Nabble - math-teach

Input from Physics teachers

2009-01-07T14:52:33Z

First published to Python community list, this edition
slightly improved (fixed some typos). Note that once
you have strong type awareness (below) then you're ready
to ask intelligent questions about what operates with
what (legally), the basis of abstract algebra
explorations, with small finite groups a focus (given
public key cryptography, built in to every web browser,
is a topic in senior high, working backwards from that
goal).

====================

I had a productive meeting with Dr. Bob Fuller,
University of Nebraska, emeritus, yesterday, a
long time associate on that First Person Physics
proposal to NSF (close, no cigar). He's working
on the Karplus legacy, in turn stemming from
Piaget.

http://controlroom.blogspot.com/2009/01/physics-update.html

Science teaching went through a more successful
transformation to "constructivist" (in the sense of
student-centered, construct your own model of reality)
than USA math teaching managed (talking later 1900s),
as the latter was mostly a panic response to Sputnik
(so-called SMSG) and it's been a backlash ever since
("back to basics" to the point of near extinction of
the subject, in terms of attracting fresh thinking).

I'm not sure how it went in the UK, other Anglophone
cultures. Others on edu-sig will have more place-based
stories of curriculum writing (the evolution thereof)
in your respective necks of the woods.

Anyway, the physics community has been interested in
video games as teaching devices right from the get go,
with museum-grade simulators (like the ones pilots train
in) representing a kind of high end state of the art
(people actually get sick in those, given the realism).

Speaking of getting sick, you'll find in my Vilnius
slides, other places, a strong emphasis on "grossology"
when working with kids. That's a part of kid culture I've
always found missing from Squeak, which seems too squeaky
clean, not sufficiently demented. For example, if using
a system language and defining a function, you'll likely
encounter strong type awareness, meaning every type
is declared *and* in a specific order e.g. f(int x, str
y) and g(str y, int x) are quite strict about what they
"eat" (function parentheses = open mouth) and if you send
them the wrong args, they will "barf" (has to be OK to
say that, or you lose a lot of would-be attenders).

The "type awareness" we want to induce is very
traditional and follows that time-honored sequence:
N, W, Z, Q, R, C. You might not think in quite those
terms (namespaces differ) but we're talking natural,
whole, integer, rational, real and complex respectively.

These are types, and there's an historical narrative
explaining the drive to expand to new horizons, starting
with simple geometric ratios such as the body diagonal
of a cube (math.sqrt(3)) or of the 1 x 2 rectangle
(math.sqrt(5)). Given the historical dimension, it's
quite appropriate to give these primitive geometric
relationships a somewhat neolithic spin i.e. some talk
of "cave people". This helps anchor some data points for
later, when we get into trigonometry and navigation
techniques (over desert, over sea).

http://www.flickr.com/photos/17157315@N00/sets/72157612202599023/
(gnu math teacher Glenn Stockton, expert in neolithic
tool making, including for astronomical purposes)

You get these simple expressions with surds (e.g. phi,
math.sqrt(2)) right out of the gate, with compass and
ruler, scribing in sand (on a spherical surface, so only
locally Euclidean -- "close enough for folk music" as
we say in geography class, zooming in on Greece in
Google Earth maybe). Pi, unlike phi, is transcendental,
not just irrational. I agree with posters here that
Ramanujan is a great source of generators (in the
Pythonic sense), plus I like playing that epic song.

http://worldgame.blogspot.com/2008/02/reflective-fragment.html

The complex numbers get added by those in the Italian
peninsula, seeking to solve Polynomial Puzzles (Pisa a
center for this kind of game playing, lots of betting,
not unlike cockfighting). Fractals ala the Mandelbrot
pattern, scribed in the complex plane, come latter ("phi
is the first fractal" -- a mnemonic we use).

However, given this is alpha-numeric literacy i.e.
string-oriented as well as numerical, we don't stop with
a recap of basic algebra. We need those regular
expressions (good for URL parsing) and Unicode studies.
Fine if the language arts teachers want to pick up the
story at this point, take it away from the algebra
teachers. We're talking DOM (Document Object Model),
XML... what became of "the outline" in Roman times
(structured thinking, rhetoric).

I'd like to thank Ian Benson of Sociality / Tizard for
confirming my impression that R0ml is correct in his
approach, with strong emphasis on Liberal Arts (in
healthy doses at OSCONs -- the guy is simply brilliant).
'Godel Escher Bach' is another trailblazing work, in
making sure we keep the string games going, don't
propagate the misinformation that "number crunching" is
all that we're about. Knuth called 'em *semi*-numerical
algorithms for a reason.

But the question remains, if you *are* committed to
keeping regular expressions within math: where to put
them? I think the answer is pretty obvious: students
need to work as a team to maintain some kind of Django
web site, could be exclusively in-house (not public),
with time line data, events in math history, adding and
morphing over time.

Actually parse URLs, triggering real SQL behind the
scenes. This is all completely topical, very job market
oriented. Yet we're in a constructivist realm, giving
imaginations free play and lots of open-ended exploration
time.

I continue with the "gnu math" and "computer algebra"
labeling, adding the Bucky stuff as a "secret sauce" --
spices it up to have something a little questioning of
authority, especially in a math learning context, where
some adults are accustomed to unchallenged authority.
No longer, rest assured.

Kirby

Re: multiplicand, multiplier, etc.

2009-01-07T12:45:13Z

I was looking for an answer to this question and Google found this old forum discussion. A math book we use at our small school states that in the statement 2 X 4 = 8 (horizontal format) 2 is the multiplier (logical if X means "times"). Written vertically with the 2 on top, the book gives 4 as the multiplier (logical if X means "multiplied by"). A wikipedia article states that this distinction between multiplier and multiplicand is "old school" and not used currently.

Math is not my specialty, but it seems to me it is the real-world application of math that I want students to grasp. Thus, if multiplication deals with 1) number of groups/sets 2) number in each group/set and 3) total, then as long as the student/"problem solver" understands these 3 elements then he/she can assign a given number any significance he desires - multiplier or multiplicand.

If a student, given the problem 2 X 4, (written vertically or horizontally) asks me which is the multiplier and which is the multiplicand, I think I might say, "you decide". Or I might ask them to give me a word problem seeking a total where 2 and 4 are known quantities, and then decide for themselves which is the multiplicand and which is the multiplier and how they want to write it. Would I be wrong?

Similarly with division, if I know the "total" (dividend), then the divisor can be either the number of groups or the number in each group - it would depend on what problem I was solving: do I want to know how many boxes I need or how many chocolates I should put in each box?

Steven Leinwand: "Moving Mathematics Out of Mediocrity"

2009-01-07T07:38:47Z

Published Online: January 5, 2009
Published in Print: January 7, 2009

<http://tinyurl.com/8pxoqt>

Commentary
Moving Mathematics Out of Mediocrity
By Steven Leinwand

The logic for the importance of improving school mathematics programs
is reasonably unassailable. The country’s long-term economic security
and social well-being are clearly linked to sustained innovation and
workplace productivity. This innovation and productivity rely, just as
clearly, on the quality of human capital and equity of opportunity
that, in turn, emerge from high-quality education, particularly in the
areas of literacy, mathematics, and science. Applying the if-then
deductive logic of classical geometry puts a strong K-12 mathematics
program at the heart of America’s long-term economic viability.

But the problems with mathematics in the United States are just as
clear. A depressingly comprehensive, yet honest, appraisal must
conclude that our typical math curriculum is generally incoherent,
skill-oriented, and accurately characterized as “a mile wide and an
inch deep.” It is dispensed via ruthless tracking practices and
focused mainly on the “one right way to get the one right answer”
approach to solving problems that few normal human beings have any
real need to consider. Moreover, it is assessed by 51 high-stakes
tests of marginal quality, and overwhelmingly implemented by
undersupported and professionally isolated teachers who too often rely
on “show-tell-practice” modes of instruction that ignore powerful
research findings about better ways to convey mathematical knowledge.

For 20 years, we have tinkered at the margins, merely adjusting parts
of the system while ignoring the fact that the basic structure has
remained largely intact and underperforming. During those 20 years,
we’ve raised achievement a little and narrowed gaps a bit. But even as
the need for broader and deeper mathematical literacy has grown, our
traditional approach still rarely works for more than a third of our
students, and it fails even more when it comes to critical-reasoning
and problem-solving skills. It shouldn’t be all that surprising that
on the 2006 Program for International Student Assessment, or
PISARequires Adobe Acrobat Reader, 15-year-old U.S. students placed an
unacceptable 25th out of 30 countries tested.

