Is the sky still falling?
Dom
http://www.ams.org/notices/200901/tx090100020p.pdf
provides more evidence of the continuing pseudo-education of American
students.
Barb Knox
<9d0bb15d-7bb0-4c48...@f20g2000yqg.googlegroups.com>,
Dom <DR...@teikyopost.edu> wrote:
I was particularly shocked and stunned by footnote 2:
Introductory level includes College Algebra, Precalculus,
and Math for Liberal Arts. Calculus level is Calculus I
through Differential Equations, Linear Algebra, and Discrete
Math. Advanced is everything above calculus level
including Introduction to Proofs.
So even an *introduction* to mathematical proofs is considered Advanced
university material these days. But hasn't "proof" been the WHOLE
BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
proof = what??
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
Mensanator
> In article
> <9d0bb15d-7bb0-4c48-aad6-d684aef35...@f20g2000yqg.googlegroups.com>,
>
> �Dom <DR...@teikyopost.edu> wrote:
> > The article by David M. Bressoud at:
>
> >http://www.ams.org/notices/200901/tx090100020p.pdf
>
> > provides more evidence of the continuing pseudo-education of American
> > students.
>
> I was particularly shocked and stunned by footnote 2:
>
> � �Introductory level includes College Algebra, Precalculus,
> � �and Math for Liberal Arts. Calculus level is Calculus I
> � �through Differential Equations, Linear Algebra, and Discrete
> � �Math. Advanced is everything above calculus level
> � �including Introduction to Proofs.
>
> So even an *introduction* to mathematical proofs is considered Advanced
> university material these days. �But hasn't "proof" been the WHOLE
> BLOODY POINT of mathematics since at least Euclid??!? �Mathematics minus
> proof = what??
Calculators.
Bob LeChevalier
>So even an *introduction* to mathematical proofs is considered Advanced
>university material these days. But hasn't "proof" been the WHOLE
>BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>proof = what??
Getting the right answer on the test. Just as is true for every other
subject.
lojbab
Bob LeChevalier - artificial linguist; genealogist
loj...@lojban.org Lojban language www.lojban.org
Angus Rodgers
<s...@sig.below> wrote:
>In article
><9d0bb15d-7bb0-4c48...@f20g2000yqg.googlegroups.com>,
> Dom <DR...@teikyopost.edu> wrote:
>
>> The article by David M. Bressoud at:
>>
>> http://www.ams.org/notices/200901/tx090100020p.pdf
>>
>> provides more evidence of the continuing pseudo-education of American
>> students.
>
>I was particularly shocked and stunned by footnote 2:
>
> Introductory level includes College Algebra, Precalculus,
> and Math for Liberal Arts. Calculus level is Calculus I
> through Differential Equations, Linear Algebra, and Discrete
> Math. Advanced is everything above calculus level
> including Introduction to Proofs.
>
>So even an *introduction* to mathematical proofs is considered Advanced
>university material these days. But hasn't "proof" been the WHOLE
>BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>proof = what??
Reading this article:
<http://fcis.oise.utoronto.ca/~ghanna/pme96prf.html>
Gila Hanna, "The Ongoing Value of Proof"
which was referred to in this sci.math thread from February 2002:
<http://groups.google.co.uk/group/sci.math/browse_frm/thread/b360c57e8add0379/>
"Trig Ideas for Teaching Proofs"
makes me wonder if an irrational revolutionary fad for Bourbakism
in the teaching of mathematics at secondary level (the "New Math")
was followed by an even less rational counterrevolutionary fad for
making mathematics appear as informal as possible. (This is just a
thought I had, a moment ago - not based on any actual experience!)
Quotes from Hanna's article:
"Let us now turn to the difficulties that may arise in applying
Lakatos' ideas to the classroom. While Lakatos may have chosen,
perhaps with good reason, to state some of his ideas over-
dramatically, some mathematics educators have taken many of them
literally and sought to translate them directly into classroom
practice. He dismissed certainty and infallibility with the rather
dramatic statement 'we never know, we only guess', for example, and
this has led some educators to present all mathematical knowledge as
provisional. (One cannot but wonder if they would be prepared to fund
a research project with the goal of finding the largest prime number
or a counter-example to the Pythagorean theorem.) As well, the
concepts of informal falsifiers and the fallibility of mathematics
seem to have led many mathematics educators to believe that we should
eliminate any reference to formal mathematics in the curriculum and
in particular that we should downplay formal proof (Dossey, 1992;
Ernest, 1991)."
"In the minds of many mathematics educators the status of proof has
also been called into question by the claim put forward, primarily
by other educators, that it is a key element in an authoritarian
view of mathematics (Confrey, 1994; Ernest, 1991; Nickson, 1994).
This claim owes much to Lakatos (1976), who not only challenged the
Euclidean programme for an authoritative, infallible, irrefutable
mathematics, as noted, but also wrote of the dangers of elitism in
mathematics.
What supporters of this claim would add is that the Euclidian view
is in conflict with the present values of society, which dictate
that one not defer to authority and not regard any knowledge as
infallible or irrefutable. They appear to see proof in general,
and rigorous proof in particular, as a mechanism of control wielded
by an authoritarian establishment to help impose upon students a
body of knowledge that it regards as infallible and irrefutable.
Now, it may be true that mathematics has sometimes been presented
as infallible and taught in an authoritarian way, but one could
hardly maintain that there has been a recent consensus among
educators that it should be. Whatever the case, one can only find
it strange that proof should have become the main target of what in
the end may be no more than a misguided desire to impose a sort of
political correctness on mathematics education.
It is not easy to understand, in the first place, what it means to
say that mathematics or a mathematical proof is authoritative.
Certainly a proof offered by a very reputable mathematician would
initially be given the benefit of the doubt, and in that sense the
fact that this mathematician is considered an authority by other
mathematicians would play some role in the eventual acceptance of
the proof. But the claim seems to be that the very use of proof is
authoritarian, and this claim is hard to fathom.
In fact the opposite is true. A proof is a transparent argument,
in which all the information used and all the rules of reasoning
are clearly displayed and open to criticism. It is in the very
nature of proof that the validity of the conclusion flows from the
proof itself, not from any external authority. Proof conveys to
students the message that they can reason for themselves, that they
do not need to defer authority. Thus the use of proof in the
classroom is actually anti-authoritarian."
References:
Confrey, J. (1994). A Theory of Intellectual Development.
For the Learning of Mathematics, 14(3), 2-8.
Dossey, J. (1992). The Nature of Mathematics: Its Role and Its
Influence. In D. Grows (Ed.), Handbook of Research on Mathematics
Teaching and Learning, 39-48. New York: Macmillan.
Ernest, P. (1991). The Philosophy of Mathematics Education.
London: Falmer.
Lakatos, I. (1976). Proofs and Refutations.
Cambridge: Cambridge University Press.
Nickson, M. (1994). The Culture of the Mathematics Classroom: An
Unknown Quantity. In S. Lerman (Ed.), Cultural Perspectives on
the Mathematics Classroom, 7-36. Dordrecht: Kluwer.
--
Angus Rodgers
mike3
> Barb Knox <s...@sig.below> wrote:
> >So even an *introduction* to mathematical proofs is considered Advanced
> >university material these days. But hasn't "proof" been the WHOLE
> >BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
> >proof = what??
>
> Getting the right answer on the test. Just as is true for every other
> subject.
>
That's really what education seems to have degenerated to in this
country,
isn't it. Just to score high on some dumb test, not to actually
_learn_ anything,
where you actually get _understanding_.
William Elliot
Social engineering. Do, don't think, just agree with the corporate
government's nation destroying and world ruinining stupidies, Criticial
thinking is bad for your career, bad for your country and bad for you.
Into the valley of malls drove the 400...
Their's was not to ask or reason why.
Their's was but to do and buy.
One nation under debt with perks and tax breaks for the rich.
Gib Bogle
<snip>
> It is not easy to understand, in the first place, what it means to
> say that mathematics or a mathematical proof is authoritative.
> Certainly a proof offered by a very reputable mathematician would
> initially be given the benefit of the doubt, and in that sense the
> fact that this mathematician is considered an authority by other
> mathematicians would play some role in the eventual acceptance of
> the proof. But the claim seems to be that the very use of proof is
> authoritarian, and this claim is hard to fathom.
>
> In fact the opposite is true. A proof is a transparent argument,
> in which all the information used and all the rules of reasoning
> are clearly displayed and open to criticism. It is in the very
> nature of proof that the validity of the conclusion flows from the
> proof itself, not from any external authority. Proof conveys to
> students the message that they can reason for themselves, that they
> do not need to defer authority. Thus the use of proof in the
> classroom is actually anti-authoritarian."
I agree strongly. Mathematical proofs have a universality that is hard
to match in any other intellectual activity. A proof submitted by a
Chinese peasant has as much validity as one submitted by a Field's
medalist from UC Berkeley. I was about to use the word "democratic",
but had second thoughts, since democracy in practice is about the
ascendancy of the majority view.
Angus Rodgers
<bo...@ihug.too.much.spam.co.nz> wrote:
>Angus Rodgers wrote:
><snip>
>> It is not easy to understand, in the first place, what it means to
>> say that mathematics or a mathematical proof is authoritative.
>> [...] Thus the use of proof in the
>> classroom is actually anti-authoritarian."
>
>I agree strongly.
I also agree, but, just for the record, the passage quoted above was
from the article:
<http://fcis.oise.utoronto.ca/~ghanna/pme96prf.html>
Gila Hanna, "The Ongoing Value of Proof"
--
Angus Rodgers
Herman Rubin
Barb Knox <s...@sig.below> wrote:
>In article
><9d0bb15d-7bb0-4c48...@f20g2000yqg.googlegroups.com>,
> Dom <DR...@teikyopost.edu> wrote:
>> The article by David M. Bressoud at:
>> http://www.ams.org/notices/200901/tx090100020p.pdf
>> provides more evidence of the continuing pseudo-education of American
>> students.
>I was particularly shocked and stunned by footnote 2:
> Introductory level includes College Algebra, Precalculus,
> and Math for Liberal Arts. Calculus level is Calculus I
> through Differential Equations, Linear Algebra, and Discrete
> Math. Advanced is everything above calculus level
> including Introduction to Proofs.
>So even an *introduction* to mathematical proofs is considered Advanced
>university material these days. But hasn't "proof" been the WHOLE
>BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>proof = what??
The attitude of the educationists, and of most humanists
and many social scientists, and I regret many scientists
as well, is that mathematics is merely a collection of
methods for solving problems.
Even this would allow for proofs, but as the current
emphasis in education is teaching facts and procedures,
proofs go out.
There is an even more important part of mathematics for
those who have difficulties, and even those who do not.
That is the concepts, and the use of mathematics as a
language. The engineer needs to understand the concepts
of calculus, so the differential equations formulated are
those appropriate to the problem, regardless of whether
there is an easy solution. Those with statistical problems
in all fields need to understand probability and statistics
as languages, so they can at least present their problems
to statisticians properly formulated, so that good methods
can be found or devised, and the latter is often the case.
Proofs can be properly taught in elementary school; it has
been done. But can they be taught to those who have gone
through the curriculum? The outlook here is not too good.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>Barb Knox <s...@sig.below> wrote:
>>So even an *introduction* to mathematical proofs is considered Advanced
>>university material these days. But hasn't "proof" been the WHOLE
>>BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>>proof = what??
>Getting the right answer on the test. Just as is true for every other
>subject.
Anyone who takes that view is opposed to education.
Education must be for the future, where the problems
are not that well formulated, and where formulating
the problem is the most important thing. Multiple
choice tests cannot test this
Donna Metler
college level, and no, I didn't take Calc II. I was a music and psychology
major, and had no reason to take it, and, in fact, it wouldn't have counted
as anything but a general math course. So, instead I took Discrete math,
which was just plain fun, and the Calc-Based Stat sequence (which counted
for my psych program).
