Herman Rubin wrote:

In article <1132511689.460296.250450@g49g2000cwa.googlegroups.com>,
<> The rule needed to solve most of the problems is:
<> Equals can be substituted for equals,
<> without changing the result.
<I saw that in a fairly recent algebra book, and it
<does explain why you perform the manipulations.
<When I learned algebra in the 70's, however, I was
<not taught it, and I'm not sure that I needed it for
<elementary problems. Understanding the concept
<of variable is necessary, though, in order to apply
<algebra to real life. Also, word problems and factoring
<require more immediate understanding, I think.
You might be able to figure out how to do elementary
problems mentally without using algebraic methods.

Actually, I was just doing the algebra in a very
mechanical way without questioning why it was so, e.g.
3x+12 = 36
3x = 36-12 = 24
x = 24/3 = 8
In the more recent textbook I mentioned, they would
give an example as such:
3x+12 = 36
3x+12-12 = 36-12
3x = 24
3x/3 = 24/3
x = 8
using the equality rule you mentioned. And indeed that
does give the whole explanation as to why the method
works, but I think it confuses some too, who might prefer
to just do things the mechanical way until they are more
mature mathematically and can understand the whole
picture.

Also, and this might be a different issue, but I was always
more comfortable if a math teacher would give one or
several simple examples rather than just describing
some theorem or idea. It would help in understanding
the more abstract idea.

Algebra does not help with computations, although it
can be, and has been, used to find better ways of
computing. But it is a way to reduce problems so
that they then become just carrying out computations.

SNIP
. I would hope that a math teacher teaching

<arithmetic would be fairly good at it.
Suppose you were teaching people to do base 60 arithmetic.
Would you necessarily have to have any particular competence
with it? Not at all, provided those you were teaching knew
what it meant. The current teachers do not know what
addition and multiplication mean, just how to do it.

I personally would have a very hard time lecturing about
something unless I understood what I was talking about.
I think you are suggesting that some teachers just teach
a mechanical method without giving the reasons behind it,
as I was demonstrating above, and you don't like that.

So, how could algebra be taught so as to communicate
more than just the mechanical method, but without
confusing those who are not ready for more?

<> The teachers are mainly trained, or rather one
<> should say mistrained, in algebra. They know
<> how to do certain types of problems, but not
<> the fundamentals, and students so brought up
<> have major problems in understanding. I have
<> been in the position of having them in my
<> classes, and many of them have hit a stone wall
<> because they can only proceed by memorizing.
<I agree that the teachers should understand those
<concepts extremely well, and be able to communicate
<them. The question is, perhaps, whether the students
<who will likely become math and engineering majors
<should be in the same class as those who might
<become art majors or janitors, and whether everyone
<needs to learn algebra well. (And what about other
<categories, such as social science majors--they
<need some of it.)
There are many aspects to algebra. They may or may not
need to learn group theory or other similar aspects, but
they do need to learn how to "speak" algebra so they can
communicate precisely. It is POSSIBLE to do this
(sometimes) without knowing algebraic language, but
usually clumsy.
A rather large number of social science majors use
statistics, or other data processing methods.
Understanding what these do requires knowing algebra.
These problems even arise in the humanities, although
not as often.

I'd be interested to know--what are some of those problems in the
humanities? The only thing I have run across is analysis of texts
to determine authorship. Is there anyone in the humanities who
does a good job with statistics?

<If a problem comes up for a cashier, for example, is
<that person really using algebra, or is it more just a
<need to be analytical and logical and thorough (and
<of course polite), using arithmetical skills in order to
<make sure the accounts are in order.
You can TRAIN people to do things without understanding.
But cashiers are not trained to use arithmetic, but
rather some tricks.

Yes, that is true.
The problem given above with the tickets and prices I
would have thought of as an arithmetic (division) problem
rather than as an algebra problem. Perhaps knowing
algebra makes it easier if you're not comfortable
with the concept of division. (That sounds very
negative on my part, I think.)
4590/25.50 versus 25.50x = 4590

C.

In article <DZWdnUyOpOyrKO7enZ2dnUVZ_tSdnZ2d@comcast.com>,
Gary Schnabl <forwarded@LivernoisYards.com> wrote:

"Herman Rubin" <hrubin@odds.stat.purdue.edu> wrote in message
news:dkjql4$4h4u@odds.stat.purdue.edu...

In article <ankpm1l3hqhptqjpaltir1n8taht7ialak@4ax.com>,
sf <sfpipeline_at_gmail.com> wrote:

On 4 Nov 2005 16:29:33 -0500, Herman Rubin wrote:

What should be required for graduation is the ability
to formulate word problems of arbitrary complexity,
and to be able to interpret answers, however obtained.
The above problems have only one variable, and even
starting with this is a mistake, as it makes it harder
to understand the concepts.

Since when do you need a great depth of mathematical understanding to
operate a cash register, pump gas or push a broom? We want high
school graduates to demonstrate a rudimentary ability to add/subtract
and read. Independent thinking is usually a negative, not a plus for
high school graduates... nor is it an on the job requirement of most
college students. Most of us are just cogs that keep the wheel
moving.

If you do not understand mathematical notation, you will
not be able to make a decision involving taking into
account facts in a non-trivial manner. You will not
be able to decide which action to take, or make an
intelligent decision to advise someone else when, as is
almost always the case, quantitative information must
be used. You will definitely NOT be able to use statistics
intelligently, or to communicate your problem to one who
can help you solve it.

