From harelb Sun Nov 21 22:18:01 EST 1993
To: cwoodbur,bservat,beverly
CC: orlandi,gargova,harelb
Subject: ENCLOSURE: Draft Proposal for Curriculum Committee


(Yes, I *know* we don't have to be so formal. So I had fun with the
word "whereas" -- so sue me... Harel   :-)

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			   D r a f t   # 1
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WHEREAS:

Several graduate students including Harel Barzilai, Maria Gargova, and
Lisa Orlandi have taken an active interest in calculus reform at
Cornell; several have already undertaken limited experimentation in
their sections as instructors (students working in groups); were
interested in Brigitte Servatius' and Cynthia Woodburn's presentations
at the Occasional Seminar on Undergraduate Teaching regarding reformed
calculus courses at Worcester Poly and New Mexico State; have found,
along with other graduate students and Professors (see below) the
111-112 sequence as it is currently structured to be problematic in
several respects and centering on the need to cover a virtually
impossibly long list of topics in the syllabus at (therefore) not
enough depth per topic;

WHEREAS:

Louis Zulli, who graduated last spring and who had taught both 111 and
112 many many times, last semester (Spring 93), before leaving,wrote a
letter to then Math-112-Czar Prof Marshall Cohen, which Cohen posted
to the "math 112" email-list politely complaining at length about the
need to re-structure 112 due to the incredibly over-stuffed syllabus.
Cohen said he would forward that to the Curriculum Committee, and
suggested that any of us who wanted to could also email our comments.
Several of us wrote/emailed Cohen, concurring.

And whereas Professor Gross and the rest of the instructors of 112
this semester find themselves in essentially the same situation as
last semester vis a vis keeping up with the syllabus (Professor Gross
did by the way know of and read Zulli's letter but now finds himself
the same boat)


WE WOULD RECOMMEND, and would request that the Curriculum Committee
study the implementation of a reformation of the 111-112 sequence
along the lines of, the following:

*1* That the Syllabi, especially for 112, be reduced substantially with
the aim of fewer topics covered at greater depth so most students at
the end of the semester honestly understand what was covered, which is
not presently the norm. Note that the bright students who *do*
understand most/all of the topics at the current size/rate dictated by
the syllabus would be able, were they to learn 1/2 to 2/3 of the
topics in real depth, to teach themselves or quickly pick up in future
classes, related techniques; e.g. four other techniques of integration
in addition to the  four covered in depth in class; or, four other
Tests for Series' Convergence in addition to the four covered in depth
in the course; or, formulas for surface area in Parametric or Polar
form after having learned an understood area or arc-length in
Parametric form, say.

At present, students encounter a formula for shells, then a formula
for washers, then a formula for arc-length, then a formula for
arc-length for Parametric equations, then another formula for
surface-area, then another formula for area in Polar coordinates, then
another formula, then another, and another... When these future
tax-paying citizens vote on funding for mathematics, will they have
memories of the beauty and joy of mathematics, or will most of them
have uneasy memories of mathematics being a bag of Tricks and Formulas
(thrown at them at a rate comparable to trying to drink from a fire
hose)? At present, we do little to disabuse students and precisely
reinforce the kind of "Tricks and Formulas" stereotypes that pervade
in this society about mathematics and which do not serve mathematics,
or society, well.

To accommodate other departments while still implementing substantial
(especially for 112) cuts in the number of topics in the syllabus,
different sections (or "varieties of the course") could be designed to
cover different topics, to be tailored to the needs of groupings of
other departments, e.g. for Bio and Chem, for Econ, etc.

*2* That the course be structured in a way allowing for and enhancing:

-- In-depth learning of a limited number of *fundamental ideas* of
calculus, including motivation, general techniques (e.g., "approximate
the continuous with the linear"), the underlying geometry, and,
crucially, the beauty of the subject in particular and of mathematics
in general. Insofar as what students get out of the course, both in
their later studies, and their impressions as adult citizens, is it
not worth sacrificing a formula or two later, to spend the time in
class on a beautiful proof or two of the formula for 1+2+...+N when
investigating this had just been motivated by Riemann-sums?

-- Cooperative learning and learning in groups, including having to
explain concept to fellow students.

-- Working on reasonably-sized but somewhat "open-ended" projects,
possibly including presentations in class

-- Writing assignments -- minimally, for the projects, possibly also
in the context of smaller-scale "home work" assignments.

--

Finally, we would like to note that there are simultaneously two
reasons for the reforms we suggest: The problems with the syllabus
that *1* attempts to address are nicely dealt with by the ideas in
*2*, i.e., a changeover from very many topics covered very quickly, to
fewer topics covered in more depth. On that basis alone, *2* could
help substantially improve the calculus sequence. In addition, we feel
that the "calculus reform" ideas outlined in *2* are good in and of
themselves as aids to better learning (as noted at the outset, several
of us have experimented with them, within the time-limitations
dictated by the current format of the course) and hence would enhance
any (or, many times of) courses even were they *not* courses whose
current structures presented difficulties which we independently
wanted to address.