Only 45% of the students were prepared for math

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Topic:	Sociology > Education
User:	"Dom"
Date:	22 Mar 2006 04:29:36 PM
Object:	Only 45% of the students were prepared for math

Los Angeles Times Mar 15, 2006 Page B9
CSU Freshmen Face Challenges; Only 45% of the students were prepared
for math and English studies at college level, report says
Author(s): Cynthia H. Cho
Document URL:
http://proquest.umi.com/pqdweb?did=1003262501&Fmt=3&clientId=16778&RQT=309&VName=PQD
CORRECTION: SEE CORRECTION APPENDED; Remedial classes -- An article
Wednesday in California about the California State University system
described the number of students who benefited from English and math
remedial classes. The article said, "Of the freshmen who enrolled in
the fall of 2004 and needed remediation, 84% -- 22,004 out of 38,859 --
became proficient in both subjects before their second year of
college." The article should have said that 84% of the 22,004 freshmen
who took remedial classes became proficient in English and math. The
total number of freshmen is 38,859.
Less than half of the freshmen currently in the California State
University system were ready for college-level math and English courses
upon enrollment -- a figure significantly below the goal established by
the system's trustees a decade ago -- a new report said.
University officials told the Board of Trustees on Tuesday that 45% of
students who entered the Cal State system in the fall were prepared for
college-level work, a mere 2% increase from the previous year. In 1996,
the Board of Trustees said it wanted 90% of students starting college
in the fall of 2007 to be proficient in mathematics and English.
"Obviously, these figures are lower than what we would hope to be in
this particular year if we are to achieve the goals set for 2007," said
Gary Reichard, executive vice chancellor and chief academic officer.
"We don't pretend otherwise," he said.
But administrators for the 23-campus system also said they anticipated
noticeable gains over the next two years, as they began to see results
from a new assessment test for high school juniors that was first
administered last spring.
When 11th-grade students take the mandatory California Standards Tests,
they may now add a voluntary exam that includes 15 additional math
questions, 15 additional English questions and an essay that make up
the Early Assessment Program. The voluntary test helps them find out
whether they are ready for college-level courses.
Last spring, 119,000 juniors took the voluntary math exam and 185,000
students took the English exam, Cal State officials said.
Cal State faculty and high school teachers are working together to
create 12th-grade courses for students whose performance on the
voluntary tests indicate that they are not prepared for college- level
instruction.
"We have trained more than 700 teachers and are in the process of
training thousands more," said Trustee Roberta Achtenberg.
But William G. Tierney, director of the Center for Higher Education
Policy Analysis at USC, said educators can't just rely on high schools
to prepare students for higher education. He said that colleges should
assume some of the responsibility and try to serve high school students
in creative ways -- after school, on weekends and during summers.
"The community colleges, colleges and universities need to be more
involved -- not simply assessing the quality of students but working
with them to prepare them for college," Tierney said.
Of the 43,005 current freshmen, 36% needed to take remedial classes in
math, down 1% from a year ago, and 45% needed remedial English classes,
down 2%.
Since 1998, when Cal State began tracking student performance, math
proficiency has increased 18 percentage points. English proficiency has
increased only 2 percentage points, "unsatisfactory from any point of
view," Achtenberg said.
Pointing to current sophomores, Cal State officials touted the success
of their remedial instruction programs. Of the freshmen who enrolled in
the fall of 2004 and needed remediation, 84% -- 22,004 out of 38,859 --
became proficient in both subjects before their second year of college.
Of those who needed remediation, 10% did not complete their courses and
were not allowed to re-enroll.
.

User: "Pubkeybreaker"
Title: Re: Only 45% of the students were prepared for math	28 Mar 2006 11:37:50 AM

Dom wrote:

<snip>
Those needing remedial courses should be required to take them at
a community college, and NOT at a state university that is supported
by tax money.
These students should not have been accepted in the first place.
And I doubt that the problem is restricted to just math and English.
I strongly suspect that most of these students simply lacked the
dedication and intellectual maturity required at any college in any
subject.
Too many students go to college just to party and have a good time.
Maybe if they had to pay for their own education, rather than just
partying at mommy's and daddy's expense, they would actually dedicate
themselves to LEARNING.
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	28 Mar 2006 07:21:33 PM

In article <1143567470.815114.36250@i40g2000cwc.googlegroups.com>,
Pubkeybreaker <Robert_silverman@raytheon.com> wrote:

Dom wrote:

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

Those needing remedial courses should be required to take them at
a community college, and NOT at a state university that is supported
by tax money.

They should be required to take them at a place which gives
quality remedial courses, not dumbed down.

These students should not have been accepted in the first place.

True, but there is no way the colleges can know that they are
needed. Reading transcripts, there is no way to tell what
material was covered in a course at any level.

And I doubt that the problem is restricted to just math and English.
I strongly suspect that most of these students simply lacked the
dedication and intellectual maturity required at any college in any
subject.

Much of it is due to the fact that the material needed was not
presented to the student. One of my colleagues told me he was
mentoring a minority student having difficulties, often with
easy problems. He was; nobody had told him that he could use
symbols to formulate problems.
The problem is that most remedial courses are taught on the
assumption that the student was not good enough to get it
the first time, and therefore it should be at a weak pace.
The results are what you would expect.

Too many students go to college just to party and have a good time.

The solution for that is clear; the colleges need to maintain
standards. They do not; they continue the elhi strategy of
teaching what they thing the weak students in the class can
manage.

Maybe if they had to pay for their own education, rather than just
partying at mommy's and daddy's expense, they would actually dedicate
themselves to LEARNING.

