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Pubkeybreaker
Guest

Posted: Mon Jul 14, 2008 3:44 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

On Jul 13, 7:57 pm, Bob LeChevalier <loj...@lojban.org> wrote:

Quote:

"Larry Hewitt" <larryh...@comporium.net> wrote:

Herman is a statistics professor of significant repute, and he seems
not to consider that sort of stat class to be proper statistics.
"Cookbook" statistical algorithms are frequently used without
understanding, leading to misleading results (sometimes intentionally
misleading).

My legislature demands that I teach this, failure to do so will result in my
termination.

Herman doesn't think that legislatures should have the right to decide
curriculum - only subject matter academicians.

And he is correct. The idea that the legislature, who knows nothing
abouth math, chemistry, physics, literature, sociology, etc. should
decide course content is absurd.... The inmates are running the
asylum.

Quote:

Other countries that outperform us in science and math tests are also
noted for being envious of American initiative and creativity. It
could very well be that there is a tradeoff between rigor and
creativity for all but the most intelligent (and maybe even them).

However, it is precisely the "most intelligent" who are responsible
for the creativity. And creativity in math/science DEMANDS rigor.

We have the 3rd largest population in the world. And the most of any
*developed* country. (China & India are getting there. I expect
China
to overtake the U.S. in science and technology during this century).
It is not surprising that we would have more creative people than most
other countries.

Perhaps what we need to do (this is VERY politically incorrect)
is to separate those who can from those who can not very early on
in school. Provide training (in the sense under discussion) for those
who can't, and provide education (in the sense under discussion) for
those
who can. It is my understanding that at least some countries already
do
this (e.g. Germany). One might argue that this will lead to a 2-
tiered
society, but I would argue that this is what we already have. The
widening
income gap in this country is driven by (IMO) the gap in education
between
the poor and the economically well-to-do. The poor are poor
precisely because
they have below average intelligence and hence get "training" instead
of
"education".

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Herman Rubin
Guest

Posted: Mon Jul 14, 2008 8:21 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

In article <gpik741k3jilkcgk64o9klmm9hmkirt5jo@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:

Quote:

hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:
In article <l7ak74pmuogf13ieglu6v1gksmvb92lrgc@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:

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

Quote:

Memorization and routine lead to automatization, which is required on
a timed test.

You continue to stress the trivia. You seem unable to tell
the difference between education and training; memorization
and routine is training.

And "We the people" want the animals (children) to be trained into
useful and self-supporting members of society. I suspect that this is
far more important to most people than any idealized concept of
"education", which is why the system has evolved in the direction that
it has.

And this is why we need to import graduate students in the
sciences, and people to teach college and graduate science,
to do non-trivial programming, etc. Sure, we can turn out
clerks, assembly line workers, and the like, but not those
who can use their brains and innovate.

The US has consistently led the world in CS, but it has not
produced many Americans in that; that many of the imports
have become Americans is to the good.

When "we the people" attempt to decide things involving the
nature of the universe, only bad things can result. In the
19th century, there was curriculum, and while understanding
has never been directly taught, some was introduced. "We
the people" did not interfere when Johnny had to repeat a
grade, and did not ask a teacher to lower the level of a
course to accommodate the weak students. Nor did they object
when Jane was skipped a grade, unless they though Joe was
smarter, and usually not even then.

--
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

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Herman Rubin
Guest

Posted: Mon Jul 14, 2008 10:03 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

In article <l7ak74pmuogf13ieglu6v1gksmvb92lrgc@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:

Quote:

hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:

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

Quote:

Look at the above. It is a matter of normalization of the
equation of a line and nothing more. The rule of equality
covers this quite well.

Further complicating the decision is a certaintity that just becaue he could
do the problem correctly ont he board yesterday does not mean he could do it
today.

One of my former colleagues told me of a student in a
graduate course complaining about a B. On one problem,
the student started correctly, but did not finish.
The colleague then told him of this and asked how he
would proceed. The student then answered that the
course was over two weeks before, and he had forgotten
that. This happens frequently when there is too much
memorization.

Quote:

STOP concentrating on memorization and routine. Minimize them.

Memorization and routine lead to automatization, which is required on
a timed test.

And even if the automation is retained, it cannot be
used for anything other than that explicit type of
problem. Understanding may not give quick answers,
but it enables students to put things together when
the case has not been covered.

If we want clerks or assembly line workers, training
may be appropriate. If we want any better, start
educating early; training will even come more quickly.
--
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

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Bob LeChevalier
Guest

Posted: Mon Jul 14, 2008 10:09 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

Pubkeybreaker <pubkeybreaker@aol.com> wrote:

Quote:

On Jul 13, 7:57 pm, Bob LeChevalier <loj...@lojban.org> wrote:
My legislature demands that I teach this, failure to do so will result in my
termination.

Herman doesn't think that legislatures should have the right to decide
curriculum - only subject matter academicians.

And he is correct. The idea that the legislature, who knows nothing
abouth math, chemistry, physics, literature, sociology, etc. should
decide course content is absurd.... The inmates are running the
asylum.

Democracy - the worst form of government - except for all the rest.

Quote:

That is questionable.

In any event, in other countries, the most intelligent don't have so
much creativity.

Quote:

They may have a larger economy, but we will likely still be the
innovators, because China does not reward innovation, and indeed often
punishes it.

Quote:

It is not surprising that we would have more creative people than most
other countries.

