[no subject]

Strong objections!

1. There is a serious conceptual, pedagogical and developmental danger in
seeing fundamental components of calculus in opposition to each other as in:
"The primary
objects of study ARE the functions, NOT the calculus tools used to
investigate the functions." There is an equally serious epistemological
obstacle facing the math ed profession from the statements as above, the
obstacle which doesn't permit to design the calculus courses where one
studies functions TOGETEHR with the study of the calculus tools. As if
linear space wouldn't come with the algebra of linear operators on it or a
group of elements without the related group of automorphisms!
2. The limiting attitude of "one as opposed to other" made already its
negative imprint on the results of mathematics reform in school instruction
through the immense confusion which exists in the profession on the question
of the relationship between procedural and conceptual. Witness the disaster
in NYCity with 77% failure rate in 8th grade state exam!
3. That arbitrary separation of the conceptual whole, which the objects of
study constitute together with its operators, results in the significant
lowering of the cognitive level reached our students without giving them the
opportunity to reach meta-cognitive knowledge hidden in the relationships
between the two components, and consequently to strengthen the calculus
schema.
4. Therefore, why even try in that direction?

Part 2

In the statement below there are several related issues which need thorough
examination.
1. " More generally an algebraic quadrature of the parabola means that the
assumption that Riemann sums and limits are necessary for determining the
area of all regions with curvilinear boundaries is simply wrong." No one
ever assumes that Riemann sums are necessary to calculate that area.
Archimedes did it by exhaustion principle, Cavalieri and Wallis by
indivisibles, Fermat by rectangles. Both Riemann Sums and indivisibles
require the notion of the limit, because both techniques relay on very
unusual methods of treating the infinity with the help of finite steps,
through the concept of the limit. It is precisely in the emphasis on the
relationship between the finite and infinite where lies the depths and
importance of limits in calculus. Because it lifts the discourse to the
higher cognitive level, higher level of schema development. The "new
standard for mathematical literacy", that you, Bill, propose, in resting the
calculus on purely algebraic methods is simply too low because it is doesn't
conduce nor challenges the students for reaching upper levels of their Zone
of Proximal Development, generally possible on this level,
I am very suspicious of the repeating attempts to argue against the
development of the limit concept in that course. Admittedly, there are
difficulties in teaching it well documented in the literature. Does this
mean that we chicken, stop investigating how to teach it adequately and
introduce algebraic methods which don't use this terrible concept, or do we
find out how to treat it so that it, the limit, can find itself "within
reach of all
students at both the secondary and undergraduate level" ? Maybe its time to
make understanding of the pedagogy of limits as the central research goal of
ARUME?
Broni Suave
(to be contiued)

----- Original Message -----
From: William Crombie <BCrombie@AOL.COM>
To: <ARUME-LIST@ENTERPRISE.MAA.ORG>
Sent: Tuesday, January 01, 2002 7:46 AM
Subject: [RUME] "Revising the Calculus Curriculum"


>Welcome everyone to yet another year.
>
>The November 2001 issue of FOCUS raises an issue which I think is  worth
>discussing. I'm referring to the article entitled "An Experiment that
Worked:
>Revising the Calculus Curriculum."
>
>I certainly applaud the direction of the changes described in the article.
>The only reservation I hold is that the conceptual reorganization of the
>calculus described in the article did not go far enough. My reservation is
>rooted in what I have come to see as a failure of understanding on the part
>of both supporters and detractors of college calculus reform. Both sides
have
>tacitly assumed that the historically derived form of the calculus is
>immutable - almost as if it were a law of nature or delivered from a
burning
>bush. This tacit assumption constitutes a failure to examine and understand
>what might be called the deep structure of the elementary calculus. And
this
>oversight ultimately manifests itself as a failure to clearly grasp the
>changes both needed and possible in the reformation of the calculus as an
>institution.
>
>I believe the basic theoretical thrust of the calculus reformation at
Centre
>College was in the right direction. The authors tell us that "The primary
>objects of study are the functions, not the calculus tools used to
>investigate the functions; indeed, the heart of this revision is this
>paradigm shift from a calculus course designed around calculus tools to a
>sequence of calculus courses designed around functions." From a modern
>perspective to understand a concept algebraically is,  in many instances,
to
>understand the function which characterizes the concept. But the approach
>taken never actually functionalized the tools of the calculus. In a sense
>they never turned these tools, the derivative and the integral, into
objects
>of study - the slope function and the area function. Instead the course
>returned to the conventional  sequence of limits, derivatives and
integrals.
>It appears that theses concepts and their relations were not fundamentally
>restructured. More so, they were merely delayed. From the evidence of
student
>numbers and responses I do not doubt that these changes made the calculus
>more accessible to more students. While this change is surely admirable I
>would not label it a paradigm shift. This effort could more accurately be
>described as a continuous deformation of the traditional calculus course
than
>a  Kuhnian shift with all the discontinuities in perspectives and practice
>that accompany a restructuring of an underlying paradigm.
>
>Let me be more specific. A deep reform, a true paradigm shift would
>constitute a  fundamental restructuring of the relationships among the
>concepts of limit, derivative and integral. Surprisingly just this type of
>transition can be most easily seen in the case of the simplest functions
>consider by the calculus - polynomials. It has long been known that the
>results of the differential calculus of polynomial functions require
neither
>infinitesimals nor limits. Purely finite, algebraic methods suffice. It has
>not been so widely known that the same is true for the integral calculus of
>polynomial functions. As a matter of fact all that is required for the
>quadrature of the parabola, or any polynomial function for that matter, is
an
>application of the algebra of polynomial functions and the geometry of
>similar figures. Even the Fundamental Theorem of Calculus (FTOC)  is
>illuminated in new and unique ways when viewed from the perspective of the
>algebraic relationship between slope and area functions. The difference
>between the algebraic version and the limit-based version of this theorem
is
>a classic example of what Richard Skemp has referred to as relational
>understanding versus instrumental understanding.  The limit-based version
of
>the FTOC is instrumental in the sense that limits certainly produce the
>desired results but they do not illuminate the underlying geometric and
>algebraic relationships which constitute these results.
>
>More generally an algebraic quadrature of the parabola means that the
>assumption that Riemann sums and limits are necessary for determining the
>area of all regions with curvilinear boundaries is simply wrong. It is
>mistaken. Limits are not needed for regions with polynomial boundaries. It
>means that we neither understood the theoretical strength of simple algebra
>nor did we understand the truly necessary role that limits play in the
>elementary calculus. From an instructional perspective an algebraic
>quadrature of the parabola means that it is now entirely possible to place
>the elementary calculus, both differential and integral, within reach of
all
>students at both the secondary and undergraduate level. It offers a new
>standard for mathematical literacy for all students at the beginning of
this
>new century.
>
>I realize that this is a rather dense presentation of a set of issues
around
>the calculus. I'm hoping, if there is interest, that we can unpack at least
a
>few of these concerns and positions.
>
>Bill Crombie