Fortunately, the solutions are as clear as the problems. The answers
do not revolve around costly new initiatives. Moving beyond mediocrity
does not have to mean new textbooks and supplemental programs, or a
slew of new calculators and computers, or jumping on the latest
bandwagon of benchmark assessments. Instead, our attention needs to
focus on how effectively existing programs are implemented, how
available technology is integrated and used to enhance the learning of
skills and concepts, and why assessments that steal valuable
instructional time must provide relevant information that is actually
put to use to inform revisions and reteaching.

In short, it’s time to turn to the real basics of what we expect
students to learn, how we convey that, how we measure student
learning, and how we support teachers and reduce their isolation.

We need first to recognize that most of our major economic
competitors, and nearly all of the highest-scoring countries on
international assessments, have a national set of mathematics
standards that guarantees a degree of coherence, focus, and alignment
absent in the patchwork of state standards in the United States. A
nationally mandated curriculum isn’t the answer. But a broadly
accepted, strongly recommended set of world-class national mathematics
standards for grades K-12 is. Such standards would provide informed
guidance and attract widespread interest, yet would not fall under the
antiquated rubric of “local control.”

If we took this route, textbooks could be revamped to cut redundancy
and add depth and balance between procedural and conceptual
understanding. The recommended math standards could delineate sensible
and reasonable expectations for students at each grade level and in
each course. Curriculum sequences and objectives could be crafted so
that all students would reach key elements of algebra in 8th grade and
leave high school with sufficient understanding of both calculus and
statistics. These skills would help them thrive in the workplace and
at postsecondary institutions.

Second, we need to examine what common sense, observation, and
research tell us about instructional practices that make significant
differences in student achievement. Such practices can be found in
high-performing schools across the country. There, we see teachers
making “Why?” a classroom mantra to support a culture of reasoning and
justification. We see cumulative review being incorporated daily. We
see deliberately planned lessons that skillfully employ alternative
approaches and multiple representations that value different ways to
reach solutions to real problems. We see teachers relying on relevant
contexts and using questions to create language-rich mathematics
classrooms.

Good mathematics instruction is hard, but it isn’t quantum physics.
Yet few vehicles are currently used to model and institutionalize
these techniques that make a difference. That is why compassionate,
collegial, and yet candid coaching and supervision, guided by a
compelling vision of high-quality mathematics instruction, can make
such a tremendous difference in how much students learn and how
teaching skills are strengthened.

Third, we need to address the current mishmash of assessments that has
emerged from implementation of the federal No Child Left Behind Act.
How can one expect instruction to focus on conceptual understanding,
or communicating one’s thinking or reasoning through a complex
problem, when tests hold students accountable for only low-level
skills and multiple-choice answers? Accountability isn’t the problem.
The problem rests with the instruments being used to hold the system
accountable.

We should look at what characterizes student assessments in other
countries. Most of Singapore’s tests, for example, consist of problem-
oriented, constructed-response items. PISA’s items are set in
realistic contexts and require thinking and reasoning about
substantive mathematics, as opposed to recall and regurgitation of
tangential content. Moving forward, we must look to the federal
government and its research-and-development muscle and investment to
create high-quality national assessments of mathematics at the ends of
grades 4, 8, and 10. Until this happens, we will continue to muddle
through multiple and meaningless standards with mixed signals and
continued mediocrity.

Establishing a set of high-stakes, high-quality, annually released
national assessments will drive improvement, reduce the current
hodgepodge of state assessments, and move the United States toward a
rational alignment between what is taught and what is tested.

Finally, we need to address professional isolation among teachers. It
is the nature of the profession that most educators practice their
craft behind closed doors. They usually go about their work unobserved
and undersupported. Far too often, teachers revert to how they were
taught, not how their effective colleagues are teaching. Common
problems are often solved individually rather than collaboratively.

Successful enterprises don’t tolerate such conditions. We must change
the professional culture of teaching. Principals must develop
innovative ways to facilitate professional sharing and interaction.
Middle and high school math departments must become true communities
of learners.

In an example of this strategy in motion, teachers in one enterprising
district I have visited regularly share and discuss their videotaped
lessons. After two years of their doing so, the district finds that
marginal teachers have become good teachers, and good teachers have
become even better. Simultaneously, classroom practices have become
far more transparent and discussions now focus on specific
instructional strategies. Common problems are approached and solved
collaboratively.

It is time to recognize math education as a critical component of
America’s economic infrastructure. National interest supports a
military for the country’s defense and an interstate highway system
for effective commerce. Now, we must support—and demand—a national
K-12 mathematics program that far better serves our students, our
economy, and our national interest.

Steven Leinwand is a principal research analyst at the American
Institutes for Research, in Washington. He is the author of the
forthcoming book Accessible Mathematics: Ten Instructional Shifts That
Raise Student Achievement (Heinemann).

SAYINGS XX

2009-01-07T06:22:40Z

*****************************
NOTE: The following quotations have been collected from a variety of
sources ... from e-mail notes, bulletin boards, the ASCD SmartBrief
and from other sources, as well, over some time.
*****************************
"To lead people, walk beside them ... As for the best leaders, the
people do not notice their existence. The next best, the people honor
and praise. The next, the people fear; and the next, the people hate
... When the best leader's work is done, the people say 'We did it
ourselves!'"

(Lao-Tsu)

"However beautiful the strategy, you should occasionally look at the results."

(Winston Churchill, British prime minister)

"The great teacher is not the one who supplies the most facts, but
the one in whose presence we become different people."

(Ralph Waldo Emerson)

"Why not go out on a limb? Isn't that where the fruit is?"

(Frank Scully, writer and columnist)

"If you want children to keep their feet on the ground, put some
responsibility on their shoulders."

(Abigail Van Buren, advice columnist)

"Under certain circumstances, profanity provides a relief denied even
to prayer."

(Samuel "Mark Twain" Clemens, writer)

"Authority without wisdom is like a heavy axe without an edge; fitter
to bruise than polish."

(Anne Bradstreet, American poet)

"Doing nothing is very hard to do ... you never know when you're finished."

(Leslier Nielsen, Canadian-American actor)

"Innovation distinguishes between a leader and a follower."

(Steve Jobs)

"Be a yardstick of quality. Some people aren't used to an environment
where excellence is expected."

(Steve Jobs, CEO of Apple)

"Wisdom is the reward you get from a lifetime of listening when you'd
have preferred to talk."

(Dough Larson)

"America cannot solve its problems using the same minds that created them."

(Albert Einstein)

"Strength is the capacity to break a chocolate bar into four pieces
with your bare hands -- and then eat just one of the pieces."

(Judith Viorst, author and journalist)

"Not all who wander are lost."

(J.R.R. Tolkien) [From the St. Olaf College website]

"Luck is the residue of design."

(Branch Rickey, the GM of the then-Brooklyn Dodgers, who signed
Jackie Robinson) [NOTE: This is a correction from an earlier
posting, by Norman Webb.]

"Education is what survives when what has been learned has been forgotten."

(B. F. Skinner, psychologist, inventor and author)

"You know children are growing up when they start asking questions
that have answers."

(John J. Plomp, author) [Sent by Susan McNally]

"I believe people run marathons every day of their lives in one way
or another, and we need to remember to give ourselves the finishers'
medals we deserve."

(Zoe Koplowitz, author and marathon competitor)

"The saddest thing in life is the waste of talent."

(Unknown)

"We cannot sustain democracy unless education is universally shared."

(Thomas Jefferson)

*********************************************
- --
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244
E-mail: jbecker@...

Mathematics textbooks for junior high and high school in Massachusetts?

2009-01-07T05:40:47Z

Hello
Does anyone know which textbooks students in junior high school and high school use in Massachusetts or bay area? Does anyone know of a comprehensive list of books used?

Per

Re: Are limits always necessary for calculus?

2009-01-06T13:31:33Z

Clifford J. Nelson wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6557238

> I get many hits when I search for many things that
> contradict most of what I have known all my life going
> back to 1946. I just don't trust the Internet very much,
> especially when I read "this has been known for some time".

This can be true for a web search, which is why I focused
on google-book searches. Those pull up actual publications,
and the hits I was using them for are scanned digital
copies of texts and journals for the time periods I stated.

