"Herman Rubin" <hrubin@[EMAIL PROTECTED]
> wrote in message
news:g5d3av$8bf2@[EMAIL PROTECTED]
> 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.
>
This is relatively easy to do at the college level, but in 8th grade
forget
it
Many problens, for ex., require only a few steps, too few to really
develop
a cluie to te student's thinking.
Not to mention the difficulty in getting the kids to separate out
relatively
smple operations, like a two step addition, into separate steps.
Yes, partial credit can be given, and often is.
But then we have to address what is really being taught.
How do you handle an error that develops but all of the stpes were not
written down? How do you handle a correct answer wtthout the steps? Do you
fail a student who got all ofthe correct answeres but failed to show her
work (usually the more intelleginet students)? Or do you enforce the
requirement arbitrarily?
>> And should sloppiness be punished?
>
> Not heavily. But someone is not going to be a good scientist,
> and I include the biological and psychological and economic
> sciences, if there is sloppiness.
>
First, very few of my students wee going to be scientists. A girl with her
heart set on being a beautician or a boy who want to be a mason souldn't
care less.
And how much is a little?
10%? 5%?
Bear in mind that, for some inexplicable reason, my state legislature has
decided that 7 points separate grades, no 10%. so a refusal to follow
directions, or even a charitable misunderstannd if instrsuction, can mean
a
letter grade difference.
Also beasr in mind that in my state, and others, there is a minimum grade
for 8th grade students to get academic credit for algebra. That is, (over
teh strenuous objections of teachers) 8th graders must score a minimum 85%
to get credit for completing algebra I (they get credit for graduating 8th
grade, but must retake algebra in 9th grade for an 84% or less).
Politics. Blah.
>>How does a teacher determine that an incorrectly set up equation in a
word
>>problem is the result of another transcription error, a reading
>>comprehension problem, or a misunderstanding of the underlying math?
>
> This is not as likely to be difficult as you think.
>
This is far more difficualt than you assume, given the many reading
comprehension problems of many 8th graders.
>>And then how does a teacher justfy what is no more than a subjective
guess
>>to angry parents and administrtors, explaining why Joey got credit and
>>Zooey
>>didn't?.
>
> The same holds for English composition.
>
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 im****tant.
After all, in public school we are teaching what our legislature has told
us
to teach, the rules and structure of our subjects.
>>> Often, the teacher grades on whether the problem is done as
>>> indicated in the textbook recipe.
>
>>Because this is what has been taught, and this is what a student is
>>expected
>>to knwo.
>
> And this is NOT what should be taught. Understand what methods
> can be applied, and apply whichever
>
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 *****s 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?
>>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.
> 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
>>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
*****sed, anot to write the equatoin in simplist form, a common error in
algebra I
So do you reward the rote math f simplifying despite the fact that the
student did not answer the question asked?
..
> one way to solve a system of linear equations using
>>hte elimination method,
>
> Where did you get that idea? If there are n equations,
> there are usually n! ways of doing this.
>
I meant the technique is fixed. The order of opertion _may_ be commutable,
but the technique is fixed, and it isthe technique that is being *****sed.
And again, os a process that takes 12 steps as good as one that tkes 4?
Not according to our standards--- efficiency is an im****tant *****sement
of
ability.
> one way to set up a box and whiskers statistical
>>chart.
>
> This is mechanical, and has no mathematical content, nor
> statistical content except descriptive.
>
Wrong.
There are calculations to determine quartiles, and an *****sment of
understanding of statistical concepts.
This chart is an im****tant foundation for further study of statistics. In
fact, my first college stat class mentioned it as a quick and easy way to
demonstrate skewed data.
>>Yes, there are other ways to "solve" the problem or display the info,
but
>>these specific algorithsm are what are being yested and knowlege of them
>>is
>>needed in future courses.
>
> Are they? In practice, solving systems of equations is
> done by computer. Understanding of the algorithms can
> be im****tant, 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 im****tant. If you haven;t learned
them
in algebra, you are at a disadvantage.
> Try reducing a system of equations over the integers to
> row echelon form. Or more so, proving it can be done.
>
A subject not taught until college linear algebra.
I has 2 weeks to teach my entire course content on linear algebra,
starting
with the definition of a linear equation. Matrices had not yet been taught
and matrix operations used to get the matrix in row echelon form were
what
was taught.
>>So how would you grade a student who uses outstanfing toechnique to
>>rpesent
>>linear eq. in point-slope form when the question alled for the
>>slope-intercept form?
>
>>Did he just not follow instructions, and shouldn;t that be punished?
>
> I would be unlikely to ask the question. I am not even sure
> that I would give such, except as how to normalize the equation
> of a line for certain purposes, and leave it at that. Memorizing
> trivia is not that im****tant.
>
My legislature demands that I teach this, failure to do so will result in
my
termination.
And, quite frankly, it is an im****tant consept used in later courses. You
pooh pooh memorization, but retention is required and testing that
retention
is the name of the game.