It probably is too late. They learned in elhi that socialization
comes before learning.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "Guess who"
Title: Re: Only 45% of the students were prepared for math	22 Mar 2006 05:00:16 PM

On 22 Mar 2006 14:29:36 -0800, "Dom" <DRosa@teikyopost.edu> wrote:

CSU Freshmen Face Challenges; Only 45% of the students were prepared
for math and English studies at college level, report says

Not surprising. Even statistics of percentages of freshmen/women who
graduate won't be comparable from times past. Back then not everybody
and their pet cat went to university in order to say in their resume
that they'd been there. Not everyone back then made it through 1st
year either; it was common to lose at least 1/3 in their first year.
Perhaps some "more traditional" methods of teaching might work to give
better results? ...and perhaps the computer/hand-held calculator are
not the magic pill as stated?
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	23 Mar 2006 10:33:53 AM

In article <g5l322hg1buf0bf6uu67136i38pcde8kno@4ax.com>,
Guess who <notreally.here@here.com> wrote:

On 22 Mar 2006 14:29:36 -0800, "Dom" <DRosa@teikyopost.edu> wrote:

CSU Freshmen Face Challenges; Only 45% of the students were prepared
for math and English studies at college level, report says

It is not the computer/hand-held calculator which is
causing the problem, and giving it up would not make
that much difference. It is the emphasis on teaching
facts and methods which is the problem, rather than
teaching concepts and structure.
The educationists cannot understand the importance of
concepts and structure, and consider these taught by
definitions again. Grammar is highly deemphasized in
English classes in favor of "free expression", and
the emphasis is mathematics classes in computing
answers, rather than being able to ask question,
understanding that things have to be proved, and also
understanding integers and real numbers. The remedial
courses really do not do an adequate job of remediation;
almost none of the concepts get across.
Concepts and structure need to come EARLY, so the students
can know why, and not just how. This also means that the
emphasis on relevance needs to go out; education is for
the distant future, not the current present.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "Guess who"
Title: Re: Only 45% of the students were prepared for math	23 Mar 2006 12:23:13 PM

On 23 Mar 2006 11:33:53 -0500,

(Herman
Rubin) wrote:

Concepts and structure need to come EARLY, so the students
can know why, and not just how.

Not so; certainly not necessarily so, and far too sweeping a
generalisation. You lose almost everyone if you pontificate. The
young are generally not ready for theory simply due to the fact that
they are very young, but might grasp it later when they have more
detail to put to that theory.
I know and taught both fact and theory, depending on the age and the
level. It's part and parcel of what you do. The order is important
for reasons other than you suggest. You might read up on the general
learning capabilities and capacity of different age groups[ as in
Piaget's principles].
With those clearly showing exceptional talent, they can be prepared
for competition level mathematics. Check out the olympiad
competitions and others for that level of required competence.
Others, by far the majority, need hands-on, "show me how to do it then
leave me to do it." Others in between can have it one way or another;
first practice then theory or vise versa. Some simply do not have the
ability to assimilate both theory and practice, having sufficient
problem handling simple examples one after another.
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 10:22:36 AM

In article <m4p522tsf03cd3aq84fm1spqnugkge7oac@4ax.com>,
Guess who <notreally.here@here.com> wrote:

On 23 Mar 2006 11:33:53 -0500,

(Herman
Rubin) wrote:

Concepts and structure need to come EARLY, so the students
can know why, and not just how.

Do you mean you cannot teach grammatical structure to
someone who has not learned a language? Nonsense.
You do not understand concepts. They are NOT the same as
theory; one can learn the theory and have no understanding
of the concepts, and vice versa. A concept is not understood
by going through the motions; it is only understood when it
can be used. Also, a general concept is often misunderstood
when it is given as a generalization of a special concept.
A former student told me that the biggest problem he had with
general topology was that he had learned metric topology,
which is a specialization of it, but that the details of the
specialization hid the general concept.
Note that there was no explicit conceptualization of the
integers before the late 19th century or theory, and the
first axiomatization was somewhat later. One of the problems
here is that there are two totally distinct concepts, which
happen to coincide in operational practice. In my opinion,
and it was my opinion then, they chose the apparently easier
one, which actually is harder. They both should be taught
somewhat together, and it is not necessary to give all the
proofs to teach the concepts. A few should be given to
teach what a proof is.
Also, the concept of variable is often misunderstood. For
one thing, it is taught for numbers only initially, instead
of being used for anything, and not just as a noun or pronoun.
For another, the presentation attempts to do things with one
variable, which further causes confusion. Variables are a
part of "mathematical linguistics", and belong with beginning
reading. One does not have to build up to it.
There was a science fiction story in which there were the
slogans, "The Wistik dufels the Moraddy.", and "The
Moraddy dufels the Wistik." At the end of the story, the
protagonist only knows that Wistik and Moraddy are proper
nouns, and dufels represents a third person singular verb.
This is really a use of variables as far as he knew.

I know and taught both fact and theory, depending on the age and the
level. It's part and parcel of what you do. The order is important
for reasons other than you suggest. You might read up on the general
learning capabilities and capacity of different age groups[ as in
Piaget's principles].

Piaget never got to the point of understanding concepts.

With those clearly showing exceptional talent, they can be prepared
for competition level mathematics. Check out the olympiad
competitions and others for that level of required competence.

That is the use of theory and ingenuity, not concepts.

Others, by far the majority, need hands-on, "show me how to do it then
leave me to do it."

When these get to college, they are almost mentally dead.
One can sometimes teach them theory, but not concepts.
I have the regrettably too rare ability to recognize that
I may know how to prove the theorems, and use the material
for computation and representation, and NOT know what is
really going on. The schools are deliberately attempting
to destroy this.
Others in between can have it one way or another;

first practice then theory or vise versa. Some simply do not have the
ability to assimilate both theory and practice, having sufficient
problem handling simple examples one after another.

I have found in my decades as a professor that the one who
learns practice and then theory is usually unable to apply
the theory to the practice, because both teach HOW. The
concepts teach why, and need to be extremely general to be
understood. A typical "methods" course can be taught in
10% of the time to someone who understands the concepts,
often if much of the theory is not known.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "The World Wide Wade"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 01:37:50 PM

In article <e016cc$1rvs@odds.stat.purdue.edu>,

(Herman Rubin) wrote:

A former student told me that the biggest problem he had with
general topology was that he had learned metric topology,
which is a specialization of it, but that the details of the
specialization hid the general concept.

That's hardly believable, as the transition from metric spaces to
general topology is quite natural. And if you haven't seen metric
spaces, the axioms for point-set topology will appear like some
arbitrary abstract nonsense from another planet. You keep
repeating this little story as if it implied something for
pedagogy; it doesn't.
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 07:59:01 PM

In article <waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>,
The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote:

In article <e016cc$1rvs@odds.stat.purdue.edu>,
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:

The transition is not natural; the use of the metric is
what is actually confusing. Also, sequences are confusing;
the extension to nets requires getting rid of the idea that
a subsequence can be obtained from a sequence by striking
out terms. The classic example of a compact Hausdorff space
which is not sequentially compact shows where the problem
is, as well as many other points.
I myself did not really understand the topology of probability
distributions and random variables until these were extended
to non-metric spaces, where even the standard definition is
unclear, and has different versions.
As for "arbitrary abstract nonsense", this is what it takes
to understand mathematics, and even to be able to apply it
to "concrete" situations. Abstract ideas are NOT abstractions
of more concrete ones, but have a real meaning otherwise.
The process of generalization is difficult, requiring unlearning.
Specialization does not require anything of the sort.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "The World Wide Wade"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 07:08:53 PM

In article <e02855$4ig0@odds.stat.purdue.edu>,

(Herman Rubin) wrote:

In article <waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>,
The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote:

In article <e016cc$1rvs@odds.stat.purdue.edu>,

(Herman Rubin) wrote:

The transition is not natural; the use of the metric is
what is actually confusing.