For an example, here's a 1730 English translation of
L'Hopital's famous textbook in calculus, considered
to be the first calculus text:

http://books.google.com/books?id=FiA6AAAAMAAJ

Here's the original 1716 French version:

http://books.google.com/books?id=KDQVAAAAQAAJ

As for your other comments and the 27 May 2003
Historia Mathematica post, if you're especially
interested in possible historical antecedents to
the roots of unity polynomial differentiation idea,
I'd recommend writing to some mathematical historians
who specialize in the work of the three people I
previously mentioned: Cayley, Hamilton, and Sylvester.
John H. Conway of Princeton is also someone who
might be able to help, at least by pointing you
to someone who can.

Dave L. Renfro

Re: limit question poll

2009-01-06T13:22:53Z

Adrian wrote:
>
> You are saying that "Let f be defined on an open
> interval about x-naught," is the same as saying
> "Suppose f was defined at some point on every open
> interval about x-naught".

No, not quite. I'm saying that, because "defined on" (by HarperCollins) and "an" (by English grammar) do in fact have more than one meaning, the clause "Let f be defined on an open interval containing x-naught (except possibly x-naught itself)," *taken on its own* (meaning apart from clarification by an author who gives it as to what that author means), *can be* the same as saying "Let f be defined at some point not equal to x-naught in every open interval containing x-naught (and let f be defined also possibly at x-naught)."

> In a word: No. Even if
> we simply replaced "an" with "any" in "Let f be
> defined on an open interval about x-naught," we would
> STILL not arrive at your reinterpretation.
> At this
> rate, I totally should have tried to lawyer my
> homework and tests more -- "You see, I actually
> didn't get a single one of these wrong!"
>
> This has gone from saying something like "There are
> those that interpret a few key words to render the
> expression critically differently," to you just
> arbitrarily rewriting what should otherwise be plain
> English. Yes, you certainly could do that if you get
> to be as arbitrary as you wish. You could just as
> well "interpret" all of Thomas and Finney as actually
> just being War and Peace -- "You know, in so many
> words...."

We would arrive at the intended interpretation, and there is nothing arbitrary about it. You didn't address the fact that the usual calculus meaning of "Let f be defined on an open interval containing x-naught (except possibly x-naught itself)," this usual calculus meaning being "Let f be defined at every point not equal to x-naught in some open interval containing x-naught (and let f be defined also possibly at x-naught)," actually implies the clause "Let f be defined at some point not equal to x-naught in every open interval containing x-naught (and let f be defined also possibly at x-naught)." And this implied clause itself implies that x-naught is a limit point (as does the usual calculus meaning, of course), that every neighborhood of x-naught contains some non-x-naught point at which f is defined. So it meets your objection that it all boils down to x-naught needing to be a limit point, and so does arrive at the intended interpretation. And it's consistent with the ana!
lysis meaning in that there's an implied subset E of the real numbers R with limit point x-naught). So since it's implied by the usual calculus meaning, since it provides a limit point, and since it's consistent with analysis, it's definitely not arbitrary.

I'm not reinterpreting Thomas and Finney or any other specific calculus author. I said the above "*taken on its own* (meaning apart from clarification by an author who gives it as to what that author means)" for a reason. All this was started by the question "Is the limit of [sin(1/x)]/[sin(1/x)] as x approaches zero equal to 1?" That's it; no context, no "according to Thomas and Finney" or according to some other author, no according to what is usually found in calculus vs. analysis. It was asked in the most general sense. The intent in not giving such context may have been to elicit contradictory replies, and if so, it worked. Lou said that the limit doesn't exist and Dave said that the limit is 1, and since Lou is a specialist in analysis and since (as I understand it) so it Dave, it really worked. But sometimes there can be more one correct answer, and I think this is an example, that in this most general sense, they are both right. And I think that that since in the gen!
eral sense undefined ambiguous terms allow for a different meaning than the usual meaning, especially if this different meaning is implied by the usual meaning, if there is no context given to these ambiguous terms to eliminate the ambiguities (such as no restricting the context to a given textbook with its own clarifications), then there is nothing wrong with going with this different meaning to give a different but still correct answer.

Paul A. Tanner III

Re: Are limits always necessary for calculus?

2009-01-06T12:32:17Z

On Jan 6, 2009, at 11:24 AM, Dave L. Renfro wrote:

> Dave L. Renfro wrote (in part):
>
> http://mathforum.org/kb/message.jspa?messageID=6557014
>
>>> First, the relevant properties of the roots of unity
>>> were not known when Newton did his work (see [1]).
>
> Clifford J. Nelson wrote:
>
> http://mathforum.org/kb/message.jspa?messageID=6557095
>
>> The Newton book usually just called "Fluxions" was not
>> published in the U.S.A. for about 250 years, according
>> to "The Mechanical Universe", a PBS TV series. Newton
>> did not publish it at all. It was published after he died.
>> Acupuncture was not known here until after Nixon visited
>> China, but I think it is very old. Stainless steal was
>> a WWI secret, but the public was told it was new when
>> it was used in a car bumper some time around 1922.
>
> The use of the term "fluxions" was quite common in mid 1700s
> to mid 1800s higher math books and journals -- I've seen
> it literally hundreds of times in books and journals from
> this period that I've checked out and looked through from
> the local university library the past few years. Also, now
> we have google-books, so you can see them for yourself if
> you don't have easy access to a university library:
>
> 1205 full-view hits for "fluxions" in publications published
> on or before 1810:
>
> http://books.google.com/books?as_brr=1&q=fluxions+date%3A1-1810
>
> I'm pretty sure I heard/read about acupuncture before Nixon
> became president. In any event, I get 991 full-view hits for
> "acupuncture" in publications published on or before 1890:
>
> http://books.google.com/books?as_brr=1&q=acupuncture+date%3A1-1890
>
> Clifford J. Nelson wrote:
>
>> What problem would I have been trying to solve with the
>> roots of unity method for a polynomial if I did not know
>> a little calculus? Why would anybody want to rotate and
>> scale a bunch of vectors and add them up according to
>> a polynomial?
>
> I have no idea, not being something I know much about,
> but I'd be extremely surprised if the answer to your
> question isn't somewhere in Hamilton's, Sylvester's,
> or Cayley's extensive mathematical writings.
>
> Dave L. Renfro

Tom Apostol did not know about the roots of unity method for finding
the derivative. He was the math advisor to " The Mechanical Universe".
See:
http://mathforum.org/kb/message.jspa?messageID=1185362&tstart=0

I get many hits for study methods that use some of the same letters of
"the Survey- 3QR method" I leaned in a phycology 1A course at
Riverside City Collage in 1969. Most of them say to do the worst
things you can do according to what I was taught. I get many hits when
I search for many things that contradict most of what I have known all
my life going back to 1946. I just don't trust the Internet very much,
especially when I read "this has been known for some time".

Cliff Nelson

Dry your tears, there's more fun for your ears,"Forward Into The Past"
2 PM to 5 PM, Sundays,California time,
http://www.geocities.com/forwardintothepast/
Don't be a square or a blockhead; see:
http://mysite.verizon.net/cjnelson9/index.htm
http://library.wolfram.com/infocenter/search/?search_results=1;search_person_id=607j

Re: Are limits always necessary for calculus?

2009-01-06T11:24:03Z

Dave L. Renfro wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6557014

>> First, the relevant properties of the roots of unity
>> were not known when Newton did his work (see [1]).

Clifford J. Nelson wrote:

http://mathforum.org/kb/message.jspa?messageID=6557095

> The Newton book usually just called "Fluxions" was not
> published in the U.S.A. for about 250 years, according
> to "The Mechanical Universe", a PBS TV series. Newton
> did not publish it at all. It was published after he died.
> Acupuncture was not known here until after Nixon visited
> China, but I think it is very old. Stainless steal was
> a WWI secret, but the public was told it was new when
> it was used in a car bumper some time around 1922.

The use of the term "fluxions" was quite common in mid 1700s
to mid 1800s higher math books and journals -- I've seen
it literally hundreds of times in books and journals from
this period that I've checked out and looked through from
the local university library the past few years. Also, now
we have google-books, so you can see them for yourself if
you don't have easy access to a university library:

1205 full-view hits for "fluxions" in publications published
on or before 1810:

http://books.google.com/books?as_brr=1&q=fluxions+date%3A1-1810

I'm pretty sure I heard/read about acupuncture before Nixon
became president. In any event, I get 991 full-view hits for
"acupuncture" in publications published on or before 1890:

http://books.google.com/books?as_brr=1&q=acupuncture+date%3A1-1890

Clifford J. Nelson wrote:

> What problem would I have been trying to solve with the
> roots of unity method for a polynomial if I did not know
> a little calculus? Why would anybody want to rotate and
> scale a bunch of vectors and add them up according to
> a polynomial?