Just because the top 15% of high school students take calculus doesn't mean
that they're all destined to be STEM majors. A lot of them are actually
Liberal Arts, Humanities, or even Fine Arts majors who take calculus in high
school because they have to have four years of math and they might as well
have something challenging in the sequence, or because their high school
guidance counselor has told them they need it for college admissions to get
into a competitive school (which is the primary reason for taking AP
classes), or simply because they were tracked into the algebra in middle
school, Geometry in 9th, Advanced Algebra in 10th, Trigonometry in 11th, and
Calculus in 12 sequence when they moved from elementary school, and there
was no reason to diverge from it.
A better question should be-is Calculus in high school a reasonable
requirement for college bound students NOT going into STEM careers, and if
not, what should these students take instead. If it's not helpful for
students in non-STEM programs, then maybe calculus in high school should
only be taken by those intending such careers.
Rowley
<snip>
>
> A better question should be-is Calculus in high school a reasonable
> requirement for college bound students NOT going into STEM careers, and if
> not, what should these students take instead. If it's not helpful for
> students in non-STEM programs, then maybe calculus in high school should
> only be taken by those intending such careers.
Problem is (IMO) not too many of the *typical* high school students have
a clue what their intended career might be.... a few might, but I'm
guessing I'm pretty safe saying that most don't. From what I saw during
my time as a vocational high school teacher - the system's
one-size-fits-all goal is to assume that every student is going on to
college get a four-year degree, and figuring out careers is something
the students will do after that achievement. Back in the days when I was
a high school student (1974-1978) we actually had someone on staff who's
job was to provide some form of vocational/career based counseling...
Didn't see much evidence during my stint at teaching that sort of
position carried forward to today. The councilors could talk for days on
how to get into college and/or what college was best... but ask one
something about a "career" and most times they wouldn't have anything to
say.
One of the reasons that I'm not teaching these days - things here (TX)
changed to the point that there wasn't any reasonable way that most
students could afford to take my sort of classes... The system here
added another credit requirement for math, and over the years they have
added enough AP classes to the point that most students couldn't fit any
electives into their schedules - except maybe a fine arts and/or
athletics (which are pretty much mandatory for college applications).
Martin
>
>
>
>
>
Bob LeChevalier
That is what "we the people" have chosen to have our education system
measured by, so of course that is what our education system will do.
You can call it "degeneration", but our education system is doing
precisely what we are paying it to do, focus on maximizing test
scores.
I doubt if 10% of the population cares whether people learn to do
proofs in mathematics, so long as the cashier counts their change
correctly.
Bob LeChevalier
>In article <h77ol4lp7ulrpd9ou...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>Barb Knox <s...@sig.below> wrote:
>>>So even an *introduction* to mathematical proofs is considered Advanced
>>>university material these days. But hasn't "proof" been the WHOLE
>>>BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>>>proof = what??
>
>>Getting the right answer on the test. Just as is true for every other
>>subject.
>
>Anyone who takes that view is opposed to education.
Anyone who takes that view is a pragmatist that realizes that "he who
pays the piper calls the tune". "We the people" don't give a damn
about your version of education, and "we the people" pay for the
schools.
You can possibly get what you consider a good education if you pay for
it yourself.
>Education must be for the future,
Education is for whatever the people paying for it want it to be for.
Donna Metler
"Rowley" <industry...@yahoo.com> wrote in message
news:gjj06...@news2.newsguy.com...
Well, in my case, I was pretty much set on music from age 12 or so, and took
the advanced science, math and the like only because it was either take an
easy, boring class, or a harder, slightly more interesting one. You won't
find many college fine arts majors who weren't set on fine arts before high
school, simply because if you're not focused on it and studying outside of a
normal high school's offerings, you're probably not going to be good enough
to get into a university or conservatory program.
If a student isn't really focused on a career, does it really make sense to
make them take courses designed for preparation for STEM fields, when that
student might be better served by taking a wider variety of coursework and
discovering that he/she might find a career?
Rowley
I'd be interested to know just how high a percentage the average /
typical public school high school student population that might also fit
that profile. Still, I feel where I was teaching - a student like that
would probable be more accommodated by the system than ignored.
Even so... in my own experience, I knew from an early age that I wanted
to be an artist and actively pursued that goal educational both in
school and out. But somehow during my sophomore year of high school - I
found myself unable to get into an art class that year. The course
counselor recommended I sign up for a technical drawing class (drafting)
instead, and when a space opened up in the art class I could transfer.
Later when that happened - and there was an opening - I declined to take
it as I had discovered that I really loved drafting - something about
the precision and technical aspect of it appealed to me. So instead of
going to college to study art - I ended up studying architecture and
engineering. And then while doing that developed an interest in
electronics and programming.
> If a student isn't really focused on a career, does it really make sense to
> make them take courses designed for preparation for STEM fields, when that
> student might be better served by taking a wider variety of coursework and
> discovering that he/she might find a career?
But that is the problem (IMO) - most kids today (that I know and have
interacted with) don't have a focus on a career - to the point of not
even imagining one. As a vocational teacher I tended to inquire about
such things and I can tell you most don't have a clue as to what is even
available as a choice. Personally I hate it when I later see kids who
were like that (drifting along, not considering a career) working as a
stocker / cashier at the local big box store.....
Martin
>
>
>
Dickey
Take chat to a whole new level 3D - Chat
http://imvu.com/catalog/web_invitation.php?userId=23576002&from=power-email
.
"Dom" <DR...@teikyopost.edu> wrote in message
news:9d0bb15d-7bb0-4c48...@f20g2000yqg.googlegroups.com...
> The article by David M. Bressoud at:
>
>
Robert H. Lewis
> >>proof = what??
>
> >Getting the right answer on the test. Just as is
>> true for every other subject.
>
> Anyone who takes that view is opposed to education.
>
> Education must be for the future, where the problems
> are not that well formulated, and where formulating
> the problem is the most important thing. Multiple
You and I believe this absolutely, but how many other "educated" Americans do? How many university and college administrators do? It really seems that very few people in America under the age of 40 have any idea what we are talking about. Most of those who do have learned to be wary of such "elitist ideas."
Robert H. Lewis
Fordham University
http://www.fordham.edu/mathematics/whatmath.html
Herman Rubin
>>> Barb Knox <s...@sig.below> wrote:
>>> >So even an *introduction* to mathematical proofs is considered Advanced
>>> >university material these days. But hasn't "proof" been the WHOLE
>>> >BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>>> >proof = what??
>>> Getting the right answer on the test. Just as is true for every other
>>> subject.
>>That's really what education seems to have degenerated to in this
>>country, isn't it. Just to score high on some dumb test, not to actually
>>_learn_ anything, where you actually get _understanding_.
>That is what "we the people" have chosen to have our education system
>measured by, so of course that is what our education system will do.
This is why we must get rid of government-run schools.
Even many educationists realize that this is not teaching
the important material.
Teaching trivial pursuit and testing it will not produce
people who can use even what has been taught.
>You can call it "degeneration", but our education system is doing
>precisely what we are paying it to do, focus on maximizing test
>scores.
AFAIK, no graduate school is willing to use the test
scores and grades of its students as evidence that they
have or have not learned the material well enough for
a degree.
>I doubt if 10% of the population cares whether people learn to do
>proofs in mathematics, so long as the cashier counts their change
>correctly.
This is weak elementary school arithmetic, and even this is
fading out. Most registers now say how much change is due,
and some even dish it out.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <h77ol4lp7ulrpd9ou...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
>>>Barb Knox <s...@sig.below> wrote:
>>>>So even an *introduction* to mathematical proofs is considered Advanced
>>>>university material these days. But hasn't "proof" been the WHOLE
>>>>BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>>>>proof = what??
>>>Getting the right answer on the test. Just as is true for every other
>>>subject.
>>Anyone who takes that view is opposed to education.
>Anyone who takes that view is a pragmatist that realizes that "he who
>pays the piper calls the tune". "We the people" don't give a damn
>about your version of education, and "we the people" pay for the
>schools.
>You can possibly get what you consider a good education if you pay for
>it yourself.
>>Education must be for the future,
>Education is for whatever the people paying for it want it to be for.
You are giving excellent reasons why the government
should get out of education, and most everything else.
That the "people" are paying for the public schools is
what is making education hard to get. It is not just
a matter of paying for it, but finding it to pay for.
The government must not provide what it does not have
to, or to hinder those who can provide it from doing so.
The laws of nature, unlike the laws which legislatures
pass, are not determined by popular opinion. The laws
of nature limit what CAN be done.
Herman Rubin
Donna Metler <dmme...@nospam.net> wrote:
>"Rowley" <industry...@yahoo.com> wrote in message
>news:gjj06...@news2.newsguy.com...
>> Donna Metler wrote:
>> <snip>
>>> A better question should be-is Calculus in high school a reasonable
>>> requirement for college bound students NOT going into STEM careers, and
>>> if not, what should these students take instead. If it's not helpful for
>>> students in non-STEM programs, then maybe calculus in high school should
>>> only be taken by those intending such careers.
The present calculus courses do little good, and may do
much harm. What is important in calculus is not the
calculation of derivatives and antiderivatives, but in
what things mean. Even those going into STEM careers
need to know the concepts, and some of the tricks; only
the ones doing research on the details even need to know
the gory details.
First, one needs algebra as a language; variables can
stand for ANYTHING. This is basic, and not difficult
when taught in this manner, but it was not easy to get
to this point. Now we are there, so use it.
Then one needs to learn the structure of the non-negative
integers and rational numbers. One should introduce limits
at this point, and point out that not all Cauchy sequences
of rationals have limits, and complete the system by introducing
the real numbers as the completion.
You say who needs it? Consider the idea of the tempered
scale, made important by Bach. The ratio of the pitches
of consecutive half-tones in this scale is the twelfth
root of 2; this shows the use of a geometric series.
Logarithms can be introduced here; not the computation,
but the use.
Also, measure and integration. The oldest integration with
respect to a measure is the computation of a bill; so many
"bushels" of wheat, and so many shekels per bushel, so many
bottles of wine at so many shekels per bottle, etc. We have
records of these more than 4000 years ago.
An important one is the area under a curve. At least in my
geometry course, the idea of using small squares to cover the
area, with possibly some overlap, and counting them was
introduced. This is also included in integration, which is
a sum of products of a value with respect to a measure, or a
limit of such. This simple idea of integration is certainly
important, even for such things as deciding how much carpeting
to get.
The important part of calculus is the use of rates of change;
velocity is the rate of change of position, acceleration that
of velocity. But one can consider the rate of change of
temperature or pressure with respect to time, etc. Understanding
these will enable the correct formulation of problems requiring
the use of calculus. Computing them is another matter.
>> Martin
--
Donna Metler
"Herman Rubin" <hru...@odds.stat.purdue.edu> wrote in message
news:gjlpqd$13...@odds.stat.purdue.edu...
that way. Heck, I don't, and one of my research papers is ON the mathematics
of Bach! It's not helpful to think of a tempered scale as a series of
geometric ratios when what you're worried about is how the notes fit
together in chords (Harmonic Analysis and Structure) or how notes fit
together in melodic forms and sequences (Form and Analysis). Since music
theory/composition comprises about 25% of the coursework taken by a music
major, it's a lot more important to be able to identify a Neopolitan 6th
chord than to recognize that logarithms can be used to explain how a scale
works.
By pushing every single at all academically qualified student to take
algebra in middle school and calculus in high school, at minimum, not to
mention everything else which is being pushed down, what happens is that
students end up only taking the academic core classes, with no time to
explore other fields or actually prepare for a major. As a high school
student, the actual REAL study that prepared me for a university music
program wasn't in a high school classroom, but in private lessons, working
on my own in the university library, and eventually, university level
classes taken on my own time extraneous to my high school career. Many of
the classes I was slotted into due to having been smart on standardized
tests in elementary school were interesting, but I can't say they were
particularly useful in college, let alone after graduation. And now, what I
was slotted into as the best and the brightest, who qualified for a magnet
program is what a general college bound student is told is necessary. And,
in many cases, what a student who wants to cut hair, repair cars, or build
cabinets is told is necessary, too.