A metric is a natural abstraction of distance. You study metric
spaces, you learn quickly about open sets and all the wonderful
things they can do for you. The transistion to topological spaces
seems very natural to me.

Also, sequences are confusing;

They certainly are to calculus students. Let's teach them nets!

the extension to nets requires getting rid of the idea that
a subsequence can be obtained from a sequence by striking
out terms. The classic example of a compact Hausdorff space
which is not sequentially compact shows where the problem
is, as well as many other points.
I myself did not really understand the topology of probability
distributions and random variables until these were extended
to non-metric spaces, where even the standard definition is
unclear, and has different versions.
As for "arbitrary abstract nonsense", this is what it takes
to understand mathematics, and even to be able to apply it
to "concrete" situations.

Abstract, yes. Arbitrary, no.

of more concrete ones, but have a real meaning otherwise.
The process of generalization is difficult, requiring unlearning.
Specialization does not require anything of the sort.

User: "James Dolan"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 03:31:58 PM

in article <waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>,
the world wide wade <waderameyxiii@comcast.remove13.net> wrote:
|In article <e016cc$1rvs@odds.stat.purdue.edu>,
|

(Herman Rubin) wrote:
|
|> A former student told me that the biggest problem he had with
|> general topology was that he had learned metric topology, which is
|> a specialization of it, but that the details of the specialization
|> hid the general concept.
|
|That's hardly believable, as the transition from metric spaces to
|general topology is quite natural.
it's entirely believable. if you hadn't already demonstrated your
limitations in other discussions it would seem amazing that you don't
know that _some_ (not all) people, including an especially high
proportion of strongly creative mathematicians, experience a revulsion
towards anything that smacks of the baroque complication of "classical
analysis", and are pleasantly surprised to discover in contrast the
austere simplicity of its topological underpinnings.
|And if you haven't seen metric
|spaces, the axioms for point-set topology will appear like some
|arbitrary abstract nonsense from another planet.
grossly false; again you might have managed a true statement if you'd
confined yourself to describing the experiences and preferences of
those who share your own limitations.
|You keep repeating
|this little story as if it implied something for pedagogy; it
|doesn't.
agreed, but the details of the specialization hides the general
concept that herman keeps repeating all of his little stories as if
they implied something; they don't.
--
[e-mail address jdolan@math.ucr.edu]
.

User: "The World Wide Wade"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 06:32:15 PM

In article <e01oge$4u9$1@glue.ucr.edu>,

(James Dolan) wrote:

in article <waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>,
the world wide wade <waderameyxiii@comcast.remove13.net> wrote:

|In article <e016cc$1rvs@odds.stat.purdue.edu>,
|

LOL, I see you're still smarting over being called a
"preposterous gasbag". You need to get over that.

it would seem amazing that you don't
know that _some_ (not all) people, including an especially high
proportion of strongly creative mathematicians,
experience a revulsion
towards anything that smacks of the baroque complication of "classical
analysis", and are pleasantly surprised to discover in contrast the
austere simplicity of its topological underpinnings.

Even if a huge proportion of strongly creative mathematicians are
"revulsed" by classical analysis in your imaginary world, they
should, by virtue of their big brains, be able to skate from
metric spaces to topological spaces with ease.

|And if you haven't seen metric
|spaces, the axioms for point-set topology will appear like some
|arbitrary abstract nonsense from another planet.

grossly false; again you might have managed a true statement if you'd
confined yourself to describing the experiences and preferences of
those who share your own limitations.

Right, I couldn't possibly understand a genius like yourself.

|You keep repeating
|this little story as if it implied something for pedagogy; it
|doesn't.

agreed, but the details of the specialization hides the general
concept that herman keeps repeating all of his little stories as if
they implied something; they don't.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 08:18:11 PM

In article <waderameyxiii-324A1C.16321525032006@comcast.dca.giganews.com>,
The World Wide Wade <waderameyxiii@comcast.remove13.net> wrote:

In article <e01oge$4u9$1@glue.ucr.edu>,
jdolan@math-cl-n03.math.ucr.edu (James Dolan) wrote:

in article <waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>,
the world wide wade <waderameyxiii@comcast.remove13.net> wrote:
|In article <e016cc$1rvs@odds.stat.purdue.edu>,
|

(Herman Rubin) wrote:
|> A former student told me that the biggest problem he had with
|> general topology was that he had learned metric topology, which is
|> a specialization of it, but that the details of the specialization
|> hid the general concept.
|That's hardly believable, as the transition from metric spaces to
|general topology is quite natural.
it's entirely believable. if you hadn't already demonstrated your
limitations in other discussions

LOL, I see you're still smarting over being called a
"preposterous gasbag". You need to get over that.

I question whether most people working in classical
analysis, in any of its forms, now even learn general
topology. This was much less the case 30 years ago,
when there was far more emphasis on getting a basic
graduate education; it is almost back to the old prewar
specialization.

No, the skating is not with ease, Situations which are
unique in metric spaces become very much not uniques in
general spaces, and can even not be so. On the other
hand, where they are similar, the lack of the metric
tricks make the proof much easier, even in metric spaces.

|And if you haven't seen metric
|spaces, the axioms for point-set topology will appear like some
|arbitrary abstract nonsense from another planet.

Not at all if properly presented. Precision does not
require obfuscation.

grossly false; again you might have managed a true statement if you'd
confined yourself to describing the experiences and preferences of
those who share your own limitations.

Right, I couldn't possibly understand a genius like yourself.

|You keep repeating
|this little story as if it implied something for pedagogy; it
|doesn't.
agreed, but the details of the specialization hides the general
concept that herman keeps repeating all of his little stories as if
they implied something; they don't.

