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Herman Rubin
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 Posted: Thu Jul 10, 2008 7:00 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
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In article <2b9a7bdc-1977-43de-b93b-7afb3a978758@m45g2000hsb.googlegroups.com>,
charles q <q.charles132@gmail.com> wrote:
.................
| Quote: |
Thats about the most rediculous thing that i have ever heard.I do not
see how calling algebra,algebra could posibly scare anybody at all.I
guess then calling english and history is scaring students also. LOL
|
English and history are "ordinary" English words, although
most of what is taught in English is not about the English
language but is "literature". These subjects also scare
students.
The subject of algebra started out (barely) with Diophantus,
but was somewhat developed by the Muslims in the Middle Ages.
There were two terms used: al-muqaballah (sp?), the process
of collecting terms, and al-jabr, obtaining the answer.
Algebra has now advanced beyond this, starting with European
developments in the 16th century, but the term remains. See
my other posting in this group for the ideas (but not the
details) expressed simply; it should be started very early.
The details and practice can come later, and are far more likely
to be understood, and as something not mysterious.
--
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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Beliavsky
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| Posted: Thu Jul 10, 2008 9:14 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
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On Jul 10, 1:32 pm, Pubkeybreaker <pubkeybrea...@aol.com> wrote:
| Quote: |
On Jul 10, 12:31 pm, Barbara <mom_2_...@hotmail.com> wrote:
On Jul 10, 9:41 am, hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
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.
My general suspicion is that the phenomenon isn't unique to 'algebra'.
I suspect that the students afraid of 'algebra' are also afraid of
'physics', 'chemistry', 'literature'... etc. They are afraid of
any subject that requires that they <gasp!> might have to study
in order to learn. Too many are simply intellectually LAZY.
|
Charles Murray has said that algebra *is* hard for people with IQ's <100, and I think he is right:
http://www.opinionjournal.com/extra/?id=110009535
What's Wrong With Vocational School?
Too many Americans are going to college.
by CHARLES MURRAY
Wednesday, January 17, 2007 12:01 a.m. EST
The topic yesterday was education and children in the lower half of
the intelligence distribution. Today I turn to the upper half, people
with IQs of 100 or higher. Today's simple truth is that far too many
of them are going to four-year colleges.
Begin with those barely into the top half, those with average
intelligence. To have an IQ of 100 means that a tough high-school
course pushes you about as far as your academic talents will take you.
If you are average in math ability, you may struggle with algebra and
probably fail a calculus course. If you are average in verbal skills,
you often misinterpret complex text and make errors in logic.
These are not devastating shortcomings. You are smart enough to engage
in any of hundreds of occupations. You can acquire more knowledge if
it is presented in a format commensurate with your intellectual
skills. But a genuine college education in the arts and sciences
begins where your skills leave off.
In engineering and most of the natural sciences, the demarcation
between high-school material and college-level material is brutally
obvious. If you cannot handle the math, you cannot pass the courses.
In the humanities and social sciences, the demarcation is fuzzier. It
is possible for someone with an IQ of 100 to sit in the lectures of
Economics 1, read the textbook, and write answers in an examination
book. But students who cannot follow complex arguments accurately are
not really learning economics. They are taking away a mishmash of half-
understood information and outright misunderstandings that probably
leave them under the illusion that they know something they do not. (A
depressing research literature documents one's inability to recognize
one's own incompetence.) Traditionally and properly understood, a four-
year college education teaches advanced analytic skills and
information at a level that exceeds the intellectual capacity of most
people. |
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Donna Metler
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| Posted: Thu Jul 10, 2008 10:55 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
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"Pubkeybreaker" <pubkeybreaker@aol.com> wrote in message
news:20e2a46d-92aa-40ba-a42a-3143bc0d936e@d45g2000hsc.googlegroups.com...
On Jul 10, 12:31 pm, Barbara <mom_2_...@hotmail.com> wrote:
| Quote: |
On Jul 10, 9:41 am, hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
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.
|
My general suspicion is that the phenomenon isn't unique to 'algebra'.
I suspect that the students afraid of 'algebra' are also afraid of
'physics', 'chemistry', 'literature'... etc. They are afraid of
any subject that requires that they <gasp!> might have to study
in order to learn. Too many are simply intellectually LAZY.
---
What I find humorous is that I was in a teaching store the other day, and
saw a series of books starting at 1st grade, labeled "Algebra" (which looked
to me like reasonable extention activities for a child who had a good grasp
on basic operations, but nothing that hasn't been taught in math classes at
that age level, at least to the higher groups, for decades). So, apparently
the word Algebra is considered too scary for high school kids to have on a
book, but not for a 6 yr old! |
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Banty
Guest
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| Posted: Fri Jul 11, 2008 1:36 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
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In article <ZZOdnahBhIvexOvVnZ2dnUVZ_oLinZ2d@comcast.com>, Donna Metler says...
| Quote: |
"Pubkeybreaker" <pubkeybreaker@aol.com> wrote in message
news:20e2a46d-92aa-40ba-a42a-3143bc0d936e@d45g2000hsc.googlegroups.com...