Puppet_Sock
[snips]
> Mathematics minus
> proof = what??
Way easier homework assignments in 2nd year algebra.
Socks
Herman Rubin
>> >Getting the right answer on the test. Just as is
>>> true for every other subject.
>> Anyone who takes that view is opposed to education.
>> Education must be for the future, where the problems
>> are not that well formulated, and where formulating
>> the problem is the most important thing. Multiple
>> choice tests cannot test this.
>You and I believe this absolutely, but how many other "educated" Americans do? How many university and college administrators do? It really seems that very few people in America under the age of 40 have any idea what we are talking about. Most of those who do have learned to be wary of such "elitist ideas."
What is elitist about this, except that good education
MUST be elitist. Anyone can see that different people
have different abilities. Otherwise, select the members
of the athletic teams at random.
Herman Rubin
Donna Metler <dmme...@nospam.net> wrote:
I am not advocating putting algebra in middle school, but
in primary school, with the practice in solution in middle
school or earlier. Similarly, the other mathematical aspects
could be done no later than middle school.
By doing things in logical order, rather than the way that
those without any understanding of the logical structure
of mathematics are teaching it, one can accomplish this
without confusing the children as our current system does,
and giving the ideas so that future problems can at least
be properly formulated for a machine or a consultant.
The present numerical drill actually makes it harder; there
should be some computational drill, but the student should
produce the addition and multiplication tables from the
characterizations (definitions, but there are many ways to
do this), so they will be understood, not just memorized,
or even memorized.
Teach from the concepts, and the facts and algorithms can
be reconstructed, if necessary. Teach the facts and
algorithms without this, and they can be forgotten, and
there is nothing to rely upon.
amy666
yeah.
1 / 3 = 0.333333333 => x 3 = 0.9999999999
Bob LeChevalier
>In article <sa6ql4pb2mmbbm96n...@4ax.com>,
>Bob LeChevalier <loj...@lojban.org> wrote:
>>>That's really what education seems to have degenerated to in this
>>>country, isn't it. Just to score high on some dumb test, not to actually
>>>_learn_ anything, where you actually get _understanding_.
>
>>That is what "we the people" have chosen to have our education system
>>measured by, so of course that is what our education system will do.
>
>This is why we must get rid of government-run schools.
As long as the government pays for schools, they will be government
run schools.
>Even many educationists realize that this is not teaching
>the important material.
"We the people" think it is, and "we the people" are paying the bills.
>Teaching trivial pursuit and testing it will not produce
>people who can use even what has been taught.
"We the people" think it will, and "we the people" are paying the
bills.
>>You can call it "degeneration", but our education system is doing
>>precisely what we are paying it to do, focus on maximizing test
>>scores.
>
>AFAIK, no graduate school is willing to use the test
>scores and grades of its students as evidence that they
>have or have not learned the material well enough for
>a degree.
"We the people" don't care what graduate schools are willing to do,
and if "we the people" are paying the bills for that graduate school,
as we are for 80% of the universities in the country, than that
graduate school will do what "we the people" tell it to do or "we the
people" will cut off their funding.
>>I doubt if 10% of the population cares whether people learn to do
>>proofs in mathematics, so long as the cashier counts their change
>>correctly.
>
>This is weak elementary school arithmetic, and even this is
>fading out. Most registers now say how much change is due,
>and some even dish it out.
That's nice. But the bottom line is that most people don't accept
your definition of mathematics, and what you or I think doesn't count
any more than what they think.
This doesn't mean that you have to change what you think, but
pragmatism suggests that you ask for something a little closer to what
"we the people" are willing to accept.
Herman Rubin
Bob LeChevalier <loj...@lojban.org> wrote:
>hru...@odds.stat.purdue.edu (Herman Rubin) wrote:
>>In article <sa6ql4pb2mmbbm96n...@4ax.com>,
>>Bob LeChevalier <loj...@lojban.org> wrote:
...................
>"We the people" don't care what graduate schools are willing to do,
>and if "we the people" are paying the bills for that graduate school,
>as we are for 80% of the universities in the country, than that
>graduate school will do what "we the people" tell it to do or "we the
>people" will cut off their funding.
If "we the people" are going to say what a subject is,
we may as well eliminate that subject from the public
schools, because it seems that they do not want anyone
to know more than they know. This is sometimes called
"mangerism", from the parable.
>>>I doubt if 10% of the population cares whether people learn to do
>>>proofs in mathematics, so long as the cashier counts their change
>>>correctly.
>>This is weak elementary school arithmetic, and even this is
>>fading out. Most registers now say how much change is due,
>>and some even dish it out.
>That's nice. But the bottom line is that most people don't accept
>your definition of mathematics, and what you or I think doesn't count
>any more than what they think.
The schools have seen to it that the people do not have any
understanding of mathematics or physics or chemistry or
any other science. The operation of the universe is not
subject to the vote.
Thousands of laws people have spoken;
A handful the Creator sent.
The former are frequently broken;
The latter can't even be bent.
>This doesn't mean that you have to change what you think, but
>pragmatism suggests that you ask for something a little closer to what
>"we the people" are willing to accept.
Let those who want social schools, where little is taught,
have them. But do not cripple the education of those willing
and able to learn.
William Elliot
>> You say who needs it? Consider the idea of the tempered scale, made
>> important by Bach. The ratio of the pitches of consecutive half-tones
>> in this scale is the twelfth root of 2; this shows the use of a
>> geometric series. Logarithms can be introduced here; not the
>> computation, but the use.
A good musician when playing an unfretted or unkeyed instruments such as
the strings, trombone or drums can play in perfect pitch for the key. No
amount of math will ever describe this accomplished comprehension of the
tempered scale. A question for the musician in the thread, is how common
is the practice of perfect pitch playing?
> It may show it, but I daresay you won't find many musicians who think of
> it that way. Heck, I don't, and one of my research papers is ON the
> mathematics of Bach!
When I listen to Bach, I hear math. Not any specific math, but the
precision of math, the temperament of math. What math did you see in Bach?
> It's not helpful to think of a tempered scale as a series of geometric
> ratios when what you're worried about is how the notes fit together in
> chords (Harmonic Analysis and Structure) or how notes fit together in
> melodic forms and sequences (Form and Analysis).
> Since music theory/composition comprises about 25% of the course work
> taken by a music major, it's a lot more important to be able to identify
> a Neapolitan 6th chord than to recognize that logarithms can be used to
> explain how a scale works.
>
I was outrage by the music theory course I took for it included no ear
training. Is no ear training to be expected in music theory. As for
composition, just play, just sing. What is there to teach or learn? Oh
the other hand, what I hear in my mind, especially what orchestrations I
hear, I've noway of putting on the staffs. Is that what you learn in
composition class, how to write on the staffs what you hear and/or
improvise. On the other hand, does classical music training stifle
improvising. It seems to have with my mother.
Do musician ever improvise? Do they ever jam? Has there ever been a
symphonic or quartet jam session? I've hear one pianist who improvise or
jam in the mode of Bach. Has a symphony or quartet ever jammed in the
style of Bach, Bethovan or Sebalius? Oh the other hand folk and jazz
musicians constantly improvise in their jam sessions and they know and use
music theory by sound and name without ever having taken a class.
Donna Metler
"William Elliot" <ma...@rdrop.remove.com> wrote in message
news:2009010303...@agora.rdrop.com...
> On Fri, 2 Jan 2009, Donna Metler wrote:
>
>>> You say who needs it? Consider the idea of the tempered scale, made
>>> important by Bach. The ratio of the pitches of consecutive half-tones
>>> in this scale is the twelfth root of 2; this shows the use of a
>>> geometric series. Logarithms can be introduced here; not the
>>> computation, but the use.
>
> A good musician when playing an unfretted or unkeyed instruments such as
> the strings, trombone or drums can play in perfect pitch for the key. No
> amount of math will ever describe this accomplished comprehension of the
> tempered scale. A question for the musician in the thread, is how common
> is the practice of perfect pitch playing?
has perfect pitch or so good relative pitch that it's almost
indistinguishable, and can tell that a note is off in such a fine degree
that it's next to impossible. If you can't do it, you're not going to be a
professional jazz or classical musician, and probably not folk music,
bluegrass, gospel, or the less technological versions of country, either.
It's a combination of training and talent, and I tend to feel it's more on
the training side, because I work with the youngest children, and from what
I've seen, by age 5, children who have had consistent musical exposure are
pretty good on pitch within their singing range already, while those who
have not, are not.
>
>> It may show it, but I daresay you won't find many musicians who think of
>> it that way. Heck, I don't, and one of my research papers is ON the
>> mathematics of Bach!
>
> When I listen to Bach, I hear math. Not any specific math, but the
> precision of math, the temperament of math. What math did you see in
> Bach?
>
I focused on sequences from Bach cello suites (which I was also using for an
orchestration project, so I was tearing them apart note by note and really
focusing on them) and the mathematical structures they
represented-basically, looking at the structures, expressing them as
variables, then graphing them. Quite pretty, and each one is unique, but not
a big help when it comes to actually playing the Violoncello suite #2 and
making it sound musical.
>> It's not helpful to think of a tempered scale as a series of geometric
>> ratios when what you're worried about is how the notes fit together in
>> chords (Harmonic Analysis and Structure) or how notes fit together in
>> melodic forms and sequences (Form and Analysis).
>
>> Since music theory/composition comprises about 25% of the course work
>> taken by a music major, it's a lot more important to be able to identify
>> a Neapolitan 6th chord than to recognize that logarithms can be used to
>> explain how a scale works.
>>
> I was outrage by the music theory course I took for it included no ear
> training. Is no ear training to be expected in music theory. As for
> composition, just play, just sing. What is there to teach or learn? Oh
> the other hand, what I hear in my mind, especially what orchestrations I
> hear, I've noway of putting on the staffs. Is that what you learn in
> composition class, how to write on the staffs what you hear and/or
> improvise. On the other hand, does classical music training stifle
> improvising. It seems to have with my mother.
>
Ear training is a separate class, and is part of the theory core. One
semester of ear training/solfege for each semester of theory. I don't know a
single music school where this isn't the case, but no, if you enroll JUST
for theory, you don't do ear training. Usually 6 credit hours in each of the
first four semesters is theory and ear training, with 3 credits each
semester after that.
Composition is exactly that-how do you put what you hear in your head on the
staff, in a way that is specific to each instrument, because each one has
it's own limits, benefits, and musicians at different levels have different
skills-and transpositions. If you've never written an orchestral score, you
don't have any idea of the complexity involved.
> Do musician ever improvise? Do they ever jam? Has there ever been a
> symphonic or quartet jam session? I've hear one pianist who improvise or
> jam in the mode of Bach. Has a symphony or quartet ever jammed in the
> style of Bach, Bethovan or Sebalius? Oh the other hand folk and jazz
> musicians constantly improvise in their jam sessions and they know and use
> music theory by sound and name without ever having taken a class.
Even in the Renaissance and Baroque era, every single style of music allows
places for improvisation-in fact, this is what gave the baroque era it's
name. Another thing to keep in mind is that improvisation isn't always
obvious. Most people don't think of a church organist as improvising, for
example, and it certainly doesn't sound like a jazz improvisation, but most
church organists and pianists can easily take a one page hymn and make it
last 10 minutes, never repeating the same thing twice. That's improvisation,
and very, very good improvisation, at that.
>
Uh, Jazz musicians DO study theory. Take a look at the courses required for
a Jazz Studies degree, and you'll see that if anything, jazz musicians take
MORE theory than classical ones do.
As far as folk or self-taught jazz musicians, yes, they study theory too,
in an apprentice-master sort of way. Believe me, a Jazz musician knows the
difference between a D minor and a D diminished chord, and can tell you and
show you exactly what it is, and how it needs to be handled differently.
Just as music can be passed down via aural tradition, so is music theory.
