========
PROLOGUE
========

As a graduate student in mathematics, during an oral exam for a
Masters in Mathematics, the author was momentarily stuck on a
particular question relating to a certain space (a K(G,1) space), then
realized  that if another "name" for the same space is used (a
projective space) the answer is almost immediate. Upon answering the
question, and adding "it's embarrassing how much the notation I use
affects my ability to see what's going on!" the author was somewhat
surprised to hear one of the two examiners, among the leading
researchers in his field, reply "the same thing happens to me too!"

============
INTRODUCTION
============

This line of inquiry began with our gradual observation over the last
ten years of teaching calculus courses that certain types of student
mistakes resurfaced each semester, were made repeatedly by certain
students, and were made by a large cross section of students. These
mistakes were first noted informally, and later were recorded in
writing in an (informal) catalogue of sample student mistakes.

The roots of our present inquiry and report began in '96-'97 when
certain structures came to our attention in this catalogue of
student mistakes, and in particular when common roots appeared to
emerge among a variety of catalogue items which are superficially
unrelated. These mistakes and commonalities turned out to be
notationally based, and further analysis revealed conceptual issues
behind the notational ones, which appear to be amenable to more
theoretical research in undergraduate mathematics education (RUME).

These recurring calculus and precalculus mistakes (RCMs) were
observed in the classroom, such as during student cooperative learning
in groups (working on activities together), during office hours, in
tests, and in drafts handed in by students of long-term calculus
projects (such as those which appear in MAA's Student Research Projects
in Calculus).

The RCMs were observed in traditional calculus students as well as in
reformed projects-based calculus (where they were, however, more
quickly detected and thus addressed) at Cornell University where the
author helped lead a reform initiative (See _Graduate Student
Initiated Calculus Reform, Part I_, PRIMUS, March 1999, and Part II,
to appear in PRIMUS, June 1999); at the University of Minnesota, a
large, research oriented public university, and most recently at
Lynchburg College, a small, teaching oriented private institution.

====================
GOALS AND MOTIVATION
====================

Our initial motivation was entirely pragmatic: that RCMs were worth
finding, understanding, and classifying to help us, as mathematics
educators, identify the "notational banana peels" upon which students
slip, semester after semester. One (minimalist) goal then, is to arm
mathematics teachers with RCM catalogs, so that they can adjust their
pedagogy appropriately, whether through warning their students,
changing presentations, or indeed changing the way textbooks are
written, due to increased awareness of these banana peels -- these
notational or visual puns.

As noted below, the analysis and classification of RCM leads to
understandings of "*conceptual* banana peels", i.e., of the complexity
or ambiguity of certain mathematical concepts, thus giving rise to a
second pragmatic (but deeper) path, one towards the goal of improving
the teaching and learning of mathematics.

Thirdly, it is hoped that fellow mathematics educators who are better
versed with and knowledgeable about existing and emerging theoretical
frameworks will either in parallel or in collaboration with the author
broaden the present analysis so that it may both inform and be
informed by theoretical frameworks and models of the learning of
mathematics. A fourth goal, that of expanding the present work to
mathematics subjects outside of calculus and precalculus, seems likely
to take place in part through success in attaining the third goal.

=======
SUMMARY
=======

Several of the RCMs considered are mistakes which one might otherwise
be tempted to classify under the "All functions are Linear"
fallacy. Nevertheless, evidence can be presented to suggest notational
roots, which in turn have deeper conceptual roots. This then is one of
the findings results of this study which bear in a very specific way
on teaching practice, indicating it is not a "false assumption of
linearity" but notational-conceptual misunderstandings with different
roots, upon which the instructor should focus (for example, many if
not most instances of "cos(x+y) = cos(x)+cos(y)": seem to be attempts
by students to apply the *distributive* law, rather than due to a
false assumption of linearity of the cosine function!)

The deeper conceptual issues include the "dual nature of functions"
as both objects and operators, and the variable-versus-constant
ambiguity perhaps best captured in the author's high school chemistry
teacher's reference to "constants which vary" or the notion of a
parameter, which embodies the constant-acting-as-variable
ambiguity. These two central issues, the dual nature of functions and
the constant/variable ambiguity turn out to be intertwined.

For example, one RCM relating to both arises from the observation
that, of the two standard definitions for the derivative f'(a), only
one "works" -- without banana peel -- when the student tries to
substitute "x" for "a" in both sides of the equation. Other examples
are much more familiar, for instance, the "real versus dummy variable"
confusion which is perhaps the most obvious sin in calculus books
which regularly define F(x) to be the integral from c to x of "f(x)dx".

Though we might suggest an alternative system of notation for calculus
to be provocative -- i.e., to provoke general thought and discussion
about the notational system we use -- it seems that no simple and easy
solution to the notational banana peel problems exist (we stress,
simple and easy), in large part due to deeper, underlying mathematical
conceptual difficulties which perhaps should be a regular part of what
is put on the table for discussion and analysis in pedagogical,
mathematics eduction (ME) and RUME circles.

Further arguments that reform of notation should be considered include
the observation that, working late at night, mistakes such as
switching "its" with "it's" are more likely to happen even if the
author is perfectly well aware of the distinction and the meaning of
each -- the "visual pun" alone suffices as banana peel, even when the
underlying concept *is* known/understood. A fortiori, mathematical
notational pitfalls of this type are particularly dangerous to
students who, in addition to being subject to such visual banana
peels, also do not necessarily understand the underlying concepts.
Questions raised therefore include: Should such notations be replaced?
Or else, how can mathematics educators and teachers best "make peace"
with them?