They don't make understanding easier? Take convergence in
probability on metric spaces; the usual definition, using
the metric, is often quite difficult to use, and does not
seem to generalize even to uniform spaces. On the other
hand, the form X_n -> Y in measure if for any open set U,
m({a: Y(a) \in U and X_n(a) \notin U}) -> 0. which is
equivalent if m is finite, the usual situation, is easier
to work with in the metric case. BTW, uniformities, which
are the extension of metrics, are in my opinion much easier
to understand.
Unless the metric is a "natural" one, such as in a normed
space, that a space is metrizable may be important, but
finding the metric not.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "Dave Rusin"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 02:38:03 AM

In article <e01oge$4u9$1@glue.ucr.edu>,
James Dolan <jdolan@math-cl-n03.math.ucr.edu> wrote:

in article <waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>,
the world wide wade <waderameyxiii@comcast.remove13.net> wrote:
|And if you haven't seen metric
|spaces, the axioms for point-set topology will appear like some
|arbitrary abstract nonsense from another planet.

grossly false;

OK, I have a bright student uncorrupted by point-set topology before me.
I present the definition of a topology; the collection of open sets
is closed under _finite_ intersections but _arbitrary_ unions. The
student asks why on earth one would take such an asymmetrical set of
axioms. Your answer?
A bit later we have to decide what the morphisms of the category are.
I hope you don't consider it a corrupting previous specialization that
the student has already encountered homomorphisms of groups and rings.
So now we have topological spaces: sets and preferred subsets. We
define the topological maps to be: functions between the sets that, um,
do what?! you define the appropriate morphisms so that the INVERSE images
of the preferred sets in Y are preferred sets in X? Why on earth would
you do that? Your answer?
I'm all about not dwelling on minutiae that hide rather than highlight
What's Really Going On. That's cool. On the other hand, I can't imagine
hiding the origins of the definitions, the conjectures that prompted the
theorems, etc. I don't know about you but I'm not interested in preparing
desert-island mathematicians who have discovered all the consequences of
a set of axioms that no one cares about in the least. My students all
want to be part of a larger culture -- at least a mathematical one --
and they want to know why the goofy axioms I present might possibly
be connected to anything else at all. So I always spend some time on
historically-important special cases.
Guess I'm just doing it wrong then.
dave
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 02:19:55 PM

In article <e02vhb$q1a$1@news.math.niu.edu>,
Dave Rusin <rusin@vesuvius.math.niu.edu> wrote:

In article <e01oge$4u9$1@glue.ucr.edu>,
James Dolan <jdolan@math-cl-n03.math.ucr.edu> wrote:

Give examples. In fact, give many of the other characterizations
of topological spaces; I do not call any of the definitions, since
they are equivalent.
If you are going to do one-dimensional topology first, do
it with intervals rather than a metric. The difference is
a great improvement in understanding, and a total lack of
reliance on the irrelevant arithmetic properties of the
real numbers. Unless algebraic properties are used, the
precise metric is an irrelevancy.

A bit later we have to decide what the morphisms of the category are.

Morphisms in category theory are not the same as morphisms
dealing with object and sets. Isomorphism is a general
principle, and does not even depend on which system is used;
a bijection preserving all the properties being considered.
And a topological isomorphism (homeomorphism) between metric
spaces need not be an isometry, and usually is not.

I hope you don't consider it a corrupting previous specialization that
the student has already encountered homomorphisms of groups and rings.

Not every student who does analysis has done abstract algebra.
It is not a prerequisite.

So now we have topological spaces: sets and preferred subsets. We
define the topological maps to be: functions between the sets that, um,
do what?! you define the appropriate morphisms so that the INVERSE images
of the preferred sets in Y are preferred sets in X? Why on earth would
you do that? Your answer?

There is more than one; one possibility is in using limits
of nets. A similar situation occurs in analysis, where
measure has the same property. Also, few examples will
show why one gets too much by using "open" rather than
"continuous".

I'm all about not dwelling on minutiae that hide rather than highlight
What's Really Going On. That's cool. On the other hand, I can't imagine
hiding the origins of the definitions, the conjectures that prompted the
theorems, etc. I don't know about you but I'm not interested in preparing
desert-island mathematicians who have discovered all the consequences of
a set of axioms that no one cares about in the least. My students all
want to be part of a larger culture -- at least a mathematical one --
and they want to know why the goofy axioms I present might possibly
be connected to anything else at all. So I always spend some time on
historically-important special cases.

None of this is left out. The historically important special
case may be pedagogically confusing. This is the case with
measure and integration, where the key spaces are discrete
spaces and what results from them by the limit process. The
first integration was computing a merchant's bill, and the
ancient Greek method of approaching area was using this idea
plus the idea of limit.

Guess I'm just doing it wrong then.

One can have rigor without hiding the essential ideas.
However, the historical approach often makes things very
difficult, by introducing irrelevancies.

dave

--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "James Dolan"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 04:04:10 AM

in article <e02vhb$q1a$1@news.math.niu.edu>,
dave rusin <rusin@vesuvius.math.niu.edu> wrote:
|In article <e01oge$4u9$1@glue.ucr.edu>,
|James Dolan <jdolan@math-cl-n03.math.ucr.edu> wrote:
|>in article
|><waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>, the
|>world wide wade <waderameyxiii@comcast.remove13.net> wrote:
|
|>|And if you haven't seen metric
|>|spaces, the axioms for point-set topology will appear like some
|>|arbitrary abstract nonsense from another planet.
|>
|>grossly false;
|
|OK, I have a bright student uncorrupted by point-set topology before
|me. I present the definition of a topology; the collection of open
|sets is closed under _finite_ intersections but _arbitrary_
|unions. The student asks why on earth one would take such an
|asymmetrical set of axioms. Your answer?
|
|A bit later we have to decide what the morphisms of the category are.
|I hope you don't consider it a corrupting previous specialization
|that the student has already encountered homomorphisms of groups and
|rings. So now we have topological spaces: sets and preferred
|subsets. We define the topological maps to be: functions between the
|sets that, um, do what?! you define the appropriate morphisms so that
|the INVERSE images of the preferred sets in Y are preferred sets in
|X? Why on earth would you do that? Your answer?
|
|I'm all about not dwelling on minutiae that hide rather than
|highlight What's Really Going On. That's cool. On the other hand, I
|can't imagine hiding the origins of the definitions, the conjectures
|that prompted the theorems, etc. I don't know about you but I'm not
|interested in preparing desert-island mathematicians who have
|discovered all the consequences of a set of axioms that no one cares
|about in the least. My students all want to be part of a larger
|culture -- at least a mathematical one -- and they want to know why
|the goofy axioms I present might possibly be connected to anything
|else at all. So I always spend some time on historically-important
|special cases.
|
|Guess I'm just doing it wrong then.
i have to question your reading comprehension if you think that your
questions here are somehow responsive to something i wrote. however
if we disregard the issue of what inspired your questions and just
consider them as dropped out of thin air for no particular reason then
i don't mind spending a minute or two answering a couple of them.
first consider what happens if you omit the cardinality restriction in
the definition of topology. namely, there's an elegant lemma (with
proof probably shorter than the statement of the lemma) that such
topologies on a set are precisely equivalent to pre-orders (which are
structures of a lower level of complexity, more directly accessible to
the intuition), and that a map is pre-order-preserving precisely in
case the inverse images of open sets are open.
it's then obvious that topologies in general are ideal refinements of
pre-orders, and that continuous maps in general are ideal refinements
of pre-order-preserving maps, and this provides the appropriate
geometric intuition for understanding topological spaces and
continuous maps as tools for studying "cohesion" in a context where
pre-orders are refined more and more finely without limit.
there's a lot more that can be said to help students understand the
details as well as the broad currents of ideas here and anyone who'd
like to pay me to say more of it is welcome to make an offer.
--
[e-mail address jdolan@math.ucr.edu]
.

