The Tyranny of Mathematics
Onar Aam
enormous position in education. Mathematics is in many ways a game
for intellectuals, even so it has managed to become one of the most
important subjects in school. I want to explore why this is so.
To understand this we have to look at the historical development of
the education system. The basic structure of the system was shaped
about 150 years ago in response to an ever increasing demand for
infrastructure. At the time Adam smith's division of labour
principle was the main philososphy of business. All organization
took place in the spirit of this philosophy, including the education
system. The system was constructed by dividing education into many
separate "boxes". The idea was that when these boxes where
assembled, they made up a holistic education. School was divided
into primary school, secondary school etc. Each of these separate
boxes were divided into new boxes, grades. And each grade was
further divided into more boxes: classes. Each class was divided
into boxes in time and space. Each class lasted for 1 hour, and
consisted of X pupils, and the X pupils were divided into a
seemingly natural entity, the individual. Similarly, the curriculum
was divided into subjects which were spread across the classes. By
adding all these fragments together one created Education, one
proclaimed. Today such a division is called the Principle of
Linearity. And as we all know, linearity is a highly cherished thing
in mathematics. The extremely structured and linear nature of
mathematics is probably the reason that this discipline has got its
position in education. Because it is so linear by nature it is easy
to fit into the education system. This is where the tyranny starts.
Because the structure of the education system and the structure of
mathematics are isomorphic by nature it gives mathematics an immense
power. It is so rigidly attached to the system that very little can
shake its position. Mathematics exploits this by forcing its own
culture upon the students. While the mathematical culture is a must
for mathematicians, it is a horror to students. Few things in school
are as unpedagogic as mathematics. In fact, pedagogy is completely
overthrown to bathe the students in its own social structure.
Mathematics _could_ have been taught pedagogically, but it wouldn't
be "right" according to the mathematical culture. So instead of
easing the burdon of learning for students, which they should,
mathematicians choose the easy path. If things are not done
_their_ way then it is _wrong_. This chauvenism has been allowed to
live for ages in the education system, and I say it is time to break
down this tyranny. The culture of mathematics is a game and should
be treated like it. This culture deserves no more place in school
than religion.
PS: I love mathematics, hate its culture
Onar.
Manley Perkel, Dept of Math & Stat, Wright State U, Dayton, OH 45435, (513)-873-2276
> It is absolutely amazing how mathematics has managed to build up its
> enormous position in education. Mathematics is in many ways a game
> for intellectuals, even so it has managed to become one of the most
> important subjects in school. I want to explore why this is so.
>
> To understand this we have to look at the historical development of
> the education system. The basic structure of the system was shaped
> about 150 years ago in response to an ever increasing demand for
> infrastructure
+ lots of other lines deleted
I have to reject your premises about how mathematics attained its role
in education.
I am sure you have done a lot of reading on this and have thought about it
a lot so I am sure it ocurred to you that MAYBE the education system
has evolved ( and was not created in some sort of compartmentalized
fashion as you seem to infer).
Just MAYBE, would you not agree, mathematics attained its position in
education as a response to the demands of society as commerce and technology
began to get more and more sophisticated over the last 100 or so years.
Of course, if you reject the need for all this commerce and technology,
then we have nothing to discuss.
But if you see a need for it then the role of mathematics follows quite
naturally in the scheme of education. After all, most of the mathematics
taught in higher education, is in the form of what are called "service"
courses to other disciplines. Maybe the experts in these other disciplines
need to be asked why they want their students to have a mathematical
background. This may answer your question.
As a prediction (nothing like a prediction to get everyone hot under the
collar) I would guess that as computers and associated technologies inject
themselves more and more into our lives, we will see a day when related
computer science (or whatever it will come to be called) will also be
an enormous part of our education system and that computer-based service
courses will be required of nearly all disciplines. To some extent
this has already begun and in a few dozen years you will be asking the
same question, but "mathematics" will be replaced in your question by
the words "computer studies".
Again, education EVOLVES as a response to societal needs and demands.
Wait and see!!
Dimitrios Diamantaras
[long discussion of education deleted]
: for mathematicians, it is a horror to students. Few things in school
: are as unpedagogic as mathematics. In fact, pedagogy is completely
: overthrown to bathe the students in its own social structure.
: Mathematics _could_ have been taught pedagogically, but it wouldn't
: be "right" according to the mathematical culture. So instead of
: easing the burdon of learning for students, which they should,
: mathematicians choose the easy path. If things are not done
: _their_ way then it is _wrong_. This chauvenism has been allowed to
: live for ages in the education system, and I say it is time to break
: down this tyranny. The culture of mathematics is a game and should
: be treated like it. This culture deserves no more place in school
: than religion.
: PS: I love mathematics, hate its culture
OK, write your Elements and we'll come back to see if you improved on
Euclid's in, say, 2000 years -:)
More seriously, who says that math should be easy to learn? If there is no
royal road to geometry, then there is no other easy road to it either (and
King Ptolemy had to accept such a statement; to his credit he did not
behead the messenger for this message). Everybody CAN learn math, but they
HAVE to make the effort. Enough already of this stuff about making
students feel good; it's gone too far and now produces uneducated adults
who cannot use their minds to find out what's plaguing them (and so couch
potato(e)s, addicts of alcohol and so on, and people feeling stuck at dead
ends in their jobs). We have no better friend than our own mental ability,
which we must be forced to exercise when young, for our own later
(immense) benefit.
: Onar.
--
Dimitrios Diamantaras
Department of Economics
Temple University
********************************************************************
* This .sig says: avoid self-referential statements. *
********************************************************************
Donald (Don) Specht
I don't agree with your word choice of "tyranny," however I
fully support a reform movement in mathematics - the NCTM
Curriculum, Evaluations, and Teaching Standards and the effort
by many localities to embrace these guidelines and restructure
math ed.
Recently I posted an essay that I think goes to the heart of
changing the "culture." Below is a portion of this essay. I
solicit your comments.
------------------------------------------------------------------------
Pillars and Gateways
With the demise of the "new math" movement came an emphasis in
a return to the "basics." The emphasis was on mastery of
skills, predominantly arithmetic symbol manipulation, and those
skills were ordered sequentially. Thus each skill was a
gateway that the student had to unlock.
Arithmetic, algebra, statistics, and geometry are
fundamental *pillars* upon which the students build
mathematical literacy. *Each* year of their schooling must
include direct instruction and experiences pertaining to *all*
of these pillars. Mix with this a classroom pedagogy that
emphasizes problem solving as the genesis of instruction, many
forms of communication with and about mathematics, reasoning
skills to think logically and explain results and conclusions,
and constant attempts to connect topics within and outside of
mathematics. Add a large pinch of technology; calculators
should *always* be available, with graphing utilities and
computers available when needed. And don't forget the
manipulatives; beginning instruction in the concrete world
firmly grounds the student as he sets sail for the land of the
abstract. The resulting mathematically rich environment will
go a long way toward guaranteeing the success of all students.
I believe that Mathematical Literacy must take a position along
side Whole Language on the restructuring agendas of every
school system whose goal is continuous improvement. We teach
children, not courses. Vertical pillars must replace
horizontal gateways.
-----------------------------------------------------------------------
In conclusion, I believe that my sig file reflects the view of
many k12 math educators. I look forward to continuing this
discussion.
---
Don Specht <dsp...@mwc.vak12ed.edu>
Clarke County High School
"Math is a language used to explain,
interpret, and predict events
in the real world."
David Smith
: Mathematics exploits this by forcing its own
: culture upon the students. While the mathematical culture is a must
: for mathematicians, it is a horror to students. Few things in school
: are as unpedagogic as mathematics. In fact, pedagogy is completely
: overthrown to bathe the students in its own social structure.
: Mathematics _could_ have been taught pedagogically, but it wouldn't
: be "right" according to the mathematical culture. So instead of
: easing the burdon of learning for students, which they should,
: mathematicians choose the easy path. If things are not done
: _their_ way then it is _wrong_. This chauvenism has been allowed to
: live for ages in the education system, and I say it is time to break
: down this tyranny. The culture of mathematics is a game and should
: be treated like it. This culture deserves no more place in school
: than religion.
On the contrary, I think that the so-called "burdon [sic] of learning" that
you talk about doesn't exist. Due to TV, movies, and the media, people have
come to expect that they will do poorly in mathematics. (How many movies
have you seen with a lovable goof--also a noted math retard--as its main
character?) Women are discouraged from entering the field--not because they
can't learn it, but because it's not "ladylike" (although parents and
educators now use a different PC word, I'm sure). This is a load of B.S.
The reason that most of the human population are math idiots is because they
weren't taught enough mathematics in school. High school students are
expected to know almost zero math by the time they graduate. Elementary
schoolers know even less, and think even less. As a result, we live in a
society where most people can't balance a checkbook and continually spend
money that they don't have (credit cards--the average American owes *1800
DOLLARS* on his or her credit cards), and look up to people who can't even
add and subtract (presidents, congressmen), let alone multiply or divide.
The "culture of mathematics" is not a game; it's one of the few aspects of
the universe that we can ever hope to get our hands on. Not only should it
not be taken out of school, it should be taught there with more rigor, with
greater preciseness, and a lot more good old-fashioned ENCOURAGEMENT.
(This happens to bring up a lot of issues on teacher education, but that, I
think, belongs in another article.)
From: David Smith : "Don't let your life be the butt of a joke
Temple University : Wrap your lips round a cool black Pepsi Coke"
dsm...@astro.ocis :
.temple.edu : --Jesus and Mary Chain
john baez
>[...] Mathematics exploits this by forcing its own
>culture upon the students. While the mathematical culture is a must
>for mathematicians, it is a horror to students. Few things in school
>are as unpedagogic as mathematics. In fact, pedagogy is completely
>overthrown to bathe the students in its own social structure.
>Mathematics _could_ have been taught pedagogically, but it wouldn't
>be "right" according to the mathematical culture. So instead of
>easing the burdon of learning for students, which they should,
>mathematicians choose the easy path. If things are not done
>_their_ way then it is _wrong_. This chauvenism has been allowed to
>live for ages in the education system, and I say it is time to break
>down this tyranny. The culture of mathematics is a game and should
>be treated like it. This culture deserves no more place in school
>than religion.
>PS: I love mathematics, hate its culture
You never explain exactly what this culture of mathematics is.
I think you are really referring to the culture of mathematics
EDUCATION. Mathematics education all too often says "There is one right
way to do the problem, and the teacher knows it. Don't question the
definitions or what one is seeking to achieve with them, just learn how
to manipulate things according to god-given syntactic rules.
Mathematics can be studied independent of all other disciplines. Subjects
must be learnt in a specific sequence. Etc." Actually, this is the
most true at the earliest stages of mathematics education; by the time
one gets to advanced undergraduate course or graduate courses -- IF one
gets to them, not having been repulsed -- one sees that these rules
are a sham.
Namely: Mathematicians spend much of their time trying to come up
with new ways to do the same old problems; there is never just one way.
Only by questioning and improving the definitions can mathematics become
clearer, and one can only do this by keeping clearly in mind the (many
conflicting) things one is seeking to achieve. Mathematics frequently
gets new and interesting ideas from outside mathematics, and some of the
best ideas come when people deliberately break the rules. Mathematics
is a complicated weave that cannot be presented in any linear sequence.
All good mathematicians know these things!! Mathematical culture
rewards those who recognize these things and exploit them.
Unfortunately, for some mysterious reason, mathematics is never taught
this way at the lower levels. I think your explanation for *why* is a
good one but I would reverse it. Mathematicians are relatively
powerless in society compared to politicians, lawyers, businessmen etc.;
it's the latter folks, who like society to be well-organized and
disciplined, who have set up the education process to be what it is, and
fund it. (Earlier, it was the Catholic church who started the European
university system.) Mathematics has been tyrannized by such folks -- of
course, usually quite willingly (since we all like to be paid to
do mathematics, and who will pay if one doesn't play along?) -- and
mathematics is sadly distorted in the process. These days, the NSF will
pay you to do pure mathematics if you can explain ahead of time what
theorems you are going to prove. Hah! Of course anyone with any savvy
figures out how to get around this: first you do your work, *then* you
apply for a grant saying that you will do it. However, this kind of
charade is not harmless. It fosters the illusion that the world is a
place that can be run according to plan, that we pretty much know what
the world is all about now and can proceed calmly in our dull little
lives from here on out, etc..
I don't know what to do about this problem, but understanding it clearly
is probably a good start -- and I'm not claiming I do. The point is of
course not to assign blame here or there so much as to figure things out.
Herman Rubin
>In article <1994Feb9.0...@hsr.no>, on...@hsr.no (Onar Aam) writes:
>> It is absolutely amazing how mathematics has managed to build up its
>> enormous position in education. Mathematics is in many ways a game
>> for intellectuals, even so it has managed to become one of the most
>> important subjects in school. I want to explore why this is so.
We do not have to look far to see why this SHOULD be the case. Mathematical
communication is, in precise situations, what ordinary verbal communication
is in highly imprecise situations. It is almost in the position occupied
by reading and writing.
What I mean by mathematics here is formulating problems and understanding
formal arguments. Manipulations, like arithmetic and the computation of
derivatives and integrals, are doing mechanics, and while they can be
useful, have nothing basic in them.
......................
--
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
Phone: (317)494-6054
hru...@snap.stat.purdue.edu (Internet, bitnet)
{purdue,pur-ee}!snap.stat!hrubin(UUCP)
Neil Rickert
>enormous position in education. Mathematics is in many ways a game
>for intellectuals, even so it has managed to become one of the most
>important subjects in school. I want to explore why this is so.
I would say it differently. Namely
Mathematics is one of the most important, practical and useful
subjects in school. It is absolutely amazing how educators have
managed to distort it in such a way that many students now see
it as no more than a game for intellectuals.
Donald (Don) Specht
>
> We do not have to look far to see why this SHOULD be the case. Mathematical
> communication is, in precise situations, what ordinary verbal communication
> is in highly imprecise situations. It is almost in the position occupied
> by reading and writing.
>
Mr. Rubin and I are on the same wavelength. He often posts
about concepts preceding facts as the proper format for
instruction in k12 math ed. (Am I right, Herman?) I believe
math is a language that works with the communicator's native
tongue, giving that person a greater ability to converse.
Problem solving, symbol manipulation ... all of those things we
have come accustomed to perceive as school math are only mental
gymnastics to enhance our ability to communicate
quantitatively, as well as qualitatively.
(BTW Herman, you are right on target in insisting that the
place to begin improvement of math ed. is in the early grades.
Many elementary teachers assume a formative role in a child's
math ed. with only a minimal understanding of anything beyond
arithmetic symbol manipulation. But I am not without hope.
There are many good programs already online in our schools,
such as _Math Their Way_, and VA's K-8 math lead teacher
program.)
John Novak
>All good mathematicians know these things!! Mathematical culture
>rewards those who recognize these things and exploit them.
>Unfortunately, for some mysterious reason, mathematics is never taught
>this way at the lower levels.
I think at least part of the problem is that, quite simply, if a
student comes up with some novel (for him or her) approach to
doing a particular problem, many grade school mathematics
teachers are not knowledgable enough to figure out whether the
trick works in all situations, or just a few special cases.
As a (rather trivial and amusing) case in point, lets suppose
we're doing fractional reductions, like reducing 12/48 to the
more familiar 1/4. Suppose further that the unfortunate teacher
puts up the problem of 16/64 on the blackboard, and some witty
young child immediately sees that, as 16/64 = 1/4, then
_obviously_ we're just cancelling sixes.
Does 13/39 then equal 1/9?
Of course not. But can the typical grade school teacher explain
why it works once in a while, and not in others?
Certainly not all of them. And no one likes to be asked a
question they can't answer, so the reaction is to label the child
a trouble-maker.
I am constantly amazed that, in many curricula, it takes
significantly less math to get a math teacher's certificate than
it does to become a math major.
Its certainly not the _whole_ problem, though. I have, in fact,
had one or two decent math teachers before I got to college-level
work, so they're not _all_ bad. And of course, there are the
perennial problems about the kids simply not seeing the relevance
of mathematics to their later lives, and even the _parents_ not
seeing the relevance.
"Ah, Jimmy's football is way more important than that stupid old
math stuff. Jimmy's gonna be a football star in college, and if
not, he can always get a factory job like I did."
(Yeah. Tell me how many factory jobs will be left in twenty
years.)
--
John S. Novak, III
j...@cegt201.bradley.edu
j...@camelot.bradley.edu
Filipp Anthony Sapienza
>In article <1994Feb9.0...@hsr.no> on...@hsr.no (Onar Aam) writes:
>>[...] Mathematics exploits this by forcing its own
>>culture upon the students. While the mathematical culture is a must
>You never explain exactly what this culture of mathematics is.