China has more "most intelligent" people than we do, by a factor of 4,
unless you assume that their population has a significantly different
bell curve. So if creativity is just another word for high
intelligence, they would have more creative people. But they don't.

Quote:

Perhaps what we need to do (this is VERY politically incorrect)
is to separate those who can from those who can not very early on
in school.

The United States is not that sort of country. If that is what we
"need" to do, we will find a different way, or choose a different
goal, because to do as you suggest would fundamentally violate our
cultural identity.

Quote:

Provide training (in the sense under discussion) for those
who can't, and provide education (in the sense under discussion) for
those who can.

Many who "can" don't *want*. And many of their parents don't "want",
either.

Quote:

It is my understanding that at least some countries already
do this (e.g. Germany).

And Germany doesn't have nearly the innovation levels that we do. They
have some superb craftsmen, though, because they value that sort of
thing.

Quote:

One might argue that this will lead to a 2-tiered
society, but I would argue that this is what we already have.

Not legally.

Quote:

George Bush was economically well off. What's your explanation for
him?

Those Asian immigrants who come over tend to arrive fairly poor. What
is your explanation for their success?

Clearly socioeconomics does provide benefits and handicaps. It does
in Germany and in that ultimate meritocracy Singapore. But the pride
of our system is our social mobility. Bill Clinton, whatever you
think of his politics, went from broken family poverty to the highest
office in the land. Obama likewise started rather low on the
socioeconomic totem pole.

It's a little harder to find such examples in the sciences, but one of
the 2006 Physics Nobelists was the son of a traveling salesman. Robert
Grubbs, one of the 2005 Chemistry Nobelist, grew up in rural Kentucky.

lojbab
Bob LeChevalier - artificial linguist; genealogist
lojbab@lojban.org Lojban language www.lojban.org

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toto
Guest

Posted: Tue Jul 15, 2008 12:40 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

On Mon, 14 Jul 2008 08:44:21 -0700 (PDT), Pubkeybreaker
<pubkeybreaker@aol.com> wrote:

Quote:

The poor are poor precisely because they have below average
intelligence and hence get "training" instead of "education".

Children of the poor vary as much in intelligence as the rest of the
population. We need to provide real education for those who have the
ability even if their parents are poor.

--
Dorothy

There is no sound, no cry in all the world
that can be heard unless someone listens ..

The Outer Limits

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Larry Hewitt
Guest

Posted: Tue Jul 15, 2008 1:50 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

"Herman Rubin" <hrubin@odds.stat.purdue.edu> wrote in message
news:g5fqts$9394@odds.stat.purdue.edu...

Quote:

In article <gpik741k3jilkcgk64o9klmm9hmkirt5jo@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:
In article <l7ak74pmuogf13ieglu6v1gksmvb92lrgc@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:

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

Memorization and routine lead to automatization, which is required on
a timed test.

You continue to stress the trivia. You seem unable to tell
the difference between education and training; memorization
and routine is training.

And "We the people" want the animals (children) to be trained into
useful and self-supporting members of society. I suspect that this is
far more important to most people than any idealized concept of
"education", which is why the system has evolved in the direction that
it has.

And this is why we need to import graduate students in the
sciences, and people to teach college and graduate science,
to do non-trivial programming, etc. Sure, we can turn out
clerks, assembly line workers, and the like, but not those
who can use their brains and innovate.

We do not import workers becaue US citizens cannot perform the work, we
import workers because they are willing to work for far less than the
prevailing wages.

According to the US BLS the unemploymern rate for college graduates is
currently about 60% higher than it was in the last decade of the last
century.

They also report that the number of college graduates working outside of
their major is more than twice what it was in that period.

IOW, there are thousands of US citizens with degrees in math, engineering,
chemistry, computer science, and other technical fields not working in those
fields beccuse of a shortage of jobs _or_ depressed salary levels mking
non-tecnical (esp finance) jbs more attractive.

Larry

Quote:

The US has consistently led the world in CS, but it has not
produced many Americans in that; that many of the imports
have become Americans is to the good.

When "we the people" attempt to decide things involving the
nature of the universe, only bad things can result. In the
19th century, there was curriculum, and while understanding
has never been directly taught, some was introduced. "We
the people" did not interfere when Johnny had to repeat a
grade, and did not ask a teacher to lower the level of a
course to accommodate the weak students. Nor did they object
when Jane was skipped a grade, unless they though Joe was
smarter, and usually not even then.

--
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

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Herman Rubin
Guest

Posted: Tue Jul 15, 2008 3:38 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

In article <pI-dnXOHJoj97OfVRVn_vwA@comporium.net>,
Larry Hewitt <larryhewi@comporium.net> wrote:

Quote:

"Herman Rubin" <hrubin@odds.stat.purdue.edu> wrote in message
news:g5d3av$8bf2@odds.stat.purdue.edu...
In article <NfWdnUqt8J2EleXVRVn_vwA@comporium.net>,
Larry Hewitt <larryhewi@comporium.net> wrote:

"Herman Rubin" <hrubin@odds.stat.purdue.edu> wrote in message
news:g58rn5$4gjs@odds.stat.purdue.edu...
In article <173d74hoh9ejhv26tari2uq56lv4hekj4r@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:
Barbara <mom_2_one@hotmail.com> wrote:
On Jul 10, 9:41am, hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
In article <486f7172.10114...@news.datemas.de>,
Way Back Jack <Rela...@home.net> wrote:

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

Quote:

Herman doesn't consider "basic high school-level algebra" to include
the "basic mathematical concepts" that he is talking about, which are
theoretical and abstract. He thinks that "basic high school-level
algebra" is mostly plug and chug recipes for solving problems, and
rote memorization of terminology, and he considers neither of these to
be real "mathematics".