In the main, where music is taught K-12, music education in the US has
improved over the last 50 years. There has been a lot of work at the
grassroots level on teaching music, and a lot of research at the ivory tower
level on how the brain works to back it up. Music is one of the few subjects
where there really is a lot of research-based instruction going on, and I'm
convinced it's largely because music has always kept it's teacher training
in-house and made sure it's teachers were musicians first, and left the
research to research-based disciplines. The best music research centers in
the world aren't in schools of music. They're in psychology, neurology, and
neuroscience programs. (Northwestern University, for example, has pretty
extensively researched early childhood music education and long-term music
education and it's relationship to language learning. Daniel Levitin's team
at McGill University is another example).
William Elliot
>>
Modern symphonic fads are deteriorating the quality of the art.
Over use of audio equipment is most annoying. Instead of hearing
a symphony, one clearly hears the instruments in the lead with
the rest of the orchestra being diminished out of the listener's
ear. The result is a series of solos with some sort of background
sounds that no longer carry the continuity and fullness of the
composition.
The second is to down tempo. Yes you hear individual notes more clearly
but the gestalt is lacking. A brilliant Bethovan sonata wilts into strings
carefully played notes, lacking in the vim and vigor, the brilliance and
excitement of the runs of notes and cascades of cords. I concluded it was
a musicians' slow down strike to lessen the number of practice hours.
>> When I listen to Bach, I hear math. Not any specific math, but the
>> precision of math, the temperament of math. What math did you see in
>> Bach?
>>
> I focused on sequences from Bach cello suites (which I was also using
> for an orchestration project, so I was tearing them apart note by note
> and really focusing on them) and the mathematical structures they
> represented-basically, looking at the structures, expressing them as
> variables, then graphing them. Quite pretty, and each one is unique, but
> not a big help when it comes to actually playing the Violoncello suite
> #2 and making it sound musical.
>
Indeed. Yes a good composer will never allow a refrain
no matter how many times it's repeated. to become stale.
> Composition is exactly that-how do you put what you hear in your head on
> the staff, in a way that is specific to each instrument, because each
> one has it's own limits, benefits, and musicians at different levels
> have different skills-and transpositions. If you've never written an
> orchestral score, you don't have any idea of the complexity involved.
>
I've enough of an idea to know to not even try it.
> Even in the Renaissance and Baroque era, every single style of music allows
> places for improvisation-in fact, this is what gave the baroque era it's
> name. Another thing to keep in mind is that improvisation isn't always
> obvious. Most people don't think of a church organist as improvising, for
> example, and it certainly doesn't sound like a jazz improvisation, but most
> church organists and pianists can easily take a one page hymn and make it
> last 10 minutes, never repeating the same thing twice. That's improvisation,
> and very, very good improvisation, at that.
>
Yes. It's the quality of a good musician. African drummers are a master
of keeping a beat with refrains and adding a second beat with varieties.
On the other hand white drummers learn a beat but they don't know it.
It's not in their bones. What is annoying about new age music and modern
popular music is that once they latch onto a neat sound or refrain, they
beat it to dead, repeating it unchangingly ad nasium. They call it music.
I call it boredom. They need to listen to the Bolero where one theme
exclusively is repeated without ever a dull moment.
> As far as folk or self-taught jazz musicians, yes, they study theory too,
> in an apprentice-master sort of way. Believe me, a Jazz musician knows the
> difference between a D minor and a D diminished chord, and can tell you and
> show you exactly what it is, and how it needs to be handled differently.
> Just as music can be passed down via aural tradition, so is music theory.
>
When they teach music theory, it's with sound; unlike the music theory
I took.
> In the main, where music is taught K-12, music education in the US has
> improved over the last 50 years. There has been a lot of work at the
> grassroots level on teaching music, and a lot of research at the ivory
> tower level on how the brain works to back it up. Music is one of the
> few subjects where there really is a lot of research-based instruction
> going on, and I'm convinced it's largely because music has always kept
> it's teacher training in-house and made sure it's teachers were
> musicians first, and left the research to research-based disciplines.
Don't we wish the sciences did the same. Just imagine a music theory
course taught by a mathematician or an education course taught by a
education theorist.
Riddle of the day.
If all you've learned is education, what have you to teach?
Jeffrey Turner
> Dom <DR...@teikyopost.edu> wrote:
>
>> The article by David M. Bressoud at:
>>
>> http://www.ams.org/notices/200901/tx090100020p.pdf
>>
>> provides more evidence of the continuing pseudo-education of American
>> students.
>
> I was particularly shocked and stunned by footnote 2:
>
> Introductory level includes College Algebra, Precalculus,
> and Math for Liberal Arts. Calculus level is Calculus I
> through Differential Equations, Linear Algebra, and Discrete
> Math. Advanced is everything above calculus level
> including Introduction to Proofs.
>
> So even an *introduction* to mathematical proofs is considered Advanced
> university material these days. But hasn't "proof" been the WHOLE
> BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
> proof = what??
Do you want kids to learn or do you want them to take tests that are
machine scoreable?
--Jeff
--
I learned that ... the most grinding
poverty is a trifling evil compared
with the inequality of classes.
--William Morris
Jeffrey Turner
I was STEM, but I found calculus very helpful in understanding Economics
101. That was before the Chicago School ideologues took over the
economics departments, but I've heard the pendulum is quickly swinging
back to reality-based economics.
Barb Knox
Jeffrey Turner <jtu...@localnet.com> wrote:
> Barb Knox wrote:
> > Dom <DR...@teikyopost.edu> wrote:
> >
> >> The article by David M. Bressoud at:
> >>
> >> http://www.ams.org/notices/200901/tx090100020p.pdf
> >>
> >> provides more evidence of the continuing pseudo-education of American
> >> students.
> >
> > I was particularly shocked and stunned by footnote 2:
> >
> > Introductory level includes College Algebra, Precalculus,
> > and Math for Liberal Arts. Calculus level is Calculus I
> > through Differential Equations, Linear Algebra, and Discrete
> > Math. Advanced is everything above calculus level
> > including Introduction to Proofs.
> >
> > So even an *introduction* to mathematical proofs is considered Advanced
> > university material these days. But hasn't "proof" been the WHOLE
> > BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
> > proof = what??
>
> Do you want kids to learn or do you want them to take tests that are
> machine scoreable?
Well, at university level they're not really kids (in the K-12 sense),
so I would expect that anyone who can handle (say) university-level
Discrete Math can also cope with actual proofs, even formal ones.
Also, there can in fact be sophisticated machine-scoreable questions. I
agree that most mechanically scored questions are of the mindless
memorisation variety, but I have taken (and produced) mechanically
scored tests that are much more sophisticated than that. Off the top of
my head, one kind of mechanically scoreable question about proofs would
be to present a proof containing some errors, and a list of 10-20
comments, and have the students tick off the comments that apply to the
proof. There would also need to be some human-scored questions, but
hey, that's what Teaching Assistants are for....
Herman Rubin
Barb Knox <s...@sig.below> wrote:
>In article <f9GdndugnO8nVP3U...@posted.localnet>,
> Jeffrey Turner <jtu...@localnet.com> wrote:
>> Barb Knox wrote:
>> > Dom <DR...@teikyopost.edu> wrote:
>> >> The article by David M. Bressoud at:
>> >> http://www.ams.org/notices/200901/tx090100020p.pdf
>> >> provides more evidence of the continuing pseudo-education of American
>> >> students.
>> > I was particularly shocked and stunned by footnote 2:
>> > Introductory level includes College Algebra, Precalculus,
>> > and Math for Liberal Arts. Calculus level is Calculus I
>> > through Differential Equations, Linear Algebra, and Discrete
>> > Math. Advanced is everything above calculus level
>> > including Introduction to Proofs.
>> > So even an *introduction* to mathematical proofs is considered Advanced
>> > university material these days. But hasn't "proof" been the WHOLE
>> > BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
>> > proof = what??
>> Do you want kids to learn or do you want them to take tests that are
>> machine scoreable?
>Well, at university level they're not really kids (in the K-12 sense),
>so I would expect that anyone who can handle (say) university-level
>Discrete Math can also cope with actual proofs, even formal ones.
I am afraid I have to disagree with you. You have no idea
about how weak university level courses are; they have been
dumbed down so the high school products can handle them,
without thinking about WHY. All they can do is memorize
formulas and methods of computation, with no understanding.
>Also, there can in fact be sophisticated machine-scoreable questions. I
>agree that most mechanically scored questions are of the mindless
>memorisation variety, but I have taken (and produced) mechanically
>scored tests that are much more sophisticated than that.
They are somewhat more sophisticated, but they still do not
get at the important material.
At a mathematics meeting decades ago, there was a discussion
of whether one could test the formulation of word problems
by multiple choice tests. My son, not quite nine yet, but
well able to follow mathematics, made the observation that
all that a multiple choice test could do is to see if they
could recognize a correct formulation. Alas, for those who
will not be mathematicians, and this includes scientists,
the important part is formulation and the use of mathematical
concepts to translate between their subject and mathematical
models, which will be used so mathematical results can get
more insight into the subject.
Off the top of
>my head, one kind of mechanically scoreable question about proofs would
>be to present a proof containing some errors, and a list of 10-20
>comments, and have the students tick off the comments that apply to the
>proof. There would also need to be some human-scored questions, but
>hey, that's what Teaching Assistants are for....
A machine can check a formal proof. A proof is a sequence
of statements, each following from the previous ones by one
of a small number of rules. A proof establishes the theorem
proved, but very often gives no insight. I have often come
up with such proofs myself, but I do not like them.
Barb Knox
Angus Rodgers <twi...@bigfoot.com> wrote:
> On Thu, 01 Jan 2009 10:55:49 +1300, Barb Knox
> <s...@sig.below> wrote:
>
> >In article
> ><9d0bb15d-7bb0-4c48...@f20g2000yqg.googlegroups.com>,
> > Dom <DR...@teikyopost.edu> wrote:
> >
> >> The article by David M. Bressoud at:
> >>
> >> http://www.ams.org/notices/200901/tx090100020p.pdf
> >>
> >> provides more evidence of the continuing pseudo-education of American
> >> students.
> >
> >I was particularly shocked and stunned by footnote 2:
> >
> > Introductory level includes College Algebra, Precalculus,
> > and Math for Liberal Arts. Calculus level is Calculus I
> > through Differential Equations, Linear Algebra, and Discrete
> > Math. Advanced is everything above calculus level
> > including Introduction to Proofs.
> >
> >So even an *introduction* to mathematical proofs is considered Advanced
> >university material these days. But hasn't "proof" been the WHOLE
> >BLOODY POINT of mathematics since at least Euclid??!? Mathematics minus
> >proof = what??
>
> Reading this article:
>
> <http://fcis.oise.utoronto.ca/~ghanna/pme96prf.html>
> Gila Hanna, "The Ongoing Value of Proof"
>
> which was referred to in this sci.math thread from February 2002:
>
> <http://groups.google.co.uk/group/sci.math/browse_frm/thread/b360c57e8add0379/
> >
> "Trig Ideas for Teaching Proofs"
>
> makes me wonder if an irrational revolutionary fad for Bourbakism
> in the teaching of mathematics at secondary level (the "New Math")
> was followed by an even less rational counterrevolutionary fad for
> making mathematics appear as informal as possible. (This is just a
> thought I had, a moment ago - not based on any actual experience!)
>
> Quotes from Hanna's article:
>
> "Let us now turn to the difficulties that may arise in applying
> Lakatos' ideas to the classroom. While Lakatos may have chosen,
> perhaps with good reason, to state some of his ideas over-
> dramatically, some mathematics educators have taken many of them
> literally and sought to translate them directly into classroom
> practice. He dismissed certainty and infallibility with the rather
> dramatic statement 'we never know, we only guess', for example, and
> this has led some educators to present all mathematical knowledge as
> provisional. (One cannot but wonder if they would be prepared to fund
> a research project with the goal of finding the largest prime number
> or a counter-example to the Pythagorean theorem.) As well, the
> concepts of informal falsifiers and the fallibility of mathematics
> seem to have led many mathematics educators to believe that we should
> eliminate any reference to formal mathematics in the curriculum and
> in particular that we should downplay formal proof (Dossey, 1992;
> Ernest, 1991)."