>I think you are really referring to the culture of mathematics
>EDUCATION. Mathematics education all too often says "There is one right
>way to do the problem, and the teacher knows it. Don't question the
>one gets to advanced undergraduate course or graduate courses -- IF one
>gets to them, not having been repulsed -- one sees that these rules
>are a sham.
I can think of a couple of instances throughout my schooling which
in part verify this. I remember once having to do long division
in grade school, and the teacher wanted us to do it his way.
Some students were doing division a little differently, including
myself, and though we got the right answer, were marked wrong.
(We were not using calculators or anything)
Another time I remember doing an algebra problem and coming up
with the answer x/y. The problem involved variables of r and s,
so I should have put r/s. However, I lost full credit for this
mistake (and others like it) not because of my ability to
discern the proper relationships between the variables, but
because I accidentally substituted x/y for r/s.
Godfrey Degamo
: More seriously, who says that math should be easy to learn? If there is no
: royal road to geometry, then there is no other easy road to it either (and
: King Ptolemy had to accept such a statement; to his credit he did not
: behead the messenger for this message). Everybody CAN learn math, but they
: HAVE to make the effort. Enough already of this stuff about making
: students feel good; it's gone too far and now produces uneducated adults
okay.
: who cannot use their minds to find out what's plaguing them (and so couch
: potato(e)s, addicts of alcohol and so on, and people feeling stuck at dead
: ends in their jobs). We have no better friend than our own mental ability,
: which we must be forced to exercise when young, for our own later
: (immense) benefit.
You must be joking about these statements?
I know some math students who are drunkards and "party animals"
It's been my experience, that if a depression strikes a math student it
makes him a "better" mathematician. (They start taking the questions of
"Existence" to a new gut level.) At least with a few that I've talked to.
(They also give up the crazy notion that they will have fun in life.)
As far as mental ability is concerned, I don't have the exact citing of
the source where I have read it, but I can get it for you, it states that
success at academics does not correlate with success in the work force.
Those successful in acedemics were found to be worse off at work, and more
prone to depression, and basically don't like life.
I read this as, the successful student, after completing his degree, and
having such high aspirations, the work force becomes so anti-climactic as
if there was "something big" waiting at the end of the "rainbow":
"This is it? So this is what I've worked my butt off and sacrificed every
darn weekend to study for? This? A 9-5 job?"
but, you are right, Reasoning is a great friend. Though, at times, it is
not applied in areas where it could be. Because a person is a couch
potato does not mean he doesn't work or use his brain. I know a mathematician
who tried to take his life. How do you know he's just not burned out?
Maybe if reasoning was applied to our social lives, we might be better off?
Who knows?
Furthermore, hard work in mathematics? I do not like to use that
phrase. Any approach with the "Just Do It" attitude in mathematics,
to me, is bound to fail. I've seen plenty of students non-math and
the like, who spend several hours trying to understand a mathematical
concept. The hard work is certainly there. But after hours of trying
and trying, (the "Just Do It" attitude toward mathematics) they give
up and become discouraged and end up hating mathematics. They wasted
time which could have been spent on something else. It is usually the
case that they just "fixated" on the wrong details. or the "why
didn't I get that before?", "how could I be so stupid" phenomena. As
Descartes points out on the very first page of *Discourse*, that
people have an equal amount of common sense, but just tend to focus on
different details.
(Well, to save my butt, I'm not an expert, and this is speaking from experience
and could be distorted.)
: : Onar.
Herman Rubin
>Onar,
>I don't agree with your word choice of "tyranny," however I
>fully support a reform movement in mathematics - the NCTM
>Curriculum, Evaluations, and Teaching Standards and the effort
>by many localities to embrace these guidelines and restructure
>math ed.
>Recently I posted an essay that I think goes to the heart of
>changing the "culture." Below is a portion of this essay. I
>solicit your comments.
>------------------------------------------------------------------------
>Pillars and Gateways
>With the demise of the "new math" movement came an emphasis in
>a return to the "basics." The emphasis was on mastery of
>skills, predominantly arithmetic symbol manipulation, and those
>skills were ordered sequentially. Thus each skill was a
>gateway that the student had to unlock.
The demise of the "new math" was SOLELY because the teachers could
not understand a mathematical structure which they had not been
indoctrinated in, and see that it provided the basis for arithmetic.
I doubt if anyone who is not PhD material ever got a concept from
the meaningless manipulations of arithmetic. Some mathematicians,
and I am one of them, are good at arithmetic. Other quite good
mathematicians should do everything with a calculator.
>Arithmetic, algebra, statistics, and geometry are
>fundamental *pillars* upon which the students build
>mathematical literacy. *Each* year of their schooling must
>include direct instruction and experiences pertaining to *all*
>of these pillars.
You really want to make it difficult for them to understand. You
give them an alphabet soup; eating alphabet soup does little to
increase literacy. Learning to do arithmetic operations without
PREVIOUSLY knowing what the operations mean is at best neutral as
a means of teaching, and at worst leading to real confusion. There
is nothing to be gained by not starting with algebraic notation in
its full form, and not restricted to numbers, in first grade, and
then this becomes familiar. The structure of the non-negative,
or if you prefer positive, integers is the structure of counting;
it is understandable by first graders but apparently not by college
students or schoolteachers. All of the principles of arithmetic
follow from these logically, and this logic can be understood by
first graders. Children with this approach will be able to see
the utility of learning procedures which make it possible to compute
much faster than by counting, and will understand what they are doing.
I am a statistician. The biggest problem in dealing with clients is
that people like you have mistaught statistics. Again, learning how
to compute answers gets in the way of asking questions. You can teach
how to compute, but very definitely not why. Without building up a
rather large framework, including probability modeling, abstract
integration, and the idea of risk, I could not teach why.
What do you mean by geometry? Do you mean the relatively unimportant
mass of geometric facts? Or do you mean the quite important ideas of
logical argument, which involves formal logic, and the quite simple
ideas of theorem and formal proof? I believe this should be started
in second or third grade, but I do not know of materials available
before fifth grade. However, the ideas of formal argument can be
introduced to help in the understanding of arithmetic structure.
Mix with this a classroom pedagogy that
>emphasizes problem solving as the genesis of instruction, many
>forms of communication with and about mathematics, reasoning
>skills to think logically and explain results and conclusions,
>and constant attempts to connect topics within and outside of
>mathematics.
All of this comes easily if done logically. Much of problem solving
consists of merely formulating a problem as a mathematical problem,
and then applying previously learned techniques, using the properties
of equality. This goes all the way up; at the higher levels there is
the art form of deciding which techniques to use, and when.
The same applies to all reasoning situations. Mathematics is mainly
universal grammar; one models the other situation, and then applies
mathematics. Modeling reduces a problem to a mathematical problem;
all that one knows of mathematics can then be applied to the model;
the results then translate back to the external problem. The formulation
is the use of algebraic notation, as is the translation of the solution.
The reasoning capabilities are a combination of pure structure and the
artistic ability to reason efficiently; this last can be evoked but by
no means taught.
` Add a large pinch of technology; calculators
>should *always* be available, with graphing utilities and
>computers available when needed. And don't forget the
>manipulatives; beginning instruction in the concrete world
>firmly grounds the student as he sets sail for the land of the
>abstract. The resulting mathematically rich environment will
>go a long way toward guaranteeing the success of all students.
The land of the abstract is the easy part. Abstraction is not
grounded in "concrete". It is the other way; and abstract idea
can be quite easily learned if this has not been knocked out of
the student. Young children learn abstract structures far more
easily than high school students, and if they wait until college,
it is a real struggle.
>I believe that Mathematical Literacy must take a position along
>side Whole Language on the restructuring agendas of every
>school system whose goal is continuous improvement. We teach
>children, not courses. Vertical pillars must replace
>horizontal gateways.
The "whole language" approach is doing for understanding of the
structure of language essentially what has been done to destroy
mathematical understanding. The idea that a language is a
vocabulary on top of a grammatical structure is deliberately
blocked. It IS possible to learn the grammatical structure
with little of the vocabulary, possibly even with none. It
speeds up learning the vocabulary greatly, as derived forms
are learned with the roots.
Benjamin J. Tilly
> Onar,
>
> I don't agree with your word choice of "tyranny," however I
> fully support a reform movement in mathematics - the NCTM
> Curriculum, Evaluations, and Teaching Standards and the effort
> by many localities to embrace these guidelines and restructure
> math ed.
>
Which raises the obvious question. What do you, or others, think of the
article in the Forum section of the Notices of the AMS?
> Recently I posted an essay that I think goes to the heart of
> changing the "culture." Below is a portion of this essay. I
> solicit your comments.
>
> ------------------------------------------------------------------------
>
> Pillars and Gateways
>
>
> With the demise of the "new math" movement came an emphasis in
> a return to the "basics." The emphasis was on mastery of
> skills, predominantly arithmetic symbol manipulation, and those
> skills were ordered sequentially. Thus each skill was a
> gateway that the student had to unlock.
>
I agree that you need to build on a foundation. I personally did not
enjoy drills, and I was bad at them as a kid. OTOH I am saddened when I
am dealing with math students at Dartmouth that did not learn their
long division well enough to be able to divide one polynomial by
another when they are trying to factor something...
> Arithmetic, algebra, statistics, and geometry are
> fundamental *pillars* upon which the students build
> mathematical literacy. *Each* year of their schooling must
> include direct instruction and experiences pertaining to *all*
> of these pillars. Mix with this a classroom pedagogy that
Say what? Personally I think that it is important to *focus* on
something to learn it. Therefore there may be a clear point to not
trying to simeultaneously cover every topic every single year.
> emphasizes problem solving as the genesis of instruction, many
> forms of communication with and about mathematics, reasoning
> skills to think logically and explain results and conclusions,
> and constant attempts to connect topics within and outside of
> mathematics.
Not a bad idea that, but there are problems with it. The main one is
that I am personally convinced that a majority of teachers do not
really understand the math that they are supposed to be teaching. This
is based on personal experience, conversations with people who have
been or are teachers, conversations with others who have been students,
knowledge of how future teachers tend to do in math and science courses
in university, and a number of threads on the net. The fact is that if
the teachers do not really understand math then they will *really* mess
up when it comes to teaching it. The more creative they get, the worse
it will be. And if you ask them to teach topics, such as probability
theory, that they are not that familiar with, then they will *really*
be lost. This is a bad situation and, IMO, the reason why the "new
math" movement was such a disaster.
I believe that the coming changes will be another disaster.
Add a large pinch of technology; calculators
> should *always* be available, with graphing utilities and
> computers available when needed. And don't forget the
> manipulatives; beginning instruction in the concrete world
> firmly grounds the student as he sets sail for the land of the
^^
> abstract. The resulting mathematically rich environment will
> go a long way toward guaranteeing the success of all students.
>
Gee. PC, but not PC enough to have learned how to be gender neutral in
writing. Secondly I believe that technology is only a tool. It is easy
to go overboard with it, but let us stay grounded in reality. Computers
are useful, but they have limitations. Secondly there is a point (as in
long division above) to having the students actually learn the
mechanics. And that means boring practice.
> I believe that Mathematical Literacy must take a position along
> side Whole Language on the restructuring agendas of every
> school system whose goal is continuous improvement. We teach
> children, not courses. Vertical pillars must replace
> horizontal gateways.
>
Nice images, no substance. Let me mention that my mother was a teacher
her whole life and covered most grade levels from K-12, as well as some
special education programs and a stint as a principal. I have a sister
who is a school teacher. My brother teaches English in Taiwan. I am a
grad student in math who is interested in teaching. I have talked to a
number of people who are or have been teachers. The consensus that I
have heard is that the whole language system is a disaster. The problem
is that there is a drive to be on the "cutting" edge in teaching. This
leads to programs such as "whole language" becoming accepted and wide
spread before they have really been tested. In fact with no evidence
that they do better than other methods. (In fact it was explained to me
how the whole language method actually *creates* problems suchg as
dyslexia.)
In fact our students have one of the worst education systems in the
industrialized world in every comparison that I have ever heard about.
I believe that given this we would be well advised to change our
attitudes and focus on those who have managed to do well. Such as Jaime
Escalante (the teacher in Stand and Deliver). His opinion on the new
teaching standards is that it looks like it was written by a PE
teacher. When people like him who have demonstrated their ability to
teach, and who know their subject, have doubts, then the rest of us
should think about it.
Ben Tilly
Neil Rickert
>In article <1994Feb09....@mwc.vak12ed.edu>
>dsp...@mwc.vak12ed.edu (Donald (Don) Specht) writes:
>
>> Onar,
>>
>> I don't agree with your word choice of "tyranny," however I
>> fully support a reform movement in mathematics - the NCTM
>> Curriculum, Evaluations, and Teaching Standards and the effort
>> by many localities to embrace these guidelines and restructure
>> math ed.
>>
>Which raises the obvious question. What do you, or others, think of the
>article in the Forum section of the Notices of the AMS?
Since you didn't give a full reference, I will. The article (actually
more like a letter to the editor) was entitled "The Coming Disaster in
Science Education in America." It was by John Saxon. Page 103 of AMS
Notices, Feb 1994.
The article was quite scary. Unfortunately I don't know enough of the
NCTM plan to evaluate its accuracy. I was amused by Saxon's statement:
Only in American mathematics education do people with a track
record of abject failure arrogate the title of "expert".
I am inclined to agree with Saxon's assessment of past experience.
This morning, while driving to work, I heard a discussion of teaching
on a local public radio station. In this case they were discussing
the teaching of history, rather than mathematics. Apparently the latest
idea is that history text books should be abandoned. Instead the students
do projects of conducting opinion surveys, going to libraries to examine
original documents, etc. To make sure the students are well motivated
they get to choose their own projects from a selection. The radio
discussion explained what a marvellous method this was. Being the cynic,
I couldn't help thinking what marvelous euphomisms they presented. But
it seemed to me they were talking about dumbing down the curriculum.
Now that johnny can't read well enough to cope with text books we solve
the problem by eliminating the text books.
>> I believe that Mathematical Literacy must take a position along
>> side Whole Language on the restructuring agendas of every
>> school system whose goal is continuous improvement. We teach
>> children, not courses. Vertical pillars must replace
>> horizontal gateways.
As part of a posting supporting the NCTM plan, this is the type of
thing that tends to make be a supporter of Saxon's critique. As
Ben Tilly said:
>Nice images, no substance.
America, once a super power, voluntarily takes the path toward becoming
a third world nation. And all because we are too polically correct to
set standards.
Donald W. Fausett
>_their_ way then it is _wrong_.
Can you provide an example to illustrate your point here?
+ Don Fausett <dfau...@zach.fit.edu> + ___ __ ____ +
| Department of Applied Mathematics | /_ / / |
| Florida Institute of Technology | / _/_ / |
+ ************************************** + ************************** +
Piano Man
: royal road to geometry, then there is no other easy road to it either (and
: King Ptolemy had to accept such a statement; to his credit he did not
: behead the messenger for this message). Everybody CAN learn math, but they
: HAVE to make the effort.
I'll say. And for what? Since the age of about eight, I've known that I
wanted to be a writer "when I grew up". Math, beyond the basic operations
required to budget money, is entirely useless to me.
I have been composing music since I was four, and don't let anyone tell
you that music is related to mathematics in any way. I can do music. I
can't do math.
Same goes for computers. Sure, the underlying systems are chock full of
math, but how many people can program something like WordPerfect?
: We have no better friend than our own mental ability,
: which we must be forced to exercise when young, for our own later
: (immense) benefit.
I agree. But who says this ability should have anything to do with numbers?
I have always detested mathematics because there is no room for creativity
in it. You do what the symbols say. You get what the numbers tell you. And
it's right or it's wrong--there are damned few gray areas. Life isn't like
that. Tell your friend who is going through an emotional crisis what is
the "right" way to handle it and you'll like as not get nowhere.
--
Ken Breadner Wilfrid Laurier University, ##############
brea...@mach1.wlu.ca Waterloo, Ontario, Canada-for-now ##############
(the BREADbox) --------------------------------- ##############
------------------------Scratch here to reveal your prize---> ##############
John Novak
>In article <1994Feb09....@mwc.vak12ed.edu>
>dsp...@mwc.vak12ed.edu (Donald (Don) Specht) writes:
> Add a large pinch of technology; calculators
>> should *always* be available, with graphing utilities and
>> computers available when needed. And don't forget the
>> manipulatives; beginning instruction in the concrete world
>> firmly grounds the student as he sets sail for the land of the
^^
>> abstract. The resulting mathematically rich environment will
>> go a long way toward guaranteeing the success of all students.