I have no idea, not being something I know much about,
but I'd be extremely surprised if the answer to your
question isn't somewhere in Hamilton's, Sylvester's,
or Cayley's extensive mathematical writings.

Dave L. Renfro

Re: Are limits always necessary for calculus?

2009-01-06T09:20:14Z

On Jan 6, 2009, at 7:37 AM, Dave L. Renfro wrote:

> Clifford J. Nelson wrote (in part):
>
> http://mathforum.org/kb/message.jspa?messageID=6556864
>
>> Maybe Newton didn't use infinitesimals or limits to
>> compute the derivative of a polynomial f(x), because
>> it can be done with the nth primitive root of unity
>> U(n) = e^(2*Pi*i/n), or if n is a prime [...]
>
> First, the relevant properties of the roots of unity
> were not known when Newton did his work (see [1]).

The Newton book usually just called "Fluxions" was not published in
the U.S.A. for about 250 years, according to "The Mechanical
Universe", a PBS TV series. Newton did not publish it at all. It was
published after he died. Acupuncture was not known here until after
Nixon visited China, but I think it is very old. Stainless steal was a
WWI secret, but the public was told it was new when it was used in a
car bumper some time around 1922.

>
> Second, the relevant properties of the expression
> e^(2*Pi*i/n) were not known until well after Newton
> died (see [2]). Third, derivatives of polynomials
> were not particularly problematic during Newton's
> time (see [3]). Fourth, Lagrange's "Calculus as Algebra"
> approach (late 1700s), in response to the numerous
> concerns over the rigor of fluxions (see [4]), especially
> as given in Berkeley's "The Analyst: A Discourse
> Addressed to an Infidel Mathematician" (see [5]),
> went well beyond what you seem to be describing.

Thanks for the links.

What problem would I have been trying to solve with the roots of unity
method for a polynomial if I did not know a little calculus? Why would
anybody want to rotate and scale a bunch of vectors and add them up
according to a polynomial?

>
>
> [1] http://www.google.com/search?q=cote's-theorem
> http://www.google.com/search?q=cotes-theorem
> http://books.google.com/books?q=cotes-theorem&as_brr=1
> http://books.google.com/books?q=cote's-theorem&as_brr=1
>
> [2] http://www.google.com/search?q=Euler's-formula+imaginary
> http://books.google.com/books?q=Euler's-formula+imaginary&as_brr=1
>
> [3] My gut feeling.
>
> [4] http://www.google.com/search?q=Lagrange+calculus-as-algebra
> http://books.google.com/books?q=Lagrange+calculus+algebra&as_brr=1
>
> [5] http://en.wikipedia.org/wiki/The_Analyst
>
> Dave L. Renfro

Cliff Nelson

Dry your tears, there's more fun for your ears,"Forward Into The Past"
2 PM to 5 PM, Sundays,California time,
http://www.geocities.com/forwardintothepast/
Don't be a square or a blockhead; see:
http://mysite.verizon.net/cjnelson9/index.htm
http://library.wolfram.com/infocenter/search/?search_results=1;search_person_id=607j

Re: Are limits always necessary for calculus?

2009-01-06T07:37:07Z

Clifford J. Nelson wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6556864

> Maybe Newton didn't use infinitesimals or limits to
> compute the derivative of a polynomial f(x), because
> it can be done with the nth primitive root of unity
> U(n) = e^(2*Pi*i/n), or if n is a prime [...]

First, the relevant properties of the roots of unity
were not known when Newton did his work (see [1]).
Second, the relevant properties of the expression
e^(2*Pi*i/n) were not known until well after Newton
died (see [2]). Third, derivatives of polynomials
were not particularly problematic during Newton's
time (see [3]). Fourth, Lagrange's "Calculus as Algebra"
approach (late 1700s), in response to the numerous
concerns over the rigor of fluxions (see [4]), especially
as given in Berkeley's "The Analyst: A Discourse
Addressed to an Infidel Mathematician" (see [5]),
went well beyond what you seem to be describing.

[1] http://www.google.com/search?q=cote's-theorem
http://www.google.com/search?q=cotes-theorem
http://books.google.com/books?q=cotes-theorem&as_brr=1
http://books.google.com/books?q=cote's-theorem&as_brr=1

[2] http://www.google.com/search?q=Euler's-formula+imaginary
http://books.google.com/books?q=Euler's-formula+imaginary&as_brr=1

[3] My gut feeling.

[4] http://www.google.com/search?q=Lagrange+calculus-as-algebra
http://books.google.com/books?q=Lagrange+calculus+algebra&as_brr=1

[5] http://en.wikipedia.org/wiki/The_Analyst

Dave L. Renfro

Are limits always necessary for calculus?

2009-01-06T06:06:33Z

Some of Newton's work about calculus was not published for years after
he died. Maybe some of it was never published.

Maybe Newton didn't use infinitesimals or limits to compute the
derivative of a polynomial f(x), because it can be done with the nth
primitive root of unity U(n) = e^(2*Pi*i/n), or if n is a prime
number, U(n) =RotateLeft[Bucky_Number_Unity[n]] which uses exact
rational arithmetic (see: http://mysite.verizon.net/cjnelson9/
index.htm) like this:

f'(x) = Sum[(f(x+U(n)^p)-f(x-U(n)^p))(U(n)^(-p)/(2*n),{p,1,n}] =
df(n,x) where n is an integer greater than half of the highest
exponent of x in f(x).

df(n,x) is a purely geometrical method of finding f'(x). It is the
average of the average rates of changes of f(x) when the changes to x
are the n nth roots of unity and the roots of unity are vectors on the
plane on the unit circle when complex numbers are used, and vectors to
the vertices of a regular simplex of edge length n in n-1 dimensions
when Bucky numbers are used using Synergetics coordinates.

df(n,x) does not divide by a non zero variable and then set the
variable to zero later like all the Calculus textbooks do when they
derive f'(x).

The method was known before I discovered it as part of complex analysis.

Cliff Nelson

Dry your tears, there's more fun for your ears,"Forward Into The Past"
2 PM to 5 PM, Sundays,California time,
http://www.geocities.com/forwardintothepast/
Don't be a square or a blockhead; see:
http://mysite.verizon.net/cjnelson9/index.htm
http://library.wolfram.com/infocenter/search/?search_results=1;search_person_id=607j

Re: limit question poll

2009-01-05T15:08:13Z

>
> In the clause "let f be defined on an open interval
> containing c (except possibly at c)," the term "an"
> is not explicitly defined in terms of quantification.
> (It's used in the Wikipedia article on limit, which
> gives the Weierstrass definition.) To see that "an"
> rather than "some" may be more common, one can do a
> Google search with the two phrases below (with
> quotation marks)
>
> limit "defined on an open interval"
>
> limit "defined on some open interval"
>
> I got the result of 3,030 hits when "an" is used and
> 586 hits when "some" is used. When "any" was used
> there were only 2 hits, and no hits with "each" or
> "every." By these hits, it's clear that the
> interpretation of "defined on" used in this phrase is
> not the HarperCollins definition, at least not with
> the maximal and essential domains being different
> sets. (It can still be the HarperCollins definition
> with the maximal and essential domains being the same
> set.) Other combinations of words and phrases yielded
> for me similar results in that there were many more
> hits with "an" being used rather than "some."
>
> Concerning the whole clause in question ("let f be
> defined on an open interval containing c (except
> possibly at c)"), this above definition of "defined
> on" can give a contextual meaning of existential
> quantification to "an" so that one can interpret the
> clause as "let f be defined at any point not equal to
> c in some open interval containing c (and let f be
> defined possibly at c as well)." But one could go the
> other way: The HarperCollins definition of "defined
> on" can give a contextual meaning of universal
> quantification to "an" so that one can interpret the
> clause as "let f be defined at some point not equal
> to c in any open interval containing c (and let f be
> defined possibly at c as well)."
>
> In the second interpretation, where x varies over the
> points at which f is defined, x approaches c, and so
> one can test for limits (including one-sided limits
> if one needs to). (So rather than going with the
> first interpretation and having to give Lou's answer
> that the limit of the function in question at the
> given point does not exist, one can go with the
> second and give Dave's answer that the limit is 1.)
>

You are saying that "Let f be defined on an open interval about x-naught," is the same as saying "Suppose f was defined at some point on every open interval about x-naught". In a word: No. Even if we simply replaced "an" with "any" in "Let f be defined on an open interval about x-naught," we would STILL not arrive at your reinterpretation. At this rate, I totally should have tried to lawyer my homework and tests more -- "You see, I actually didn't get a single one of these wrong!"