On Jul 10, 12:31 pm, Barbara <mom_2_...@hotmail.com> wrote:
On Jul 10, 9:41 am, hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
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.
My general suspicion is that the phenomenon isn't unique to 'algebra'.
I suspect that the students afraid of 'algebra' are also afraid of
'physics', 'chemistry', 'literature'... etc. They are afraid of
any subject that requires that they <gasp!> might have to study
in order to learn. Too many are simply intellectually LAZY.
---
What I find humorous is that I was in a teaching store the other day, and
saw a series of books starting at 1st grade, labeled "Algebra" (which looked
to me like reasonable extention activities for a child who had a good grasp
on basic operations, but nothing that hasn't been taught in math classes at
that age level, at least to the higher groups, for decades). So, apparently
the word Algebra is considered too scary for high school kids to have on a
book, but not for a 6 yr old!
|
This kinda reminds me of the town of Ossining, New York.
It's the site of Sing Sing prison. Ossining used to be called "Sing Sing",
hence the prison name, so Ossining changed their name to be more like the
original Indian name.
A decade or so ago, Sing Sing was renamed the "Ossining Correctional
Institution".
Heh.
This renaming thing is majorly dumb. Only a non-mathematical (i.e. - verbally
inclined) person woudl have thought of it.
Banty |
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Bob LeChevalier
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| Posted: Fri Jul 11, 2008 3:41 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
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Barbara <mom_2_one@hotmail.com> wrote:
| Quote: |
On Jul 10, 9:41 am, hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
In article <486f7172.10114...@news.datemas.de>,
Way Back Jack <Rela...@home.net> wrote:
COMMENT: Yeah, that's the ticket, call it "Yomomma," or "Slam Dunk."
That should create interest in the dumbed-down set.
___________
Would name change help algebra students?
Posted by Dave Murray | The Grand Rapids Press July 03, 2008 11:09AM
As Shakespeare wrote, a rose by any other name would smell as sweet.
But would Algebra 2 be as difficult if it were called something else?
State Sen. Wayne Kuipers, R-Holland, recently told a gathering of Kent
County school board members that he believes more students would pass
the upper-level math course if it were called something less scary.
His theory is that students have convinced themselves that algebra is
too difficult and that they throw in the towel before giving it a
chance.
Students have not convinced themselves of this; teachers
who do not understand algebra, and this includes most
algebra teachers and an even higher proportion of elementary
school teachers, think it is difficult.
Well, whenever you post this, I ask for some peer-reviewed or other
reliable studies supporting your claims that most math teachers in the
United States do not understand basic mathematical concepts such as
basic high school-level algebra (other than the fact that your
repeated statements that this is so).
|
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.
| 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.
|
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.
lojbab
Bob LeChevalier - artificial linguist; genealogist
lojbab@lojban.org Lojban language www.lojban.org |
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Herman Rubin
Guest
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| Posted: Sat Jul 12, 2008 4:52 am Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
|
|
In article <173d74hoh9ejhv26tari2uq56lv4hekj4r@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:
| Quote: |
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.
Often, the teacher grades on whether the problem is done as
indicated in the textbook recipe. 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.
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.
| 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.
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.
--
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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Larry Hewitt
Guest
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| Posted: Sat Jul 12, 2008 6:25 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:g58rn5$4gjs@odds.stat.purdue.edu...
| Quote: |
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:
..............
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 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.
|
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?.
| Quote: |
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
| Quote: |
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?
| 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".
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.
| 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.
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
| Quote: |
--
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: Sun Jul 13, 2008 7:26 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
|
|
In article <NfWdnUqt8J2EleXVRVn_vwA@comporium.net>,
Larry Hewitt <larryhewi@comporium.net> wrote:
| Quote: |
"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.
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: |
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: |
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: |
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: |
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: |
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.
| Quote: |
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
| Quote: |
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
This is mechanical, and has no mathematical content, nor
statistical content except descriptive.
| Quote: |
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.
Try reducing a system of equations over the integers to
row echelon form. Or more so, proving it can be done.
| Quote: |
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: |
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: |
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: |
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.
| Quote: |
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: |
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.
| 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: |
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.
| Quote: |
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
| Quote: |
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.
| Quote: |
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@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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Bob LeChevalier
Guest
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| Posted: Sun Jul 13, 2008 9:26 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
|
|
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:
| Quote: |
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.
|
That is what the state standards expect him to know. If you don't
teach it, you get fired.
| Quote: |
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.
|
The state test will ask the question, and you as the teacher will be
blamed if he cannot solve it in the required manner.
| Quote: |
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.
|
It is, when the state tests ask questions about trivia. Which they
do.
| 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.