>
> "In the minds of many mathematics educators the status of proof has
> also been called into question by the claim put forward, primarily
> by other educators, that it is a key element in an authoritarian
> view of mathematics (Confrey, 1994; Ernest, 1991; Nickson, 1994).
> This claim owes much to Lakatos (1976), who not only challenged the
> Euclidean programme for an authoritative, infallible, irrefutable
> mathematics, as noted, but also wrote of the dangers of elitism in
> mathematics.
>
> What supporters of this claim would add is that the Euclidian view
> is in conflict with the present values of society, which dictate
> that one not defer to authority and not regard any knowledge as
> infallible or irrefutable. They appear to see proof in general,
> and rigorous proof in particular, as a mechanism of control wielded
> by an authoritarian establishment to help impose upon students a
> body of knowledge that it regards as infallible and irrefutable.
>
> Now, it may be true that mathematics has sometimes been presented
> as infallible and taught in an authoritarian way, but one could
> hardly maintain that there has been a recent consensus among
> educators that it should be. Whatever the case, one can only find
> it strange that proof should have become the main target of what in
> the end may be no more than a misguided desire to impose a sort of
> political correctness on mathematics education.
>
> It is not easy to understand, in the first place, what it means to
> say that mathematics or a mathematical proof is authoritative.
> Certainly a proof offered by a very reputable mathematician would
> initially be given the benefit of the doubt, and in that sense the
> fact that this mathematician is considered an authority by other
> mathematicians would play some role in the eventual acceptance of
> the proof. But the claim seems to be that the very use of proof is
> authoritarian, and this claim is hard to fathom.
>
> In fact the opposite is true. A proof is a transparent argument,
> in which all the information used and all the rules of reasoning
> are clearly displayed and open to criticism. It is in the very
> nature of proof that the validity of the conclusion flows from the
> proof itself, not from any external authority. Proof conveys to
> students the message that they can reason for themselves, that they
> do not need to defer authority. Thus the use of proof in the
> classroom is actually anti-authoritarian."
An interesting paper. I particularly like the conclusion:
With today's stress on teaching meaningful mathematics, teachers
are being encouraged to focus on the explanation of mathematical
concepts and students are being asked to justify their findings
and assertions. This would seem to be the right climate to make
the most of proof as an explanatory tool, as well as to exercise
it in its role as the ultimate form of mathematical
justification. But for this to succeed, students must be made
familiar with the standards of mathematical argumentation; in
other words, they must be taught proof.
Teaching students to both recognize and produce valid
mathematical arguments is certainly a challenge. We know all too
well that many students have difficulty following any sort of
logical argument, much less a mathematical proof. We cannot
avoid this challenge, however. We need to find ways, through
research and classroom experience, to help students master the
skills and gain the understanding they need. Our failure to do
so will deny us a valuable teaching tool and deny our students
access to a crucial element of mathematics.
[snip references]
Herman Rubin
>> >> http://www.ams.org/notices/200901/tx090100020p.pdf
>> Reading this article:
>> <http://groups.google.co.uk/group/sci.math/browse_frm/thread/b360c57e8add0379/
They need to be taught proof, and even more they need to
be taught concepts. With the concepts, and the formal
language of mathematics, which is that of variables and
belongs with beginning reading, they can FORMULATE
problems, even if they cannot solve them. This is what
the engineer and scientist, and the architect and
merchant, need most. Teaching people to solve simple
problems gets nowhere if more complicated situations
occur, and those without real ability will not be able to
formulate them so that they can be solved if they do not
have the concepts.
What are the key concepts? The first is the use of
temporary names, variables. This seems so straightforward
that most do not think of it, and in principle one does
not need them, but try doing anything non-trivial without
them. Elementary school word problems become trivial
with variables; one can see what needs to be done. The
key method of solving problems is the ONE rule of equality;'
the same operation performed on equal entities gets the
same result.
Then there are the other linguistic concepts. Among these
are well-formed formulas, propositions, proof methods, and
theorems. They should be aware of all of these; the
first-order predicate calculus, which is what is needed
for mathematical proofs, has been successfully taught
to fifth graders, and I believe that modifying my late
wife's logic book could co it for third graders. I would
be willing to work with others on this.
Then we have the mathematical concepts. Here we have
non-negative integer, integer, rational number, real
number, (complex number), and the standard geometric
concepts. Starting with some version of the Peano
Postulates, these could be taught early and rigorously,
with not necessarily all proofs presented. We also
need to include the cardinal version of finite sets,
but the ordinal version makes it easy, and I know of
no way to even precisely define finite set without at
some point using ordinals.
> Teaching students to both recognize and produce valid
> mathematical arguments is certainly a challenge. We know all too
> well that many students have difficulty following any sort of
> logical argument, much less a mathematical proof.
Following a formal logical argument is easy; it is the
informal arguments which are difficult. A formal proof
consists of a sequence of lines, each following from
the preceding lines by one of a small finite set of rules.
A machine can follow this.
We cannot
> avoid this challenge, however. We need to find ways, through
> research and classroom experience, to help students master the
> skills and gain the understanding they need. Our failure to do
> so will deny us a valuable teaching tool and deny our students
> access to a crucial element of mathematics.
See my remarks above. I was involved with the creation
of the rules of natural deduction in Suppes logic book,
and in my wife's textbooks. Again, if you or someone
you know is interested in working on these, I have thought
about many of the problems. However, I am not a writer.
>[snip references]
>--
>---------------------------
>| BBB b \ Barbara at LivingHistory stop co stop uk
>| B B aa rrr b |
>| BBB a a r bbb | Quidquid latine dictum sit,
>| B B a a r b b | altum viditur.
>| BBB aa a r bbb |
>-----------------------------
spudnik
of the entire scale to "A=44 Hertz" in the late '30s
-- therein lies a tale -- for the sake of "brilliance;"
it hurts.
classically, more-or-less if not an adopted standard
per se, Verdi's tuning of middle-C = 256 cycles/second is a kind
of nice. so, why is is that "C" is the cannonical major scale --
so that Am is the cannonical minor scale?
> Indeed. Yes a good composer will never allow a refrain
> no matter how many times it's repeated. to become stale.
thus:
I think it is fairly safe to say,
Man could always count *with* coconuts;
just don't hang-out underneath even one!
> man could not even count coconuts, and there are many open questions
thus:
truly, they were the most-obvious-ever-Druids-award kind
of Druids, but that doesn't mean, they were really fake, and
lots of kids have used better-quality 2-by-9s;
farmer's are good business men, two. D&D,
not the world's greatest carpenters!
> I will guess that Wookiepoopya has no rational explanation
> for cropcircles, other than "Doug and Dave
> with *the world's smallest pile* of 2by9s."
thus:
groovier than a pile of rocks,
just by comparison with the Face and Genitals ... a-hem;
Stranger in "a" Strange Land territory.
thus:
dood was totally quanta; I mean, if
you're going to say "it's a singleton -- yeah;
that's what it is!" well, OK;
what spin does it partake of -- is it a fermion?
I know, there are those of you who'd jump
on the answer, "weellll, if there's only one?" -- and
so would I, myself & what's-his-name; me, not just so as
to not fall into the Many Universes default; except,
when I'm "@Nancago.edU of." -- death to the triangle!
any way, Stanton F. is higher than a pile of rocks, iff
you listen to Art Bell et sequentia Al Nouri;
is his first name really Albert?... does Bell parlay
of distance at a spooky action, huge media conglome,
semiotic highway signage?... oh, my;
they are actually better than cropcircles, if
you can just get the slant on the "correct side;"
even a mobius strip has at least one!
thus:
did Copernicus expose Ptolemy's hoax?... and, if not,
how'd you know that it was wrong --
did you know that the sky is round?
spudnik
you know in what direction.
I'm sure that Andrew was not the only one,
who asked Doctor Spock, if that was entirely logical,
Captain Crunch!
thus:
Ribet'd be a great escrow agent; stipulate,
he keeps the money, if you're wrong:
he's at UCBerkeley, I think.
> Cf.: K. Ribet's
> ``From the Taniyama-Shimura conjecture to Fermat's last theorem"
> (1990),http://www.numdam.org/numdam-bin/item?id=AFST_1990_5_11_1_116_0
thus:
there was an arbitrary shift of temperament
of the entire scale to "A=44 Hertz" in the late '30s
-- therein lies a tale -- for the sake of "brilliance;"
it hurts.
classically, more-or-less if not an adopted standard
per se, Verdi's tuning of middle-C = 256 cycles/second is a kind
of nice. so, why is is that "C" is the cannonical major scale --
so that Am is the cannonical minor scale?
> Indeed. Yes a good composer will never allow a refrain
> no matter how many times it's repeated. to become stale.
thus:
I think it is fairly safe to say,
Man could always count *with* coconuts;
just don't hang-out underneath even one!
> man could not even count coconuts, and there are many open questions
thus: truly,
they were the Guinsees-Book-obvious-ever-Druids-award kind
of Druids, but that doesn't mean, they were perfectly fake, and
lots of kids have used better-quality 2-by-9s, since
they retired -- a-hem; farmer's are good business men, two....
Michael Press
Barb Knox <s...@sig.below> wrote:
> Teaching students to both recognize and produce valid
> mathematical arguments is certainly a challenge. We know all too
> well that many students have difficulty following any sort of
> logical argument, much less a mathematical proof. We cannot
> avoid this challenge, however. We need to find ways, through
> research and classroom experience, to help students master the
> skills and gain the understanding they need. Our failure to do
> so will deny us a valuable teaching tool and deny our students
> access to a crucial element of mathematics.
Start young; it gets late early.
<http://www.cs.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf>
-------------
Abstract
Learning to program is notoriously difficult. A substantial
minority of students fails in every introductory programming
course in every UK university. Despite heroic academic
effort, the proportion has increased rather than decreased
over the years. Despite a great deal of research into
teaching methods and student responses, we have no idea
of the cause.
It has long been suspected that some people have a natural
aptitude for programming, but until now there has been
no psychological test which could detect it. Programming
ability is not known to be correlated with age, with sex,
or with educational attainment; nor has it been found to
be correlated with any of the aptitudes measured in
conventional 'intelligence' or 'problem-solving-ability'
tests.
We have found a test for programming aptitude, of which
we give details. We can predict success or failure even
before students have had any contact with any programming
language with very high accuracy, and by testing with the
same instrument after a few weeks of exposure, with extreme
accuracy. We present experimental evidence to support our
claim. We point out that programming teaching is useless
for those who are bound to fail and pointless for those
who are certain to succeed.
-------------
The aptitude test is a bunch of questions one would find
in the first week's homework assignment. The people who
could not answer the aptitude test questions before the
course could not answer them at the _end_ of the course.
--
Michael Press
Herman Rubin
>-------------
> Abstract
------------
>The aptitude test is a bunch of questions one would find
>in the first week's homework assignment. The people who
>could not answer the aptitude test questions before the
>course could not answer them at the _end_ of the course.
--
>Michael Press
The problem is again linguistic. One thing which throws
programmers off, and is a problem in programming languages,
is that of variables, which in programming are NOT the same
as in mathematics.
Essentially what happens is that any computer operates
on the contents of locations, real or virtual, and the
contents of a location is a variable. This is never
pointed out. When one writes "x = 3", this means that
"3" is put in the location assigned to x, and this
assignment may be quite complicated, but rarely does
the user have to know this.
Another problem is that they start out with a high level
language; I have never found one which has been reasonably
done. I believe that a good version of an assembler
language can be produced, which does not take much longer
to program in than a HLL. At this level, even the most
sophisticated architecture is easy to understand.
Assemblers rarely use overloaded operators, which makes
them very slow to use.