User: "James Dolan"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 06:07:45 PM

in article <e034iq$ecg$1@glue.ucr.edu>,
james dolan <jdolan@math-rs-n03.math.ucr.edu> wrote:
|in article <e02vhb$q1a$1@news.math.niu.edu>, dave rusin
|<rusin@vesuvius.math.niu.edu> wrote:
|
||In article <e01oge$4u9$1@glue.ucr.edu>,
||James Dolan <jdolan@math-cl-n03.math.ucr.edu> wrote:
||>in article
||><waderameyxiii-E2E22B.11375024032006@comcast.dca.giganews.com>, the
||>world wide wade <waderameyxiii@comcast.remove13.net> wrote:
||
||>|And if you haven't seen metric spaces, the axioms for point-set
||>|topology will appear like some arbitrary abstract nonsense from
||>|another planet.
||>
||>grossly false;
||
||OK, I have a bright student uncorrupted by point-set topology before
||me. I present the definition of a topology; the collection of open
||sets is closed under _finite_ intersections but _arbitrary_
||unions. The student asks why on earth one would take such an
||asymmetrical set of axioms. Your answer?
||
||A bit later we have to decide what the morphisms of the category
||are. I hope you don't consider it a corrupting previous
||specialization that the student has already encountered
||homomorphisms of groups and rings. So now we have topological
||spaces: sets and preferred subsets. We define the topological maps
||to be: functions between the sets that, um, do what?! you define the
||appropriate morphisms so that the INVERSE images of the preferred
||sets in Y are preferred sets in X? Why on earth would you do that?
||Your answer?
||
||I'm all about not dwelling on minutiae that hide rather than
||highlight What's Really Going On. That's cool. On the other hand, I
||can't imagine hiding the origins of the definitions, the conjectures
||that prompted the theorems, etc. I don't know about you but I'm not
||interested in preparing desert-island mathematicians who have
||discovered all the consequences of a set of axioms that no one cares
||about in the least. My students all want to be part of a larger
||culture -- at least a mathematical one -- and they want to know why
||the goofy axioms I present might possibly be connected to anything
||else at all. So I always spend some time on historically-important
||special cases.
||
||Guess I'm just doing it wrong then.
|
|i have to question your reading comprehension if you think that your
|questions here are somehow responsive to something i wrote. however
|if we disregard the issue of what inspired your questions and just
|consider them as dropped out of thin air for no particular reason
|then i don't mind spending a minute or two answering a couple of
|them.
|
|first consider what happens if you omit the cardinality restriction
|in the definition of topology. namely, there's an elegant lemma
|(with proof probably shorter than the statement of the lemma) that
|such topologies on a set are precisely equivalent to pre-orders
|(which are structures of a lower level of complexity, more directly
|accessible to the intuition), and that a map is pre-order-preserving
|precisely in case the inverse images of open sets are open.
|
|it's then obvious that topologies in general are ideal refinements of
|pre-orders, and that continuous maps in general are ideal refinements
|of pre-order-preserving maps, and this provides the appropriate
|geometric intuition for understanding topological spaces and
|continuous maps as tools for studying "cohesion" in a context where
|pre-orders are refined more and more finely without limit.
to amplify a bit:
a nice space is triangulable. associated to a triangulation is the
pre-order "p belongs to the smallest simplex that q belongs to". the
topology is the ideal coarsest common refinement of the sequence of
pre-orders associated to a sequence of triangulations which are
subdivided more and more finely without limit. no metric is involved
and introducing one into the conceptual development would be worse
than useless for most sensible purposes. continuous maps have
simplicial approximations, which can be interpreted as
pre-order-preserving maps. the seemingly weird asymmetry between
unions and intersections is explained by the fact that the simplest
way of repairing the asymmetry in fact yields the simpler and more
useful and more beautiful concept of "pre-order", of which the concept
of "topology" is a deliberate generalization meant to include examples
arising as ideal coarsest common refinements of sequences of
increasingly fine pre-orders which naively have only the trivial
pre-order as their coarsest common refinement. the possibility of
repairing the asymmetry between unions and intersections in a less
beautiful way can be explored by anyone with the motivation to study
the resulting more general objects.
metric spaces are actually somewhat interesting objects, interpreted
as enriched categories of a sort, but they don't have much to do with
topology since the natural kinds of morphisms between them are pretty
different in character from the continuous maps.
students with different preferences can try different approaches to
topology. i was not promoting any one-size-fits-all approach; rather
i was pointing out how silly it was for wade ramey to doubt the
existence of _some_ students who are quite happy grasping the point of
topology without bothering with a detour through the red herring of
metric spaces. historical studies can be interesting and useful to
mathematicians but mostly with a good dose of -as-it-should-have-been
and metric spaces could arguably stand considerable de-emphasis.
--
[e-mail address jdolan@math.ucr.edu]
.

