difficult to write a logic
text which could be used by average elementary
school children with a good teacher; with some
modifications, my late wife's book can almost
do this, although it was written for juniors
in college. If someone is interested in making
the modifications, I would be glad to cooperate,
including introductory material on variables
which could be used with beginning reading.
I am NOT a writer, and I know how much work
is involved in writing a book. In this case,
there are good templates to work from, so it
will be easier.


Well, one piece of advice I can give you
if you decide to do this (and if you want my
advice), is to do it in small pieces, and
get feedback from your audience (some
exceptionally gifted children) as you are
working on it. I'm sure many in these newsgroups
would be interested in discussing it with you too.

It sounds challenging, however, to go from a
junior level college text to something for
early elementary school. You might want
someone extra (perhaps a mathematician who
has taught elementary school?) to help you.

I think I have found some references to your
wife's books (sorry to be nosy), so when I
get a chance, I'll take a look at them just
to see what it is that you are talking about.

The books that my son studied to learn logic are
available, but I doubt that many primary school
children can handle them. The one he learned
algebra from might be a possibility for someone
who has already learned to read.

The book which I recommend for "adults" to learn
the ordinal approach to the integers was definitely
written for those who were adept at using algebra
and had some idea of a proof, and who were willing
to see that the details could be filled in. It is
not appropriate for beginning teaching, and in fact
it is even inadequate in its treatment of the integers,
as it does not go into positional notation. But
making sense of positional notation requires ordinals.

I recall learning about different bases in elementary
school--it didn't seem like a big issue. Wasn't that
part of the 'new math?'

It is a minor part. The important aspect of this is
that there are lots of representations for numbers,
and it is mainly a matter of convenience which to use.
I am not that familiar with the Mayan representation,
but most of the representations did not use the same
characters in different positions, which requires the
"zero" character. It does not make that much difference,
until one gets into numbers with large numbers of digits,
which is relatively modern.


I remember learning about the Egyptian and Roman
numerals. I very much enjoyed learning history
and math at the same time, though that was
likely meant to be a side issue.

Those who produced the new math materials did not start
with a finished product. We can to some extent go back
to the old structured materials we had before the idea
that children should always be with their age groups
was imposed, and the idea of "relevance" instead of
learning for the distant future, and doing things
in a manner removing repetition, were the rule.


So, you mean to start more from scratch than
what those who designed 'new math' started
with?

In my opinion, and it was my opinion more than
50 years ago, the new math made the mistake of
concentrating on the cardinal representation of
numbers. It is deceptively easy, but there are
major difficulties, including what is a finite
number. The only fully adequate "definition"
of this is one which can be counted and the
counting terminates. This brings in the ordinal
counting process.

The ordinal approach is self-contained, and
the definitions are clearly definitions. One
should do both; the cardinal (how many) and
the ordinal (count in order) concepts describe
the same finite objects, but give different ways
of looking at them.



--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558

"laraine" wrote in message
ps.com...
Perhaps the issue with such courses now is
that the homework can take quite a bit of time,
and homework seems to be a no-no these days.

I don't know where you live, but it's so the opposite where I am. The kids
are overwhelmed with homework.

In article om,
laraine wrote:
Herman Rubin wrote:
In article om,
laraine wrote:
Herman Rubin wrote:
In article . com,
laraine wrote:
Herman Rubin wrote:

...............

No, I haven't assumed that--I don't yet feel
knowledgeable enough to make a judgment
about it.


Consider that when those who understood the concepts
taught it, the original "new math" was hardly a
failure, or it would never have been adopted.

I think that I was taught 'new math' in elementary
school in the 70's. In any case, I personally found
the textbooks excellent, and I don't recall too many
people having trouble with them. My recollection is
that people did reasonably well until they got to
algebra. The 'new math' was not as complex as
what Tom Lehrer makes fun of, though, when
I studied it.

This was not that common; the big problem was the
teachers.

Many new math books had considerable use of
variables. As I keep pointing out, it is the
linguistic use of these which is the important
one, and do not try to skimp on how many are
used. Most algebra books discuss solution of
one equation in one unknown, and then give many
problems which should not even be tried with
one unknown. Use as many as you want, and use
the rule of equality to eliminate.

What I didn't like was the year and a half
I had in 6th and part of 7th grade when
there were no math lectures, and we
essentially worked independently. It would
have been better, I think, if a computer could
have given us each the right kinds of
problems geared to our individual abilities,
but we were essentially on our own (with
a teacher who would privately answer
questions).

Also,
we have evidence that strong logic can be taught in
elementary school,

I recall learning a lot of elementary set theory.
That seems similar.

No, this is merely sentential logic. Very little
can be done with it; what is needed is predicate
logic, with quantification only over individuals.
This is definitely more complicated, but not that
much, in its rules, and when it comes to using it,
it is much harder. It is adequate for all of
mathematics, and any question in sentential
calculus can be automatically answered.

and that a good proportion of
high school students could handle "Euclid" geometry.
In my time, in Chicago the high school enrollment
was quite high.

We also know that 60 years ago (and earlier) the
standard high school curriculum for going to college
required a good proof-oriented geometry course.


It's funny--I have met people who have told me
that they found geometry much easier than
algebra. Now, I found algebra much easier
than geometry. I don't know the reason for that.

Which geometry? The "Euclid" course is rarely
given now, except as an honors course. But
Euclidean geometry is a totally different subject
than algebra; high school level algebra essentially
deals with real numbers, while geometry deals with
idealized points, lines, circles, etc. It does
contain some algebra, in that it uses special cases
of the rule of equality.