Quote:

The following includes essentially all of algebra, except
for technical terms not used at the high school level:

A variable is a temporary name for something,
which must maintain its meaning in a given context.

The same operation performed on equal entities
yields equal results.

I respectfully disagree. For whatever reason, the term *algebra* has
taken on some mythical status as something extremely difficult and
fear-inducing.

The reason, as I learned from raising two kids who got that attitude,
is that *algebra* IS extremely difficult and fear-inducing.

All other subjects (except the more mathematical sciences) use the
normal English language, where words have fuzzy meanings that can be
gleaned from context, and there is some overlap with the methodology
that they use in solving non-academic problems.

Mathematical language is first and foremost *precise*. Misspell a
word and people will understand you. Fail to remember a word in most
subjects, and you can talk around the word and show that you
understand. But in mathematics, every step must be followed
rigorously, and the most minor error means that you are totally and
irrecoverably wrong, unless you notice the error and start over or
backtrack. Nothing else in a kid's life works like that. Life allows
for some amount of sloppiness. Mathematics does not. Teachers don't
know how to teach this (if they realize that this is the essential
difference) and kids see it as "difficult" and ultimately not
kid-like.

Unfortunately, teachers who do not know better grade on the
answer. One should grade on understanding what is to be done,
and as in English, errors should be corrected and pointed out
to the student.

Quote:

Nice in theiry, difficcult to imposssible in real life.

Easy in real life, if one concentrates on understanding,
and asks steps to be given. Do you remember your geometry
course, where you had to give proofs with statements and
reasons? If you did not have that course, which is common
now, you have probably not had a real mathematics course
in high school. The important :

Quote:

How does a teacher determine, for example. whether an error in a
computation with negative numbers is lack of understanding, a simple
arithemtic error, or a transcription error indropping a sign whe copying
from a work sheet.

By having the student put down the work, rather than just
the answer. I am the "czar" of our department's qualifiers,
and I can assure you that most students make errors on
most of the type of problems we assign. We give partial
credit, and once the faculty see how to do this, there is
not much disagreement on scores.

Quote:

This is relatively easy to do at the college level, but in 8th grade forget
it

Many problens, for ex., require only a few steps, too few to really develop
a cluie to te student's thinking.

Not to mention the difficulty in getting the kids to separate out relatively
smple operations, like a two step addition, into separate steps.

Yes, partial credit can be given, and often is.

But then we have to address what is really being taught.

How do you handle an error that develops but all of the stpes were not
written down? How do you handle a correct answer wtthout the steps? Do you
fail a student who got all ofthe correct answeres but failed to show her
work (usually the more intelleginet students)? Or do you enforce the
requirement arbitrarily?

And should sloppiness be punished?

Not heavily. But someone is not going to be a good scientist,
and I include the biological and psychological and economic
sciences, if there is sloppiness.

Quote:

First, very few of my students wee going to be scientists. A girl with her
heart set on being a beautician or a boy who want to be a mason souldn't
care less.

And how much is a little?

10%? 5%?

Bear in mind that, for some inexplicable reason, my state legislature has
decided that 7 points separate grades, no 10%. so a refusal to follow
directions, or even a charitable misunderstannd if instrsuction, can mean a
letter grade difference.

Also beasr in mind that in my state, and others, there is a minimum grade
for 8th grade students to get academic credit for algebra. That is, (over
teh strenuous objections of teachers) 8th graders must score a minimum 85%
to get credit for completing algebra I (they get credit for graduating 8th
grade, but must retake algebra in 9th grade for an 84% or less).

Politics. Blah.

How does a teacher determine that an incorrectly set up equation in a word
problem is the result of another transcription error, a reading
comprehension problem, or a misunderstanding of the underlying math?

This is not as likely to be difficult as you think.

Quote:

This is far more difficualt than you assume, given the many reading
comprehension problems of many 8th graders.

And then how does a teacher justfy what is no more than a subjective guess
to angry parents and administrtors, explaining why Joey got credit and
Zooey
didn't?.

The same holds for English composition.

Quote:

As noted above, in many states there are elevated grade requirements for
algebra I in middle school.

And to be perfectly honest, mu English peers moved away from subjective
grading, too, using purely objective measures like counting spelling and
grammar errors. The logic is that we are not training novelists so content
are less important.

After all, in public school we are teaching what our legislature has told us
to teach, the rules and structure of our subjects.

Often, the teacher grades on whether the problem is done as
indicated in the textbook recipe.

Because this is what has been taught, and this is what a student is
expected
to knwo.

And this is NOT what should be taught. Understand what methods
can be applied, and apply whichever

Quote:

Wrong.

We are thaching hte methods.

We cannot allow a student to choose on particular method that he has become
comfortable with, ignoring all others.

First, we are mandated to teach and assess his ability on _all_ methods.

Second, a studnet is incapable of stermining whter or not the method he
ignores will be nedded in later courses.

Third, this isentirely contrary to your desire to teach an understadning of
underlying conceepts. The ability to contrast and compare different
operations, and to determine which is most
efficient/accurate/easiest/reliable in a given situation is basic. If a
student does not use an alternative technifue how do we determine if it is
lack of knowledge, lack of comprehension, lack of logical ability, or just
plain orneriness?