This has gone from saying something like "There are those that interpret a few key words to render the expression critically differently," to you just arbitrarily rewriting what should otherwise be plain English. Yes, you certainly could do that if you get to be as arbitrary as you wish. You could just as well "interpret" all of Thomas and Finney as actually just being War and Peace -- "You know, in so many words...."

Re: Because They Can

2009-01-05T12:27:09Z

> Kirby, you've missed the point, and I think it very
> probably a valid one. There aren't many women wanting
> to get a "Google appengine" going, not because they
> don't want to get IT's permission, but because they'd
> rather do something else. The ones that have the math
> ability equal to the IT geeks tend also to be competent
> verbally and socially, and so are attracted to other
> professions that aren't filled with so many geeks.
>
> Can you blame them?
>
> - -Greg
>

I was making much the same point Greg. If women just
work with each other, and don't call it IT, then they
drop off the map as far as many of these CS departments
are concerned. They still show up at OSCONs however,
to discuss their alternative approaches:

http://worldgame.blogspot.com/2008/07/women-and-foss.html

Forgive me if I assume I have better sources of
information than most math teachers on this subject, or
even than InfoWorld in some cases, about the state of
my art.

Kirby

Re: New Post at RME: Lisa's Method: A New Arrangement For Calculating Slope

2009-01-05T11:57:37Z

Michael Paul Goldenberg wrote (in part):

http://mathforum.org/kb/message.jspa?messageID=6555877

> Read the entire post at: <http://tinyurl.com/8kxzs6>

Excerpt from the blog post:

> When I asked if she knew what to do next, she said, "I'm
> not really certain." I wrote Dy/Dx = (y2 - y1)/(x2 - x1)
> and asked her if that made sense. At that point, she said,
> "Oh, now I see why I keep getting the problems wrong: I'm
> putting the x's on the top."

I've always been more fond of methods/tricks that rely
on useful-to-know-concepts, rather than, for example,
placing numbers in an arrary of boxes as is sometimes given
in textbooks for the "working together at different
constant rates" problems (which I prefer to set up
in a "distance = rate*time" format), or at least which
rely on background knowledge (not necessarily math,
but preferably math if possible) that the student already
knows. On the other hand, any device/method/mnemonic
that a particular student comes up with that works
is almost certainly going to work well for that
particular student (but often not for other students
who are simply told of the device/method/mnemonic).

The method your student came up with appears to be
a good example of this last part. It's probably going
to work well for her, but I suspect it's not going to
work all that well for others who are simply told about
her method.

Here's how I would try to fix the problem she initially
had about which way you're to divide, although past
experience tells me that mathematically weak students
aren't helped all that much and mathematically strong
students are already using a method like this (or
they've somehow managed to memorize which way to
divide, such as rise/run). I'd remind her that she
already know that a flat line (i.e. horizontal line)
has zero slope (because it doesn't rise or fall as you
go along the line), so it's just a matter of seeing
which way of dividing gives you a zero slope for a
horizontal line. Of course, you'll probably have to show
how to pick two points on a horizontal line, such as
(4,8) and (7,8) -- two points that are at the same
"y-level" when graphed.

Here are some related comments about using what you know
to settle issues like which way you're supposed to divide
or subtract.

***********************************************************
***********************************************************

math-teach (15 February 2005)
http://mathforum.org/kb/message.jspa?messageID=3664795

Pam <Pamkgm@...>
[math-teach: 15 February 15, 2005 08:20:14 -0500 (EST)]
http://mathforum.org/epigone/math-teach/kramstordrand

wrote (in part):

> Student to Miss Pam: "Can I have a calculator?"
>
> Miss Pam to student: "You don't need a calculator to find
> the divisors of 24. I have confidence in you!"
>
> Student to Pam: "But why? My teacher in school wants us to
> use the calculator all the time. That's how we find factors."

I don't doubt a student said this, but I'm sure we all realize
that some students will distort things so much that you almost
find yourself feeling sorry for them because one day they're
going to tell a spouse or a boss something just as far from
the truth when the spouse or boss knows otherwise. I can also
imagine the student's teacher allowing the use of calculators
almost all of the time and then saying something like "well,
use your calculator to see if 3 divides into 24" to a slower
student who doesn't yet know the times table like they should,
and then this being misinterpreted by the other student
(who might have only heard the teacher and thus missed
the context).

The flip side of this is that I often see students
who don't know how (or sometimes, I suspect, don't bother)
to use their calculators to double-check their work.
Here are some examples from the past few weeks:

- ----------------------------

Student: "I forgot ... when I solve x^2 = 5, is it
supposed to be plus and minus the square root?"

Me: "Well, you can use your calculator to see if the
number you're not sure about satisfies the equation."

- ----------------------------

Student: "I understand how to do the problem" (find
an equation of the line passing through a given point
and perpendicular to a given line), "but I'm having
trouble with something. I know I have to find the
slope of 4x - 2y = 6." (Student looks at me for
non-verbal signs that she's on the right track.)
"Well, I have" -- she shows me on her paper the equation
y = (6-4x)/(-2) -- "but I don't know if this is supposed
to be" -- she shows me two possibilities, one of which
is not correct.

Me: "Well, you can use your calculator to plug two or three
numbers into (6-4x)/(-2) and into the possibilities you
have to see which one gives you the same results."

- ----------------------------

Student: "When I'm adding 1/3 and 1/4, do I have to get
a common denominator?"

Me: "Well, you can use your calculator to find what 1/3 + 1/4
is as a decimal number, and then see which of the ways
you have of doing this gives you the same decimal number."

- ----------------------------

Student: "When I found the derivative I got" (student
points to a correct symbolically evaluated derivative),
"but the answer at the back of the book is different"
(points to what looks like it could easily be an
algebraic simplification/rewriting of the student's
expression).

Me: "Well, you can plug x = 0, 1, 2 into both your answer
and the book's answer and see if you get the same values.
This won't prove they're the same, but if you don't get
the same values then you know for sure they're not the
same." (The connection with calculators is that both
expressions had square roots, log's, etc. in them.)

- ----------------------------

Obviously, I'd prefer that they know to include both square roots,
be able to correctly distribute multiplication/division over
addition/subtraction, know to find common denominators when
adding/subtracting fractions, and know how to algebraically
rewrite and simplify algebraic expressions (especially if they
already know what they want to end up with). But this doesn't
bother me nearly as much as when they don't recognize how
easily their calculators can be used to get past the things
they don't know. Yes, I realize this too is a skill and
one they don't have a good command of. Personally, I think
this ability -- being able to make full use of things available
to you in general -- is more important than specific mathematical
knowledge, and I'm always looking for ways to improve it
in my students.

***********************************************************
***********************************************************

misc.education, sci.math (5 May 2008)
http://groups.google.com/group/misc.education/msg/313dbbad22280f46

michalc...@... wrote:

> You may be right about at least some kids getting
> mislead by this technique. My main problems is that
> wwhat you suggest is really not practical. For example
> a common mistake kids make is accidently or mistakenly
> cancelling part of a numerator or denominator as in
> a^2/(a+1) --> a/1 --> a. There is no reversable step
> that they can use to test. Or sometimes the reverse
> step is just way too hard for the kid such as in
> (2x^2 + x + 2) (x^2-2x+3) = 2*x^4-3*x^3+6*x^2-x+6.
> Like I said, going over the probelm again is not
> goos since people are prone to make the same mistake
> twice. If you can come up with a better alternative
> i would be grateful.

I don't like your -3 test for two reasons. One reason
is that the computations involved with plugging in -3
are often not immediate, especially for 8th and 9th
grade students. Another reason is that you're missing
an opportunity to reinforce some important concepts
(e.g. dividing causes exponents to subtract, etc.)

What you're looking at are things I called "safety
checks" in my classes. The most important two safety
checks are plugging in 0 and looking at leading terms.
Of course, you can't always use 0, so use 1 or -1,
although now the process isn't so immediate.

For example, I know (2x^2 + x + 2)(x^2 - 2x + 3)
has a leading term of (2x^2)(x^2) = 2x^4 and a
constant term of (2)(3) = 6, the latter obtained
either by plugging in x = 0 or by knowing the
lowest order term is obtained by appropriately
combining the lowest order terms in the inputs.

Also, I know (x^3 + 3x^2 - 4x - 12)/(x - 2)
can't be x + 3 for two reasons. One is that
the 'x' in x + 3 is not equal to x^3/x, and
the other is that the '3' in x + 3 is not
(-12)/(-2).