>Gee. PC, but not PC enough to have learned how to be gender neutral in
>writing. Secondly I believe that technology is only a tool. It is easy
>to go overboard with it, but let us stay grounded in reality. Computers
>are useful, but they have limitations. Secondly there is a point (as in
>long division above) to having the students actually learn the
>mechanics. And that means boring practice.
I agree entirely.
I cannot see the point in showing students how to do division on
a calculator before they learn to do division on scratch paper.
Human beings are notoriously lazy creatures-- show someone the
'easy' way to do something, and they will have no motivation to
do learn to do things the difficult way.
Aside from the basic fact that showing someone how to use a
machine to do division goes a long way toward damaging their
idea of long division. Teaching children the ideas of graphing
and the notions of a function using first a computer strikes me as
s recipe for ensuring that no student ever bothers to _think_
about what's going on.
And the nightmare of generalizing from rectilinear to polar
graphs or (God forbid) three dimensional rectilinear to spherical
to cylindrical coordinates (to name a few imporant in electrical
engineering) gives me cold sweats.
I'm a big, big fan of computers and calculators speeding up
practical applications, the same way I'm a fan of using a
dictionary or a spellchecker while writing an important paper.
But in the same way as I would never allow children to use a
dictionary during a spelling test, I am loathe to allow
calculators or computers to illustrate the basics of mathematics,
unless you're _real_ careful about it. Computers don't typically
illustrate the fundamentals, they disguise them, hide them, and
make them invisible.
>Nice images, no substance. Let me mention that my mother was a teacher
>her whole life and covered most grade levels from K-12, as well as some
>special education programs and a stint as a principal. I have a sister
>who is a school teacher. My brother teaches English in Taiwan. I am a
>grad student in math who is interested in teaching. I have talked to a
>number of people who are or have been teachers. The consensus that I
>have heard is that the whole language system is a disaster. The problem
>is that there is a drive to be on the "cutting" edge in teaching. This
>leads to programs such as "whole language" becoming accepted and wide
>spread before they have really been tested. In fact with no evidence
>that they do better than other methods. (In fact it was explained to me
>how the whole language method actually *creates* problems suchg as
>dyslexia.)
Just for grins, could someone explain to me just exactly what the
"Whole language" idea of teaching is? I've heard Herman Rubin
railing against it, but I couldn't even tell you if I was taught
in that manner or not.
john baez
>I have been composing music since I was four, and don't let anyone tell
>you that music is related to mathematics in any way. I can do music. I
>can't do math.
They *are* related, but more mathematicians seem to be musically
talented than vice versa.
David B. Chorlian
impression I got from teaching from the standard textbooks was that
everyone, both student and teacher, was immensely bored by them. I
love mathematics and I love communicating interesting ideas, but
the mathematics I found interesting was not to be found in the
textbooks. I tried, with varying degrees of success, to put it into
the classroom. To make an analogy with the teaching of English, it
is as if we spent all our time on spelling and diagramming sentences,
and only occasionally read or wrote an essay or short story or poem.
We need the poetry of mathematics in the classroom!
David B. Chorlian
Neurodynamics Lab SUNY/HSCB
chor...@sp1p.neurodyn.hscbklyn.edu
dav...@panix.com
What is now proved was once only imagined. Wm. Blake
--
David B. Chorlian
Neurodynamics Lab SUNY/HSCB
chor...@sp1p.neurodyn.hscbklyn.edu
dav...@panix.com
a...@laphroig.mch.sni.de
>I'll say. And for what? Since the age of about eight, I've known that I
>wanted to be a writer "when I grew up". Math, beyond the basic operations
>required to budget money, is entirely useless to me.
You seem to be utterly confused about the difference between mathematics
and doing calculations. Of course, not everybody can become a
methematician, but you should be able to appreciate the beauty of mathematics.
>I have been composing music since I was four, and don't let anyone tell
>you that music is related to mathematics in any way. I can do music. I
>can't do math.
Sure, it is related! See you compose music, not everyone can do that.
But everyone should be able to enjoy good music. So it is with every art
form and so it is with mathematics, which just happens to be an art that
also has useful applications in other areas.
>But who says this ability should have anything to do with numbers?
Let me tell you once again that math is a lot more than numbers. Just
think of geometry.
>I have always detested mathematics because there is no room for creativity
>in it. You do what the symbols say. You get what the numbers tell you. And
>it's right or it's wrong--there are damned few gray areas. Life isn't like
>that. Tell your friend who is going through an emotional crisis what is
>the "right" way to handle it and you'll like as not get nowhere.
For me in mathematics is there is a lot of room for creativity; in fact
it is the sole area where I can be creative, because I lack the skill
for painting or composing. Though I enjoy looking at a good painting or
listening to good music.
Andreas "I love number theory" Eder
Onar Aam
>In article <1994Feb9.0...@hsr.no> on...@hsr.no (Onar Aam) writes:
>> ... If things are not done
>>_their_ way then it is _wrong_.
>
>Can you provide an example to illustrate your point here?
That should be the least of problems. School is very concerned with testing. To
get good scores on tests is (supposedly) a motivation for learning. But students
don't learn for the sake of learning, they learn in order to do well on tests.
Getting the answer right isn't what is always of importance, it is _how_ you get
them. This motivates recipee-learning rather than understanding. Instead of
striving towards understanding, the students are forced to strive towards the
_norms_. Understanding does in no way assure you good grades, following the norms
does. In other words, getting good grades is about adaption to the norms, getting
good grades is about complying with the culture. True, this is not typical for
mathematics. In fact, the entire education system is built up on such enforcement.
Onar.
Onar Aam
>Teaching children the ideas of graphing
>and the notions of a function using first a computer strikes me as
>s recipe for ensuring that no student ever bothers to _think_
>about what's going on.
Congratulations! You have actually plunged right into the heart of the
mathematical tyranny, the cultural belief that thinking MUST be done in particular
ways, the idea that automatation of thought is inferior to REAL thinking. Or
rather what mathematics VALUES as real thinking. But reality is quite another.
Everyone who has studied the processes of thinking knows that automatition is an
absolutely necessary basis. Without it, no thinking can occur. There is very
little difference between any kind of automatition, whether it occurs in your
brains or on the computer screen. But of some mysterious reason, mathematicians
believe that mental automatition is somehow superior to external automatition.
Can you please tell me why? I believe I know the answer: tradition. Mathematics
was founded in a time when computers where not available. Therefore it has not
become a valued tool in the math culture.
>I cannot see the point in showing students how to do division on
>a calculator before they learn to do division on scratch paper.
Mathematics _was_ however founded when pen and paper was around. Therefore these
tools are accepted in the mathematical culture. You even stated so yourself. My
question to you is: why on earth do you accept pen and paper? If you are so
afraid of the students ability to think, shouldn't they learn to do long division
in their heads? This would surely make everyone better thinkers according to your
logic.
>show someone the
>'easy' way to do something, and they will have no motivation to
>do learn to do things the difficult way.
Exactly. This is why most students don't do arithmetics in their heads, they do
it on paper.
>But in the same way as I would never allow children to use a
>dictionary during a spelling test, I am loathe to allow
>calculators or computers to illustrate the basics of mathematics.
What a lousy example! First, you are justifying the mathematical culture by
referring to an equally tyrannic culture. Who on earth said that spelling tests
are the right way to do it? Second, when _learning_ to spell you are allowed to
use as many dictionaries you like. Why shouldn't the same apply to mathematics?
What on earth does testing have to do with learning?
>Computers don't typically
>illustrate the fundamentals, they disguise them, hide them, and
^^^^^^^^^^^^
>make them invisible.
What fundamentals? The ones _dictated_ by the mathematical culture??? Are you
suggesting that there are only one way of understanding mathematics?
The cultural chauvenism of mathematics permeates your mind.
Onar.
Onar Aam
Likewise, the math loving students who are tyrannized by the mathematical culture
often becomes tyrants themselves. The only way to break this vicious cycle is to
break down the entire mathematical culture. This is not easily done. This culture
is a powerful discourse. Its members become completely addicted to it, believing
that it is the right way to do things. (Note that this has nothing to do with
mathematics itself, rather its culture.) And in the course of time this culture
has in education evolved into pure chauvenism because its power position has
remained unquestioned. Therefore I strongly urge mathematicians with a position
in education to scrutinize their culture. However, I fear that the culture is
such an integrated part of the mathematical community that mathematicians may be
incapable of such a self-scrutinization.
Onar.
Scott Brown
>John Novak writes:
>>Teaching children the ideas of graphing
>>and the notions of a function using first a computer strikes me as
>>s recipe for ensuring that no student ever bothers to _think_
>>about what's going on.
>Congratulations! You have actually plunged right into the heart of the
>mathematical tyranny, the cultural belief that thinking MUST be
>done in particular
>ways, the idea that automatation of thought is inferior to REAL thinking.
Have you ever tried to teach algebra to people who can't
do arithmetic because they never learned to do it
without a calculator?
>Or, rather what mathematics VALUES as real thinking. But
>reality is quite another.
>Everyone who has studied the processes of thinking knows that
>automatition is an
>absolutely necessary basis. Without it, no thinking can occur.
>There is very
>little difference between any kind of automatition, whether
>it occurs in your
>brains or on the computer screen.
That strikes me as totally absurd. The difference between
learning something yourself and having a machine do it for
you is tremendous.
>But of some mysterious reason,
>mathematicians
>believe that mental automatition is somehow superior to external
>automatition.
>Can you please tell me why?
For the same reason that learning to play an instrument is
a superior way to learn music theory than buying a CD at the store,
or jogging is a better way to get exercise than driving a car,
or learning to read is more useful than hiring someone to read
aloud to you.
--
Onar Aam
>You never explain exactly what this culture of mathematics is.
>I think you are really referring to the culture of mathematics
>EDUCATION. Mathematics education all too often says "There is one right
>way to do the problem, and the teacher knows it. Don't question the
>definitions or what one is seeking to achieve with them, just learn how
>to manipulate things according to god-given syntactic rules.
>Mathematics can be studied independent of all other disciplines. Subjects
>must be learnt in a specific sequence. Etc." Actually, this is the
>most true at the earliest stages of mathematics education; by the time
>one gets to advanced undergraduate course or graduate courses -- IF one
>gets to them, not having been repulsed -- one sees that these rules
>are a sham.
>
>Namely: Mathematicians spend much of their time trying to come up
>with new ways to do the same old problems; there is never just one way.
>Only by questioning and improving the definitions can mathematics become
>clearer, and one can only do this by keeping clearly in mind the (many
>conflicting) things one is seeking to achieve. Mathematics frequently
>gets new and interesting ideas from outside mathematics, and some of the
>best ideas come when people deliberately break the rules. Mathematics
>is a complicated weave that cannot be presented in any linear sequence.
I think we are on the same wavelength here. I am indeed referring to the
_education_ culture of mathematics. You capture my point entirely.
Onar.
Onar Aam
Oh? The importance of pure mathematics is questionable. Which proofs will you
need in engineering for instance? In what way is (pure) mathematics useful to a
musician or a writer? And do you actually question the playfullness of math? I
love math, it's a wonderful game to play on your own premises, but when that game
is forced upon people it turns into a nightmare.
Onar.
Scott Brown
>Congratulations! You have actually plunged right into the heart of the
>mathematical tyranny, the cultural belief that thinking MUST be done
>in particular
>ways, the idea that automatation of thought
This expression bugs me. I think you are using it to refer
to automating a procedure that would otherwise be done by
mental work. The "automation" eliminates the "thinking".
You're trying to make it sound like the automation just
makes the thinking more efficient, when in reality it
replaces thought with mechanical procedure.
>is inferior to REAL thinking. Or rather what mathematics VALUES
>as real thinking. But reality is quite another.
>Everyone who has studied the processes of thinking knows that
>automatition is an absolutely necessary basis.
>Without it, no thinking can occur.
'Zat so?
Hmm.
Scott Brown
--
Onar Aam
I am pleased to say that your ideas entirely coincide with my
understanding of how education should be conducted. I could
write pages about the above topic, but you have so beautifully
summarized the key to proper math education that I feel no need
to do so. Thankyou for your insightful words.
Onar.
Herman Rubin
.............................
>I have always detested mathematics because there is no room for creativity
>in it. You do what the symbols say. You get what the numbers tell you. And
>it's right or it's wrong--there are damned few gray areas. Life isn't like
>that. Tell your friend who is going through an emotional crisis what is
>the "right" way to handle it and you'll like as not get nowhere.
It is quite true that there are rigid rules in mathematics. There are
also rigid rules in what a musical instrument can do. But there is
plenty of room for creativity.
You have to do what the symbols allow. But one of the hardest things to
get across is that there can be many ways of doing a problem, all of which
are correct. This is even the case in elementary school arithmetic. We
have had postings about elementary teachers marking things wrong when they
were not done the teacher's way. Such teachers should at least be severely
chastised. This happens in other subjects as well.
As someone who has been paid for being creative in mathematics for quite
some time, I can tell you that there are still lots of important results
which nobody has stumbled upon yet.
Neil Rickert
This is just silliness. I say that mathematics is practical and useful,
and you respond by putting "pure" in front of "mathematics" wherever
you can. If you want to disagree with me, at least disagree with
something I said rather than with your own invention.
If you had actually read what I said you might have noticed the implication
that I think mathematics is badly taught. If you don't agree with this,
then you must think mathematics is well taught. If you think mathematics
is well taught, and if -- as you claim -- you love math, what on earth
are you griping about?
As for engineers, I can't argue whether or not they need proofs. But
I sure don't want to fly in an airplane designed by an engineer who is
so mathematically inept that he cannot understand mathematical proofs.
John Novak
>Congratulations! You have actually plunged right into the heart of the
>mathematical tyranny, the cultural belief that thinking MUST be done in particular
>ways, the idea that automatation of thought is inferior to REAL thinking. Or
>rather what mathematics VALUES as real thinking. But reality is quite another.
>Everyone who has studied the processes of thinking knows that automatition is an
>absolutely necessary basis. Without it, no thinking can occur. There is very
>little difference between any kind of automatition, whether it occurs in your
>brains or on the computer screen. But of some mysterious reason, mathematicians
>believe that mental automatition is somehow superior to external automatition.
>Can you please tell me why? I believe I know the answer: tradition. Mathematics
>was founded in a time when computers where not available. Therefore it has not
>become a valued tool in the math culture.
Slow down, cowboy.
First off, figure out how your terminal works, so you can put
some line feeds at the end of your lines. You'll look much more
readable.
Second, and on to the actual meat of the above text, you are
flatly in error in your claim that anyone who has studied the
process of thinking knows that automatition is an absolutely
necessary basis. Because _I_ have studied the process of
thinking, and I know no such thing. I might be helped, however,
if you defined automatition-- I'm not familiar with the word.
I take extreme exception, however, to the notion that whether
work is done by a computer, or work is done by a human brain, it
makes no difference in the matter of learning. But more on that
later.
>>I cannot see the point in showing students how to do division on
>>a calculator before they learn to do division on scratch paper.
>Mathematics _was_ however founded when pen and paper was around. Therefore these
>tools are accepted in the mathematical culture. You even stated so yourself. My
>question to you is: why on earth do you accept pen and paper? If you are so
>afraid of the students ability to think, shouldn't they learn to do long division
>in their heads? This would surely make everyone better thinkers according to your
>logic.
Quite simply, because there are vast differences between using a
pencil and paper, and using a computer while learning a subject.
Qualitatively, a pen and paper, like counting on one's fingers,
are an aid to concentration, not a physical means to an end. A
computer does not help one concentrate, unless the computer is
being used for the care and maintenance of drudgework which is
_already_ known. A case in point example is professional
computer aided design for radio frequency circuits. I know how
each individual part functions, and I know the physics and the
calculus and the differential equation for each part of the
problem in design. In theory, I could go ahead and solve that
problem by hand. It would just take me a rather long time to do
it. But the CAD software makes things easy for me, _because I
already understand the processes which the computer is handling
for me_.
A friend recently asked me to show him how to use the RF design
software I use. But he doesn't know anything about the
associated engineering concepts (he's never studied radio
frequency engineering) and doesn't know nearly as much about the
mathematics. Showing him the software is pointless, because he
doesn't understand why RF circuits are designed one way (with
distributed elements) as opposed to low frequency circuits (with
lumped elements.) He doesn't know what properties we design for
(scattering parameters, etc.) It would be utterly pointless to
introduce him to the software.