Angus Rodgers
[Cross-posting removed, because I'm ranting about a subject
I don't know a lot about, and sci.math is probably at least
wearily accustomed to my occasional ramblings by now, but I
can't expect other NGs to be as tolerant.]
On Thu, 08 Jan 2009 21:28:37 -0800, Michael Press
<rub...@pacbell.net> wrote:
>In article <see-8B0FC2.2...@feeder.motzarella.org>,
> Barb Knox <s...@sig.below> wrote:
>
>> Teaching students to both recognize and produce valid
>> mathematical arguments is certainly a challenge. We know all too
>> well that many students have difficulty following any sort of
>> logical argument, much less a mathematical proof. We cannot
>> avoid this challenge, however. We need to find ways, through
>> research and classroom experience, to help students master the
>> skills and gain the understanding they need. Our failure to do
>> so will deny us a valuable teaching tool and deny our students
>> access to a crucial element of mathematics.
>
>
>Start young; it gets late early.
><http://www.cs.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf>
I've written a parser generator for regular right part attribute
grammars in C++ (whose multiple levels of recursion made my eyes
water, and which was appallingly badly written, but which I got
to work, after a fashion), and programmed in FORTRAN IV, Algol
60, assembly language, BASIC, PL/I, Miranda, SML, and I forget
what else, but I would probably fail any introductory programming
course given by Richard Bornat, just because he is such a scary
bloke! (If you think David Ullrich is intimidating, think again!)
I shall have to read this (somewhat sceptically, and nervously).
My impression of Prof. Bornat's introductory programming book
was that he expected the reader to jump through /exactly/ the
same mental hoops as he does himself, whereas I believe there
should be room for people to work in their own individual style,
so long as they observe some rational standards, and are managed
well. (But then, I never actually did have a job as a programmer,
so I don't really know what it takes to do it professionally.
Also, I don't know if I still have my notes on Prof. Bornat's
book, and my memory of it and other things is unreliable.)
Quote:
"The remaining 8% refused to answer all or almost all of the
questions. We call this the blank group."
I haven't even read the paper yet, and I already want to join
the blank group! :-)
If I have an objective point to make here (off the top of my
head), it is that program code has human qualities, inherited
from its creator (this is also true of programming languages,
as a special case - and also true of programming textbooks,
as I've already all but implied - and it doesn't presuppose
or imply any beliefs about "artificial intelligence"), and
therefore the question of how programming is best done is a
very human question, not a merely technical one with a single
best answer. Programmers should not be programmed. I don't
think there are any special rules about large programming
projects other than those that ought to govern any large
cooperative human activities.
I think Prof. Bornat is rightly concerned that programming,
like mathematics, should be a highly /rational/ activity,
but with him, as with many theoretical computer scientists,
I have the impression that rationality is confused with an
adherence to prescribed formal standards - a confusion that
mathematicians are less likely to make.
I have been meaning for a long time to try to learn to program
properly (all my efforts have been amateurish), and I would be
interested to learn of ways (such as Knuth's "literate programm-
ing", perhaps?) in which programming can be made more rational
(or in another word, scientific), without becoming too formally
controlled. Certainly some sort of correctness proofs should
be involved (and I think Prof. Bornat is rightly very concerned
about this), but I am open-minded (or vague, if that seems more
honest) as to what form they should take - /except/ that they
/must/ be human-readable (even if some or all of them are also
carried in some machine-readable form).
There must be many sci.math readers who (unlike me) program or
have programmed professionally. How do they believe it is best
for people to communicate their reasonings as to why they think
their programs "work"? What progress has been made in recent
decades to make programming /genuinely/ more like mathematical,
scientific, and/or engineering activity (all of which it should
be, but which I think it has too often been falsely advertised
as being)?
--
Angus Rodgers
Cary Kittrell
Historical curiosity: Pascal's assignment statement:
x := 3
is alleged to have been meant to suggest an arrow,
and thus make the difference between assignment
and equality:
x = 3
more obvious.
-- cary
Angus Rodgers
>On Thu, 08 Jan 2009 21:28:37 -0800, Michael Press
><rub...@pacbell.net> wrote:
>
>>Start young; it gets late early.
>><http://www.cs.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf>
>
>[...]
>
>My impression of Prof. Bornat's introductory programming book
>was that he expected the reader to jump through /exactly/ the
>same mental hoops as he does himself [...]
>
>[...] Programmers should not be programmed. [...]
>
>I think Prof. Bornat is rightly concerned that programming,
>like mathematics, should be a highly /rational/ activity,
>but with him, as with many theoretical computer scientists,
>I have the impression that rationality is confused with an
>adherence to prescribed formal standards - a confusion that
>mathematicians are less likely to make.
>
>[...] Certainly some sort of correctness proofs should
>be involved (and I think Prof. Bornat is rightly very concerned
>about this) [...]
Reading a little way into the paper, I see that Prof. Bornat
himself has come to [at least some of] the same conclusions,
the hard way:
"Programming teachers, being programmers and therefore formalists,
are particularly prone to the ‘deductive fallacy’, the notion that
there is a rational way in which knowledge can be laid out, through
which students should be led step-by-step. One of us even wrote a
book [8] which attempted to teach programming via formal reasoning.
Expert programmers can justify their programs, he argued, so let’s
teach novices to do the same! The novices protested that they didn’t
know what counted as a justification, and Bornat was pushed further
and further into formal reasoning. After seventeen years or so of
futile effort, he was set free by a casual remark of Thomas Green’s,
who observed “people don’t learn like that”, introducing him to the
notion of inductive, exploratory learning."
--
Angus Rodgers
Michael Press
Angus Rodgers <twi...@bigfoot.com> wrote:
I have no ideas, good or bad, on pedagogy. My approach
to programming has always been "This is what I want to
do, here is a computer, here is a language. How do I
translate my formulation of my solution into a set of
computer instructions?" I only ever took one formal
computer class, and it was taught by a gent of wide renown.
The class was good, and I did well.
--
Michael Press
Angus Rodgers
>If I have an objective point to make here (off the top of my
>head), it is that program code has human qualities, inherited
>from its creator (this is also true of programming languages,
>as a special case - and also true of programming textbooks,
>as I've already all but implied - and it doesn't presuppose
>or imply any beliefs about "artificial intelligence"), and
>therefore the question of how programming is best done is a
>very human question, not a merely technical one with a single
>best answer. Programmers should not be programmed. I don't
>think there are any special rules about large programming
>projects other than those that ought to govern any large
>cooperative human activities.
>
>[...]
>
>I have been meaning for a long time to try to learn to program
>properly (all my efforts have been amateurish), and I would be
>interested to learn of ways (such as Knuth's "literate programm-
>ing", perhaps?) in which programming can be made more rational
>(or in another word, scientific), without becoming too formally
>controlled. Certainly some sort of correctness proofs should
>be involved (and I think Prof. Bornat is rightly very concerned
>about this), but I am open-minded (or vague, if that seems more
>honest) as to what form they should take - /except/ that they
>/must/ be human-readable (even if some or all of them are also
>carried in some machine-readable form).
>
>There must be many sci.math readers who (unlike me) program or
>have programmed professionally. How do they believe it is best
>for people to communicate their reasonings as to why they think
>their programs "work"? What progress has been made in recent
>decades to make programming /genuinely/ more like mathematical,
>scientific, and/or engineering activity (all of which it should
>be, but which I think it has too often been falsely advertised
>as being)?
Here's a concrete suggestion (again off the top of my head, and
again based on intuition, rather than professional experience):
Programming should be considered as much as possible as an act of
human communication. The programmer should be encouraged as much
as possible to forget that his/her instructions will ultimately be
carried out by a machine, with no understanding of their meaning.
This makes self-deception harder, by encouraging peer review in
the same way as preparing a mathematical or scientific article
for publication. Indeed, a computer program should always be
considered first and foremost as a publication - which, however,
will usually be of limited circulation - although the existence
of the Internet (and vast modern data storage capacities) opens
up the possibility of archives for public access.
Presumably the open source movement has already made a lot of
progress along these lines? (Pardon me for being out of touch.)
The slogan ought to be something like: "Don't try to convince
the stupid machine; and don't try to convince yourself; don't
try to construct a mechanically formal proof of correctness,
either; instead, try first to convince /me/!"
How practical is this suggestion?
--
Angus Rodgers
William Elliot
> The problem is again linguistic. One thing which throws programmers
> off, and is a problem in programming languages, is that of variables,
> which in programming are NOT the same as in mathematics.
Nor is
16 mod 5
anything like
1 = 16 (mod 5).
> Essentially what happens is that any computer operates on the contents
> of locations, real or virtual, and the contents of a location is a
> variable. This is never pointed out. When one writes "x = 3", this
> means that "3" is put in the location assigned to x, and this assignment
> may be quite complicated, but rarely does the user have to know this.
More vivid is the contrast between
x := x + 3
and
x = x + 3.
> Another problem is that they start out with a high level language; I
> have never found one which has been reasonably done. I believe that a
> good version of an assembler language can be produced, which does not
> take much longer to program in than a HLL. At this level, even the most
> sophisticated architecture is easy to understand.
Is knowing computer language first a handicap for learning math?
Somehow these computer geeks think that they know math because
they know computerese. The reverse seems more the likely, that to learn
math requires unlearning what math they think they geek know.
> Assemblers rarely use overloaded operators, which makes them very slow
> to use.
My system analysis friends are nearly mathematically illiterate. It's no
handicap to their work. They ask me how to make efficient a mathematical
routine. Unlike geeks, they know what they don't know.
What's weird is the pressure being put upon community college teachers
to teach algebra and other math courses with graphic calculators being the
major approach to the subject. Isn't that teaching mathematical
illiteracy? Are college teachers and professors also pressured toward
teaching calculators foremost?
Barb Knox
> In article <see-8B0FC2.2...@feeder.motzarella.org>,
> Barb Knox <s...@sig.below> wrote:
>
> > Teaching students to both recognize and produce valid
> > mathematical arguments is certainly a challenge. We know all too
> > well that many students have difficulty following any sort of
> > logical argument, much less a mathematical proof. We cannot
> > avoid this challenge, however. We need to find ways, through
> > research and classroom experience, to help students master the
> > skills and gain the understanding they need. Our failure to do
> > so will deny us a valuable teaching tool and deny our students
> > access to a crucial element of mathematics.
>
>
> Start young; it gets late early.
> <http://www.cs.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf>
A very interesting paper. It is some comfort to know that the majority
of those 1st-year programming students whom I failed to reach were
probably unreachable to begin with.
That's not the result. The result is that the students who initially
answered the various questions in a CONSISTENT manner (even if those
answers were wrong) were capable of learning programming, whereas those
who used no consistent method at all (not even a wrong one) were not
capable.
Herman Rubin
The answer is probably yes. Using flow charts might help, but the
tricky languages which have all their restrictions leave out the
flexibility of mathematics, as well as the fact that mathematical
notation has often been preempted in computer languages.
Oh, be careful of optimizing compilers. They often "optimize"
out the part which is of interest.
>> Assemblers rarely use overloaded operators, which makes them very slow
>> to use.
>My system analysis friends are nearly mathematically illiterate. It's no
>handicap to their work. They ask me how to make efficient a mathematical
>routine. Unlike geeks, they know what they don't know.
Are they mathematically illiterate? Can they formulater their
problems; if they can do so they are using the "literate" part
of mathematics. Being unable to solve problems is not being
illiterate. The mathematical analog of simple native language
literacy is being able to formulate a sizable problem, using
however many variables are convenient.
>What's weird is the pressure being put upon community college teachers
>to teach algebra and other math courses with graphic calculators being the
>major approach to the subject. Isn't that teaching mathematical
>illiteracy? Are college teachers and professors also pressured toward
>teaching calculators foremost?
As a means of solving well-formulated problems, there is nothing
wrong with using calculators and computers, provided one understands
the concepts well enough to know what is being done.
As a means of gaining understanding, computers do not cut it.
