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[Phys-L] Animated Math - comments

Hi all-
Dan MacIsaac has invited me to comment on what appears to be a
webcourse on introductory calculus: http://mathanimated.com/. The
origins are described by the author as follows, in this e-mail from a
retired Israeli Physics Professor.:
Finally I decided to dedicate my time and effort to education of
University and college students. Since I had a long experience in
teaching math for first year students of physics, I decided to write a
courseware on that subject. The courseware is based on XML
technologies, recommended by the WEB consortium ( http://www.w3.org/
),
namely MathML - for mathematical expressions and SVG - for animated
and
interactive graphics. Although the aim of the courseware is college
and
university undergraduate students of physics and engineering, the
animations and interactivity make the courseware suitable as a
teaching.

I welcome this opportunity to relieve myself of some long-festering
thoughts on math teaching . especially the teaching of calculus.

I taught introductory calculus physics in a community college for several
years in the .80.s. A few years ago I was invited to teach an
introductory calculus course in another community college. By this time
my teaching had become informed by Arnold Arons. book, <A Guide to
Introductory Physics Teaching > which sets out some cardinal principles,
based upon a couple of decades of observation of student learning (at the
first year college level). I summarize some of these principles:
1. Most of the time, the average student in the average course doesn.t
know what you.re talking about, so discuss with the students . repeat,
with the students - the definitions of the words you use;
2. Concept before name;
3. Don.t introduce a concept until you need it;
4. Your students don.t understand arithmetical reasoning, let alone
algebra.
5. Shut up and listen! You.ll learn wonderful new ways that the students
have not understood what you.re talking about.
To these 4, I.ve added my own:
A. Most students taking calculus can barely do algebra.
B. Almost no students have any idea of how to prove a proposition in
mathematics; mostly, students equate .proof. with .example..
C. Learning math is not a linear process; we don.t learn in the logical
sequence presented in a classical math text (Bourbaki is no help to the
math teacher).
D. After three semesters of calculus, most students don.t understand the
geometric meanings of derivative and integral (slope and area).

When I looked at the Calculus text that I was given to teach from, I could
see that its authors were innocent of Arons. teachings, so I set about a
search of available calculus texts. I doubt that you could name one that
I didn.t look at. They were, to a large extent, clones where the authors
couldn.t decide whether they were teaching calculus or an advanced course
in real analysis. Arons. principles had never reached the math teaching
profession (happily, shortly before his death he was an invited visitor to
the math department at the University of Illinois in Chicago) I went ahead
with the assigned text, but, shortly thereafter, started one of my own..

The titles of the first few sections of <Calculus>, by Osterbee and Zorn,
are typical:
I. Functions in Calculus
a. Functions, calculus style.
b. Graphs.
c. Machine Graphics
d. What is a function?
e. A Field Guide to Elementary Functions.
And so on .
The Opening Words of the Text are::
.Calculus is a branch of Mathematical Analysis . the study of functions..

This introduction takes me back to my first calculus class at MIT in the
fall of 1941. We were required to have read the first section of
Phillips. <Calculus> with a rigorous epsilon-delta definition of a
derivative. I knew some of the Greek letters from high school trig, but
the rest of it might as well have been written in Sanskrit. Impressed?
Wow! Taught anything about calculus? Hardly. I did, however, learn the
New England pronunciation of the Greek letters: .Alpher., .Bater.,
.Gammer., ., a lesson that has never left me..

When I taught the Calculus course I had already discovered the
introduction to Calculus in the Feynman-inspired Cal Tech text, <The
Mechanical Universe> (please, the regular edition, not, not, not the
Advanced Edition). There is a set of videos on CD which are part of the
high school version of the course, which animate mathematical operations
. including the derivative . as well as illustrate physics concepts. I
found these well received by students, when viewed under carefully
supervised conditions. Thus, when I got Dan.s invitation to look at
MathAnimated, I had a preconception of what I hoped to see.

What I did see, was this:
CHAPTER 1 . DIFFERENTIATION
1. Real Numbers
Rational Numbers
Countability
Irrational Numbers
Exercises
2. Operations
Basic operations on real numbers
.
The opening sentence of this calculus text is:
.Rational numbers are defined as a ratio of integers.. (All of the above
is apparently considered by the author to be so original that it seems to
be impervious to copying off the net.). I remark in passing that the
Exercises at the end of the first section include the following:
.2. Prove that all the rational numbers r in the interval 0<=r<=1 are
countable. Give the first 12 of them.. (I hope that this is intended as
a trick question, but I fear not Hint: show that there is no upper or
lower bound to the rational numbers in the open interval (0,1); 0 and 1
are, of course, not in the interval.).

A couple of years ago I attended a conference on the teaching of
introductory physics at the University of Illinois, Chicago campus. I
heard proposals by people who were preparing new introductory physics
texts. I got some enthusiastic applause from the audience for remarking,
substantially as follows:
It seems to be that most physics teachers are endowed with an
inability to listen. Every physics faculty member and his dog has a
.new. proposal for a physics text, but nobody listens to anybody else.s
proposal. There is no dialogue that starts with a concept for a text and
seeks peer agreement on ways to improve the concept. There is, in fact no
dialogue. So what we end up with is a collection of shabby clones.
I think that the same remarks apply to the teaching of elementary
calculus.

I am currently working on Chapter 14 of a 30 chapter elementary calculus
text. The text begins:

...................................
The introduction and first three chapters are on the net for anyone to
view., comment on and, hopefully, steal if you think that you can do a
better job of the remainder than I can. It tries to incorporate the
lessons taught by Arons and my own experiences with struggling students.
(http://www.hep.anl.gov/jlu/ index.html click on either .ps or .pdf under
.book.)

I think that that Animated Math would benefit greatly if the author would
familiarize himself with the many ways that we are not understood by our
students, the attempts at teaching calculus that are presently available,
modern research into the pedagogy of teaching mathematics and/or physics,
and then decide whether he wants to publish a calculus text or a text on
real analysis. He should also try to characterize his intended audience.

Regards,
Jack Uretsky . July 2005
--
"Trust me. I have a lot of experience at this."
General Custer's unremembered message to his men,
just before leading them into the Little Big Horn Valley
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