Quantitatively (yes, there is a quantitative difference) the pen
and paper are vastly different from the computer as well. The
computer _performs_ operations. It does work, in a mathematical
sense. A good engineering software package will tell you
approximately how many floating point operations it performed, so
that you can have an idea of how much work it performed, and how
a similar application will fare when ported over to the mainframe
next door. But a pen and paper do _not_ in and of themselves, do
operations. If you're missing so basic and obvious a distinction
of that, then I question the diligence of your self-proclaimed
study of the process of thinking.
>>Computers don't typically
>>illustrate the fundamentals, they disguise them, hide them, and
> ^^^^^^^^^^^^
>>make them invisible.
>What fundamentals? The ones _dictated_ by the mathematical culture??? Are you
>suggesting that there are only one way of understanding mathematics?
I'm suggesting that the only way to understand anything at all is
to _do_ something. Explanations are wonderful, but I've seen
many a college student go belly up because, although he or she
understood the lectures perfectly (or thought so, at any rate)
they could not perform the material. As simple as that.
Hell, my first year and a half, I fell into that trap, as well.
"Ah, I understood Prof. so-and-so's electronics lecture, today,
and he doesn't collect the homework, so why bother?" It caught
up with me later, and then, through diligence and hard work, I
caught back up with it, so to speak.
Its kinda like sports. Just because you've seen a player swing a
bat and hit a home run doesn't mean _you_ can do the same, the
first time you try.
>The cultural chauvenism of mathematics permeates your mind.
<Snort>
John Novak
>Oh? The importance of pure mathematics is questionable. Which proofs will you
>need in engineering for instance? In what way is (pure) mathematics useful to a
>musician or a writer? And do you actually question the playfullness of math? I
>love math, it's a wonderful game to play on your own premises, but when that game
>is forced upon people it turns into a nightmare.
I can only speak for engineering, and electrical engineering at
that.
For research engineers, the need for rigorous mathematical
understanding is patently obvious-- you're doing something that
no one has done before, and when you're done, you need to _prove_
that you got a particular result for a particular reason, and you
need to be able to _prove_ that using the same technique will get
you similar answers for certain classes of problems. (Ideally.)
For practicing engineers, the situation may be a little less
clear. But it is my opinion, as an engineer, that if you
understand the proofs which lead up to your area of expertise,
you will end up making a fool of yourself much less. When your
supervisor asks you to justify a technique, you can do it. If
you need to adapt a technique from down the hall at R&D into
practice, you can understand their work, bring it down from the
Ivory Tower, and either make it work, or reject it _with
reasons_.
In other words, if you can prove what you say, you'll be taken
seriously.
Of course, electrical engineering is a rather mathematical
subject, no matter how much some of my fellow students would like
to deny it... :-/
Marc Olschok
In <CL155...@mach1.wlu.ca> brea...@mach1.wlu.ca (Piano Man) said:
> [...]
> [...] And for what? Since the age of about eight, I've known that I
> wanted to be a writer "when I grew up". Math, beyond the basic operations
> required to budget money, is entirely useless to me.
So what ?
In a similar way literature and music are useless to most people, but that
should not prevent them from reading books or visiting a concert.
It is high time to reemphasize the distinction between useful and valuable.
> I have been composing music since I was four, and don't let anyone tell
> you that music is related to mathematics in any way. I can do music. I
> can't do math.
>
> [...]
>
> : We have no better friend than our own mental ability,
> : which we must be forced to exercise when young, for our own later
> : (immense) benefit.
>
> I agree. But who says this ability should have anything to do with numbers?
I agree. But who says that mathematics has to do anything with numbers ?
> I have always detested mathematics because there is no room for creativity
> in it. You do what the symbols say. You get what the numbers tell you. And
> it's right or it's wrong--there are damned few gray areas. Life isn't like
> that. Tell your friend who is going through an emotional crisis what is
> the "right" way to handle it and you'll like as not get nowhere.
May I have a guess ? Your opinions about mathematics were formed by school experience.
Imagine a student who has to endure a very dull and boring music education. If this
student has no exposure to the creative part of music (e.g. by playing an instrument)
he may as well post in later years:
And for what? Since the age of about eight, I've known that I
wanted to be a programmer "when I grew up". Music is entirely useless to me.
I have always detested music because there is no room for creativity
in it. You do what the notes say. You get what the sounds tell you. And
it's right or it's wrong--there are damned few gray areas. Life isn't like
that.
I am afraid when You were a student You were not presented mathematics but
mindless calculations. This is not Your fault.
Of course mathematics is not to everyone's taste. But it is depressing to see
that most people turn away from mathematics without ever actually having seen it.
> Ken Breadner Wilfrid Laurier University,
> brea...@mach1.wlu.ca Waterloo, Ontario, Canada-for-now
> (the BREADbox) ---------------------------------
> ------------------------Scratch here to reveal your prize--->
^^^^^^^^^^
lost again!
I am just too stupid.
/-------------------------\
|Marc Olschok |
|ols...@acsu.buffalo.edu |
|SUNY at BUFFALO |
\-------------------------/
steve.c...@daytonoh.ncr.com
dimi...@astro.ocis.temple.edu (Dimitrios Diamantaras) says, in part:
>Onar Aam (on...@hsr.no) wrote:
>: ...So instead of
>: easing the burdon of learning for students, which they should,
>: mathematicians choose the easy path. If things are not done
>: _their_ way then it is _wrong_. This chauvenism has been allowed to
>: live for ages in the education system, and I say it is time to break
>: down this tyranny. The culture of mathematics is a game and should
>: be treated like it. This culture deserves no more place in school
>: than religion. ...
>
>...More seriously, who says that math should be easy to learn? If there is no
>royal road to geometry, then there is no other easy road to it either (and
>King Ptolemy had to accept such a statement; to his credit he did not
>behead the messenger for this message). Everybody CAN learn math, but they
>HAVE to make the effort. Enough already of this stuff about making
>students feel good; it's gone too far and now produces uneducated adults ...
Like many of you, I've been through what Onar Aam describes on several
mind-bending occasions. In one early case, I had a different way of subtracting
mixed numbers which worked fine for me. My math teacher frankly didn't
understand it and was totally resistant to learning about such alternatives
from her students. Next thing I knew, I was sitting in a teacher / parent
conference hearing about my attitude problem.
I don't think Onar was suggesting any math learning ought to be sacrificed,
only that the culture needs to be changed. Like the young woman on that
10 PM news magazine show said this week, the way we've taught math
traditionally has felt to many students like being in a "shark tank."
Steve Chenoweth, Technology & Development
AT&T Global Information Solutions (NCR)
1700 S. Patterson Blvd., Dayton, OH 45479
steve.c...@daytonOH.ncr.com
richh
You'd be surprised. In the liner notes to "Dirty Mind" Prince proves
FLT for exponent 4 without using infinite descent.
And they say Mariah Carey has a nifty, if not altogether rigorous
proof that there is a vertex in every complex polygon in the Euclidean
plane which has no three equidistant vertices.
But you can't dance to it, so Sony won't publish it.
RICHH
Tony W H Lai
>X-Nntp-Posting-Host: broremann.hsr.no
Is hsr.no a Norwegian site? Is mathematics taught by actual mathematicians
in schools in Norway? If not, are the curricula and methods of teaching
decided by mathematicians, rather than teachers or administrators?
BTW, could you please use lines with fewer than 80 columns?
Moses Klein
> Math, beyond the basic operations
>required to budget money,
Which isn't really math
>is entirely useless to me.
>
>I have been composing music since I was four, and don't let anyone tell
>you that music is related to mathematics in any way. I can do music. I
>can't do math.
They are related. You do not perceive the mathematical nature of a lot of
music because you are unmathematical. The way a person understands a
piece of music says as much about the person as about the music. A
mathematician can often appreciate on a mathematical level what an
unmathematical person can still appreciate on a different level. In my
view, this subtlety is one of the things that can make music so
fascinating and beautiful.
>I agree. But who says this ability should have anything to do with numbers?
So math = arithmetic. This misunderstanding is the most common result of
everything that is wrong with most mathematics education.
>I have always detested mathematics because there is no room for creativity
>in it.
Correction: there is no room for creativity in mathematics AS IT IS
USUALLY TAUGHT. "Math" education usually involves standard techniques
which are practiced until thoroughly mastered, and the students are only
expected to be able to apply them in precisely the way they have been
shown. In other words, following recipes.
Ask any mathematician about this. Once you know that a problem can be
done routinely by some recipe or other, we refer to it as "trivial". Real
mathematics does involve creativity.
Moses Klein (kl...@math.wisc.edu)
Donald (Don) Specht
[and Don Specht has liberally edited]
> I am a statistician.
And a lovable old curmudgeon. :-)
> ... people like you have mistaught statistics.
How do you know what I'm like from two postings and one essay?
> Mathematics is mainly universal grammar...
This is our common ground for compromise. Please don't let
your blood pressure get too high that a high school teacher
actually agrees with you.
> Abstraction is not grounded in "concrete".
I agree to disagree with you on this point. Maybe I should
restate the idea. Learning is more powerful and lasting if
initiated with concrete experiences prior to the abstract
"general case."
---
Don Specht <dsp...@mwc.vak12ed.edu>
Clarke County High School
"Math is a language used to explain,
interpret, and predict events
in the real world."
Donald (Don) Specht
[and Don Specht deleted a bunch of stuff]
DS > mathematical literacy. *Each* year of their schooling must
DS > include direct instruction and experiences pertaining to *all*
DS > of these pillars. Mix with this a classroom pedagogy that
>
> Say what? Personally I think that it is important to *focus* on
> something to learn it.
And to focus on the connections.
DS > firmly grounds the student as he sets sail for the land of the
^^
> Gee. PC, but not PC enough to have learned how to be gender neutral in
> writing.
I didn't deserve that.
> I believe that technology is only a tool.
Yup. Along with paper and pencil, estimating, and mental
arithmetic. The student who learns the appropriate use for
each tool and when to use that tool has learned a powerful
lesson.
DS > I believe that Mathematical Literacy must take a position along
DS > side Whole Language on the restructuring agendas of every
DS > school system whose goal is continuous improvement. We teach
DS > children, not courses. Vertical pillars must replace
DS > horizontal gateways.
> Nice images, no substance.
I hope you will avail yourself of the "substance" provided.
> I believe that given this we would be well advised to change our
> attitudes and focus on those who have managed to do well. Such as Jaime
> Escalante (the teacher in Stand and Deliver). His opinion on the new
> teaching standards is that it looks like it was written by a PE
> teacher. When people like him who have demonstrated their ability to
> teach, and who know their subject, have doubts, then the rest of us
> should think about it.
I'm sorry to hear that you believe that Mr. Escalante feels
that way. There is no doubt that he was successful, and there
is no doubt that his method was successful.
All students can learn if they have the ganas.
I believe this and my students love it when we watch _Stand and
Deliver_ together.
I look forward to continuing this discussion.
john baez
brea...@mach1.wlu.ca) and got to the bottom, where Piano Man writes:
>I have always detested mathematics because there is no room for creativity
>in it. You do what the symbols say. You get what the numbers tell you. And
>it's right or it's wrong--there are damned few gray areas. Life isn't like
>that.
Mathematics isn't like that, either. You simply haven't done
mathematics -- though you may have done calculations. Some people take
piano lessons and get fed up with practicing scales, and never get
around to improvising. That's understandable. However, they should at
least realize that music is not really about playing scales, and you
should at least realize that mathematics is not about doing long
division or any other kind of problem where you simply follow the rules.
Imagine someone who thought that literature had no room for creativity
in it because one had to spell the words just like the teacher said.
That someone would be letting the mechanics interfere with his view of
the real point of literature. Probably you just had bad teachers.
gsm...@uoft02.utoledo.edu
> Have you ever tried to teach algebra to people who can't
> do arithmetic because they never learned to do it
> without a calculator?
One way to do that would be to have them do the algebra by machine
also, of course. At this rate we might have to teach concepts
and not just techniques, which has a certain attractiveness to
it. Of course, students would hate this new "math culture" far
more than they do the one we have now.
> That strikes me as totally absurd. The difference between
> learning something yourself and having a machine do it for
> you is tremendous.
Serious computation generally means machine computation. Anyone who
does much of it comes to feel that the computer is almost an extension
of the self. If we start implanting chips, this might really end
up being true, and how's that for a horrible prospect?
--
Gene Ward Smith/Brahms Gang/University of Toledo
gsm...@uoft02.utoledo.edu
Joseph K. Ball
of the discovery of a record Mersenne prime (roughly 10^450,000 ) by
some folks at Cray Research. What I was wondering is:
-- Is there a published report of this discovery, and where
would I look for it? Alternatively, contact with the researchers would
be good too.
-- Who out there does searches for Mersenne primes? I am
interseted in knowing what ranges of candidates have been searched,
and general correspondence.
-- Where art thou, David Slowinski? I would like to find
David's whereabouts, since he has found several Mersene primes
himself.
Any answers to these questions or any leads on this information would
be greatly appreciated. Please send e-mail to k-b...@uchicago.edu
Keith Ball, Dept. of Physics, Univ. of Chicago
--
-----------------------------------------------------------------------
AF: I met Andy Warhol at a really chic party.
DM: Blow it out your hairdo, 'cause you work at Hardee's!
Dick Adams
>> It is absolutely amazing how mathematics has managed to
>> build up its enormous position in education. Mathematics
>> is in many ways a game for intellectuals, even so it has
>> managed to become one of the most important subjects in
>> school. I want to explore why this is so.
ric...@mp.cs.niu.edu (Neil Rickert) writes:
> I would say it differently. Namely
> Mathematics is one of the most important, practical and
> useful subjects in school. It is absolutely amazing how
> educators have managed to distort it in such a way that
> many students now see it as no more than a game for
> intellectuals.
I would classify the comments of my colleague Neil Rickert as
an accurate representation of one of the problems that exist
in primary and secondary education.
The literature is replete with studies which support the claim
that women who do well in mathematics-based subjects suffer
significantly less from gender-based economic discrimination
than do women who did not do well in mathematics.
Perhaps someone could explain how any well-educated person could
possibly perceive the position of mathematics in education as one
of overemphasis. IMHO, it is one of underemphasis.
==================================================================
| Richard D. Adams, CPA | E-Mail: eaj...@ube.ub.umd.edu |
| Accounting Faculty | Office: (410) 837-5115 |
| University of Baltimore | Home: (410) 465-6362 |
| 1420 N. Charles Street | Fax: (410) 837-4899 |
| Baltimore, MD 21201-5779 | "MY BLOOD RUNS CAROLINA BLUE!!" |
==================================================================
| Reading, statistics and foreplay are what life is about. |
| Reading opens up your life. Statistics opens up your mind. |
| Foreplay opens up your soul. |
==================================================================
john baez
>I can only speak for engineering, and electrical engineering at
>that.
>Of course, electrical engineering is a rather mathematical
>subject, no matter how much some of my fellow students would like
>to deny it... :-/
Electric engineering is, on the one hand, the basis of much of the
everyday technology on which our modern lifestyle depends, and on the
other had, quite mathematical. Let's not forget that Raoul Bott, the
renowned topologist and geometer, started out in electrical engineering.
From Kirchoff's laws it's just a skip and a jump to the cohomology theory
of 1-complexes, and then you're off and running.... I really hear that's
how he got going! On the other hand, I'm teaching complex analysis to
some physicists here, and looking for some nice practical applications I
checked out a few books on digital signal processing and circuit theory.
It's great! Check out the exercises in Signals in Linear Circuits by
Cruz and Valkenburg, for example. There's a list of projects at the
end. The first one: "Solve a set of simultaneous algebraic equations..
Use one fo the following: Gauss elimination, Gauss-Jordan method,
Gauss-Seidel method." I don't even know what those last two are! (Or
maybe I do but don't realize it.) Moving on a bit: "Find the roots of
an algebraic equation. Study interval halving, secant method,
Newton-Raphson methods, Lin-Barstow method." Hmm, I can guess what the
first one is, know the next two, but not the last. Near the end:
"Fast Fourier Transform (FFT). Use a computer program package for the
FFT. For a given non-periodic signal, determine the magnitude and phase
spectra." And this is a pretty elementary text.
Herman Rubin
>Onar Aam writes:
................................
> Have you ever tried to teach algebra to people who can't
> do arithmetic because they never learned to do it
> without a calculator?
Which part of algebra? How to compute numerical answers? The most
important part of algebra, how to ask questions, can AND SHOULD be
taught BEFORE arithmetic. It IS possible to teach all of mathematics
with the exception of arithmetic without teaching arithmetic.
>>Or, rather what mathematics VALUES as real thinking. But
>>reality is quite another.