>>Did he not knwo the correct form? Did he start out right but lose his
>>way,
>>either taking a wrong path or end toosoon?
>
> Look at the above. It is a matter of normalization of the
> equation of a line and nothing more. The rule of equality
> covers this quite well.
>
Wrong.
It is a question of writing the equation in a format such that the student
can, by examination alone, determine certain characteristcs of the line.
It
also prepares the equation for further evaluation or calculation, such as
graphing, a subject we teach.
>>Further complicating the decision is a certaintity that just becaue he
>>could
>>do the problem correctly ont he board yesterday does not mean he could
do
>>it
>>today.
>
> STOP concentrating on memorization and routine. Minimize them.
>
That is what I taught.
Despite your experience with a genius child and genius college students,
the
kids I teach need to concentrate on th algorithm, on the methods. I _try_
to
impart a little underlying theory, but snores ripple through the room in
moments, and I really do not ahve time to stray far.
Understand , computation is a challenge for some of the students. My first
week of algebra i class was a review of calcualtions with negative
numbers.
I agree with you, to a point. Math needs to be taken more seriously, is
far
more im****tant than the way it is treated.
But also understand that eeven most college graduates do not take more
than
1 or 2 math courses, one of them calculus for the humanities student. A
"D"
is all they need to graduate, adn they aim for the "D".
I put myself through college turoring amth to business students, and some
took two or three tries to pass. Not because of how they were taught, but
because oftheir attitude. Soem even were forced to delay graduatoin for
one
omre summer schools ession because they kept blowing off the work.
I deal with what I am given.
And when, as I said, I am dealt a student body who's)and their parent's)
life expectations for math is balancing a check book, I deal with it. I
cannot force a student to learn what he does not want to learn.
>> 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.
>
Exactly.
SO how do you *****s what you cannot determine?
>>So, for example, is a process that took 12 steps to combine like terms
in
>>an
>>equation as "correct", as good an indicator of knowledge, as one that
took
>>4
>>steps?
>
> I do not expect a student to find a short method, especially on
> a test. I would rather a student figure out a method from basic
> principles, no matter how clumsy, than memorize a trick.
>
I do.
I have found that wandering, meandering processes are indiciative of a
lack
of knowedge. OOne of the things I am required to *****s is the ability to
accomplish a list of taks in a certain time frome.
College even requires this ---- you have a fixed time period to compelte
te
*****sment, no more.
>>> 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.
>
I am not tlking randomness, O am referong to your reference of precision.
The subjects I listed are inherently incapable of achieving the level of
precision required in math, physics, etc.
>>>>>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.
>
How does one *****s that for for the 112 students I typically taugh?
Does not hte *****sment of development become a major distraction?
Tell me how I can get a taxophobic electorate to pay for the *****sments.
Tell me how I can convince parents who think that what I am teaching is a
bunch of unecessary hooey that their child will never need to push the
legislature to implement this testing.
>>>>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.
>
Wrong.
Generalization requires biological advancement.
> They are incapable of understanding symbolic representation,
>
> This is utter baloney.
>
Nope. It is current understanding of hiuman development.
I do not intend to be offensive to anyone, but you _really_ need to get
dwon
into the trensches. You seem to be dealiing with the top few percent of
intellects.
I deal with the m*****, and what you propose is impossible.
>>logical sequences,
>
> They understand rules of a simple game. This is what formal logical
> sequences are.
>
But the rules are not immutable. In fact, chaging hte rules is the rule,
not
the exception. Watch kids payng a ganm, they cahnge the rules to meet
cir***stance. Their worlds are flexible, fungible, variable.
They modify the rules to meet cri***stances or to intensify enjoyment.
How many kids do you know that don;t add house rules to Monopoly, don;t
argue over the rules for hide-'n-go-seek, amke up crd games rather than
play the staid old maid or war?
Strict adherence to rules is an impediment to early childhood development,
not a goal. They are experimenting, experiencing, evaluating, learning.
Rigorous attention to rule shuts down this process.
Larry
> Now this is not what inductive inference is. Inductive inference
> should be done as statistical decision theory, which is simple to
> state, but not at all easy to carry out. I will not go further into
> this here.
>
> cause and effect.
>
> You are raising a full garbage can of worms here. Often,
> to understand cause and effect, one needs to use precise
> mathematics. This definitely applies to disease risk
> factors, including a disease I have. My conclusions, from
> reading the studies, do not agree with those of physicians,
> who seem unable to distinguish between correlation and causation.
>
> This effect was, AFAIK, first noticed by a biologist in 1919.
> Once pointed out mathematically, it becomes obvious to one who
> can think precisely. I wish our politicians could understand
> this instead of their misunderstandings of cause and effect,
> what can be done instead of what they want to legislate.
>
> They have limited vocabularies and
>>limited abilities to integrate disparate knowedge points into a whole.
>
> I do not see the abilities of adults who cannot handle precision
> as that great.
>
>>They are kids, after all, and have not reached adult stages of
>>development.
>>Some will not reach this stage until their late teens.
>
> 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
|