User: "The World Wide Wade"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 06:44:12 PM

In article <e04m0h$pqt$1@glue.ucr.edu>,

(James Dolan) wrote:

i was pointing out how silly it was for wade ramey to doubt the
existence of _some_ students who are quite happy grasping the point of
topology without bothering with a detour through the red herring of
metric spaces.

Of course I said no such thing. But don't let that get in the way
of your ongoing string of tirades.
.

User: "Dave Rusin"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 11:55:54 AM

In article <e016cc$1rvs@odds.stat.purdue.edu>,
Herman Rubin <hrubin@odds.stat.purdue.edu> wrote:

There was a science fiction story in which there were the
slogans, "The Wistik dufels the Moraddy.",

Actually "The Gostak distimms the Doshes".
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 12:25:19 PM

In article <e01bra$uqh$1@news.math.niu.edu>,
Dave Rusin <rusin@vesuvius.math.niu.edu> wrote:

In article <e016cc$1rvs@odds.stat.purdue.edu>,
Herman Rubin <hrubin@odds.stat.purdue.edu> wrote:

There was a science fiction story in which there were the
slogans, "The Wistik dufels the Moraddy.",

Actually "The Gostak distimms the Doshes".

This is a different story.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "Guess who"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 03:15:07 PM

On 24 Mar 2006 11:22:36 -0500,

(Herman
Rubin) wrote:

In article <m4p522tsf03cd3aq84fm1spqnugkge7oac@4ax.com>,
Guess who <notreally.here@here.com> wrote:

On 23 Mar 2006 11:33:53 -0500,

(Herman
Rubin) wrote:

Concepts and structure need to come EARLY, so the students
can know why, and not just how.

Do you mean you cannot teach grammatical structure to
someone who has not learned a language? Nonsense.

In what language to you teach it then?

You do not understand concepts. They are NOT the same as
theory; one can learn the theory and have no understanding
of the concepts, and vice versa.

Don't be speculative about what I do or do not know. I *taught*
concepts. I *argued* that is was lack of knowledge and understanding
of concepts that made the difference, being not surprised even when my
own daughter showed a decided lack of that knowledge in her studies in
physics.
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	24 Mar 2006 08:10:22 PM

In article <25o822pt3oh9p6jjqq70ietqgasccq929h@4ax.com>,
Guess who <notreally.here@here.com> wrote:

On 24 Mar 2006 11:22:36 -0500,

(Herman
Rubin) wrote:

In article <m4p522tsf03cd3aq84fm1spqnugkge7oac@4ax.com>,
Guess who <notreally.here@here.com> wrote:

On 23 Mar 2006 11:33:53 -0500,

(Herman
Rubin) wrote:

Concepts and structure need to come EARLY, so the students
can know why, and not just how.

Do you mean you cannot teach grammatical structure to
someone who has not learned a language? Nonsense.

In what language to you teach it then?

One needs very little of a language. Scientists have
demonstrated that little vocabulary is learned before
the idea of grammatical structure is managed; there
has been an argument, with data to back it, that
children earlier than one year, with zero vocabulary,
can comprehend grammatical ideas.
Once one has an adequate amount to use to communicate,
entirely foreign grammar can be taught.

You do not understand concepts. They are NOT the same as
theory; one can learn the theory and have no understanding
of the concepts, and vice versa.

What are the unrelated concepts of the integers, which
I have been mentioning? They do need to be tied together,
but they are totally distinct concepts. There are other
conceptual extensions, but these are very basic.
Also, the classical Euclidean geometry used intuition
only for the axioms (including some unstated ones), and
then was completely formal. This does not mean that SOME
"geometric intuition" may not be helpful; however, I saw
early that it was a mistake to rely on this, despite the
standard pedagogical saw about the importance. This is
the case even in many "geometric" situations.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "toto"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 09:58:00 AM

On 24 Mar 2006 21:10:22 -0500,

(Herman
Rubin) wrote:

Herman, you do this constantly. What scientists? How many
studies? Have these studies been replicated? Where is this
data? For heaven's sake, what parents have even allowed
their children of less than a year old to be involved in this kind
of foolishness?
--
Dorothy
There is no sound, no cry in all the world
that can be heard unless someone listens ..
The Outer Limits
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 02:53:03 PM

In article <p3qa22llgo0dogmkdnltd3fc3o72tumdpj@4ax.com>,
toto <scarecrow@wicked.witch> wrote:

On 24 Mar 2006 21:10:22 -0500,

(Herman
Rubin) wrote:

I do not recall the specific articles in _Science_.
The experiments are described, and I can easily
see that parents might be willing for their children
to engage in such.
--
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
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
.

User: "Bob LeChevalier"
Title: Re: Only 45% of the students were prepared for math	25 Mar 2006 04:37:05 PM

toto <scarecrow@wicked.witch> wrote:

On 24 Mar 2006 21:10:22 -0500,

(Herman
Rubin) wrote:

He seems to be arguing that because children who are a year old learn
correct grammar by absorption without explanation, that they could
similarly learn mathematical thinking by absorption without
explanation.
The flaw is that a two year old does not in fact "understand" any of
the rules that he follows; he just follows them, very concretely
(though the result looks like an abstraction to those who are thinking
abstractly). Mathematicians expect to understand the rules that they
follow, and learning the understanding is NOT something that young
kids tend to be able to do very easily. Young kids "understand"
grammatical structure in the same way that computers "understand" the
programs that they execute; young kids subconsciously "program
themselves" in the same way that programs running in computers can be
written to adapt to stimuli.
lojbab
.

User: "Reef Fish"
Title: Re: Only 45% of the students were prepared for math	27 Mar 2006 11:58:53 PM

Bob LeChevalier wrote:

toto <scarecrow@wicked.witch> wrote:

On 24 Mar 2006 21:10:22 -0500,

(Herman
Rubin) wrote:

First of all, Herman seems excessively gullible about one article of
which he couldn't quite recall.
The idea of young children understand grammatical ideas before
vocabulary has to have come from someone who is linguistically
challenged. The grammatical structure of Romance languages
(French, Spanish) and Germanic languages (German) and their
derivatives from Latin, such as English, are vastly different.
Children
who are less than 6 years old should have no difficulty being multi
lingual in English, French, Spanish, German, and even Chinese.
The older one gets, the more difficult it is to learn.
I was told by a Berlitz teacher that I tried to THINK too much about
the grammar of Spanish (when I already learned those in French,
German, and English). I was no match for any 6 year old.
In fact, when I was 6 years old, I was multi-lingual in four dialects
of Chinese -- each of which is as different as German, French,
and English.
Young children have a completely different way of learning a
language, by "imitation of rules" rather than abstraction of rules.
Vocabulary certainly precedes grammar.
At Yale, there is a tall science building called the Kline Tower.
The mother of a German kid mused when her son couldn't
understand why the tower was "Klein" when the word means
"small" in German. :-) The kid had no trouble speaking
English and German fluently. I had the hardest time learning
why every inanimate object is male, female, or neuter in German,
and how to look for the separable prefixes and suffixes that may
be a page or two away. Kids NEVER had that kind of problem
in learning German.
Scientist are often blinded by their own prejudices and ignorance
in conjecturing and testing the untestible, as in less than 1-yr olds.
But often they get GRANTS to do the silliness, if they can BS
enough pages in application for the grant.
-- Bob.
.