>>Everyone who has studied the processes of thinking knows that
>>automatition is an
>>absolutely necessary basis. Without it, no thinking can occur.
>>There is very
>>little difference between any kind of automatition, whether
>>it occurs in your
>>brains or on the computer screen.
>
> That strikes me as totally absurd. The difference between
> learning something yourself and having a machine do it for
> you is tremendous.
I agree with you that THINKING cannot be automated. But being both
a supposedly reasonably good mathematician and being rather good at
arithmetic, I can tell you flatly that my arithmetic abilities are
helpful in getting numerical answers, but not in understanding any
part of mathematics.
Herman Rubin
>Herman Rubin (hru...@mentor.cc.purdue.edu ) writes:
...........................
>> Mathematics is mainly universal grammar...
>
>This is our common ground for compromise. Please don't let
>your blood pressure get too high that a high school teacher
>actually agrees with you.
>
>> Abstraction is not grounded in "concrete".
>
>I agree to disagree with you on this point. Maybe I should
>restate the idea. Learning is more powerful and lasting if
>initiated with concrete experiences prior to the abstract
>"general case."
There are several reasons to disagree with this. For one thing,
it is generally difficult to have enough examples that the examples
do not have something in common other than the abstract idea. It is
even common to have only one kind of example introduced in advance.
This means that much painful unlearning must be done.
There are even cases where essentially only one example exists. The
structure of the integers is categorical. The same holds for the
real numbers. This statement means that there is only one example.
It is a real pedagogical problem to get students to throw off the
misconceptions they have developed. Those who have learned integration
with resepect to the customary Euclidean measures have difficulty in
going to the simplest kind of general integral, the discrete version.
The customary integrals are limits of this simple and not fully general
version, and understanding the general concept, but not being able to
prove the theorems, only involves understanding algebraic notation and
the properties of the real numbers.
This does not mean that examples do not help, and does not mean that
lots of examples should not be given to illustrate the general concept.
But the generality keeps the errors from getting in from the start.
Allen Knutson
...
}Let's not forget that Raoul Bott, the
}renowned topologist and geometer, started out in electrical engineering.
}From Kirchoff's laws it's just a skip and a jump to the cohomology theory
}of 1-complexes, and then you're off and running.... I really hear that's
}how he got going!
He discusses this connection in (as always) excellent contribution to
"The Mathematical Heritage of Hermann Weyl", published by the AMS.
Wonderful stuff, recommended to everyone interested in Weyl's life
and/or the topology of Lie groups. Allen K.
john baez
steve.c...@daytonOH.ncr.com writes:
>Like many of you, I've been through what Onar Aam describes on several
>mind-bending occasions.
Reading Onar Aam's posts is indeed a bit of an experience, but don't you
think you're exaggerating?
Herman Rubin
>>>_their_ way then it is _wrong_.
>>
>
>
>That should be the least of problems. School is very concerned with testing. To
>get good scores on tests is (supposedly) a motivation for learning. But students
>don't learn for the sake of learning, they learn in order to do well on tests.
This is what usually happens, and to a considerable extent it is encouraged
by the teaching methods and attitudes.
>Getting the answer right isn't what is always of importance, it is _how_ you get
>them. This motivates recipee-learning rather than understanding. Instead of
>striving towards understanding, the students are forced to strive towards the
>_norms_. Understanding does in no way assure you good grades, following the norms
>does. In other words, getting good grades is about adaption to the norms, getting
>good grades is about complying with the culture. True, this is not typical for
>mathematics. In fact, the entire education system is built up on such enforcement.
You are sooooo right! This is what comes from a system which ignores
concepts and structures. This is what comes from saying that getting
50% on a test is a very bad score, when it should be definitely passing.
This is what comes from a system which discourages that a student should
have to do a problem which does not look exactly like one seen, and put
various things together.
The surprise is that some students, even after a dozen years of this,
are still capable of thinking. It is not surprising that those who
go through the schools of education are largely not so capable.
Herman Rubin
>Onar Aam writes:
>>John Novak writes:
>>>Teaching children the ideas of graphing
>>>and the notions of a function using first a computer strikes me as
>>>s recipe for ensuring that no student ever bothers to _think_
>>>about what's going on.
>>Congratulations! You have actually plunged right into the heart of the
>>mathematical tyranny, the cultural belief that thinking MUST be
>>done in particular
>>ways, the idea that automatation of thought is inferior to REAL thinking.
> Have you ever tried to teach algebra to people who can't
> do arithmetic because they never learned to do it
> without a calculator?
Let me attack both of these viewpoints. I would not teach arithmetic
before teaching the basic ideas of algebra, which are variable, function,
and relation. Children know lots of these which are not mathematical at
all, such as mother, father, son, daughter, etc. Here we have relationships,
some of which are funtions and some of which are not. We can use variables
for what they are, NOTATION.
A function is NOT something which takes numbers into numbers; that is a
special type of function. It is true that a a function taking numbers
into numbers, if sufficiently smooth, can be given by a visual graph,
but that obscures the idea of function. There are lots of good
mathematicians who I would not trust with simple arithmetic; I am
not saying that one should not teach the manipulations, but even
these should be put in context.
> That strikes me as totally absurd. The difference between
> learning something yourself and having a machine do it for
> you is tremendous.
I then suggest you calculate pi to a billion decimal places.
You do not have to climb the Empire State building to get to the top;
you can take an elevator.
How much you need to know of details depends on what you want to
do with them. But it does no good to know how to add if you do
not know what addition means structurally. The "new math" provided
the structure, possibly not in the best manner. The teachers could
not understand that the structure stood without the manipulations.
jj, curmudgeon and sometimes slow reader, I guess
>
>The resemblance is striking. Abused children often become abusers themselves.
>Likewise, the math loving students who are tyrannized by the mathematical culture
>often becomes tyrants themselves. The only way to break this vicious cycle is to
>break down the entire mathematical culture.
Comparing mathmeticians to child abusers is utterly dishonest.
"tyrranized", "tyrants".
Mathematics answers to proof and understanding, not to emotional
diatribe, and it would behoove this Onar Aam individual to learn
what mathematics is before s/he compares mathematics to child abuse,
even by association.
--
Copyright alice!jj 1994, all rights reserved, except transmission by USENET and like facilities granted. Said permission is granted only for complete copies that include this notice. Use on pay-for-read services or non-electronic media specifically disallowed. -------
Colored lights can hypnotize, shine in someone else's eyes
-----
j...@alice.att.com Member HASA - Atheist Scum Division
Herman Rubin
>
> [and Don Specht has liberally edited]
>
>
>> I am a statistician.
>
>And a lovable old curmudgeon. :-)
>
>> ... people like you have mistaught statistics.
>
>How do you know what I'm like from two postings and one essay?
If you suggest teaching statistics in the primary grades, you cannot
but misteach it. The point of statistics is that from sample data
one can reason about the "population" or the underlying structure.
One cannot teach this without probability and much more. I suggest
that anyone wishing to look at the problem read something like
Raiffa's _Decision Analysis_. You will not learn statistics from
it, but you will see why quite a bit of machinery is needed to
have any idea of why anything is done.
It is very much harder to teach this to someone who has had
statistical recipes.
>> Mathematics is mainly universal grammar...
>
>This is our common ground for compromise. Please don't let
>your blood pressure get too high that a high school teacher
>actually agrees with you.
>
>> Abstraction is not grounded in "concrete".
>
>I agree to disagree with you on this point. Maybe I should
>restate the idea. Learning is more powerful and lasting if
>initiated with concrete experiences prior to the abstract
>"general case."
Having seen too many situations where the examples and experiences
interfere with learning the concepts, I must challenge this. Even
if a sufficient variety of more concrete instances are enough to
help the student to understand the abstract idea, it is very inefficient
from a pedagogical standpoint. One has to do a great deal of unlearning
to remove the concrete garbage; this is not easy. Also, it is very
time-consuming. Learning the abstract idea first is likely to reduce
the length of time to learn the particular applications by more than a
factor of 2, and it is very often the case that learning the abstract
concept takes longer if the special cases are known. There is a saying,
Any fool can learn from his own experience;
The wise man learns from the experience of others.
I have seen too much to accept the first, but the second is why we have
an educational system. When Dedekind finally came up with an expression
of the essential properties of the integers, he made it possible for a
young child to get a clear understanding of what mathematicians had been
trying to discover for millenia. Since quite competent mathematicians
had been unable to put it all together, as you need to have the correct
muse strike you in a way that you get it, it is not enough to search for
it by experience. Newton and Euler and Gauss and Legendre and ..., all
comparable in ability to Dedekind, did not get it. And yet it is simple
enough for a child to understand. This was not a gradual process, putting
a large number of things together, but a flash of insight.
Thomas Marlowe
Anyone who says "We teach children, not <subject matter>" has never studied
a language with a rich enough case structure. Anyone who says "We teach
<subject matter>, not children" might as well never have studied anything.
(What would you say to someone who said: "I drive home, not an
automobile"?!? "We serve customers, not meals"? (Unless it's the Alferd E.
Packer Cafeteria!))
tom marlowe
Onar Aam
>
>"tyrranized", "tyrants".
>
>Mathematics answers to proof and understanding, not to emotional
>diatribe, and it would behoove this Onar Aam individual to learn
>what mathematics is before s/he compares mathematics to child abuse,
>even by association.
I am schocked by this reply. Completely irrational. My comparison was a
sociological observation and is not dishonest at all. Mathematicians
deal with nice, abstract structures, but they are still humans and very
much behave like humans. YOU are the one being dishonest here, not me.
As long as mathematics is in school mathematics does not only answer to
proof and understanding, it answers (or at least should) to humans.
Humans cannot be treated as if they were learning machines. Mathematics
contains very rigid formal structures, but the process of understanding
them is highly organic. Knowledge and understanding is an organism. It
needs to be cuddled with and taken care of, nursed and stimulated.
Never, ever forget that.
Onar.
gsm...@uoft02.utoledo.edu
> I have seen too much to accept the first, but the second is why we have
> an educational system. When Dedekind finally came up with an expression
> of the essential properties of the integers, he made it possible for a
> young child to get a clear understanding of what mathematicians had been
> trying to discover for millenia. Since quite competent mathematicians
> had been unable to put it all together, as you need to have the correct
> muse strike you in a way that you get it, it is not enough to search for
> it by experience. Newton and Euler and Gauss and Legendre and ..., all
> comparable in ability to Dedekind, did not get it. And yet it is simple
> enough for a child to understand. This was not a gradual process, putting
> a large number of things together, but a flash of insight.
This sounds like nonsense to me. So far as I can see, Newton, Euler,
Gauss and Legendre understood the integers well enough. Dedekind no
doubt had another way of looking at the situation, by comparing the
integers to rings of integers in number fields. But doing this, or
even moving from N to Z, involves removing some important special
features N.
How, exactly, does anything that Dedekind did make it easier for children
to understand N?
Onnie Lynn Winebarger
>Humans cannot be treated as if they were learning machines. Mathematics
we wouldn't be cognizant of anything.
The only question is how they learn.
Lynn
Onar Aam
Herman Rubin writes:
>Learning the abstract idea first is likely to reduce
>the length of time to learn the particular applications by more than a
>factor of 2, and it is very often the case that learning the abstract
>concept takes longer if the special cases are known.
I am of the exact opposite experience. Examplification is a good tool in making
structures visible to people. At least this is how it is in complex
understanding. This may be different in basic math as you are dealing with simpler
and purer structures than in reality. I am therefore very interested in hearing
about Dedekind's expression of the essential properties of the integers. Could
you please state them? I agree that children very early are capable of
abstraction. But all abstractions are based upon the concrete. Absolutely no
concepts escapes the outer world. Therefore I suspect that you do not mean to the
absolutely naked structures of math (which is impossible). Some sort of
visualization is an absolute requirement, either by linguistic means or by
graphical means.
Onar.
Scott Brown
>Scott Brown writes:
>> Have you ever tried to teach algebra to people who can't
>> do arithmetic because they never learned to do it
>> without a calculator?
>Let me attack both of these viewpoints. I would not teach arithmetic
>before teaching the basic ideas of algebra, which are variable, function,
>and relation. Children know lots of these which are not mathematical at
>all, such as mother, father, son, daughter, etc.
So, how do you explain why is is correct to distribute products
but not exponents, as in a(b+c) = ab + ac but (a+b)^2 != a^2 + b^2,
to someone who can't understand arithmetic?
You give examples of some very general principles, and even
went on to point out that these extremely general ideas are not
exclusively algebraic.
I refer to teaching algebra, the specific skills needed to
manipulate mathematical equations and obtain solutions or other
useful information. Not "the basic ideas of algebra", which I
agree are general enough to approach by other means, but the
actual physical skill of solving specific problems.
I find it hard to believe you have a method of teaching how
to solve a quadratic, or find the solutions to a problem involving
absolute value, to a student who is very unfamiliar with and
uncomfortable with the fundamentals of arithmetic.
[deletia]
>> That strikes me as totally absurd. The difference between
>> learning something yourself and having a machine do it for
>> you is tremendous.
>I then suggest you calculate pi to a billion decimal places.
>You do not have to climb the Empire State building to get to the top;
>you can take an elevator.
*sigh*
You don't have to exercise either; you can sit on your a*s and
watch the discovery channel all day. You don't have to practice
the piano; you can just buy a cd and a stereo system.
Herman, I'm amazed that you made the comment about pi above.
I thought it was totally clear from the content of my post that
the "difference" I was referring to is the difference between
practicing an activity for the sake of improving one's skill
at the activity versus having a machine perform the activity
merely to get it done.
Are you suggesting that someone who types a command into
Mathematica to print out a few thousand digits of pi has
gotten the same from the act as a student who learns how
to do a series approximation with error estimate and calculates
the first 10 digits of pi that way?
I'm really a bit dumbfounded by your suggestion that
you thought I was advocating _all_ such tasks should be
carried out by hand. Of course not; that's silly, and I
really think you knew I wasn't suggesting anything of the
sort.
Scott Brown
--
John Novak
>I am schocked by this reply. Completely irrational. My comparison was a
>sociological observation and is not dishonest at all. Mathematicians
>deal with nice, abstract structures, but they are still humans and very
>much behave like humans. YOU are the one being dishonest here, not me.
You're using emotionally loaded words and concepts, and I think
you know it.
>As long as mathematics is in school mathematics does not only answer to
>proof and understanding, it answers (or at least should) to humans.
>Humans cannot be treated as if they were learning machines. Mathematics
>contains very rigid formal structures, but the process of understanding
>them is highly organic. Knowledge and understanding is an organism. It
>needs to be cuddled with and taken care of, nursed and stimulated.
>Never, ever forget that.
Human beings _are_ learning machines.
So are many organisms with nervous systems. Its what we're
_about_.
The only interesting question in town is, "How, exactly, do we
learn?"
--
John S. Novak, III
j...@cegt201.bradley.edu
j...@camelot.bradley.edu
Benjamin J. Tilly
hru...@mentor.cc.purdue.edu (Herman Rubin) writes:
> [...] But being both
> a supposedly reasonably good mathematician and being rather good at
> arithmetic, I can tell you flatly that my arithmetic abilities are
> helpful in getting numerical answers, but not in understanding any
> part of mathematics.
Suppose that you want to understand how to divide one polynomial by
another. The knowledge that I have of the procedure of long-division in
arithmetic, and the reasons for that procedure, help me to not only do
that task, but also to understand what I am doing.
Is this not at all true for you? If not then how would you tackle the
above task and how do you understand what you are doing?
Secondly I find that much of my practical arithmetic knowledge of
factoring helps me to understand mathematics. Not probability theory of
course, but mathematics none-the-less.
Ben Tilly
Benjamin J. Tilly
> Benjamin J. Tilly (Benjamin...@dartmouth.edu ) writes:
>
> [and Don Specht deleted a bunch of stuff]
>
(FAR too liberally I might add. The point of leaving in stuff is so
that people may have comments in context when they appear. This makes
responding easier and more fruitful. In spoken conversation we remember
the last few exchanges. On a much slower bulleting board conversation
we have to rely on the relevant details being left in.)
> DS > mathematical literacy. *Each* year of their schooling must
> DS > include direct instruction and experiences pertaining to *all*
> DS > of these pillars. Mix with this a classroom pedagogy that
> >
>
> > Say what? Personally I think that it is important to *focus* on
> > something to learn it.
>
> And to focus on the connections.
>
But there is little point in focusing on the connections when people do
not understand either topic. And some of these "pillars" do not tend to
connect, others do. For example probability and geometry do not have a
tremendous amount of connection at the elementary level (other than a
family resemblance that all forms of math have, and certain probability
problems that are stated in a geometrical format) but both have
intricate connections to algebra. However people will not be able to
appreciate the connection to algebra unless they actually have worked
on the algebra first.