Quote:

In algebra I there is truly little mathematically correct variation from
the
"book recipe".

Unfortunately. Also, at least 90% of the problems supposed to
be done with one variable should not, at least by beginners.

When my son was 8, and studying calculus mostly by himself from
Apostol's excellent book, too hard for most, we also had him
brush up on his algebra from an algebra 2 book. He was using
the number of variables expected, as he usually could, but was
unable to do one problem in which two variables were supposed
to be used. With the bound removed, he did it with seven.

Quote:

I have no idea what you are trying to say here.

I'm talking algegra I here, and none of what I think yoiu said is relevant.

Now if a genius, having really learned the subject, has difficulty
using the assigned number of variables, what do you expect of the
typical student? And this means that the teacher has to be able
to follow the reasoning.

Quote:

Algebra I does not have problems like this.

Quote:

Algebra I is simplfy

3e + 5t - 6y = 8y -2e

There is, for example, only one way to write a linear equation in
slope-intercept form,

But many ways to go about getting the equation.

Quote:

But only one optimal way.

And my point is that that writing the equation in that form is what is being
assessed, anot to write the equatoin in simplist form, a common error in
algebra I

So do you reward the rote math f simplifying despite the fact that the
student did not answer the question asked?
.
one way to solve a system of linear equations using
hte elimination method,

Where did you get that idea? If there are n equations,
there are usually n! ways of doing this.

I meant the technique is fixed. The order of opertion _may_ be commutable,
but the technique is fixed, and it isthe technique that is being assessed.

And again, os a process that takes 12 steps as good as one that tkes 4?

Not according to our standards--- efficiency is an important assessement of
ability.

one way to set up a box and whiskers statistical
chart.

This is mechanical, and has no mathematical content, nor
statistical content except descriptive.

Quote:

Wrong.

There are calculations to determine quartiles, and an assessment of
understanding of statistical concepts.

This chart is an important foundation for further study of statistics. In
fact, my first college stat class mentioned it as a quick and easy way to
demonstrate skewed data.

Yes, there are other ways to "solve" the problem or display the info, but
these specific algorithsm are what are being yested and knowlege of them
is
needed in future courses.

Are they? In practice, solving systems of equations is
done by computer. Understanding of the algorithms can
be important, but memorization of them no.

Quote:

Yes, they are.

In linear algebra, for ex, being able to determine the most efficient way to
solve a systme of linear equations is important. If you haven;t learned them
in algebra, you are at a disadvantage.

Try reducing a system of equations over the integers to
row echelon form. Or more so, proving it can be done.

Quote:

A subject not taught until college linear algebra.

I has 2 weeks to teach my entire course content on linear algebra, starting
with the definition of a linear equation. Matrices had not yet been taught
and matrix operations used to get the matrix in row echelon form were what
was taught.

So how would you grade a student who uses outstanfing toechnique to
rpesent
linear eq. in point-slope form when the question alled for the
slope-intercept form?

Did he just not follow instructions, and shouldn;t that be punished?

I would be unlikely to ask the question. I am not even sure
that I would give such, except as how to normalize the equation
of a line for certain purposes, and leave it at that. Memorizing
trivia is not that important.

Quote:

My legislature demands that I teach this, failure to do so will result in my
termination.

And, quite frankly, it is an important consept used in later courses. You
pooh pooh memorization, but retention is required and testing that retention
is the name of the game.

Did he not knwo the correct form? Did he start out right but lose his
way,
either taking a wrong path or end toosoon?

Look at the above. It is a matter of normalization of the
equation of a line and nothing more. The rule of equality
covers this quite well.

Quote:

Wrong.

It is a question of writing the equation in a format such that the student
can, by examination alone, determine certain characteristcs of the line. It
also prepares the equation for further evaluation or calculation, such as
graphing, a subject we teach.

Further complicating the decision is a certaintity that just becaue he
could
do the problem correctly ont he board yesterday does not mean he could do
it
today.

STOP concentrating on memorization and routine. Minimize them.

Quote:

That is what I taught.

Despite your experience with a genius child and genius college students, the
kids I teach need to concentrate on th algorithm, on the methods. I _try_ to
impart a little underlying theory, but snores ripple through the room in
moments, and I really do not ahve time to stray far.

Understand , computation is a challenge for some of the students. My first
week of algebra i class was a review of calcualtions with negative numbers.

I agree with you, to a point. Math needs to be taken more seriously, is far
more important than the way it is treated.

But also understand that eeven most college graduates do not take more than
1 or 2 math courses, one of them calculus for the humanities student. A "D"
is all they need to graduate, adn they aim for the "D".

I put myself through college turoring amth to business students, and some
took two or three tries to pass. Not because of how they were taught, but
because oftheir attitude. Soem even were forced to delay graduatoin for one
omre summer schools ession because they kept blowing off the work.

Did you get these simple concepts across? If they ever use
their business courses, they will have to FORMULATE problems,
not solve routine stuff. They need to use algebra as a language,
not as a means to get numerical answers.

This holds for students in anything which uses anything
quantitative.

Quote:

I deal with what I am given.

So do I, but I try to make a difference, not just continue
training them to be worse than robots.

Quote:

And when, as I said, I am dealt a student body who's)and their parent's)
life expectations for math is balancing a check book, I deal with it. I
cannot force a student to learn what he does not want to learn.

If he has learned the two principles above, he will know
how to balance a check book. He may need a calculator,
but so what?

