In article <NfWdnUqt8J2EleXVRVn_vwA@[EMAIL PROTECTED]
>,
Larry Hewitt <larryhewi@[EMAIL PROTECTED]
> wrote:
>"Herman Rubin" <hrubin@[EMAIL PROTECTED]
> wrote in message
>news:g58rn5$4gjs@[EMAIL PROTECTED]
>> In article <173d74hoh9ejhv26tari2uq56lv4hekj4r@[EMAIL PROTECTED]
>,
>> Bob LeChevalier <lojbab@[EMAIL PROTECTED]
> wrote:
>>>Barbara <mom_2_one@[EMAIL PROTECTED]
> wrote:
>>>>On Jul 10, 9:41am, hru...@[EMAIL PROTECTED]
(Herman Rubin) wrote:
>>>>> In article <486f7172.10114...@[EMAIL PROTECTED]
>,
>>>>> Way Back Jack <Rela...@[EMAIL PROTECTED]
> wrote:
..............
>>>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".
>>>>> The following includes essentially all of algebra, except
>>>>> for technical terms not used at the high school level:
>>>>> A variable is a tem****ary 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.
>Nice in theiry, difficcult to imposssible in real life.
>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.
> 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.
>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.
>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.
>> 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
>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.
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.
>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.
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.
one way to set up a box and whiskers statistical
>chart.
This is mechanical, and has no mathematical content, nor
statistical content except descriptive.
>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 im****tant, but memorization of them no.
Try reducing a system of equations over the integers to
row echelon form. Or more so, proving it can be done.
>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 im****tant.
>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.
>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.
> 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.
>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.
>> 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".
>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.
>>>>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 im****tant part should be taught as soon as the student
can read and produce symbols.
>>>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.
>> 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.
They are incapable of understanding symbolic representation,
This is utter baloney.
>logical sequences,
They understand rules of a simple game. This is what formal logical
sequences are.
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.
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@[EMAIL PROTECTED]
Phone: (765)494-6054 FAX: (765)494-0558
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