>
> DS > firmly grounds the student as he sets sail for the land of the
> ^^
>
> > Gee. PC, but not PC enough to have learned how to be gender neutral in
> > writing.
>
> I didn't deserve that.
>
Sorry, I was only guessing that you were PC. It is a point of fact that
you were not being gender neutral in your writing.
> > I believe that technology is only a tool.
>
> Yup. Along with paper and pencil, estimating, and mental
> arithmetic. The student who learns the appropriate use for
> each tool and when to use that tool has learned a powerful
> lesson.
>
Yup. And the student who learns on the computer and never learns how to
function off of it will never have the ability to learn how to
estimate. You see learning with paper and pencil has the advantage that
the *person* is doing the work. Once they have done that it is not hard
to teach them to do reliable estimates in their head, and to teach them
to use a computer. However if they learn the computer then it will not
help them at all when it comes to doing those things off of the
computer. And when it comes to algebra techniques (which some of them
will have to learn no matter what) they will have been shortchanged if
they have not had to face the arithmetic procedures beforehand.
In other words what I said previously (and your liberal editing has
brought out of context) was talking about the fact that in order to
learn the appropriate use for each tool you have to learn how to do
without the tools first.
>
> DS > I believe that Mathematical Literacy must take a position along
> DS > side Whole Language on the restructuring agendas of every
> DS > school system whose goal is continuous improvement. We teach
> DS > children, not courses. Vertical pillars must replace
> DS > horizontal gateways.
>
>
> > Nice images, no substance.
>
> I hope you will avail yourself of the "substance" provided.
>
I meant that there is no substance to anything to do with the Whole
Language being so great. If you had finished my quote this would have
been entirely clear since I went on about it at some length. I will
deal with the standards in seperate postings.
>
>
> > I believe that given this we would be well advised to change our
> > attitudes and focus on those who have managed to do well. Such as Jaime
> > Escalante (the teacher in Stand and Deliver). His opinion on the new
> > teaching standards is that it looks like it was written by a PE
> > teacher. When people like him who have demonstrated their ability to
> > teach, and who know their subject, have doubts, then the rest of us
> > should think about it.
>
> I'm sorry to hear that you believe that Mr. Escalante feels
> that way. There is no doubt that he was successful, and there
> is no doubt that his method was successful.
>
I was quoting what *he* says his belief is. I am sorry that you think
of this as just my personal opinion as to his belief. If you want
verification then I point you to the article in the Notices of the AMS
that I mentioned before (but you deleted).
> All students can learn if they have the ganas.
>
Agreed.
> I believe this and my students love it when we watch _Stand and
> Deliver_ together.
>
I met one of the main actors, Diamond Lou Phillips (the touch hispanic
kid who wants two texts so that nobody will see him carrying books
around...). He had some interesting things to say about Jaime
Escalante. For example they would be acting out writing the tests, and
he was not really doing the problems. Then when the take was done Jaime
would come up and tell them how to do the questions and here is what
they should have been doing...
>
> I look forward to continuing this discussion.
Cheers,
Ben Tilly
Onar Aam
This is semantic bigotry. Oxford's defines one meaning of machine
as: "person who acts automatically, without thinking." A machine
has a rigid linear structure. This means that it is an apparatus
which is assembled from simpler components doing simple tasks. It
is absolutely perverse that the linear philosophy has been allowed
to pervade the education system. Learning is a NON-linear process.
John Novak writes:
>You're using emotionally loaded words and concepts, and I think
>you know it.
Well, of course I am. But that does not make me dishonest. What on
earth is wrong with using emotionally loaded words? I made a
comparison as anattempt to explain the mechanism of the mathematical
culture. If you mean that it is wrong then refute it! Don't throw
around with empty words like "dishonest".
Onar.
Gerry Myerson
wrote:
>
> I have been composing music since I was four...
>
> I have always detested mathematics because there is no room for creativity
> in it. You do what the symbols say.
Funny --- I have always detested music for exactly the same reason. There
is no room for creativity in it. You do what the notes say.
What's that you say? The creative part comes in writing down the notes in
the first place? In the composition of the music? Do you suppose there
might be something analogous in mathematics? That some creative people
gave us the mathematics? Nah, couldn't be, mathematics was handed down
to us on Sinai, carved in stone.
Think about it.
Gerry Myerson
Newsfeed unreliable. If you post a follow-up to this article, please
send it to me by email, as well.
Herman Rubin
>Herman Rubin writes:
>
>>Scott Brown writes:
>
>>> Have you ever tried to teach algebra to people who can't
>>> do arithmetic because they never learned to do it
>>> without a calculator?
>
>>Let me attack both of these viewpoints. I would not teach arithmetic
>>before teaching the basic ideas of algebra, which are variable, function,
>>and relation. Children know lots of these which are not mathematical at
>>all, such as mother, father, son, daughter, etc.
>
> So, how do you explain why is is correct to distribute products
>but not exponents, as in a(b+c) = ab + ac but (a+b)^2 != a^2 + b^2,
>to someone who can't understand arithmetic?
The law of universal distribution is false; what one has to do is
to see in which cases operations distribute AND WHY. In fact, the
distributive law is unusual, even more so than the commutative law.
So what you have to explain is why, and from what source, we know
the distributive law above. It is not enough to compute examples;
for it to be mathematics, there must be proof. At some stages the
proof may be indicated, or even relegated to later as too difficult,
or even not given as requiring too much background. But the teacher
must realize that there is a proof before saying it.
> You give examples of some very general principles, and even
>went on to point out that these extremely general ideas are not
>exclusively algebraic.
>
> I refer to teaching algebra, the specific skills needed to
>manipulate mathematical equations and obtain solutions or other
>useful information. Not "the basic ideas of algebra", which I
>agree are general enough to approach by other means, but the
>actual physical skill of solving specific problems.
>
> I find it hard to believe you have a method of teaching how
>to solve a quadratic, or find the solutions to a problem involving
>absolute value, to a student who is very unfamiliar with and
>uncomfortable with the fundamentals of arithmetic.
What are the fundamentals of arithmetic? They are not the manipulations
of addition, subtraction, multiplican, and division. Rather, they are
the deduced laws governing these operations, deduced from the structure
of the integers or rational numbers or real numbers or complex numbers.
>
> [deletia]
>
>>> That strikes me as totally absurd. The difference between
>>> learning something yourself and having a machine do it for
>>> you is tremendous.
>
>>I then suggest you calculate pi to a billion decimal places.
>
>>You do not have to climb the Empire State building to get to the top;
>>you can take an elevator.
>
> *sigh*
>
> You don't have to exercise either; you can sit on your a*s and
>watch the discovery channel all day. You don't have to practice
>the piano; you can just buy a cd and a stereo system.
>
> Herman, I'm amazed that you made the comment about pi above.
>I thought it was totally clear from the content of my post that
>the "difference" I was referring to is the difference between
>practicing an activity for the sake of improving one's skill
>at the activity versus having a machine perform the activity
>merely to get it done.
>
> Are you suggesting that someone who types a command into
>Mathematica to print out a few thousand digits of pi has
>gotten the same from the act as a student who learns how
>to do a series approximation with error estimate and calculates
>the first 10 digits of pi that way?
But the methods used to calculate pi to many digits are not
those. And one can have Mathematica print out the steps in
the series; this will be just as good as doing it by hand.
Now I have little difficulty in doing manual arithmetic, and
with well above average speed. So what does it get me? It
does not get me any insight into anything except the way to
carry out arithmetic operations.
And suppose you teach, and use, Karatsuba-Offman to calculate
multiple precision products. For large numbers, it is faster
than the usual. But without knowing the usual method as a
formal procedure, it will not make sense. And it will make
as much sense if you never actually carry it out.
> I'm really a bit dumbfounded by your suggestion that
>you thought I was advocating _all_ such tasks should be
>carried out by hand. Of course not; that's silly, and I
>really think you knew I wasn't suggesting anything of the
>sort.
Herman Rubin
>In article <CL4Mo...@mentor.cc.purdue.edu>
>hru...@mentor.cc.purdue.edu (Herman Rubin) writes:
.............................
>Suppose that you want to understand how to divide one polynomial by
>another. The knowledge that I have of the procedure of long-division in
>arithmetic, and the reasons for that procedure, help me to not only do
>that task, but also to understand what I am doing.
Just describe the procedure symbolically. This shows what is going on,
and does not require manipulative skills.
John Novak
>This is semantic bigotry. Oxford's defines one meaning of machine
>as: "person who acts automatically, without thinking." A machine
>has a rigid linear structure. This means that it is an apparatus
>which is assembled from simpler components doing simple tasks. It
>is absolutely perverse that the linear philosophy has been allowed
>to pervade the education system. Learning is a NON-linear process.
There are far more appropriate definitions of 'machine' in my
little tiny Webster's Collegiate. One of them makes mention of
"an intricate natural system or organism, as the human body."
Your concept of a machine as a rigid structure is your own
baggage. A spring it not rigid, yet it can be a machine, or part
of a machine. Your concept of a machine as linear is your own
baggage. My stereo amplifiers are machines, and are very most
certainly non-linear.
Not all machinery is rigid, nor is it all linear. Any
conclusions based on those premises are destined for spectacular
error.
The 'semantic bigotry' seems to be yours.
>John Novak writes:
>>You're using emotionally loaded words and concepts, and I think
>>you know it.
>Well, of course I am. But that does not make me dishonest. What on
>earth is wrong with using emotionally loaded words? I made a
>comparison as anattempt to explain the mechanism of the mathematical
>culture. If you mean that it is wrong then refute it! Don't throw
>around with empty words like "dishonest".
If you are using emotionally loaded words to convince people,
then you are obviously unsure of the ability of your logic to
convince. Using emotionally charged words to convince people of
the merits of your arguments is considered bad form in many
forums, at the very least.
Its like comparing rape to stealing a pack of bubble-gum, because
they're both crime. Very very superficial.
And this argument over emotionally loaded words has nothing to do
with math or education, any more.
DP Laurie
Mathematics is like swimming. If you are afraid of it, you struggle and go
under. If you feel confident, you get great results with minimal effort.
In teaching mathematics, the most important thing is to let the student feel
that s/he is on top of the subject. I learnt that lesson from a music
teacher who extravagantly praised an inflexionless effort by an obviously
untalented pupil. When questioned about it, she said "If I praise her, she
goes home and practises, and feels good about it. If I scold her, there is
nothing she can do about it -- as you have noticed, she has no talent."
iiiiiii Dirk Laurie, Department of Mathematics and Computer Science
\ I / Potchefstroom University for Christian Higher Education
I P.O. Box 1174, Vanderbijlpark 1900, South Africa
In Thy Light wsk...@puknet.puk.ac.za Tel:(27)(16)807-3600 Fax:...-3614
Herman Rubin
>In article <CL6H8...@mentor.cc.purdue.edu>, hru...@mentor.cc.purdue.edu (Herman Rubin) writes:
.........................
>This sounds like nonsense to me. So far as I can see, Newton, Euler,
>Gauss and Legendre understood the integers well enough. Dedekind no
>doubt had another way of looking at the situation, by comparing the
>integers to rings of integers in number fields. But doing this, or
>even moving from N to Z, involves removing some important special
>features N.
>How, exactly, does anything that Dedekind did make it easier for children
>to understand N?
Dedekind certainly worked in rings, but this is not what he did as
far as contributing to the understanding of the integers. While
the others had an intuitive concept of the structure of the integers,
they did not, as far as we are aware, ever succeed in expressing it.
Dedekind's work here is, "Was sind, und wie sollen, die Zahlen." This
translates literally quite well; "What Are, and What Should Be, the Numbers."
This was slightly modified by Peano, who also proved that, in second order
logic, the integers were categorical.
Marvin Minsky
Benjamin J. Tilly
hru...@mentor.cc.purdue.edu (Herman Rubin) writes:
> >Suppose that you want to understand how to divide one polynomial by
> >another. The knowledge that I have of the procedure of long-division in
> >arithmetic, and the reasons for that procedure, help me to not only do
> >that task, but also to understand what I am doing.
>
> Just describe the procedure symbolically. This shows what is going on,
> and does not require manipulative skills.
>
But the fact that I am actually familiar with the calculation does help
me keep track of what is going on when I am trying to understand the
procedure. Perhaps I am unlike you in that a technical deescription of
a procedure, followed by a proof tends to confuse me to no end unless I
have the patience to sit down and try the procedure out to get a sense
of what is going on. Which requires computational skills. So for me
computational skills assist me in learning theoretical procedures.
I also find it interesting that you deleted my comments about purely
computational information that I have about things like factorization
properties helping me to understand theoretical math concepts. Remember
that math does not start or end with probability theory, as interesting
and useful as that branch of it may be.
Cheers,
Ben Tilly
Samuel R. Kaplan
Let a newcomer interrupt this dialogue for a moment. Aren't you guys
simply saying that sometimes we learn things from deduction and
sometimes from induction? When I have a mathematical problem to
solve, sometimes I recall the general rule and other times I abstract
it from concrete examples. For example, if I want to determine the
sum of an infinite series.
--
*************************************************************************
Sam Kaplan 804 982 5819
University of Virginia 804 982 5524 FAX
Center for Public Service kap...@virginia.edu
Wayne Radinsky
>
>Everybody CAN learn math, but they
>HAVE to make the effort.
And then you say:
>We have no better friend than our own mental ability,
>which we must be forced to exercise when young, for our own later
>(immense) benefit.
There is an enormous difference between a person making the effort to learn
math, and a person being forced by someone else to make the effort to learn
math.
Kids interested in arithmetic will learn it in six months. Kids being
forced to learn arithmetic will take six years, feel like they were
being punished for something, and hate it for the rest of their lives.
If you want kids to learn math, you can't just march in and say,
"you must learn this!" You must negotiate with them as if they are
human beings.
Sincerely,
Wayne
gsm...@uoft02.utoledo.edu
> Dedekind's work here is, "Was sind, und wie sollen, die Zahlen." This
> translates literally quite well; "What Are, and What Should Be, the Numbers."
> This was slightly modified by Peano, who also proved that, in second order
> logic, the integers were categorical.
If we end up trying to teach a second-order axiomization in grade
school, we are not going to produce a lot of children who know
what an integer is. If we do what we do now, and talk about
three oranges or three puppy dogs and then generalize to the
idea of three anything, then we have given them the idea. Of course,
it is the same idea as Euclid had, but I see nothing wrong with that.
If we confine ourselves to first-order theories, we will do just
the opposite of what you want--*mis*teach what the integers are,
so that it might need to be undone later. I find the idea of
grade-school kids thinking about induction in the context of an
axiomatic theory more than a little mind-boggling in any case.
Herman Rubin
This time I deleted nothing. You will find nothing in my writings
indicating that I considered probablity theory important, except as
being necessary to understand statistics. I have never suggested
that it be a core part of a mathematics curriculum, and I wish to
make it clear that, while I consider statistics important, this is
a separate problem. I specifically objected to the inclusion of
statistical procedures in the early grades as other than methodological
terminology.
As for requiring computational skills, would having a computer write
out all the steps which you do by hand be any less helpful at understanding
all but the manipulative skills? There is a point in acquiring manipulative
ability, as it is often much faster than going to the computer. But if you
cannot run fast, use a vehicle. And while I am personally quite good and
fast at computation, I find it also possible to understand even the
properties of computational procedures without doing any computation
for the purpose.
Herman Rubin
>In article <CL9s6...@mentor.cc.purdue.edu>, hru...@mentor.cc.purdue.edu (Herman Rubin) writes:
>> Dedekind's work here is, "Was sind, und wie sollen, die Zahlen." This
>> translates literally quite well; "What Are, and What Should Be, the Numbers."
>> This was slightly modified by Peano, who also proved that, in second order
>> logic, the integers were categorical.
>If we end up trying to teach a second-order axiomization in grade
>school, we are not going to produce a lot of children who know
>what an integer is.
The second-order feature in this case makes it easier. It is to
use sets instead of properties in the induction postulate. It is
not even clear that one has to show that there is essentially only
one version of the integers.
If we do what we do now, and talk about
>three oranges or three puppy dogs and then generalize to the
>idea of three anything, then we have given them the idea. Of course,
>it is the same idea as Euclid had, but I see nothing wrong with that.
The idea is that of doing the ordinals first, not the cardinals.