A few years ago I taught 4 or 5 sections of (college)
intermediate algebra one year, and I'd bet that 90%
of the incorrect algebraic manipulation mistakes I
found on tests could be seen immediately as incorrect
by just applying the two safety checks above.

Now these aren't going to catch everything, but that's
not the goal. It is not the goal to independently work
the problem (especially when many of the students, and
most of those in your target audience, can't work the
problem correctly in just one way), but rather to have
available 2-second "sight checks" that one can use as
safety checks.

I note that one of your examples, a^2/(a+1) --> a/1 --> a,
passes both of my safety checks. However, this computation
relies on two separate errors occurring together. The first
is cancelling an 'a' without factoring an 'a' out in the
denominator. The second error is in replacing the cancelled
'a' with 0 instead of 1. Now I don't doubt that this can
happen (and I've almost certainly seen it, given how many
papers I've graded in nearly 30 years), but as I've already
indicated, the vast majority of algebraic errors can be
detected with the two 2-second "sight checks" I mentioned.

***********************************************************
***********************************************************

Dave L. Renfro

New Post at RME: Lisa's Method: A New Arrangement For Calculating Slope

2009-01-05T08:52:01Z

Please read the latest post at RationalMathEd.blogspot.com: Lisa's
Method: A New Arrangement For Calculating Slope

Excerpt:

The National Council of Teachers of Mathematics continues to advocate
for more student-centered mathematics teaching (NCTM, 2000). One of my
dreams as a mathematics teacher and tutor is to have a student come up
with an innovative approach to doing or thinking about some
mathematics. And one of my nightmares is hearing of or seeing a
student do something creative and be ignored or slapped down by an
insensitive or ignorant instructor who fails or fears to consider that
a student’s ideas about something might be of value. One of the worst
aspects of “sage on the stage” mathematics teaching is its propensity
for minimizing opportunities for students to do any thinking in class
that strays from the teacher’s directed and generally very beaten path.

Recently, I had one of those much-desired opportunities to see a
student spontaneously come up with what was, to me at least, an
original approach to something that is easy for and familiar to many,
but distressingly hard for a significant number of students:
calculating the slope of a straight line given two points. My student,
Lisa, had been working her way through a computer-based first semester
geometry course and came to me unsure about how to find slope. As I
anticipated, her problem centered the question of where to put the x-
and y-coordinates in the slope equation. She was given the points (3,
2) and (-2, -4) with a picture of the graph, so I began by asking if
she could tell from the graph whether the slope would be positive or
negative. Once she was clear that the slope of a line that went up
from left to right had to be positive, which was indeed the case with
the line in question, I modeled for her how to draw a right triangle
by extending a vertical segment down from (3,2) and a horizontal one
to meet it from (-2, -4). I then asked her to find the lengths of the
two legs of this triangle (5, and 6, respectively): an advantage for
many students of this graphical approach is that lengths are positive
regardless of the presence of negative numbers amongst the
coordinates, and it is relatively easy for them to see or be convinced
that adding the distances left and right or above and below the axes,
respectively, as all positive numbers will get the correct lengths.

Read the entire post at: <http://tinyurl.com/8kxzs6>

Re: Because They Can

2009-01-05T08:20:44Z

Kirby, you've missed the point, and I think it very probably a valid
one. There aren't many women wanting to get a "Google appengine" going,
not because they don't want to get IT's permission, but because they'd
rather do something else. The ones that have the math ability equal to
the IT geeks tend also to be competent verbally and socially, and so are
attracted to other professions that aren't filled with so many geeks.

Can you blame them?

- -Greg

Kirby Urner wrote:

> I'd like to stick a few caveats into this thread, in that
> we're in a somewhat circular definition if we define
> "into IT" as "having a CS degree" or being in the academic
> pipeline in that way.
>
> What happened in guy world a lot is we followed Bill Gates
> in dropping out of school (Harvard in his case)
> becoming instead a vast army of VB/VBA programmers.
> I say "we" because I'm a guy, but actually didn't go the
> VB route, stayed in computers though, with some CS at
> Princeton but not enough to go around calling myself a
> guru at that point. Even today, I don't favor "guru" as
> a label.
>
> So like in GeekWorld we have this species of XX (female)
> that haunts user groups, goes to XO meetings (goes for
> G1G1), hangs out with geek XYs, but isn't really in it
> for the degree or academic credentials. An open source
> "coven" with a titular male cover, on SourceForge or
> FreshMeat, isn't necessarily university trained, all the
> key knowledge being free on the Internet. What it takes
> is a track record and in fact, many a university program
> is still stuck in the 1970s or 1980s, going over highly
> theoretical ground that doesn't impart many on-the-ground
> skills. That's a parody of course. You'll need a strong
> background in engineering if you really want to be a
> computer engineer. Stay in school. But if it's a
> Rails website you're working on, for women to sell from
> their community garden to local food outlets, or a point
> of sale device with a specific purpose, there's nothing
> to (a) keep women out of doing such work or (b) to force
> such women to show up on official IT radar in Houston or
> wherever InfoWorld is getting this data. Any girl with
> a grin can get a Google appengine going, no permission
> from "IT" either sought or required.
>
> That being said, I'm happy to agree that the pipeline is
> drying up.
>
> Kirby

Re: Because They Can

2009-01-04T21:43:24Z

> Kirby Posted: Jan 3, 2009 10:51 PM
>
> >That being said, I'm happy to agree that the
> >pipeline is drying up.
>
> Kirby,
>
> And I am happy that you are happy. And now that
> you are happy, I am sure I can count on you to
> help me explain to the gender-hustlers and assorted
> equitists that the low representation of women in IT
> (and, by extension, in mathematics) is not a
> conspiracy of white men.
>
> Thanks in advance,
>
> Haim
> Je me souviens

I'm suggesting geeks of either gender needn't show up
on any dweeby white guy radar as "IT" per se i.e. you
don't need that CS or math degree to do sysop work, set
up a network, code a point of sale device, run a
community garden, a casino (very math intensive),
found Microsoft or Apple.

These trends go together in that youngsters learn from
their high school experience that doing homework pays
off, after-hours poking around in newsgroups, reading
tutorials, equips you for the workplace. Journalist
types invent Django. Medical doctors fire up a Google
appengine.

Some of these are women, yet needn't register with
InfoWorld (or any old boy net) as being "into IT" (a
subculture / ethnicity many choose to avoid).

By analogy, measuring the average public level of
mathematical sophistication in terms of how many choose
doctoral programs at the university in that discipline,
is not necessarily smart psychometrics. Mathematicians
might think that way, but a numero-literacy specialist
would likely use other criteria.

When Mandelbrot gave us fractals and everyone started
learning complex plane math, to get those fun T-shirts,
my graphs show an uptick, even if the number of math PhDs
was in a dive.

Does that help?

Kirby

Re: limit question poll

2009-01-04T21:25:57Z

> > Regardless of how much rigor one thinks one has,
> > there are always undefined terms, making it
> possible
> > to obtain quite different consequences from the
> same
> > axioms or definitions. I believe the same is
> > happening here with the obtaining of different
> > consequences, a limit of 1 or no limit, from the
> same
> > definition. One undefined term is the inherently
> > ambiguous natural language preposition "on" in
> > phrases that speak of a function defined or not
> > defined *on* a set. And another undefined "term" is
> > x, undefined in the sense of what the values of x
> are
> > to be and are not to be, in language that uses
> > explicitly or implicitly the phrase "values of x."
> > (Dave implicitly rejected values of x such that
> f(x)
> > is undefined while Lou spoke of values of x such
> that
> > f(x) is undefined.) Having different consequences
> > from the same definition or axiom because of
> > undefined terms is a result of the Lowenheim-Skolem
> > Theorem, according to Morris Kline in his book
> > "Mathematics: The Loss of Certainty." See this link
> > to Kline's discussion on this:
> >
> >
> http://books.google.com/books?id=RNwnUL33epsC&pg=PA272
> >
> &lpg=PA272&dq=mathematics:+the+loss+of+certainty+%22un
> >
> defined+terms%22+lowenheim-skolem&source=bl&ots=P73k1X
> >
> 85Nf&sig=7Oj6Vt38aGDYmicen9BG91j4aIw&hl=en&sa=X&oi=boo
> > k_result&resnum=1&ct=result
> >
> > obtained from Googling (with colon and quotation
> > marks):
> > mathematics: the loss of certainty "undefined
> terms"
> > lowenheim-skolem
> >
>
> Morris Kline was an idiot (and far worse than that,
> for that matter), but certainly "perfect" or
> "absolute" rigor is impossible. However, that's not
> what's going on here, Paul. Even though there is
> some lawyerly reinterpretation possible, the fragile
> mechanism by which this all works cannot support just
> anything. Of course, it is possible to completely
> reinterpret everything to the point of it not even
> being recognizable anymore.