Computers are not 'brains"; they are sub-imbeciles. In many
cases, they can do by brute force what people cannot seem to
find the way, but this does not make them intelligent.
galathaea
i'm gonna type down some topics
and then try to explain their relation to your post
- reuse
- structured programming
- functional programming and validation
- object and type
- is-a / plays-a inheritance vs duck-typed genericity
(and the uses of abstraction in reuse)
- patterns and idioms of a language
and abstract structural architecture theory
- theory of formal structural architecture
(state machines and declarative transport from i/o
in relation to run-once-calculate vs interaction systems)
- domain specific languages and generative programming
- intentional programming
- process and agile programming
(evolution from waterfall into modern techniques
refactoring in architectural theory
test-driven techniques and things like pair programming)
i think much of what you ask
has some good answers in this path of topics
but it is a long path
and so i hope a fuzzy overview can be helpful
use is the start of social programming
if one person is writing code
that another person is going to use
additional constraints on output are imposed
isolated coding
where the program's source will never be used by another
need not conform to any requirements
beyond those of the programmer for their aims
the aims may be given externally
like the output of compiling must be an executable
with certain properties runtime
like being able to run on a given physical system
but the language chosen to write the program
and the structure of the solution
have wide flexibility
when use is considered
though
several additional real world constraints are usually added
usually one wants that the one who uses the code
should be able to use it more simply than writing it themselves
this is often broken down in two types of constraints:
- temporal: it should take less time to understand and use
than solving the problem domain
- space or complexity: the amount of code needed
should be smaller using the code than writing it
this is where the analysis turn to _interfaces_
which what the _user_ of the code interacts with
to use the supplied code
there is an equivalent runtime consideration and theory
but i will stick to the code part itself
now use is not something that frequently happens once
ideally
one task should only ever have to be coded once
even across entire societies
once it's done
it doesn't need to be done again unless a bug is found
reuse of an interface can be measured using a variety of metrics
the complexity of use metric mentioned earlier
is one that immediately measures something relevant
because time saved has immediate economic value
algorithmic complexity is another example
when looking at the runtime interface
computer scientists created
many formal descriptions of these interfaces
through the formalisation of languages
on models of computation
it became immediately clear
that there was a distinction between two formal styles
imperative programming used languages
to manipulate state in the manner required
using descriptions of the manipulations
to build interfaces of subroutines or procedures
because this style manipulates state
it allows for side-effects
where calling an interface
does not produce the same behavior every time
this is a very useful feature
for programs that interact and evolve with their user
but it makes it very difficult to analyse for correctness
behavior must be checked so that all assumptions
or invariants expected of the interface
are maintained by access over the entire state space
to get around some of these difficulties
programming theory developed rules
like those of the structured programming paradigm
which
if followed
made it easier to verify
goto's were considered harmful
multiple returns
a variety of structural analyses
removed loops and branches of unnecessary complexity
the other style identified was declarative
languages that provide a logical means
of describing what a program should accomplish
instead of how it does it
side effects are anathema to these types of languages
they usually do not have names
that refer to locations of state
(which can change)
instead
the names stand for instances of state
values
obeying some static specification
functional and other declarative languages
are much more open to verification
particularly since the specification
is directly related to the program
additional verifications can often be read off the source
and this can be formalised in program reductions
and a general theory of interface transformations
however
imperative programming became much more widely used
and there appear to be at least a few reasons why
- engineers tend to approach problems more procedurally
building compositionally with working state
often gives initial solutions through brainstorming
- you need state to provide for an i/o system
if it is going to interact with the user
and adapt to present optimal interfaces for the user's task
functional programming languages adopted monads
and similar state machine mechanisms
to accommodate such needs
early on
both declarative and imperative styles
learned the advantages of type theory
to the ensurance of program correctness
the data structures expected by interfaces
need to be the data structures given to them
errors of type are a large class
when interface procedures and functions
are specified to take specifically named types
the compiler gains the capability to verify use
and refuse to compile type mismatches that would cause errors
a procedure to join two trees at the root
would not have to worry about trying to join two cyclical graphs
because the interface could specify
rootJoin: Tree -> Tree -> Tree
and attempts to pass objects like Graphs would not compile
there would be a need to pass the Graph first to a
ifConvertibleReturnTree: Graph -> Tree
which specifically tests
and returns the correct type or exceptions / errs
the semantics of typed construction
allowed the compiler to interpret the syntax of a program
for better correctness verification
associating interfaces to types created this association
which exists in many informal languages
nouns and verbs
objects and their transformations
the ontology of types could replicate
the roles played in any describable world or model
now
to promote reuse
it was found that structure could be usefully inherited
to compress future specifications
a library could be written offering functionality
for anything that obeyed an abstract interface specification
even though the things could still be distinct types
this type relation allows
for instance
a Tree to be passed to a function that takes a Graph
a Tree _is_a_ Graph
even though a Graph isn't necessarily a Tree
a Tree has an adjacency matrix representation
and a representation in the vertex / edge collection form
and a (fairly trivial) dual
and ...
a List is a Tree and thus also a Graph
conceptually
the inheritance relation is often called an "is-a" relation
as the conceptual boundaries tend to follow
the standard venn-diagram usage
however
since the interfaces expected in usage
particularly in imperative languages
are not always completely specified
and not all invariants of the conceptual space are covered
the relation often amounts to
whether the type "plays-a" nother type well-enough
each of these data structures shouldn't have to reimplement
the common functionality available to interfaces
unless algorithmic complexity gains are possible
or other optional economies are present
inheritance expands the scope of reuse of libraries
the open-closed principle describes the general goal of reuse
to provide for interfaces that are closed to modification
(every piece of the code has been written
and need not be modified to work over
the whole space of it's interface)
and yet open to expansion
(it's interface uses a type inherited from
allowing much more potential capability for the future
than may be available on first use)
this inheritance type-matching is usually declared in the syntax
through an explicit relationship created between types
so an interface may be verified
according to a greatest lower bound Type it is expressed to
but there is a different kind of genericity
that instead of working with explicitly defined types
is in a sense type-generating
creating the lowest bound Type when the interface is accessed
during the translation process
languages that define potential interfaces
through generic arguments that must simply have available
the interface used by the routine
without any further constraints
are in a sense duck-typed in this genericity
if it looks like a duck
and it walks like a duck
and it sounds like a duck
it's good enough to be a duck
for the purposes of the routine
the collection of interfaces used by the routine
would define the type's glb for the interface
these two types of polymorphism
inheritance and generic
have two different kinds of concerns for verification
the former wants the invariants of the base obeyed
the base is defined in code
maybe it's just an interface
but it's explicitly called something
so other coders have an idea of what it should be
the former type really wants an is-a
and not the plays-a
because it wants the coders using it in interfaces
to be able to make assumptions of invariants
any actual invariant enforcement
occurs relative to the base class
and so is often implemented "there"
the latter genericity
wants the plays-a
it's how it offers greatest reuse
the verification here
must be that the routine uses no additional assumptions
the additional assumptions that usually arise
involve assumption on the state invariants
of the objects passed
if it is being passed values as in functional settings
it does not have these potential errors
but more generally
if referential transparency can be proven
the genericity is safe
often these two types are also separated by translation models
inheritance polymorphism can have interfaces
that are compiled seperately
whereas generic interfaces usually need
to compile the interface with the code using it
however
over the entire translationphase
the distinction is not always relevant
but because of the type exclusion in inheritance
where types with the same signature
still may not be allowed in an interface
generic relationships offer a different way to form relationships
which can affect runtime behavior of generic objects
together
they form a deep calculus for building reusable code
that can define strict verification systems
into the type check system
and being able to name types by behavior
is really what programmers are doing anyways
so trying to build intuitive ways to express this
is a basic goal of building computer languages
programmers tend to think of abstract "places" of code
with interactions between various places
thought of as calls and returns
passing information of these various types
the "places" tend to be thought of
as warehouses and factories and other job locations
where the information gets "worked" on
in ways that map to the real-world tasks being worked on
these places often have very common conceptual tasks
and it has been found that there are many
structural idioms and patterns
that are common across a language
and even across multiple languages
architectural theory has begun formalising these patterns
and many are easy to formalise correctness
(and there are great famous instances
where correctness investigations failed
see: double-checked locking in singletons)
one of the most fundamental patterns
differentiates two different kinds of programs
programs that are meant to calculate a result
possibly taking information input during execution
and producing a result
as mentioned earlier
declarative programs are always possible here
and they have advantages in verification
so from such a view they are the preferred solution
in terms of possibility of correctness proofs
programs that are meant to interact
though
will need to have an event loop of some sort
this is the event-driven programming paradigm
the event loop is what reads information off input devices
and dispatches the data to parts of the program
that will handle the information properly
layering in architecture
is a common way to factor out type relationships
there is a measure of type information dependency
called coupling
that dictates when different parts of code may be worked on
independently of other parts of the code
interface use produces coupling
and the various polymorphisms are means of reducing coupling
layering is a way to extract coupling into horizontal units
that can help isolate code into areas of "fixedness"
each layer performs it's duties
using any layer it is above
hiding it's work behind it's own interface
for higher layers
propagation of changes can be isolated better
it is a real challenge to keep changes from propagating
because real code does change
based on new requirements
bug fixes
and such
many of the challenges of reuse
reduce to minimising coupling
and on event loops it has been seen
that three natural layers are common architecturally
the communications layer is the code
that interacts with the input and output devices
keyboard input
mouse movement
serial ports
ethernet
...
these are usually libraries or modules
that have no idea what the data means
and so they just deliver incoming information
as blobs or packets deciphered only minimally
(input devices separate their data into symbols
that although meaningful to the device
like a certain key pressed or mouse button clicked
may be interpreted by the program in many different ways)
a layer above this
the transport layer
typically is used to transform this information
into the type system of the third or application layer
this transformation involves serialising the data blob
into a fully typed value or object
and dispatching the result
to the application layer handler that has registered to receive it
it also handles deserialising data to output
coming in as types from the application layer
the application layer
then resides completely in a type space
that describes it's problem domain
as far as verification goes
this allows the lower layer to be verified independently
in respect to their abstracted interfaces
the communication layer need only be verified
to the specification of their associated devices
the transport layer can be written completely declaratively
assisting verification tools
and the application layer
implements the statemachines that process the events
in a type-safe framework relative to it's domain
these application layer type constructs
help define what are called domain-specific languages
in that they use names and relations
from the real-world ontology they represent
verification of correctness must undergo
the full statespace verification
that imperative programming requires
but can be assisted by the type verification
in general
a full domain-specific language expressiveness
becomes desirable
since verification might be possible
by experts in the specific domain
accounting programs might be verified by accountants
3d graphics program might be verified
by experts in 3d graphics programming
and much of this expertise
might be encoded in the type system architecture
so less experienced programmers can still contribute
this attempt to move as much application code
into a language understood in the domain
has it's logical conclusion
in what is called intentional programming
and related technologies for reducing programming
to the abstractions used to specify it
this reduces the full issue of program verification
to verification in a specific ontology
created by the programmers for the problem
in the limit of this approach
the full domain language would be transferred
matrices would be represented as mathematicians are familiar
a theory studied in japanese
could be coded in japanese
all symbols specific to a domain
would be renderable
along with this language level evolution
there has been a concomitant evolution
in the processes used by engineers to produce the programs
agile process have looked at means
of decomposing large projects into incremental development
to allow for adaptation to new goals
during the course of the product's full development
test-drive development has looked to enforcing
the development of full invariant checks
as a means to drive more correct development
by coding the checks first
and then developing the interfaces
so that they pass the tests
this is really an extension of the type-system checks
and the common code-compile iterative method
but provides a means of runtime invariant checking
to discover semantic errors outside the syntax
pair-programming
code reviews
and similar techniques
help expose the code to peer review
and through the construction of these processes
a verification infrastructure
helps assist the construction of the code base
but you might notice
much of this discussion also applies
to mathematical symbology
every mathemagician is making it up
it's all novelty through effort
just like the progrimagineers
but with formally-driven fields of engineering
there is a theory of architecture
that formalises approaches to their symbolism
complexity and reusability theory formalises
intuitive notions of elegant proofs
what we see with category theory
infiltrating so many fields of modern mathematics
is secretly an invading army of ideas
based on the operationally verifiability
and it's intentional view of object properties
the constructive view of meaning
which is all that architecture is concerned with
computer science is mathematics
providing a specification and producing a program
is the exact same problem
as providing a statement and asking for a proof
the computer scientists
readily see the utility of operationalism and reuse
in mathematics
it is a quality of many good mathematicians
but refactoring as a process of analysis
is less commonly taught and kept more implicit
in things like graduate projects on other results
effectively
though
there is no formal distinction between the tasks
nor should there be
as the productions are formally related
through the curry-howard isomorphism
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
William Elliot
> William Elliot <ma...@rdrop.remove.com> wrote:
>> Is knowing computer language first a handicap for learning math?
