"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.
And should sloppiness be punished?
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?
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?.
> 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.
In algebra I there is truly little mathematically correct variation from
the
"book recipe".
There is, for example, only one way to write a linear equation in
slope-intercept form, onwe way to solve a system of linear equations using
hte elimination method, one way to set up a box and whiskers statistcal
chart.
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.
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?
Did he not knwo the correct form? Did he start out right but lose his
way,
either taking a wrong path or end toosoon?
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.
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?
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?
> 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.
>>>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.
>
>>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. They are incapable of understanding symbolic
representation,
logical sequences, cause and effect. They have limited vocabularies adn
limited abilities to integrate disparate knowedge points into a whole.
They are kids, after all, and have not reached adult stages of
development.
Some will not reach this stage until their late teens.
Larry
> --
> 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
|