There is a way to interpret just two terms/phrases so as to have in effect a limit point c, subset E, and x in E approaching c, and so have something that is quite recognizable and have a mechanism that supports quite a lot. See the below.

...

>
> Let's go back one more time and take a look again.
> You were saying that one could take "f is defined
> d on" in the calculus definition to mean that f is
> simply defined for some values in that interval. If
> you do it that way, then you HAVE to have the
> criteria about x-naught being a limit point of the
> set on which f is defined. I have given examples
> where you end up at absurd and clearly unintended
> interpretations of "limit" if you do not require that
> x-naught be a limit point. So, no, you cannot
> interpret *Thomas and Finney* that way. (You are
> effectively using these varying definitions of domain
> to be able to get the more general set E rather than
> just an interval out of all this. The problem with
> your whole plan of attack though is that you need not
> only the ability to have a set E, but also to capture
> implicitly the idea that x-naught is a limit point of
> E. You may even have accomplished the former, at
> least in this case, but failed to capture the latter.
> So, this cannot be an acceptable way to go around
> d interpreting things at all -- because it will lead
> to absurdity in other cases, then.)
>
> A standard real analysis text, though, will have all
> of the machinery to do it properly. And let me just
> recopy something down here that Renfro said to
> comment on real quick:
>
> > "The fact that some
> > real numbers aren't in the domain isn't relevant,
> any
> > more than the fact that certain matrices aren't in
> > the domain..."
>
> Well, it's *relevant* though it doesn't end up
> mattering in this case. If enough real numbers
> hadn't been in the domain so that x-naught would no
> longer have been a limit point of the domain, then
> you would not be able to talk about x approaching
> x-naught. That didn't happen in this case so the
> limit is 1.
>
> That's basically the whole issue in all of this.
> According to Calculus, it doesn't make sense to talk
> k about x approaching x-naught here. That's not
> really true -- it is just fine to talk about such a
> thing, in this case. Unfortunately, it takes some
> better tools than calculus to do it, though.

In part by using in the HarperCollins Dictionary of Mathematics definition of "defined on," there is a way to make it so that x approaches the constant in question, to have a limit point without explicitly talking about a limit point. This way takes advantage of what I said above, that an undefined and ambiguous term or phrase allows for more than one meaning.

Throughout mathematics, one finds the indefinite articles "a" and "an" being used such that they are not explicitly defined in terms of quantification, the existential quantifier ("some," meaning "at least one") or the universal quantifier ("any" and its variants). Since these indefinite articles are ambiguous regarding quantification, one is to gather any such quantifier meaning from the context - sometimes one meaning is relevant, sometimes the other is, but sometimes neither is.

In the clause "let f be defined on an open interval containing c (except possibly at c)," the term "an" is not explicitly defined in terms of quantification. (It's used in the Wikipedia article on limit, which gives the Weierstrass definition.) To see that "an" rather than "some" may be more common, one can do a Google search with the two phrases below (with quotation marks)

limit "defined on an open interval"

limit "defined on some open interval"

I got the result of 3,030 hits when "an" is used and 586 hits when "some" is used. When "any" was used there were only 2 hits, and no hits with "each" or "every." By these hits, it's clear that the interpretation of "defined on" used in this phrase is not the HarperCollins definition, at least not with the maximal and essential domains being different sets. (It can still be the HarperCollins definition with the maximal and essential domains being the same set.) Other combinations of words and phrases yielded for me similar results in that there were many more hits with "an" being used rather than "some."

Concerning the whole clause in question ("let f be defined on an open interval containing c (except possibly at c)"), this above definition of "defined on" can give a contextual meaning of existential quantification to "an" so that one can interpret the clause as "let f be defined at any point not equal to c in some open interval containing c (and let f be defined possibly at c as well)." But one could go the other way: The HarperCollins definition of "defined on" can give a contextual meaning of universal quantification to "an" so that one can interpret the clause as "let f be defined at some point not equal to c in any open interval containing c (and let f be defined possibly at c as well)."

In the second interpretation, where x varies over the points at which f is defined, x approaches c, and so one can test for limits (including one-sided limits if one needs to). (So rather than going with the first interpretation and having to give Lou's answer that the limit of the function in question at the given point does not exist, one can go with the second and give Dave's answer that the limit is 1.)

And note this perhaps interesting fact: The first and less general interpretation of the clause implies the second and more general interpretation of the clause, meaning that the concept of the second interpretation is not foreign to the first interpretation and so is not foreign to calculus.

(To have the more general interpretation as an alternative to the less general one, interpreting "defined on" according to HarperCollins and interpreting "an" as "any" would be small quantitative (but not necessarily small qualitative) changes in interpretation. If "some" is used explicitly in a given rendering of the definition of limit instead of "an," then since so many other renderings use "an," one could replace the former type of rendering with the latter and then make the changes. The more general interpretation gives some perhaps welcome alternatives, and as was already pointed out, it is not foreign to calculus since it is implied by the less general interpretation.)

Paul A. Tanner III

Re: Two articles in Dec 2008 MAA FOCUS

2009-01-04T20:08:06Z

Although I do stress that solutions be written in
grammatical English (or Spanish, if they prefer)
I do give credit for coherent, correct
mathematics when that rule is violated. My bad.

I have never prohibited calculators of any kind
in my mathematics classes although some of my
colleagues do. I do insist on fairly complete
solutions and exact answers, when appropriate,
and detract credit for missing key steps or
inappropriate approximate answers. For properly
prepared students, use of calculators are not
helpful, however. In fact, the reverse is true.

Wayne

At 02:46 PM 1/4/2009, Richard Strausz wrote:

>Wayne, I didn't misread anything. Your narrative
>didn't make it clear. Anyhow, how about the
>following course description/procedure excerpts
>from some classes at U of Kentucky. I don't
>think these would have been in your classroom, eh?
>
>Richard
>=====================
>Math 113 - Calculus I
>
>"In order to help you learn to write mathematic
>and present clear, well-written solutions to
>problems, there will be six written assignments.
>Your solutions to these assignments are expected
>to be carefully written in complete sentences
>and grammatically correct English. You should
>give clear reasoning and present the steps of your solution in logical order"
>"
>
>
>Math 109 - College Algebra
>
>"You may use a graphing calculator during the
>exams, but NO calculator with a Computer Algebra
>System (CAS) or a QWERTY keyboard is permitted"
>"
>
>
>Math 108R - Introductory Algebra
>
>"No laptop computer should be used in the
>classroom. Other electronic devices such as cell
>phones should be unseen and unheard. Calculators are allowed"
>"
>
>
> >
> > You appear to have misread the course descriptions.
> > Those were the
> > course descriptions of Bellarmine University, not U
> > of Kentucky,...
>
>__________ Information from ESET NOD32
>Antivirus, version of virus signature database 3231 (20080701) __________
>
>The message was checked by ESET NOD32 Antivirus.
>
>http://www.eset.com

Re: Two articles in Dec 2008 MAA FOCUS

2009-01-04T19:44:21Z

> The Dec 2008 issue of MAA FOCUS at:
>
> http://www.maa.org/pubs/dec08web.pdf

> The article by Sharon Vestal on using WebAssign for
> Calculus I homework is on Page 22.

WebAssign may be the miracle cure for getting students to do their own calculus homework, but Vestal works as a WebAssign instrutor for a company that sells the textbook she uses. A glowing review is not surprising.

The one thing I did get out of it was that students were more engaged when homework was graded.

Re: Two articles in Dec 2008 MAA FOCUS

2009-01-04T19:17:35Z

TI of newt!

Failure, thy name is printed circuits

Yon Casio has a lean and electronic look. Such things are dangerous.

Out, damned Sharp!

A plague on both your motherboards!

A course in pure math never did run smooth.

Alas, poor pencil and paper! I knew them, Horatio.

An HP! An HP! My kingdom for an HP!