Interesting... I have never thought of algebra as
dealing with real numbers--I thought that was
more of a calculus issue. I always thought that
the goal of algebra was to set up the problem,
then solve for the unknown variable (or variables),
whether they represented integers, reals, or
something else. Sorry if I am woefully ignorant
about the philosophical structure of mathematics.

If this was what you got, it was better than what
most ended up with. But as for details, the only
kinds of numbers you had were real, until you got
to a small amount of complex. Integers and
rational numbers are types of real numbers, and
mostly one stays within them.

The problem with most students in algebra, and
the courses and what they learn reflects this,
is that the emphasis is on solving formulated
problems, using only routine methods. I have
seen an outline which has quite a few rules for
handling equations, all of which are special
cases of the rule of equality, and the students
memorize them one at a time. One of my colleagues
told me about a bright minority student who was
in danger of flunking out; nobody had told him
that he could formulate word problems. I said
a bright student; for such, it is that it has
not been taught, not that it was not learned.


I did take a proof-oriented geometry course in
high school in the late 70's. Looking at Euclid's
"Elements" on the web, though, I'd have to say
that my course was taught at a more basic level.

Perhaps the issue with such courses now is
that the homework can take quite a bit of time,
and homework seems to be a no-no these days.

That would be much easier and possibly more
sobering to read than the many letters on this
newsgroup, which, while they are quite
enlightening and entertaining at times (and
a sort of literature in themselves), are also
repetitive, disorganized, and don't seem to
lead to many clear suggestions of solutions.
The existence of such books might also
encourage discussion from educators
or at least mathematicians everywhere.

The books will almost entirely have to be written.

So perhaps you are proposing that a new
textbook needs to be written specifically
for exceptionally gifted children in math.

An exceptionally gifted child can use the
existing books. My son did, with not that
much help. In fact, for logic he used two
books, with somewhat different notations,
around age 6.

It would not be difficult to write a logic
text which could be used by average elementary
school children with a good teacher; with some
modifications, my late wife's book can almost
do this, although it was written for juniors
in college. If someone is interested in making
the modifications, I would be glad to cooperate,
including introductory material on variables
which could be used with beginning reading.
I am NOT a writer, and I know how much work
is involved in writing a book. In this case,
there are good templates to work from, so it
will be easier.

Well, one piece of advice I can give you
if you decide to do this (and if you want my
advice), is to do it in small pieces, and
get feedback from your audience (some
exceptionally gifted children) as you are
working on it. I'm sure many in these newsgroups
would be interested in discussing it with you too.

The pieces cannot be too small, and that has been
considered.

It sounds challenging, however, to go from a
junior level college text to something for
early elementary school. You might want
someone extra (perhaps a mathematician who
has taught elementary school?) to help you.

I do not think it harder than the ones which
have been taught at the elementary school
level, except that the applications, which are
not in those other books, may involve unknown
material. A small amount of the vocabulary
may need to be changed, but not much.

I think I have found some references to your
wife's books (sorry to be nosy), so when I
get a chance, I'll take a look at them just
to see what it is that you are talking about.

You are likely to have GREAT difficulty with
any of the others. They are written for people
with considerable mathematical knowledge, while
this one, despite it being junior level, has no
real prerequisites.

The books that my son studied to learn logic are
available, but I doubt that many primary school
children can handle them. The one he learned
algebra from might be a possibility for someone
who has already learned to read.

The book which I recommend for "adults" to learn
the ordinal approach to the integers was definitely
written for those who were adept at using algebra
and had some idea of a proof, and who were willing
to see that the details could be filled in. It is
not appropriate for beginning teaching, and in fact
it is even inadequate in its treatment of the integers,
as it does not go into positional notation. But
making sense of positional notation requires ordinals.

I recall learning about different bases in elementary
school--it didn't seem like a big issue. Wasn't that
part of the 'new math?'

It is a minor part. The important aspect of this is
that there are lots of representations for numbers,
and it is mainly a matter of convenience which to use.
I am not that familiar with the Mayan representation,
but most of the representations did not use the same
characters in different positions, which requires the
"zero" character. It does not make that much difference,
until one gets into numbers with large numbers of digits,
which is relatively modern.


I remember learning about the Egyptian and Roman
numerals. I very much enjoyed learning history
and math at the same time, though that was
likely meant to be a side issue.

Those who produced the new math materials did not start
with a finished product. We can to some extent go back
to the old structured materials we had before the idea
that children should always be with their age groups
was imposed, and the idea of "relevance" instead of
learning for the distant future, and doing things
in a manner removing repetition, were the rule.

So, you mean to start more from scratch than
what those who designed 'new math' started
with?

Not really. We do have the theoretical materials in
a pedagogical form. The only problems are to combine
it all, extend where necessary, produce exercises, and
figure out how fast to do it. Find one person willing
to work on this with me, as I am not a good writer, and
a "self-paced" program without enough exercises can be
produced in a few months.

In my opinion, and it was my opinion more than
50 years ago, the new math made the mistake of
concentrating on the cardinal representation of
numbers. It is deceptively easy, but there are
major difficulties, including what is a finite
number. The only fully adequate "definition"
of this is one which can be counted and the
counting terminates. This brings in the ordinal
counting process.

The ordinal approach is self-contained, and
the definitions are clearly definitions. One
should do both; the cardinal (how many) and
the ordinal (count in order) concepts describe
the same finite objects, but give different ways
of looking at them.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
Phone: (765)494-6054 FAX: (765)494-0558