User: "Bob LeChevalier"
Title: Re: Only 45% of the students were prepared for math	28 Mar 2006 04:22:14 AM

"Reef Fish" <Large_Nassau_Grouper@Yahoo.com> wrote:

Bob LeChevalier wrote:

toto <scarecrow@wicked.witch> wrote:

On 24 Mar 2006 21:10:22 -0500,

(Herman
Rubin) wrote:

First of all, Herman seems excessively gullible about one article of
which he couldn't quite recall.

There have been multiple articles, but they simply did not say what
Herman thinks it said. It said that the infants had shown that they
had learned grammatical rules by some sort of pattern identification,
not that the could "comprehend grammatical ideas" which suggests that
they consciously could think about those rules.
Here is probably the article Herman meant, since it was in Science
http://www.psych.nyu.edu/gary/marcusArticles/marcus%20et%20al%201999%20science.pdf
and here are some of the multitude of commentaries on the
interpretation of the results.
http://lcnl.wisc.edu/people/marks/courses/lang&mind/6marcusLetters.html
http://cnl.psych.cornell.edu/abstracts/transfer-learning.html
and a related article by the original author
http://www.psych.nyu.edu/gary/marcusArticles/marcus%202000%20cdps.pdf
Here is another reporting on experiments with somewhat older infants.
http://www.associatedcontent.com/article/23681/artificial_grammar_learning_by_1yearolds.html

The idea of young children understand grammatical ideas before
vocabulary has to have come from someone who is linguistically
challenged.

They can *recognize* simple grammatical rules at a time fairly close
to when they start to *recognize* that particular words have meaning.
The issue is whether pattern recognition is "understanding". I didn't
even know what a differential equation was, when I "tutored" a kid who
was reading a textbook section on solving 2nd order linear
differential equations - I recognized the pattern of the quadratic
formula in the method of solution. That doesn't mean that I had any
clue as to why the quadratic formula could be used, or even what
"solution" meant other than getting the right answer to the problem,
but having recognized the pattern that the other kid had missed, HE
was able to better understand.
Thus clearly pattern recognition is a useful tool in abstraction, but
I don't think it constitutes "understanding".

Young children have a completely different way of learning a
language, by "imitation of rules" rather than abstraction of rules.

Tough call because Herman uses a vague definition of "abstraction".
Clearly They must DO some kind of abstract pattern recognition in
order to be able to imitate rules, since that seems to be how kids
learn which rules to imitate.

Vocabulary certainly precedes grammar.

At Yale, there is a tall science building called the Kline Tower.
The mother of a German kid mused when her son couldn't
understand why the tower was "Klein" when the word means
"small" in German. :-)

And yet he understood that Klein was modifying Tower and not vice
versa (there are languages where the adjective comes after the noun).

Scientist are often blinded by their own prejudices and ignorance
in conjecturing and testing the untestible, as in less than 1-yr olds.

But often they get GRANTS to do the silliness, if they can BS
enough pages in application for the grant.

But the research wasn't silly - it just wasn't testing what Herman
seems to think it was testing.
lojbab
.

User: "Herman Rubin"
Title: Re: Only 45% of the students were prepared for math	28 Mar 2006 02:56:11 PM

In article <1143525533.152262.56820@i39g2000cwa.googlegroups.com>,
Reef Fish <Large_Nassau_Grouper@Yahoo.com> wrote:

Bob LeChevalier wrote:

toto <scarecrow@wicked.witch> wrote:

On 24 Mar 2006 21:10:22 -0500,

(Herman
Rubin) wrote:

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

First of all, Herman seems excessively gullible about one article of
which he couldn't quite recall.

I can recall the article; I cannot recall the issue of
_Science_ in which it occurred. It was also referred
to in _U S News and World Report_.

The idea of young children understand grammatical ideas before
vocabulary has to have come from someone who is linguistically
challenged. The grammatical structure of Romance languages
(French, Spanish) and Germanic languages (German) and their
derivatives from Latin, such as English, are vastly different.

Vastly? Not at all. The way in which the parts of speech
are put together does differ in detail, but not much in
concept, in the Indo-European family. I am not that much
of a linguist, but I am familiar with the grammatical
structures of all the languages listed above (I can read
all of them), and somewhat with that of Russian and other
IE languages. I can also compare them to the different
structures, but still similar ideas, of the Semitic
languages, and the differences are not such as to cause as
great problems as idiomatic expressions do.

Children
who are less than 6 years old should have no difficulty being multi
lingual in English, French, Spanish, German, and even Chinese.
The older one gets, the more difficult it is to learn.

They have just as much problem with detail confusion.
In reading or listening to a language, grammatical details
are not that much of a problem; in writing or speaking,
they definitely are. I have read mathematical papers in
Latin, Italian, Portuguese, and Romanian, never having
taken any of those languages.

I was told by a Berlitz teacher that I tried to THINK too much about
the grammar of Spanish (when I already learned those in French,
German, and English). I was no match for any 6 year old.

In speaking or reading? My one-year French course gave me
a reading vocabulary larger than the speaking vocabulary of
a 6 year old native speaker. I did carry on discussions in
which the other person spoke French and I spoke English.
BTW, that course did all the grammar in less than 1/2
academic year.

In fact, when I was 6 years old, I was multi-lingual in four dialects
of Chinese -- each of which is as different as German, French,
and English.
Young children have a completely different way of learning a
language, by "imitation of rules" rather than abstraction of rules.
Vocabulary certainly precedes grammar.

I suggest you read articles by scientists. Children learn
regular rules, and then the irregularities; that is why
phonics is far better at teaching reading than the whole
word method.