I submit to you the formulated and paced subject matters of elementary
school language arts and math with lesson plans that the principals
require to be copied straight out of those books into our own "lesson
plan" book. Take a look at them and then tell me where any
independent thinking or creativity is hidden. I'm sure the publisher
didn't intend that it should be followed so formulaically! It's a
good tool for new teachers and I like the fact that a child can
transfer from school to school w/o losing any speed in subject matter,
but the way these programs are being implemented is absolutely
criminal.

The person who cannot do independent thinking is the
person who is swayed by contentless rhetoric. Those
who impose these strategies in the schools are total
enemies of education, and do an excellent job in
reducing the capabilities of those who can think.a

I doubt if 5% of the high school teachers of mathematics
understand mathematics, or that the teachers of language
understand languager as a means of rational communication.

"sf" <see_reply_address@nospam.com> wrote in message
news:m7brm1dtdp9bf70iln4h9emegmc5426j83@4ax.com...
as a means of rational communication.

Nor does it matter. The only thing that matters is if teachers *and
their students* can follow directions (make that orders). There is no
thinking required to teach anymore, nor is thinking required of high
school graduates and simpletons with a simple college diplioma.

So what's stopping you from teaching simutaneously at two or more different
levels? The Amish do this well in both English and German with high levels
of academics compared to public and private schools today. A typical block
of time for a particular task in today's classes might be, say, 45 minutes.
The bright kids could get their work done in 15 minutes or less. The
brightest might do it in 5 minutes or so. Many teachers I knew would tell
the brighter kids to read or play games on the computers in the classroom.
One 6th grade teacher lets her kids do coloring books and such when they
finished early, for example. She once told me, "My girls **love** to color."
What a waste!

I agree that it is a total waste. The bright children get 1/3
of their education or less. Or do they even get that? They
can handle material at a higher level, with even more efficient
progress.

During my three years at teaching (before moving on to better things), I
could muli-task. Either by teaching split 4th and 5th grades simultaneously
in the same classroom or by making accelerated learning available for the
roughly 20 to 35% of the brighter kids in my classes.

How can you make accelerated learning available in mathematics?
Only by giving material "normally" covered in later classes.
The same holds in science, history, geography, language, music,
art, and many others. Those bright 4th graders possibly be
doing "honors" high school or college work. The idea of keeping
children with their age group for educational purposes should be
considered a major felony; slowing the progress of a student
should require that those responsible pay for the lack of future
earnings, and also provide private tutelage to try to undo the
damage resulting from the mind-deadening "schooling".

You seem to dislike "mandates." But what constitutes "orders" to you? If a
mandate is an expected minimum performance level for academics, I would
welcome that. That minimum level then would be analygous to "building
codes."

Building codes are nothing more than a minimum mandated level of
construction performance. No sensible home owners would want their houses
built just to code. That could entail a house being sloppily built with
barely acceptable materials. So why can't you see to it that the poorer
performers in your classes are "built to code" and educate the rest to one
or more higher academic levels, based upon their competence?

From my limited experience, I would tend to agree with Herman in most
academic areas. I don't know if only 5% of American teachers are competent
in math (or science), but it's possibly darn close. I would say from what I
read from those authors or teachers who disapprove of today's teaching
methods that that figure is surely less than 20%.

From what I've seen, the older teachers are far more competent in math than
the newer crowd. I doubt that the older teachers are doing things much
differently than when they first started teaching, except for the
dumbing-down of curricula. Axiomatic geometry or calculus are rarely taught
today. However, the axiomatic methods were the norm, even mandated, when I
was in HS or college from 1957 to the mid 1960s.

This was the case in the high schools even before WWII.
While those old books have subtle errors and missing
axioms, this had very little adverse effect on the teaching
and understanding, as what was assumed was not hidden.

But when the original "new math" was introduced, the
elementary teachers then could not handle what had been
tested on tens of thousands of school children. Even in
these school children went to better schools, one would
expect the teachers to be able to manage the subject. But
alas, attempts to teach them had extremely poor success, as
did attempts to teach the high school teachers the basic
abstract undergraduate courses.

The lower salaries for IT or engineers is not the sole reason for the
rampant off-shoring of IT or engineering jobs since the 1990s. Just look at
the Americans who enter or complete college in those fields of study. Some
of them take 4 semesters of remedial courses (meaning HS level) upon
entering college. I attended the University of Wisconsin studying electrical
engineering back when it was almost 100% Americans. Today, Americans are
probably no higher than 25% or so in engineering with about the same
percentage of foreign profs. Could this be the expected result of having no
mandates during the past three or four decades of academic abuse?

A lot of this is due to the lack of teaching concepts in
courses. We have admitted many graduates of American
universities, who have what are claimed to be the basic
abstract undergraduate course on their records, but the
pressure to pass the bodies who showed up in the classes
had weakened those courses. Adding to this is the student
evaluation of teachers; whether the students are ready for
what the course should be is not taken into account.

How can you give a good undergraduate probability course if
the students do not know what a limit is, and consider an
integral to be only evaluating an antiderivative between
the endpoints? But if the teacher of the calculus course
emphasized the students understanding what limits,
derivatives, and integrals are, not just how to calculate,
the pressure from other departments whose students could
not cope, and even from the mathematics department
administrators worried about keeping their faculty if the
enrollment dropped, would get those off the people teaching
calculus.