The axioms are about counting only, not about objects. Three is
merely the number after two; the notion of "threeness" comes in
only after a few properties of counting have been developed. This
avoids one of the problems of the cardinal approach, namely, how are
the cardinals to be defined? While you seem to dismiss this problem,
it is not easily dismissed; how do we know that if we attempt to
match up objects in a different way, they will still match up? For
infinite sets, they need not, but the treatment of finiteness from
the cardinal approach is very definitely non trivial. But the
ordinal approach avoids these problems, at the expense of having
to prove that addition and multiplication have nice simple properties.
Every rigorous axiomatic approach to set theory I have seen handles the
integers in essentially this way. The second-order problems are avoided
by embedding in set theory.
The idea of induction can be taught much more easily to first graders
than it can to those who have had too much arithmetic and manipulation.
Seeing the uniqueness of the arithmetic operations coming out of simple
axioms and characterizations will give the idea of mathematical reasoning
in an easily understood context.
Herman Rubin
>>In article <1994Feb9.0...@hsr.no> on...@hsr.no (Onar Aam) writes:
>>>It is absolutely amazing how mathematics has managed to build up its
>>>enormous position in education. Mathematics is in many ways a game
>>>for intellectuals, even so it has managed to become one of the most
>>>important subjects in school. I want to explore why this is so.
>>
>>I would say it differently. Namely
>>
>> Mathematics is one of the most important, practical and useful
>> subjects in school. It is absolutely amazing how educators have
>> managed to distort it in such a way that many students now see
>> it as no more than a game for intellectuals.
>
>
>Oh? The importance of pure mathematics is questionable. Which proofs will you
>need in engineering for instance? In what way is (pure) mathematics useful to a
>musician or a writer? And do you actually question the playfullness of math? I
>love math, it's a wonderful game to play on your own premises, but when that game
>is forced upon people it turns into a nightmare.
There is mathematics which has been applied and mathematics which has not
yet been applied. The applicability of analytic function theory, linear
algebra, group theory, and number theory, among others, have been questioned
by strong mathematicians.
Scott Brown
Herman, I have a question.
How old were the youngest students you have ever taught
algebra and arithmetic to using your methods, and what
was their math background?
Scott
--
Herman Rubin
>In article <CL4nr...@mentor.cc.purdue.edu>,
>Herman Rubin <hru...@mentor.cc.purdue.edu> wrote:
>>In article <1994Feb11....@mwc.vak12ed.edu> dsp...@mwc.vak12ed.edu (Donald (Don) Specht) writes:
>>>Herman Rubin (hru...@mentor.cc.purdue.edu ) writes:
..................
>>This does not mean that examples do not help, and does not mean that
>>lots of examples should not be given to illustrate the general concept.
>>But the generality keeps the errors from getting in from the start.
>Let a newcomer interrupt this dialogue for a moment. Aren't you guys
>simply saying that sometimes we learn things from deduction and
>sometimes from induction? When I have a mathematical problem to
>solve, sometimes I recall the general rule and other times I abstract
>it from concrete examples. For example, if I want to determine the
>sum of an infinite series.
If you mean proofs by mathematical induction, those are deduction.
All real proofs are deduction.
Induction is also used for probabilistic inference. I do not believe
that this is what you mean.
As for learning concepts, and also guessing how to proceed in solving
problems, I do not believe that we know much of anything. While things
can be done to bring insight, the muse cannot be forced to light the
lamp; creativity cannot be taught. There is some evidence that a
little understanding sometimes occurs before the "aha" phase strikes.
It is important to keep in mind that, once one understands a concept,
one is likely to kick oneself for being stupid in not seeing it before.
john baez
>In article <2jr4f8$k...@dartvax.dartmouth.edu> Benjamin...@dartmouth.edu (Benjamin J. Tilly) writes:
>>Remember
>>that math does not start or end with probability theory, as interesting
>>and useful as that branch of it may be.
>This time I deleted nothing. You will find nothing in my writings
>indicating that I considered probablity theory important, except as
>being necessary to understand statistics. I have never suggested
>that it be a core part of a mathematics curriculum, and I wish to
>make it clear that, while I consider statistics important, this is
>a separate problem.
This reminds me of something odd in the latest (February) edition of the
AMS Notices. In an article entitled The Coming Disaster in Science
Education in America, John Saxon tears into the recent "National Council
of Teachers of Mathematics" (NCTM) report. He says that "This
organization has decided, with no advanced testing whatsoever, to
replace preparation for calculus, physics, chemistry and engineering
with a watered-down mathematics curriculum that will emphasize the
teaching of probability and statistics and will encourage the
replacement of the development of paper-and-pencil skills with drills
on calculators and computers." Does the NCTM stuff really emphasize
probability and statistics? I'm just curious. (Note, I am not arguing
with Tilly or Rubin here; their remarks just triggered an association in
my mind.....)
john baez
>There is mathematics which has been applied and mathematics which has not
>yet been applied. The applicability of analytic function theory, linear
>algebra, group theory, and number theory, among others, have been questioned
>by strong mathematicians.
Eh? I don't know about "strong" mathematicians, but *smart*
mathematicians surely don't doubt the applicability of these subjects.
Analytic functions are used all the time in electrical engineering,
physics etc. (indeed I was just brought in as a hired gun by the physics
dept. to teach the grad students some complex analysis). Linear
algebra, i.e. matrices, finds occaisional application here and there,
I'm sure. :-) Group theory is basic to modern particle physics,
crystallography, etc.. Number theory is very important in modern
cryptography. So I really don't get this.
james dolan
-In article <CLBp0...@mentor.cc.purdue.edu> hru...@mentor.cc.purdue.edu
- (Herman Rubin) writes:
-
->There is mathematics which has been applied and mathematics which has not
->yet been applied. The applicability of analytic function theory, linear
->algebra, group theory, and number theory, among others, have been questioned
->by strong mathematicians.
-
-Eh? I don't know about "strong" mathematicians, but *smart*
-mathematicians surely don't doubt the applicability of these subjects.
...
-crystallography, etc.. Number theory is very important in modern
-cryptography. So I really don't get this.
rubin's claim is that the applicability of these subjects was
questioned by strong mathematicians in the _past_.
john baez
>john baez writes:
>-Eh? I don't know about "strong" mathematicians, but *smart*
>-mathematicians surely don't doubt the applicability of these subjects.
>rubin's claim is that the applicability of these subjects was
>questioned by strong mathematicians in the _past_.
Whoops. Stupid me.
Herman Rubin
The NCTM report does not change the order of development; it proceeds
in the same old way by presenting manipulations and memorization. It
does not present algebra at an earlier stage, and I suspect the
"problem solving" will be, in practice, memorization of how to solve
certain explicit problems.
It does spend quite a bit of time on probability and statistics. The
statistics part, started quite early, is in graphic presentation of
data and memorizing the formulas for, and computing, the usual "cookbook"
statistics. The probability part is mainly permutations and combinations,
and applications to games of chance. That latter part is part of algebra,
but provides no insight into probability, just how to calculate probabilities.
It is not particularly useful for a decent introduction to the important
part of statistics, namely, the formalization of problems of decision
making under uncertainty, and what one can deduce from the formulation
about lines of action.
Herman Rubin
I did not say that this applicability is questioned TODAY. But it was
by such as Gauss, Cayley, Sylvester, and everyone until recently about
number theory.
Daniel P Heyman
Two years ago I chaired a committee that was formed to suggest
curriculum changes in math and science for my local school district.
The math subcommittee consisted of about a dozen people, including
about 4 PhDs. We spent some time formulating our dream curriculum,
specifying what we wanted to add and what we wanted to delete from
the math curriculum as we understood it from our past as students and
our present as parents. The supervisor of the elementary school math
program suggested that we look at the NCTM standards, and got us a copy.
The NCTM proposals for what to add and what to delete were almost a
perfect match to ours.
They want more probability and statistics, but
the statement above is a distortion of the recommendations. Since
there is virtually no probability and statistics taught now, calling
for more of it is hardly a radical change. (Truth in advertizing alert:
I make my living by doing probability and statistics.) They want to
eliminate the homework assignements of the form: 30 problems where you
add 5 6-digit numbers or multiply 2 6-digit numbers. They DEMAND that
students learn how to do arithmetic with pencil and paper, and claim
that once the principles are understood, the mechanics are best done
by a calculator so that the students can concentrate on understanding
what arithmetic has to be done, and not get bogged down doing it. This
is important in teaching statistics, because calculating simple summary
statistics of data sets is time consuming, boring, and one is prone to
make errors when doing it by hand.
Regarding the statement about preparation for calculus, there is an
element of truth there. The NCTM claims that discrete math is ignored
in the current curricula, and that there is an overemphasis on the
stuff that leads to differential calculus. For example, one of topics
they want to deemphasize is trigonometric identities. They DON'T
want to eliminate them, they just want to reduce the time spent proving
arcane identities and use the time for (e.g.) elementary graph theory.
The NCTM proposes many things that have been suggested in this
newsgroup. Among them are more word problems, and writing the solutions
to the word problems in properly constructed sentences.
I would expect that the NCTM curriculum would produce better calculus
students and better math students in general.
--
Dan Heyman d...@bellcore.com
Donald (Don) Specht
>
> This reminds me of something odd in the latest (February) edition of the
> AMS Notices. In an article entitled The Coming Disaster in Science
> Education in America, John Saxon tears into the recent "National Council
> of Teachers of Mathematics" (NCTM) report. He says that "This
> organization has decided, with no advanced testing whatsoever, to
> replace preparation for calculus, physics, chemistry and engineering
> with a watered-down mathematics curriculum that will emphasize the
> teaching of probability and statistics and will encourage the
> replacement of the development of paper-and-pencil skills with drills
> on calculators and computers." Does the NCTM stuff really emphasize
> probability and statistics? I'm just curious. (Note, I am not arguing
> with Tilly or Rubin here; their remarks just triggered an association in
> my mind.....)
>
John,
Please read the NCTM Curriculum Standards. The basic text of
all and the suggested changes in emphasis have been posted in
this group under the titles "Substance for Ben Tilly - k-4,
5-8, and 9-12." Then decide for yourself.
We here in Clarke have embraced the Standards as our basic
guidelines for k-12 math ed. How we interpret and apply them
is a continuous process of renewal. Maybe they aren't the best
thing since sliced bread, but they give us a framework for
improvement without having to reinvent the wheel.
Now if we can make this wheel roll, all children will benefit.
---
Don Specht <dsp...@mwc.vak12ed.edu>
Clarke County High School
"Math is a language used to explain,
interpret, and predict events
in the real world."
Don_Pettengill
: Eh? I don't know about "strong" mathematicians, but *smart*
: mathematicians surely don't doubt the applicability of these subjects.
: Analytic functions are used all the time in electrical engineering,
: physics etc. (indeed I was just brought in as a hired gun by the physics
: dept. to teach the grad students some complex analysis). Linear
: algebra, i.e. matrices, finds occaisional application here and there,
: I'm sure. :-) Group theory is basic to modern particle physics,
: crystallography, etc.. Number theory is very important in modern
: cryptography. So I really don't get this.
Group theory and many flavors of Riemannian geometry were around LONG
before us fizzycists got to play with 'em. The division between "pure"
and "applied" is very time-dependent. I doubt that it is safe to label
*any* mathematics as "pure for ever", in the sense that its only utility
is its beauty. But there are certainly mathematical fields that have
yet to find any real-world application. And there are many areas of
mathematics which are "pure" in the sense that, at the level of the
student, there are no real-world applications. In fact, it is at this
level that most mathematics is first encountered - or at least, this
used to be the case. And a good way to do it, too, in my opinion. Why
bound the mathematical content by the limits of applicability? Far
better to let the student discover the uses *after* they understand the
mathematics.
regards,
don
--
________________________________________________________________________
Don Pettengill E-mail: do...@hpcvitf.cv.hp.com
Hewlett Packard IJBU, 3U-R3 Telephone: (503)750-5369
1040 North Circle Boulevard Fax: (503)750-3306
Corvallis, Oregon 97330-4200
________________________________________________________________________
Herman Rubin
>john baez (ba...@guitar.ucr.edu) wrote:
>
>: Eh? I don't know about "strong" mathematicians, but *smart*
>: mathematicians surely don't doubt the applicability of these subjects.
>: Analytic functions are used all the time in electrical engineering,
>: physics etc. (indeed I was just brought in as a hired gun by the physics
>: dept. to teach the grad students some complex analysis). Linear
>: algebra, i.e. matrices, finds occaisional application here and there,
>: I'm sure. :-) Group theory is basic to modern particle physics,
>: crystallography, etc.. Number theory is very important in modern
>: cryptography. So I really don't get this.
>Group theory and many flavors of Riemannian geometry were around LONG
>before us fizzycists got to play with 'em. The division between "pure"
>and "applied" is very time-dependent. I doubt that it is safe to label
>*any* mathematics as "pure for ever", in the sense that its only utility
>is its beauty. But there are certainly mathematical fields that have
>yet to find any real-world application. And there are many areas of
>mathematics which are "pure" in the sense that, at the level of the
>student, there are no real-world applications. In fact, it is at this
>level that most mathematics is first encountered - or at least, this
>used to be the case. And a good way to do it, too, in my opinion. Why
>bound the mathematical content by the limits of applicability? Far
>better to let the student discover the uses *after* they understand the
>mathematics.
Also, from the pedagogical standpoint, it is usually easier to learn
the "pure" idea first, and then apply it. There is very little difference
in the aspects of calculus or linear algebra as seen by biologists or
economists or physicists or chemists. Applying merely involves translating
into mathematics from the applied field.
john baez
>Also, from the pedagogical standpoint, it is usually easier to learn
>the "pure" idea first, and then apply it. There is very little difference
>in the aspects of calculus or linear algebra as seen by biologists or
>economists or physicists or chemists. Applying merely involves translating
>into mathematics from the applied field.
That may be true for you, but not for me. Maybe that's because I'm just a
mathematical physicist and not able to keep interested in things very
long unless I see how I might be able to apply them to physics. For
example, as a kid I hated algebra. Normal subgroups, Sylow's theorem
and all that seemed pretty tiresome to me. Then I found out that
group theory was the basis of the gauge theory in the Standard Model!
Of course, it wasn't the silly finite groups that the course was always
going on about, but *Lie groups*, that were what counted. My interest
in algebra picked right up and it became easier to learn. This has
happened at other times, too. The abstract theory of roots and weights
as taught in my graduate Lie algebra course seemed unmotivated and
abstract. But when I recognized how it worked in *examples,* such as
how the representation theory of SU(3) gives the classification of
mesons and hadrons in Gell-Man's "eightfold way," it all became much
more fun.
So: while teaching things "purely" may be good in some ways, it is very
difficult for most people to stay interested unless they see that the
material bears some relation to what they are interested in. I have
deliberately given rather highbrow examples to show that even someone
who sort of likes mathematics may have trouble unless they see lots of
applications; I imagine it might be even worse at lower levels.
Donald (Don) Specht
>
> So: while teaching things "purely" may be good in some ways, it is very
> difficult for most people to stay interested unless they see that the
> material bears some relation to what they are interested in. I have
> deliberately given rather highbrow examples to show that even someone
> who sort of likes mathematics may have trouble unless they see lots of
> applications; I imagine it might be even worse at lower levels.
>
Well said and right on the mark, IMO. Symbol manipulation, for
example, becomes tiresome to most hs students when applications
are not forthcoming.
Then they will ask the toughest question: "What are we ever
going to use this for?"
Herman Rubin
>john baez (ba...@guitar.ucr.edu ) writes:
>
>> So: while teaching things "purely" may be good in some ways, it is very
>> difficult for most people to stay interested unless they see that the
>> material bears some relation to what they are interested in. I have
>> deliberately given rather highbrow examples to show that even someone
>> who sort of likes mathematics may have trouble unless they see lots of
>> applications; I imagine it might be even worse at lower levels.
>Well said and right on the mark, IMO. Symbol manipulation, for
>example, becomes tiresome to most hs students when applications
>are not forthcoming.
>Then they will ask the toughest question: "What are we ever
>going to use this for?"
Routine manipulations teaches nothing but those manipulations.
The structure and concepts must be emphasized; they rarely are
even stated. Nor are students any more encouraged to think in
these terms, not that it was ever done well.
It is unfortunate that students have had the practical utility
of learning drilled into them. But what they want is not just
utility of being able to use the concepts, but they want to
see it NOW. Since sensible statistics cannot be taught without
a fairly good understanding of probability concepts, not games
of chance, what do you suggest for the person who comes in with
this attitude? Or even worse, the student who does not understand
algebraic notation and wants to understand statistics. Quite a
few college students, even with calculus, do not understand this.