Quote:

There may be many ways
about doing the problem; if the second sentence is followed,
other than arithmetic errors or sloppiness, there will be
no mistake made.

But is, for exampel, a long, meadnering process that takes many more
steps
than needed an indication of knowledge or luck? Andisn;t effciincy an
indication of understanding?

Possibly and possibly not.

We try to make our exams so that thinking must be used,
and knowing routine does not work. We do not quite
succeed, and it often shows. This should be done in
first grade; by the time the students come to college,
it is very often much too late.

Quote:

Exactly.

SO how do you assess what you cannot determine?

So, for example, is a process that took 12 steps to combine like terms in
an
equation as "correct", as good an indicator of knowledge, as one that took
4
steps?

I do not expect a student to find a short method, especially on
a test. I would rather a student figure out a method from basic
principles, no matter how clumsy, than memorize a trick.

Quote:

I do.

If you do, you are harming your students.

Quote:

A lack of knowledge of tricks. The practice should come
after the understanding; memorizing the multiplication
tables teaches nothing about multiplication.

Quote:

College even requires this ---- you have a fixed time period to compelte te
assessment, no more.

Alas, too often. The faculty is generally forced to by the
administration. But one can give take-home tests, and these
can be very effective. I found from one of them that only
5 of the 21 prospective teachers in my probability class had
any understanding whatever of what integration means; they
could not even set up problems similar to those thoroughly
gone over in class. I had one engineer in that class who
had not used any math in 14 years; he understood.

Quote:

I see no such restriction. "Social science" is a stupid
term selected by the educationists who did not think
students had learned geography if they needed to consult
works to find the geographical features involved in a
history course. So they had to put them together, and
added some more weakening. The "old" curriculum expected
students to be able to use what they had learned in previous
courses, even in different subjects.

Quote:

Under this requirement medicine, sociology, economics, astronomy, and a
whole host of disciplines crrently categorixed as "science" would fail
your
test. Now this may be good or bad, accurate or inaccurate, right or wrong.
But it certianly would be disruptive and chaotic.

Wrong. Randomness is subject to mathematical precision, as is
the more complicated quantum mechanics. It is just that there
is no simple correct deterministic process. For many purposes,
one can neglect the differences, just as we can neglect the
effect of cosmic dust on the Earth-Mars trajectory.

I am not tlking randomness, O am referong to your reference of precision.

Medicine is actually an art, rather than a science. The science
is biology. There is literary economics, which might not be a
science, but quantitative economics definitely is one. Science
does not go from "simple' situations to more complicated ones;
in principle, as in Newton's Third Law, the entire universe needs
to be considered simultaneously.

In sciences where the measurements are necessarily crude, or
there are disturbances which need to be added to the model,
one is not going to get accurate predictions. In meteorology,
it was found (the start of chaos theory) that a slight change
can have considerable consequences. And what is wrong with
astronomy?

Quote:

The subjects I listed are inherently incapable of achieving the level of
precision required in math, physics, etc.

Again, you do not understand precision. If one has a system
of stochastic difference equations, even if your model is
correct, you can only predict limiting proportions correctly.

If a couple have blood genotypes AO and BO, the offspring
are equally likely to be AB, AO, BO, or OO. This is precise.
Situations with small sample sizes and randomness do not
result in precise predictions, except long-term probability,
and long can be really long.

Quote:

Yet without referring to it as *algebra* per se, the
aforementioned concepts are introduced in most math curriculums in the
4th or 5th grade (5th grade at One's school, which uses a truly awful
math curriculum). Discussion at lunch -- One's friend: *your school
is so far behind ours! WE'RE learning algebra!* One *We're not even
close to algebra. We're learning about variables.*

Of course, the answer is not to re-name the subject. Rather, the
answer is to show the students that algebra isn't that difficult.

The important part should be taught as soon as the student
can read and produce symbols.

Quote:

How does one assess that for for the 112 students I typically taugh?

In how many classes? Anyhow, heterogeneous classes are a
great mistake.

It would be easier in first grade. By fifth grade it might
be late. But do not limit variables to mathematical objects:
they can stand for anything, including the rabbits in "Peter
Rabbit". They can also disambiguate ordinary English.

Good formal logic HAS been taught to the upper half
of fifth graders. I believe that my late wife's book
could be SLIGHTLY modified for the bulk of third graders.

Quote:

Does not hte assessment of development become a major distraction?

It depends on how it is assessed. Homework should be for
learning; if a student knows before starting the homework
assignment how to do a problem, it is a wasted problem.

Quote:

Tell me how I can get a taxophobic electorate to pay for the assessments.
Tell me how I can convince parents who think that what I am teaching is a
bunch of unecessary hooey that their child will never need to push the
legislature to implement this testing.

I do not believe the public schools are salvageable.
Also, I am far from alone in believing that "certified"
teachers are desirable; one poster on the mathematics
newsgroup asked whether the education courses required
a lobotomy.

Quote:

You can't show what isn't true. Mathematics is difficult unless one
first learns to appreciate precision and rigor. That may be why
skilled musicians tend to do well in math - part of becoming skilled
is learning that precision. But most kids don't stick with music for
the same reason - hours of practice learning to produce precisely the
sound you want isn't worth it to them.

Mathematical rigor is NOT like precision of observation.
Mathematical rigor is just that one can only proceed from
assumptions by the use of a SMALL number of procedures.
My late wife's logic book has a listing of these, and as
I stated, I believe it would take little modification of
the book to use in third grade.