On Jan 4, 2009, at 5:46 PM, Richard Strausz wrote:

> Wayne, I didn't misread anything. Your narrative didn't make it
> clear. Anyhow, how about the following course description/procedure
> excerpts from some classes at U of Kentucky. I don't think these
> would have been in your classroom, eh?
>
> Richard
> =====================
> Math 113 - Calculus I
>
> "…In order to help you learn to write mathematics and present clear,
> well-written solutions to problems, there will be six written
> assignments. Your solutions to these assignments are expected to be
> carefully written in complete sentences and grammatically correct
> English. You should give clear reasoning and present the steps of
> your solution in logical order…"
>
>
> Math 109 - College Algebra
>
> "…You may use a graphing calculator during the exams, but NO
> calculator with a Computer Algebra System (CAS) or a QWERTY keyboard
> is permitted…"
>
>
> Math 108R - Introductory Algebra
>
> "…No laptop computer should be used in the classroom. Other
> electronic devices such as cell phones should be unseen and unheard.
> Calculators are allowed…"
>
>
>>
>> You appear to have misread the course descriptions.
>> Those were the
>> course descriptions of Bellarmine University, not U
>> of Kentucky,...

Should rational area and volume be taught?

2009-01-04T16:46:39Z

Here is one reason to use Synergetics coordinates StoP and PtoS
orientation to the Cartesian system. Measuring area by triangles and
tetra-volume of the "side show freaks" of polyhedra that have
irrational content if you use Cartesian coordinates.
See:
http://mysite.verizon.net/cjnelson9/index.htm

Clear[a, b, c, d, e, f]
generalTriangle = {{a, b, -a - b}, {c, d, -c - d},
{e, f, -e - f}}

generalTriangle - 1/3
{{-(1/3) + a, -(1/3) + b, -(1/3) - a - b},
{-(1/3) + c, -(1/3) + d, -(1/3) - c - d},
{-(1/3) + e, -(1/3) + f, -(1/3) - e - f}}

The area in equilateral triangles is the determinant of that matrix.

FullSimplify[Det[generalTriangle - 1/3]]
b*(c - e) + d*e - c*f + a*(-d + f)

Now for tetra-volume.
Clear[a, b, c, d, e, f, g, h, i, j, k, l]
generalTetrahedron = {{a, b, c, -a - b - c},
{d, e, f, -d - e - f}, {g, h, i, -g - h - i},
{j, k, l, -j - k - l}}

generalTetrahedron - 1/4
{{-(1/4) + a, -(1/4) + b, -(1/4) + c,
- -(1/4) - a - b - c}, {-(1/4) + d, -(1/4) + e,
- -(1/4) + f, -(1/4) - d - e - f},
{-(1/4) + g, -(1/4) + h, -(1/4) + i,
- -(1/4) - g - h - i}, {-(1/4) + j, -(1/4) + k,
- -(1/4) + l, -(1/4) - j - k - l}}

The volume measured by regular tetrahedrons is the determinant of the
four by four matrix.

FullSimplify[Det[generalTetrahedron - 1/4]]
a*f*h - a*e*i - f*h*j + e*i*j - a*f*k + f*g*k +
a*i*k - d*i*k + c*((-d)*h + e*(g - j) + h*j + d*k -
g*k) + a*e*l - e*g*l - a*h*l + d*h*l +
b*((-f)*g + d*i + f*j - i*j - d*l + g*l)

If the Synergetics coordinates are rational the area and tetra-volume
are rational. If you append a column of 1/Factorial[d-1] to a d by
(d-1) matrix of d rows of Cartesian coordinates points the determinant
is the square or cubical area or volume.

Does anybody measure area and volume by anything but squares and cubes
with coordinates?

Cliff Nelson

Dry your tears, there's more fun for your ears,
"Forward Into The Past" 2 PM to 5 PM, Sundays, California time:
http://www.geocities.com/forwardintothepast/
Don't be a square or a blockhead; see:
http://mysite.verizon.net/cjnelson9/index.htm
http://library.wolfram.com/infocenter/search/?search_results=1;search_person_id=607

Re: Two articles in Dec 2008 MAA FOCUS

2009-01-04T14:46:28Z

Wayne, I didn't misread anything. Your narrative didn't make it clear. Anyhow, how about the following course description/procedure excerpts from some classes at U of Kentucky. I don't think these would have been in your classroom, eh?

Richard
=====================
Math 113 - Calculus I

"…In order to help you learn to write mathematics and present clear, well-written solutions to problems, there will be six written assignments. Your solutions to these assignments are expected to be carefully written in complete sentences and grammatically correct English. You should give clear reasoning and present the steps of your solution in logical order…"

Math 109 - College Algebra

"…You may use a graphing calculator during the exams, but NO calculator with a Computer Algebra System (CAS) or a QWERTY keyboard is permitted…"

Math 108R - Introductory Algebra

"…No laptop computer should be used in the classroom. Other electronic devices such as cell phones should be unseen and unheard. Calculators are allowed…"

>
> You appear to have misread the course descriptions.
> Those were the
> course descriptions of Bellarmine University, not U
> of Kentucky,...

Re: Because They Can

2009-01-04T13:13:16Z

Kirby Posted: Jan 3, 2009 10:51 PM

>That being said, I'm happy to agree that the pipeline is
>drying up.

Kirby,

And I am happy that you are happy. And now that you are happy, I am sure I can count on you to help me explain to the gender-hustlers and assorted equitists that the low representation of women in IT (and, by extension, in mathematics) is not a conspiracy of white men.

Thanks in advance,

Haim
Je me souviens

Re: Because They Can

2009-01-03T19:51:07Z

I'd like to stick a few caveats into this thread, in that
we're in a somewhat circular definition if we define
"into IT" as "having a CS degree" or being in the academic
pipeline in that way.

What happened in guy world a lot is we followed Bill Gates
in dropping out of school (Harvard in his case)
becoming instead a vast army of VB/VBA programmers.
I say "we" because I'm a guy, but actually didn't go the
VB route, stayed in computers though, with some CS at
Princeton but not enough to go around calling myself a
guru at that point. Even today, I don't favor "guru" as
a label.

So like in GeekWorld we have this species of XX (female)
that haunts user groups, goes to XO meetings (goes for
G1G1), hangs out with geek XYs, but isn't really in it
for the degree or academic credentials. An open source
"coven" with a titular male cover, on SourceForge or
FreshMeat, isn't necessarily university trained, all the
key knowledge being free on the Internet. What it takes
is a track record and in fact, many a university program
is still stuck in the 1970s or 1980s, going over highly
theoretical ground that doesn't impart many on-the-ground
skills. That's a parody of course. You'll need a strong
background in engineering if you really want to be a
computer engineer. Stay in school. But if it's a
Rails website you're working on, for women to sell from
their community garden to local food outlets, or a point
of sale device with a specific purpose, there's nothing
to (a) keep women out of doing such work or (b) to force
such women to show up on official IT radar in Houston or
wherever InfoWorld is getting this data. Any girl with
a grin can get a Google appengine going, no permission
from "IT" either sought or required.

That being said, I'm happy to agree that the pipeline is
drying up.

Kirby

Re: Two articles in Dec 2008 MAA FOCUS

2009-01-03T19:32:08Z

At 02:30 PM 1/2/2009, Richard Strausz wrote:
>But Wayne (I love it when you over-generalize),

Overgeneralize? Moi?! I thought I left any conclusions, if any were
to be appropriate, to the reader?

>the first two courses at the University of Kentucky offer
>'computer-based explorations.'See below.

You appear to have misread the course descriptions. Those were the
course descriptions of Bellarmine University, not U of Kentucky, not
U of Michigan, not...

>Might they realize that the pedagogy of the 1950s needs updating?

They might think so and they might even be right but pretending that
pedagogy is content has been an egregious error of design since the
inception of the New New Math. A number of colleges of engineering
told their departments of mathematics to get back to focusing on
calculus content competence or they would be compelled to teach their
own calculus. Bellarmine University does not appear to have an
engineering program, however, so that would not a problem. And they
do have a large school of education so all is well.

Wayne

>Richard
>
> > Math. 117 Calculus I
> > Limits and continuity of functions; the concept
> > of derivative; calculating derivatives;
> > applications of derivatives such as optimization
> > and related rates; integration through the
> > Fundamental Theorem. The course includes
> > computer-based explorations. (Prerequisite: Math.
> > 116 or its equivalent.)
> >
> > Math. 118 Calculus II
> > Applications of integration such as area, volume,
> > and arc length; techniques of integration and
> > improper integrals; approximation of integrals;
> > infinite sequences and infinite series. The
> > course includes computer-based explorations.
> > (Prerequisite: Math. 117 or its equivalent.) Every
> > spring.
> >
>
>__________ Information from ESET NOD32 Antivirus, version of virus
>signature database 3231 (20080701) __________
>
>The message was checked by ESET NOD32 Antivirus.
>
>http://www.eset.com