Charles Yeomans
Donald (Don) Specht <dsp...@mwc.vak12ed.edu> wrote:
>john baez (ba...@guitar.ucr.edu ) writes:
>
>>
>> So: while teaching things "purely" may be good in some ways, it is very
>> difficult for most people to stay interested unless they see that the
>> material bears some relation to what they are interested in. I have
>> deliberately given rather highbrow examples to show that even someone
>> who sort of likes mathematics may have trouble unless they see lots of
>> applications; I imagine it might be even worse at lower levels.
>>
>
>Well said and right on the mark, IMO. Symbol manipulation, for
>example, becomes tiresome to most hs students when applications
>are not forthcoming.
>
>Then they will ask the toughest question: "What are we ever
>going to use this for?"
To which you should reply "that depends on what you are going to
do with the rest of your life".
In my experience, I can tell students about applications
all day, and they still won't be happy in general, because
they have little or no experience on which to draw.
Charles Yeomans
Thomas R. Scavo
>AMS Notices. In an article entitled The Coming Disaster in Science
>Education in America, John Saxon tears into the recent "National Council
>of Teachers of Mathematics" (NCTM) report. He says that "This
>organization has decided, with no advanced testing whatsoever, to
>replace preparation for calculus, physics, chemistry and engineering
>with a watered-down mathematics curriculum that will emphasize the
>teaching of probability and statistics and will encourage the
>replacement of the development of paper-and-pencil skills with drills
>on calculators and computers."
Not counting the brief letter to the editor of the _Notices_ the
previous month, this is the first article I've seen anywhere that
has come out against the Standards. I think Saxon's article is
way off the mark. Certainly his above paraphrasing of the
Standards is misleading and inaccurate.
>Does the NCTM stuff really emphasize probability and statistics?
Emphasize it? No, not at all, but the Curriculum Standards do
call for more probability, statistics, geometry, and other topics
currently receiving less than their fare share of exposure in the
elementary mathematics curriculum. Basically, the Standards are
telling us that we spend way too much time on arithmetic at the
expense of everything else. Kids come away with the view that
mathematics IS arithmetic, and nothing else. The Standards want
to change that.
Curriculum is only part of the message of the Standards. Another
important part of the movement is instruction, i.e., the way we
deliver the curriculum. My choice of words was intentional,
because in all too many cases, we as teachers do just that:
present the students with techniques and isolated facts which
they are then told to assimilate. The Standards make it clear
that there are many ways to organize a classroom (whole class,
small groups, individual) and that kids learn in a variety of
ways. This diversity of students and learning styles should be
reflected in our instructional methodologies.
A third aspect of the Standards is a new approach to assessment.
Teachers are being asked to recognize that there are various modes
of assessment (written, oral, computer-oriented) as well as a
broad range of assessment techniques (homework, projects, journals,
essays, dramatizations, class presentations, interviews) in addition
to the standardized objective tests which now prevail.
There are two other NCTM documents: the Professional Standards
(which addresses teacher training issues) and the Assessment
Standards (which take the ideas in the Curriculum Standards much
further). The Assessment Standards are in draft form, and the
NCTM is currently soliciting input from the community (contact:
National Council of Teachers of Mathematics, 1906 Association
Drive, Reston, Virginia 22091, 903-620-9840, nct...@tmn.com).
Imho, the Professional Standards are the weakest link in this
trio of reports, but that doesn't detract significantly from
the usefulness of the other two.
By publishing the Standards documents, teachers of mathematics
(via the NCTM) have taken control of their own destiny (unlike
the "new math" of the 60's where the curriculum was handed down
by committee). As a result, the mathematics community is the
envy of other disciplines. The NCTM is to be commended for its
effort.
--
Tom Scavo
trs...@mailbox.syr.edu
Thomas R. Scavo
>In article <2jugb4$o...@galaxy.ucr.edu> ba...@guitar.ucr.edu (john baez) writes:
>
>>This reminds me of something odd in the latest (February) edition of the
>>AMS Notices. In an article entitled The Coming Disaster in Science
>>Education in America, John Saxon tears into the recent "National Council
>>of Teachers of Mathematics" (NCTM) report. He says that "This
>>organization has decided, with no advanced testing whatsoever, to
>>replace preparation for calculus, physics, chemistry and engineering
>>with a watered-down mathematics curriculum that will emphasize the
>>teaching of probability and statistics and will encourage the
>>replacement of the development of paper-and-pencil skills with drills
>>on calculators and computers." Does the NCTM stuff really emphasize
>>probability and statistics? I'm just curious. (Note, I am not arguing
>>with Tilly or Rubin here; their remarks just triggered an association in
>>my mind.....)
>>
>The NCTM report does not change the order of development; it proceeds
>in the same old way by presenting manipulations and memorization.
Patently false. Nothing could be further from the truth.
>It does not present algebra at an earlier stage, and I suspect the
>"problem solving" will be, in practice, memorization of how to solve
>certain explicit problems.
Algebra, the way it's currently taught in middle schools and high
schools, could never be done at the lower levels. However, there
are ways to introduce algebraic concepts into the elementary curri-
culum. Cuisenaire rods, base ten blocks, and other manipulatives
(which the Standards support) immediately come to mind (see, for
example: M. Laycock and M.A. Smart. _Manipulative Interludes,
Algebra: Building Understanding wtih Base Ten Blocks._ Activity
Resources Co., Hayward, CA, 1990).
>It does spend quite a bit of time on probability and statistics. The
>statistics part, started quite early, is in graphic presentation of
>data and memorizing the formulas for, and computing, the usual "cookbook"
>statistics.
I don't know where you get this idea that the Curriculum
Standards emphasizes memorization and cookbook approaches to
problem solving. Nothing could be further from the truth.
>The probability part is mainly permutations and combinations,
>and applications to games of chance. That latter part is part of algebra,
>but provides no insight into probability, just how to calculate probabilities.
My son's fifth grade teacher and I just finished designing an
extended (2--4 week) lesson on probability centered around coin
tossing and dice throwing. Some of the activities have already
been tried in the classroom, and some have yet to be tried. The
unit incorporates many of the ideas in the Standards (group work,
conjectures, experiments, technology, writing, graphing, etc.)
and the kids seem to like it. Even the kids who have trouble
with arithmetic (which is all that's ever done in many elementary
classrooms) are getting into it. It's not clear whether they can
apply what they've learned to new situations however, we'll just
have to wait and see.
>It is not particularly useful for a decent introduction to the important
>part of statistics, namely, the formalization of problems of decision
>making under uncertainty, and what one can deduce from the formulation
>about lines of action.
We're also planning to do a unit on randomness, sampling, and
surveys. The class will try to determine prevailing attitudes
of fellow students on the playground, which is a real sore spot
in our school. We hope to integrate social science and
statistics (which some may argue is NOT mathematics) into one
coherent unit. It will have a significant written component
and there will be oral presentations (which the kids in this
class are really good at).
--
Tom Scavo
trs...@mailbox.syr.edu
Tal Kubo
>The division between "pure" and "applied" is very time-dependent.
>I doubt that it is safe to label *any* mathematics as "pure for ever",
^^^^^
>in the sense that its only utility is its beauty.
Such claims are often made, but I don't think they would withstand
scrutiny.
Of course there are some famous examples like number theory,
group theory, N-dimensional geometry, etc. The problem, as in
the posting quoted above, is that the next move after trotting
out these "usual suspects", is inevitably to make a completely
unwarranted generalization to *all* of mathematics. I don't see
how such examples make it plausible that uncountable abelian group
theory, or metrizability theory of topological spaces, would be
useful for any "applications".
Herman Rubin
>In article <2jugb4$o...@galaxy.ucr.edu> ba...@guitar.ucr.edu (john baez) writes:
>>This reminds me of something odd in the latest (February) edition of the
>>AMS Notices. In an article entitled The Coming Disaster in Science
>>Education in America, John Saxon tears into the recent "National Council
>>of Teachers of Mathematics" (NCTM) report. He says that "This
>>organization has decided, with no advanced testing whatsoever, to
>>replace preparation for calculus, physics, chemistry and engineering
>>with a watered-down mathematics curriculum that will emphasize the
>>teaching of probability and statistics and will encourage the
>>replacement of the development of paper-and-pencil skills with drills
>>on calculators and computers."
>Not counting the brief letter to the editor of the _Notices_ the
>previous month, this is the first article I've seen anywhere that
>has come out against the Standards. I think Saxon's article is
>way off the mark. Certainly his above paraphrasing of the
>Standards is misleading and inaccurate.
I do not agree with everything Saxon says; I myself have long
decired the emphasis on manipulative skills, and see nothing
wrong with using calculators for them. But the small amount
of real conceptual material is being greatly eroded in favor
of developing what are essentially parlor tricks.
>>Does the NCTM stuff really emphasize probability and statistics?
>
>Emphasize it? No, not at all, but the Curriculum Standards do
>call for more probability, statistics, geometry, and other topics
>currently receiving less than their fare share of exposure in the
>elementary mathematics curriculum.
The important part of the classical geometry course was understanding
proofs, not geometric facts. The important part of the classical
algebra course was the linguistic use of symbols, and the realization
that formal symbols (variables) have, when interpreted as such, the
properties of whatever they represent. This last is not too well
stated; but if numbers have a property, this property can be applied
to the symbolic expression of them.
As for probability and statistics, I am wondering if any of the
members of the panel understand those fields. It is possible to
learn a few manipulations, but, as someone who tries to get the
ideas into college students, what they know that ain't so really
does hurt them. And that is much of what the Standards call for.
Basically, the Standards are
>telling us that we spend way too much time on arithmetic at the
>expense of everything else. Kids come away with the view that
>mathematics IS arithmetic, and nothing else. The Standards want
>to change that.
I agree with that completely. But memorizing facts and techniques,
in arithmetic or otherwise, has essentially nothing to do with
getting any sort of understanding of mathematics.
>Curriculum is only part of the message of the Standards. Another
>important part of the movement is instruction, i.e., the way we
>deliver the curriculum. My choice of words was intentional,
>because in all too many cases, we as teachers do just that:
>present the students with techniques and isolated facts which
>they are then told to assimilate. The Standards make it clear
>that there are many ways to organize a classroom (whole class,
>small groups, individual) and that kids learn in a variety of
>ways. This diversity of students and learning styles should be
>reflected in our instructional methodologies.
Cut this edubabble out! The goal of any remotely decent educational
system must be to educate the individual student. I agree that the
means to do this are not fixed. It takes no curriculum standards
committee for this.
........................
>By publishing the Standards documents, teachers of mathematics
>(via the NCTM) have taken control of their own destiny (unlike
>the "new math" of the 60's where the curriculum was handed down
>by committee). As a result, the mathematics community is the
>envy of other disciplines. The NCTM is to be commended for its
>effort.
The "new math" was an attempt by MATHEMATICIANS, who understand
MATHEMATICS, to get the IDEAS, rather than the manipulations,
across to the children. It foundered mainly because the teachers
COULD NOT learn the concepts, but only manipulations. If anything,
the situation is worse now; those teachers who still had the
mathematics curriculum before the educationists dumbed it down
have all retired.
Nobody who has not had a strong conceptual background in mathematics,
including the rigorous abstract courses which unfortunately are not
taught before the junior-senior level in the better universities,
and who understands the difference between concepts and manipulations,
should be involved in setting standards.
Teaching memorization and recognition of geometric objects, and
the terminology of descriptive statistics, and probability
calculations from games of chance, will do nothing to enhance
understanding.
Herman Rubin
>>In article <2jugb4$o...@galaxy.ucr.edu> ba...@guitar.ucr.edu (john baez) writes:
....................
>>The NCTM report does not change the order of development; it proceeds
>>in the same old way by presenting manipulations and memorization.
>Patently false. Nothing could be further from the truth.
It is different manipulation, but still manipulation.
>>It does not present algebra at an earlier stage, and I suspect the
>>"problem solving" will be, in practice, memorization of how to solve
>>certain explicit problems.
>Algebra, the way it's currently taught in middle schools and high
>schools, could never be done at the lower levels. However, there
>are ways to introduce algebraic concepts into the elementary curri-
>culum. Cuisenaire rods, base ten blocks, and other manipulatives
>(which the Standards support) immediately come to mind (see, for
>example: M. Laycock and M.A. Smart. _Manipulative Interludes,
>Algebra: Building Understanding wtih Base Ten Blocks._ Activity
>Resources Co., Hayward, CA, 1990).
Not a single item you have above has anything to do with algebra.
What the heck does base ten have to do with mathematics? Nothing.
Algebra, properly done, starts out with the linguistic idea of using
variable symbols as additional pronouns, and constant symbols (the
numerals, +, -, =, etc.) as linguistic objects. The important part
of algebra is the linguistic aspects, and the realization that the
general works for the specific.
>>It does spend quite a bit of time on probability and statistics. The
>>statistics part, started quite early, is in graphic presentation of
>>data and memorizing the formulas for, and computing, the usual "cookbook"
>>statistics.
>I don't know where you get this idea that the Curriculum
>Standards emphasizes memorization and cookbook approaches to
>problem solving. Nothing could be further from the truth.
The powerful method of problem solving, namely, formulate the
problem in symbols and work with the symbols, treats the problem
well. Learning to do specific types of problems leads to the
memorization of tricks, and makes the general concept harder.
>>The probability part is mainly permutations and combinations,
>>and applications to games of chance. That latter part is part of algebra,
>>but provides no insight into probability, just how to calculate probabilities.
>
>My son's fifth grade teacher and I just finished designing an
>extended (2--4 week) lesson on probability centered around coin
>tossing and dice throwing.
This is the manipulative aspect which makes it much harder to
learn the ideas of probability. I see the students who cannot
get rid of the idea that "equally likely" is the basis.
...................
>>It is not particularly useful for a decent introduction to the important
>>part of statistics, namely, the formalization of problems of decision
>>making under uncertainty, and what one can deduce from the formulation
>>about lines of action.
>We're also planning to do a unit on randomness, sampling, and
>surveys. The class will try to determine prevailing attitudes
>of fellow students on the playground, which is a real sore spot
>in our school. We hope to integrate social science and
>statistics (which some may argue is NOT mathematics) into one
>coherent unit. It will have a significant written component
>and there will be oral presentations (which the kids in this
>class are really good at).
This is still mainly descriptive statistics, and again I must
emphasize that it makes the job of eventually learning how to
treat the real problems much harder. The mathematics is not
difficult, but requires an understanding of algebraic symbols,
not "mathematical facts", and the ability to look at problems
as abstractly formulated. If they are to ever treat problems
involving statistical reasoning in a sensible manner, it will
be necessary to completely overturn this method of thinking.
Small children can learn abstract concepts, if taught by those
who understand them. These same children, after they have been
taught by those who use routine drill instead of reasoning, have
great difficulty in learning the abstract concepts. A first
grader can learn how to use algebra as a language, and the
structure of the positive (or non-negative) integers. Those who
have been deprived of this by the gimmicks above have considerable
difficulty in being able to see this later.
Herman Rubin
Both uncountable abelian (and even solvable) group theory, and the
fact that topological spaces are metrizable under certain conditions,
have been applied in probability and statistics. For understanding
asymptotic results here, which are widely used, removing the metric
notions often adds to understanding as well. And one of the real
obstacles in this direction was in removing special properties of
finite-dimensional Euclidean spaces, which were obscuring the whole
problem immensely.
This is not esoterics, but the now standard approaches to the
distribution of results in the usual types of statistical
procedures used in physics, engineering, biology, and the
various social sciences.
Alberto Moreira
>
>>
>> So: while teaching things "purely" may be good in some ways, it is very
>> difficult for most people to stay interested unless they see that the
>> material bears some relation to what they are interested in. I have
>> deliberately given rather highbrow examples to show that even someone
>> who sort of likes mathematics may have trouble unless they see lots of
>> applications; I imagine it might be even worse at lower levels.
>>
>
>Well said and right on the mark, IMO. Symbol manipulation, for
>example, becomes tiresome to most hs students when applications
>are not forthcoming.
>
>Then they will ask the toughest question: "What are we ever
>going to use this for?"
>---
Fantastic. I'm into computers; my daughter is into biology;
my friend is an aeronautical engineer. I'd be bored to death
in a math class hearing teachers talk about aerofoils or
genetic processes to illustrate their math. Unless you split
classes early in the game, it ends up happenning the way it
happened to me: I had to go through semesters and semesters
of engineering and physics math that I threw away the moment
I started my professional career.
One of the things that irritate me most in a math class of
any sort is people trying to come up with contrived examples
about usage on disciplines that are far away from my interest.
Stick to math, I say. Leave the applications to the disciplines
that need it. Or else specialize math teaching alongside
professional careers.