That 2+3 = 5 is mathematical precision. It is not like
the precision in observation in physics or chemistry.
In fact, lack of precision can be an advantage at an
early stage. Chemistry would have been in difficulty
if accurate weighing had been possible in 1800; the
isotope effect, not detectable at the time, would have
messed up the ratio of weights of the various elements,
which led to constant proportion of elements in compounds.

Likewise, Kepler's Laws depended on the perturbation of
the orbit of Mars by Jupiter not being enough to mess
up the observations. One less decimal place, and he
would not have been able to distinguish between a circle
and an ellipse; one more, and Jupiter makes Mars too
irregular for it to have an elliptic orbit.

Mathematical precision is totally unlike precision in
the sciences; do not confuse them.

Quote:

Teach the appreciation of precision and rigor in first grade,
and that part of the problem will disappear. We CAN teach
precise mathematical concepts to kids, but it is difficult to
do this with adults. Stop hurting children by avoiding the
rigor which adults seem unable to understand.

Current knowledge is that children of that age are mentally incapable of
the
rigor you want.

Are they? The game _WFF N PROOF_ was marketed to such children.
They are capable of the rigor if you present it to them as such,
and not try to lead them up to it. The same holds for other concepts;
an abstract concept is NOT an abstraction of more concrete ones.

Going from general to special is easy; going from special to general
requires unlearning, which is always difficult.

Quote:

Wrong.

Generalization requires biological advancement.

Generalization requires mental advancement, not biological,
and luck. Abstract concepts are NOT the abstraction of
more concrete ones, they have their own existence. But
educationists cannot recognize it, as they do not have
those precise concepts. Learn the Peano Postulates, and
you will have the basic ordinal concepts of the integers.
The original "new math" used the cardinal concepts. There
are others, but as Dedekind stated, if it looks like the
integers, and acts like the integers, it is a version of
the integers.

Quote:

They are incapable of understanding symbolic representation,

This is utter baloney.

Quote:

Nope. It is current understanding of hiuman development.

That current understanding does not include the idea that
an abstract concept can be taught. It is like teaching
the complete rules of a game, but without all the details.

Quote:

I do not intend to be offensive to anyone, but you _really_ need to get dwon
into the trensches. You seem to be dealiing with the top few percent of
intellects.

I deal with the masses, and what you propose is impossible.

A book which I think has several shortcomings was quite
successful with the upper half of 5th graders. It is that,
my results with teaching logic to my children, and the
opinion of my late wife, who was strongly recognized as a
good teacher even by college students, which goes into this.

Quote:

logical sequences,

They understand rules of a simple game. This is what formal logical
sequences are.

Quote:

But the rules are not immutable. In fact, chaging hte rules is the rule, not
the exception. Watch kids payng a ganm, they cahnge the rules to meet
circumstance. Their worlds are flexible, fungible, variable.

SOME games. They recognize that the rules are immutable
until a specific change is made.

They have a choice of similar games, which is not the same
as changing the rules.

And some 8 year olds have done well at chess.

Quote:

They modify the rules to meet cricumstances or to intensify enjoyment.

How many kids do you know that don;t add house rules to Monopoly, don;t
argue over the rules for hide-'n-go-seek, amke up crd games rather than
play the staid old maid or war?

This did not happen in my day. Who is dumbing things down?

Quote:

Strict adherence to rules is an impediment to early childhood development,
not a goal. They are experimenting, experiencing, evaluating, learning.
Rigorous attention to rule shuts down this process.

They do not have the tools to evaluate. They do not change the
rules of arithmetic. They are able to understand that some
rules are unchangeable. The ones who can't believe in magic,
that the government can solve any problem by passing a law,
or become philosophers according to the following definition:

"A philosopher is someone who is looking for a black cat,
in a totally dark room, which isn't there, and FINDS IT."

This is also the attitude of the educationists. They have
philosophized about how people think, and have found the
black cat and curse anyone who crosses its path.

Quote:

Larry

Now this is not what inductive inference is. Inductive inference
should be done as statistical decision theory, which is simple to
state, but not at all easy to carry out. I will not go further into
this here.

cause and effect.

You are raising a full garbage can of worms here. Often,
to understand cause and effect, one needs to use precise
mathematics. This definitely applies to disease risk
factors, including a disease I have. My conclusions, from
reading the studies, do not agree with those of physicians,
who seem unable to distinguish between correlation and causation.

This effect was, AFAIK, first noticed by a biologist in 1919.
Once pointed out mathematically, it becomes obvious to one who
can think precisely. I wish our politicians could understand
this instead of their misunderstandings of cause and effect,
what can be done instead of what they want to legislate.

They have limited vocabularies and
limited abilities to integrate disparate knowedge points into a whole.

I do not see the abilities of adults who cannot handle precision
as that great.

They are kids, after all, and have not reached adult stages of
development.
Some will not reach this stage until their late teens.

Some have had their ability to achieve what children can
achieve knocked out of them by the educationists and schools.

Quote:

My son, at age 6, was a high school student in mathematics,
and at the college level in logic. Learning to think
precisely may even get more difficult with increasing age;
I would not want to try to teach most of today's teachers,
even high school mathematics teachers. My late wife had
much experience here, and it rarely made her feel good.

The original "new math" was tested on tens of thousands of
children; when taught by those who understood, it worked.
But the teachers could not learn it; they could not understand.
It is my opinion, based on decades of experience and discussion
with others, that teaching facts and methods before understanding
does not help with understanding, but those who understand can
use the facts and know what the methods are doing and WHY.