>> Somehow these computer geeks think that they know math because they
>> know computerese. The reverse seems more the likely, that to learn
>> math requires unlearning what math they think they geek know.
>
> The answer is probably yes. Using flow charts might help, but the
> tricky languages which have all their restrictions leave out the
> flexibility of mathematics, as well as the fact that mathematical
> notation has often been preempted in computer languages.
>
> Oh, be careful of optimizing compilers. They often "optimize"
> out the part which is of interest.
>
>>> Assemblers rarely use overloaded operators, which makes them very slow
>>> to use.
>
>> My system analysis friends are nearly mathematically illiterate. It's no
>> handicap to their work. They ask me how to make efficient a mathematical
>> routine. Unlike geeks, they know what they don't know.
>
> Are they mathematically illiterate? Can they formulater their
> problems; if they can do so they are using the "literate" part
> of mathematics. Being unable to solve problems is not being
> illiterate. The mathematical analog of simple native language
> literacy is being able to formulate a sizable problem, using
> however many variables are convenient.
>
Ok. Unlike many posters at sci.math, their math was clear.
What they lack is mathematical knowledge.
>> What's weird is the pressure being put upon community college teachers
>> to teach algebra and other math courses with graphic calculators being the
>> major approach to the subject. Isn't that teaching mathematical
>> illiteracy? Are college teachers and professors also pressured toward
>> teaching calculators foremost?
>
> As a means of solving well-formulated problems, there is nothing
> wrong with using calculators and computers, provided one understands
> the concepts well enough to know what is being done.
>
That's was how I was tutoring, "Turn off the computer until you understand
the math."
> As a means of gaining understanding, computers do not cut it.
> Computers are not 'brains"; they are sub-imbeciles. In many
> cases, they can do by brute force what people cannot seem to
> find the way, but this does not make them intelligent.
Unfortunately the educationists like to turn math classes into graphic
calculator classes. It's easier than teaching math especially since high
school teachers who know math are becoming as extinct as the American
middle class.
----
Brian Chandler
> In article <gk8a4r$j...@odds.stat.purdue.edu> hru...@odds.stat.purdue.edu (Herman Rubin) writes:
> > Essentially what happens is that any computer operates
> > on the contents of locations, real or virtual, and the
> > contents of a location is a variable. This is never
> > pointed out. When one writes "x = 3", this means that
> > "3" is put in the location assigned to x, and this
> > assignment may be quite complicated, but rarely does
> > the user have to know this.
>
> Historical curiosity: Pascal's assignment statement:
>
> x := 3
>
> is alleged to have been meant to suggest an arrow,
> and thus make the difference between assignment
> and equality:
>
> x = 3
>
> more obvious.
Um, not just Pascal -- the whole Algol family, and as far as I can
see, every programming language designed on the East side of the
Lesser Pond.
BCPL, (following its CPL parent, UIMM) uses :=, but somewhere on the
way to B and C, the Americans took the colon out. Hmm.
Brian Chandler
Brian Chandler
> In article <gjjar8$1de$1...@news.motzarella.org>,
> Donna Metler <dmme...@nospam.net> wrote:
>
> >"Rowley" <industry...@yahoo.com> wrote in message
> >news:gjj06...@news2.newsguy.com...
> >> Donna Metler wrote:
> >> <snip>
> You say who needs it? Consider the idea of the tempered
> scale, made important by Bach. The ratio of the pitches
> of consecutive half-tones in this scale is the twelfth
> root of 2; this shows the use of a geometric series.
Um, this is wrong, I believe. The scale you are describing is *equal
temperament*. But Bach's 24 is called Das *Wohl*temperierte Klavier,
not Das Equitemperierte Klavier. The "temperings" are variants on
*just intonation* (in which a fifth really is 2:3, a third 4:5 etc),
to smooth out the horrible jump if you save the difference between
(3/2)^12 and 2^7 for just one interval.
But of course understanding the elementary maths involved is helpful,
in that it means the student doesn't have to take statements about
just intonation intervals entirely on trust. A little practice with it
can occasionally be useful practically, if you're trying to sing a
major third that is properly in tune, because you can work out which
direction you have to push in.
Brian Chandler
Herman Rubin
In article <2009011205...@agora.rdrop.com>,
William Elliot <ma...@rdrop.remove.com> wrote:
>On Sun, 11 Jan 2009, Herman Rubin wrote:
>> William Elliot <ma...@rdrop.remove.com> wrote:
...............
>>> What's weird is the pressure being put upon community college teachers
>>> to teach algebra and other math courses with graphic calculators being the
>>> major approach to the subject. Isn't that teaching mathematical
>>> illiteracy? Are college teachers and professors also pressured toward
>>> teaching calculators foremost?
>> As a means of solving well-formulated problems, there is nothing
>> wrong with using calculators and computers, provided one understands
>> the concepts well enough to know what is being done.
>That's was how I was tutoring, "Turn off the computer until you understand
>the math."
But the educationists want to teach people how to get the
answers, not how to know what they are doing. Learn how to
ask the question first, and what the answer means.
>> As a means of gaining understanding, computers do not cut it.
>> Computers are not 'brains"; they are sub-imbeciles. In many
>> cases, they can do by brute force what people cannot seem to
>> find the way, but this does not make them intelligent.
>Unfortunately the educationists like to turn math classes into graphic
>calculator classes. It's easier than teaching math especially since high
>school teachers who know math are becoming as extinct as the American
>middle class.
To the educationists, mathematics is merely a collection
of facts and algorithms, with no logical content. These
algorithms exist, and they have no interest in how one
might find new ones, as I have often had to do for
practical problems.
The use of graphing calculators adds more algorithms, and
so it is (to the educationists) important to teach them.
To teach mathematics properly, it is necessary to teach
WHY, not HOW. If one knows why, one can often easily
deduce how, and in any case it will make it easier to
learn how, but it seems that knowing how hinders learning
why. This goes against classical educationist theory.
Robert H. Lewis
> What's weird is the pressure being put upon community
> college teachers
> to teach algebra and other math courses with graphic
> calculators being the
> major approach to the subject. Isn't that teaching
> mathematical
> illiteracy? Are college teachers and professors also
> pressured toward
> teaching calculators foremost?
Not in my experience. We almost entirely eshew calculators. I have never used a calculator in any calculus course. One of my colleagues is very vociferous in his opposition to calculators. I don't know of any colleagues who use calculators in calculus.
My impression, FWIW, is that the big fascination with calculators that started about 12 years ago has faded.
Robert H. Lewis
Fordham University
Robert H. Lewis
> considered Advanced
> > university material these days. But hasn't "proof"
> been the WHOLE
> > BLOODY POINT of mathematics since at least
> Euclid??!? Mathematics minus
> > proof = what??
>
> Do you want kids to learn or do you want them to take
> tests that are
> machine scoreable?
>
> --Jeff
No one in our department gives machine scorable tests. There are no multiple choice tests.
Learning to think is the be-all and end-all of education. Without that, education is a pathetic farce.
Robert H. Lewis
Fordham University
Mathematics
William Elliot
>>> William Elliot <ma...@rdrop.remove.com> wrote:
>
>>> As a means of solving well-formulated problems, there is nothing wrong
>>> with using calculators and computers, provided one understands the
>>> concepts well enough to know what is being done.
>
>> That's was how I was tutoring, "Turn off the computer until you
>> understand the math."
>
> But the educationists want to teach people how to get the answers, not
> how to know what they are doing. Learn how to ask the question first,
> and what the answer means.
>
correct or even if you've asked the correct question or asked the question
correctly?
>> Unfortunately the educationists like to turn math classes into graphic
>> calculator classes. It's easier than teaching math especially since
>> high school teachers who know math are becoming as extinct as the
>> American middle class.
>
> To the educationists, mathematics is merely a collection
> of facts and algorithms, with no logical content. These
> algorithms exist, and they have no interest in how one
> might find new ones, as I have often had to do for
> practical problems.
>
They are raising a collection of scientists as stupid as they.
They are raising a generation who are being trained to not think.
Critical thinking is dangerous to your political correctness.
> The use of graphing calculators adds more algorithms, and
> so it is (to the educationists) important to teach them.
>
It's social engineering to convert people into production units.
I doubt this don't-think training is limited to mathematics.
> To teach mathematics properly, it is necessary to teach
> WHY, not HOW. If one knows why, one can often easily
> deduce how, and in any case it will make it easier to
> learn how, but it seems that knowing how hinders learning
> why. This goes against classical educationist theory.
Ours is not to ask or reason why.
Ours is but to do and die.
-- The Dummy Down Dunce Dance
Forelorn am I to scorn
the country wherein I am born,
where creativiity is shorn
to fit some standard norm.
Bob LeChevalier
>On Sat, 17 Jan 2009, Herman Rubin wrote:
>>>> William Elliot <ma...@rdrop.remove.com> wrote:
>>
>>>> As a means of solving well-formulated problems, there is nothing wrong
>>>> with using calculators and computers, provided one understands the
>>>> concepts well enough to know what is being done.
>>
>>> That's was how I was tutoring, "Turn off the computer until you
>>> understand the math."
>>
>> But the educationists want to teach people how to get the answers, not
>> how to know what they are doing. Learn how to ask the question first,
>> and what the answer means.
>>
>If you don't know what you're doing, how can you know if your answer is
>correct or even if you've asked the correct question or asked the question
>correctly?
In today's exam-based world, your answer is correct if the grader says
it is, and you aren't asking questions, only answering the ones given
to you on the exam.
Unfortunately, 95% of all kids are perfectly content with that, and of
course so are the legislators who are forced by constituents into
demanding student and teacher accountability measured quantitatively
and of course cheaply.
>> To the educationists, mathematics is merely a collection
>> of facts and algorithms, with no logical content. These
>> algorithms exist, and they have no interest in how one
>> might find new ones, as I have often had to do for
>> practical problems.
>>
>They are raising a collection of scientists
They aren't raising scientists. They are raising children, no more
than 10% of which will even consider the sciences or math-related
careers.
>as stupid as they.
Most adults are more than a little scared of raising kids that are
smarter than they are.
>They are raising a generation who are being trained to not think.
Just like all prior generations.
>> The use of graphing calculators adds more algorithms, and
>> so it is (to the educationists) important to teach them.
>>
>It's social engineering to convert people into production units.
I'd broaden that to "economic units". Most adults today don't
"produce" anything. But they can sell, and process bureaucratically,
and interact personally with clients and co-workers.
>I doubt this don't-think training is limited to mathematics.
Nor is it limited to this country.
>Ours is not to ask or reason why.
>Ours is but to do and die.
Precisely.
lojbab
Bob LeChevalier - artificial linguist; genealogist
loj...@lojban.org Lojban language www.lojban.org
Herman Rubin
>> --Jeff
Alas, this is not universal. Most universities give multiple
choice tests, or parts of tests, even in mathematics.
As for being able to think, the educationists discourage the
real thing, even in their "critical thinking" classes. These
often are just classes in political correctness.
As for thinking in mathematics, educationists seem to be
unaware that it exists. This goes beyond educationists;
people are continuously bringing in reams of data, ore even
worse, their summaries of it, and asking for the state of
the universe.