At Yale, there is a tall science building called the Kline Tower.
The mother of a German kid mused when her son couldn't
understand why the tower was "Klein" when the word means
"small" in German. :-)

One can easily have a 6-footer named "Small" or a 5-footer
named "Gross". This happens in any language.
The kid had no trouble speaking

English and German fluently. I had the hardest time learning
why every inanimate object is male, female, or neuter in German,
and how to look for the separable prefixes and suffixes that may
be a page or two away. Kids NEVER had that kind of problem
in learning German.

Are you so sure? I had no problem with learning about
grammatical gender, and there are more kinds than that.
Many languages have only "masculine" and "feminine",
and their speakers are quite aware that gender is not
the same as biological sex. There may be some changes
to drop some endings, but speakers of those languages
do not object to "hurricane" because its first syllable
is pronounced the same as "her".

Scientist are often blinded by their own prejudices and ignorance
in conjecturing and testing the untestible, as in less than 1-yr olds.

I suggest you read the paper before jumping to conclusions.

But often they get GRANTS to do the silliness, if they can BS
enough pages in application for the grant.
-- Bob.

User: "Reef Fish"
Title: Re: Only 45% of the students were prepared for math	28 Mar 2006 04:54:59 PM

Herman Rubin wrote:

In article <1143525533.152262.56820@i39g2000cwa.googlegroups.com>,
Reef Fish <Large_Nassau_Grouper@Yahoo.com> wrote:

Bob LeChevalier wrote:

toto <scarecrow@wicked.witch> wrote:

On 24 Mar 2006 21:10:22 -0500,

(Herman
Rubin) wrote:

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

First of all, Herman seems excessively gullible about one article of
which he couldn't quite recall.

I can recall the article; I cannot recall the issue of
_Science_ in which it occurred. It was also referred
to in _U S News and World Report_.

Vastly? Not at all. The way in which the parts of speech
are put together does differ in detail, but not much in
concept,

Here, we are obviously not speaking of the same meanings of the
terms "detail", "concept", and "structure", linguistically speaking!
The fact that Germanic and Romance languages are vastly
different in those respect are well-known.

Children
who are less than 6 years old should have no difficulty being multi
lingual in English, French, Spanish, German, and even Chinese.
The older one gets, the more difficult it is to learn.

They have just as much problem with detail confusion.
In reading or listening to a language, grammatical details
are not that much of a problem; in writing or speaking,
they definitely are.

I was speaking of their SPEAKING ability in those languages. Not
many children read or write well at or below that age.

I have read mathematical papers in
Latin, Italian, Portuguese, and Romanian, never having
taken any of those languages.

That's because it's the TYPE of mathematical papers you read,
especially if the mathematical content is something with which you
are already familiar. In some mathematical journals, you can throw
away ALL the words and leave only the symbols and equations,
you can probably read it. In that case, naming the languages is
not even necessary. I can probably read a few of those papers
in Outer Mongolian too. ;)
L.J.Savage has written four papers in Italian, totalling over 100
pages,
on Bayesian statistics. The number of equations and mathematical
symbols in those 100+ pages total less than half a page, if that much.
I challenge you to make a coherent translation/paraphrase/summary
of any one of thoe 100+ pages!

I was told by a Berlitz teacher that I tried to THINK too much about
the grammar of Spanish (when I already learned those in French,
German, and English). I was no match for any 6 year old.

In speaking or reading?

In speaking of course. Berlitz is the "total immersion" method in
which
both teacher and students speak ONLY in the language in which the
students knew absolutely NOTHING to begin with, and reach some
proficiency after a short course of limited number of hours. Just
like
kids learn how to speak a foreign language. Thus,"reading" in Berlitz
is an oxymoron. In practice, unfortunately, the instructors cheat and
sneak in some English in class as well as Berlitz books -- which is
contradictory to the founding philosophy of Maximilien Berlitz.

My one-year French course gave me
a reading vocabulary larger than the speaking vocabulary of
a 6 year old native speaker.

Sure. No 6 year old would start his speaking with "this is a blue
pen".
That's why most of the traditional courses in foreign languages are
worthless, for ordinary or scientific use.

In fact, when I was 6 years old, I was multi-lingual in four dialects
of Chinese -- each of which is as different as German, French,
and English.

Young children have a completely different way of learning a
language, by "imitation of rules" rather than abstraction of rules.
Vocabulary certainly precedes grammar.

I suggest you read articles by scientists.

Unfortunately, I read too many of them. Now everyone calls themselves
a scientist and an engineer. A garbage pick up man is a "sanitation
engineer", and the garbage department is full of "sanitation disposal
scientists"! :-) But the behavioral scientists produce MUCH MORE
garbage than all the sanitation engineers in the world put together.
What were you saying, Herman, about scientists? ;-/

One can easily have a 6-footer named "Small" or a 5-footer
named "Gross". This happens in any language.

You missed tht point that Kline was a proper name in Kline Tower,
and the kid, in his spoken language knowledge naturally took
Kline to be an adjective, qualifying the noun Tower, hence small
tower, without having to understand any grammatical structure of
what is a noun and what is an adjective. But the word KLEIN
means "small" in whatever form of speech to the kid. Hence
vocabulary over grammatical structure.
Your example does not even FIT that analogy. If a kid hears
"Herman Small is a basketball player", he probably would understood
immediately that the speaker wasn't talking about the SIZE of Herman!

The kid had no trouble speaking

Are you so sure?

Sure in the sense that if a kid took 1/10 as long as I needed to learn
the conjugation of ordinary verbs and the dem-die-das matrix
associated with objects, they would not have learned to speak
a word before they are 10.

I had no problem with learning about
grammatical gender, and there are more kinds than that.
Many languages have only "masculine" and "feminine",
and their speakers are quite aware that gender is not
the same as biological sex. There may be some changes
to drop some endings, but speakers of those languages
do not object to "hurricane" because its first syllable
is pronounced the same as "her".

Confusion of a different kind..

Scientist are often blinded by their own prejudices and ignorance
in conjecturing and testing the untestible, as in less than 1-yr olds.

I suggest you read the paper before jumping to conclusions.

I would agree with you if it's about 6 year olds, because by that
age, I was already in the 3rd grade, having passed a written
entrance exam.
But LESS than 1-yr olds? The only ones the scientists fooled are
themselves and the gullible!

But often they get GRANTS to do the silliness, if they can BS
enough pages in application for the grant