--
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

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Posted: Tue Jul 15, 2008 4:30 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

In article <bd0n74d303ut3j65o8drchpdq2nc4jrbt9@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:

Quote:

Pubkeybreaker <pubkeybreaker@aol.com> wrote:
On Jul 13, 7:57pm, Bob LeChevalier <loj...@lojban.org> wrote:
My legislature demands that I teach this, failure to do so will result in my
termination.

Herman doesn't think that legislatures should have the right to decide
curriculum - only subject matter academicians.

And he is correct. The idea that the legislature, who knows nothing
abouth math, chemistry, physics, literature, sociology, etc. should
decide course content is absurd.... The inmates are running the
asylum.

Democracy - the worst form of government - except for all the rest.

Thousands of laws people have spoken,
A handful the Creator sent.
The former are frequently broken,
The latter can't even be bent.

If our representatives legislate in contradiction to
the laws of nature, which laws will be observed?

Congress can legislate a bound of 3 dollars a gallon
for gasoline, but this will not change the price of
oil, and nobody will get gasoline. Congress can
legislate that everyone will get good health care,
and that will mean that some bureaucrat will decide
who gets what; the resources are not there. I have
been in various medical facilities now for the last
5 weeks, and these are considered good facilities,
but today I had to wait 15 minutes for a nurse to
respond; an assistant would have been adequate.
If everyone had the medical attention Ted Kennedy
had, assuming it could be done, it would cost
several times the GDP.

The inmates are running the asylum. They assume
that everyone can learn a good amount at the same
rate in homogeneous classes, and that the bright
and gifted can learn it "deeper". In mathematics,
this cannot be done except by concentrating on
trivia. The mathematical equivalent of writing
paragraphs is the formulation of word problems;
the economists I worked with, even those with
not too great a mathematical background, could
do this, even including calculus. It is more
important that the engineer formulates the
problem correctly than that he is able to solve
it in closed form.

Quote:

It is, because many have had their ability to use
their intelligence, or even their intelligence,
weakened by the (expletive deleted) schools.

Quote:

In any event, in other countries, the most intelligent don't have so
much creativity.

Many are converted to religion and other forms of
philosophy. Zen does not prove theorems.

Quote:

We have the 3rd largest population in the world. And the most of any
*developed* country. (China & India are getting there. I expect
China to overtake the U.S. in science and technology during this century).

They may have a larger economy, but we will likely still be the
innovators, because China does not reward innovation, and indeed often
punishes it.

It is not surprising that we would have more creative people than most
other countries.

China has more "most intelligent" people than we do, by a factor of 4,
unless you assume that their population has a significantly different
bell curve. So if creativity is just another word for high
intelligence, they would have more creative people. But they don't.

Many of their best come to the US and other western countries.

Japan, which used to be highly creative, has lost it.

Israel, a small country, is highly creative.

Quote:

Perhaps what we need to do (this is VERY politically incorrect)
is to separate those who can from those who can not very early on
in school.

The United States is not that sort of country. If that is what we
"need" to do, we will find a different way, or choose a different
goal, because to do as you suggest would fundamentally violate our
cultural identity.

Nonsense. It was the policy in the 19th century, and lasted
mainly to WWII.

Quote:

Provide training (in the sense under discussion) for those
who can't, and provide education (in the sense under discussion) for
those who can.

There were the three tier high school programs when I went.
College preparative, shop, and "general". Many who could
have taken the college preparative program did not because
they did not think they could get into college or afford it;
these were the GIs who raised standards after WWII in the
colleges.

This program was put in under democracy.

Quote:

Many who "can" don't *want*. And many of their parents don't "want",
either.

This can be a problem. What do you propose to do about
it? Is depriving those who can and want an answer?

Quote:

It is my understanding that at least some countries already
do this (e.g. Germany).

And Germany doesn't have nearly the innovation levels that we do. They
have some superb craftsmen, though, because they value that sort of
thing.

Because a large number of top Germans came to the US.
Also Chinese; there are five Zhang's in our department.

Before WWII, Germany was number 1 in science, with France
and England behind, and the US possibly ahead of those.
This with the US doctoral program starting in 1876.

Quote:

One might argue that this will lead to a 2-tiered
society, but I would argue that this is what we already have.

Not legally.

The widening income gap in this country is driven by (IMO) the gap in education
between the poor and the economically well-to-do. The poor are poor
precisely because they have below average intelligence and hence get "training" instead
of "education".

George Bush was economically well off. What's your explanation for
him?

What is the problem? I have seen his IQ estimated at 128.

Quote:

Those Asian immigrants who come over tend to arrive fairly poor. What
is your explanation for their success?

They believe in education, in learning more and better.

This is in contrast to many Americans, who at most believe
in grades.

Quote:

Clearly socioeconomics does provide benefits and handicaps. It does
in Germany and in that ultimate meritocracy Singapore. But the pride
of our system is our social mobility. Bill Clinton, whatever you
think of his politics, went from broken family poverty to the highest
office in the land. Obama likewise started rather low on the
socioeconomic totem pole.

I was not from a broken family, but from poverty. I was
taught to read very early, but taught little more; I
believe arithmetic. I did learn in the schools, but not
from the teachers. It was not until just before I got to
high school that I learned anything about matheamtics, and
it was the first principle of algebra; this is all I needed
to take off. How many years earlier could I have done that?