STOP concentrating on memorization and routine. Minimize them.
|
Memorization and routine lead to automatization, which is required on
a timed test.
lojbab
Bob LeChevalier - artificial linguist; genealogist
lojbab@lojban.org Lojban language www.lojban.org |
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Herman Rubin
Guest
|
| Posted: Sun Jul 13, 2008 10:42 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:
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.
That is what the state standards expect him to know. If you don't
teach it, you get fired.
|
We are having considerable discussion on the effect of
this kind of state standard. We cannot have good education
if the educationists, who ONLY know memorization and routine,
make up the tests.
| Quote: |
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.
The state test will ask the question, and you as the teacher will be
blamed if he cannot solve it in the required manner.
|
I repeat what I have said above. There even are may
teachers who complain about the students not learning
what is important because of teaching to the test.
This continues in college, and the colleges are not
willing to back up their professors who would teach
the more important parts.
| Quote: |
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.
It is, when the state tests ask questions about trivia. Which they
do.
|
How many times have I said that the only way the public
schools can be improved is to have affordable private schools,
however it is done?
One can destroy the quality of a school quickly. It is a major
problem to even improve it, let alone restore it. Until we
have the attitude that a school should educate each student
as if he or she were the only student, and there were no such
things as being in certain grades, we can only have the present
bad turnout from the high schools, the universities, and even
the graduate schools.
| 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.
STOP concentrating on memorization and routine. Minimize them.
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. This destroys the ability to be
educated later. Education gives the ability to use the
concepts, and reconstruct the methodology when needed.
--
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
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| Posted: Sun Jul 13, 2008 11:42 pm Post subject: Re: Don't Call It "Algebra"; Call It Something Warm And Fuzz |
|
|
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:
| Quote: |
In article <l7ak74pmuogf13ieglu6v1gksmvb92lrgc@4ax.com>,
Bob LeChevalier <lojbab@lojban.org> wrote:
hrubin@odds.stat.purdue.edu (Herman Rubin) wrote:
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.
That is what the state standards expect him to know. If you don't
teach it, you get fired.
We are having considerable discussion on the effect of
this kind of state standard. We cannot have good education
if the educationists, who ONLY know memorization and routine,
make up the tests.
|
The public wants the tests, and furthermore, they want kids to learn
the mathematics (i.e. algorithmic arithmetic) that they learned as
kids. And they seem likely to vote out of office anyone who tells
them "no".
| Quote: |
The state test will ask the question, and you as the teacher will be
blamed if he cannot solve it in the required manner.
I repeat what I have said above. There even are may
teachers who complain about the students not learning
what is important because of teaching to the test.
|
Yep, and yet the tests aren't going away. This should tell you
something. We can hope that maybe NCLB goes away, which will reduce
SOME of the pressure of teaching to the test, but the tests themselves
are almost certainly here to stay, and indeed are likely to expand to
cover all grades and not just the ones under the federal mandate.
| Quote: |
This continues in college, and the colleges are not
willing to back up their professors who would teach
the more important parts.
|
Which also should teach you something about social reality. He who
pays the piper calls the tune.
| Quote: |
It is, when the state tests ask questions about trivia. Which they
do.
How many times have I said that the only way the public
schools can be improved is to have affordable private schools,
however it is done?
|
It can't be done, without making them into public schools.
| Quote: |
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.
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.
lojbab
Bob LeChevalier - artificial linguist; genealogist
lojbab@lojban.org Lojban language www.lojban.org |
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Larry Hewitt
Guest
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| Posted: Mon Jul 14, 2008 2:19 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:g5d3av$8bf2@odds.stat.purdue.edu...
| Quote: |
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:
..............
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 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.
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.
|
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?
| Quote: |
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.
|
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.
| Quote: |
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.
|
This is far more difficualt than you assume, given the many reading
comprehension problems of many 8th graders.
| Quote: |
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.
|
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.
| Quote: |
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
|
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.
|
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.
| Quote: |
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.
|
Algebra I does not have problems like this.
Algebra I is simplfy
3e + 5t - 6y = 8y -2e
| Quote: |
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.
|
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?
..
| Quote: |
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.
| Quote: |
one way to set up a box and whiskers statistical
chart.
This is mechanical, and has no mathematical content, nor
statistical content except descriptive.
|
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.
| Quote: |
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.
|
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.
| Quote: |
Try reducing a system of equations over the integers to
row echelon form. Or more so, proving it can be done.
|
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.
| Quote: |
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.
|
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.
| Quote: |
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.
|
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.
| Quote: |
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.
|
That is what I taught.
Despite your experience with a gen | | |
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