Before, I was mainly reading biography, history, and
geography; the science section available to grade school
students was of poor quality.

Quote:

It's a little harder to find such examples in the sciences, but one of
the 2006 Physics Nobelists was the son of a traveling salesman. Robert
Grubbs, one of the 2005 Chemistry Nobelist, grew up in rural Kentucky.

My father became unemployed when I was 3, and I was
12 when my parents progressed to opening a mom-and-pop
delicatessen, and later they sold it and opened a
grocery. My father must have been a genius, but had
no education. We need to search out and encourage these,
not bury them in a heterogeneous class.
--
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

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Posted: Tue Jul 15, 2008 5:03 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:

Quote:

The inmates are running the asylum. They assume
that everyone can learn a good amount at the same
rate in homogeneous classes, and that the bright
and gifted can learn it "deeper".

No. They DEFINE "a good amount" as what most kids manage to pick up
of the subject, REQUIRE that teachers attempt to teach that amount to
all students, and figure that WHATEVER the kids who are gifted can
learn above what is REQUIRED is more or less of equal importance, so
they don't much care if it is more or deeper or different, as long as
it doesn't cost too much.

Quote:

However, it is precisely the "most intelligent" who are responsible
for the creativity.

That is questionable.

It is, because many have had their ability to use
their intelligence, or even their intelligence,
weakened by the (expletive deleted) schools.

If that were really the case, then home-schooled kids, and those
educated at the few "academic private schools" would be showing up
their peers of comparable IQ who attend the public schools. I haven't
seen the evidence.

Quote:

It is not surprising that we would have more creative people than most
other countries.

China has more "most intelligent" people than we do, by a factor of 4,
unless you assume that their population has a significantly different
bell curve. So if creativity is just another word for high
intelligence, they would have more creative people. But they don't.

Many of their best come to the US and other western countries.

Even if you count all of those, they don't have more creative people.
And remember that China controls who gets to leave.

Quote:

Japan, which used to be highly creative, has lost it.

Japan was NEVER highly creative in technical fields. They built their
economy by copying western technology and producing it cheaply
efficiently.

Quote:

In the 19th century almost no one attended secondary education OR
college, regardless of their IQ.

It was money and location that determined access to education, not
intelligence.

Quote:

Many who "can" don't *want*. And many of their parents don't "want",
either.

This can be a problem. What do you propose to do about it?

Not much. Individual and parental rights are paramount in our
culture.

Quote:

Is depriving those who can and want an answer?

Those who can and want, in our culture can usually find a way. Maybe
not while they are kids. But society really doesn't place much of a
priority of meeting what kids want. Money still rules, and kids
generally don't have any.

Quote:

The widening income gap in this country is driven by (IMO) the gap in education
between the poor and the economically well-to-do. The poor are poor
precisely because they have below average intelligence and hence get "training" instead
of "education".

George Bush was economically well off. What's your explanation for
him?

What is the problem? I have seen his IQ estimated at 128.

And despite all his money, he has demonstrated that his education was
worthless. He comes across as neither well-educated NOR well-trained.

Quote:

Those Asian immigrants who come over tend to arrive fairly poor. What
is your explanation for their success?

They believe in education, in learning more and better.

Therefore it isn't about money, but about values.

American culture doesn't much value education. It values the ability
to make more money. Whichever, "education" or "training", advances
economic status, is what society values and is willing to pay for.
Those who want something else, can take what is given or pay for
something else on their own.

Quote:

This is in contrast to many Americans, who at most believe
in grades.

Because grades result in more money than "education".

Those who worship the free market have to take the good and bad that
comes with it - both of which center on measuring everything in terms
of economic value.

lojbab
Bob LeChevalier - artificial linguist; genealogist
lojbab@lojban.org Lojban language www.lojban.org

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Posted: Tue Jul 15, 2008 5:35 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz

hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:

Quote:

Nice in theiry, difficcult to imposssible in real life.

Easy in real life, if one concentrates on understanding,
and asks steps to be given. Do you remember your geometry
course, where you had to give proofs with statements and
reasons?

I remember the course, and I couldn't do a geometry proof today if I
wanted to (and I don't have any reason to want to).

Quote:

I put myself through college turoring amth to business students, and some
took two or three tries to pass. Not because of how they were taught, but
because oftheir attitude. Soem even were forced to delay graduatoin for one
omre summer schools ession because they kept blowing off the work.

Did you get these simple concepts across? If they ever use
their business courses, they will have to FORMULATE problems,

They won't. They'll make a lot of money and pay someone else a
pittance to do it.

Quote:

not solve routine stuff. They need to use algebra as a language,
not as a means to get numerical answers.

They won't use algebra at all.

Quote:

I have found that wandering, meandering processes are indiciative of a lack
of knowedge. OOne of the things I am required to assess is the ability to
accomplish a list of taks in a certain time frome.

A lack of knowledge of tricks. The practice should come
after the understanding; memorizing the multiplication
tables teaches nothing about multiplication.

It teaches them to get the answer quickly (and time is money).

Quote:

This precision in mathematics is also needed in ALL of the
sciences, and alas the public seems unable to understand that
the government cannot just legislate in violation of the laws
of nature, and achieve miracles.

This would severly restrict what can be defined as a "science".

I see no such restriction. "Social science" is a stupid
term selected by the educationists who did not think
students had learned geography if they needed to consult
works to find the geographical features involved in a
history course.

"Social science" is a term invented by academics, not by teachers, in