International Teacher Education Conference 2014 1

International Teacher Education Conference 2014
a personal level, how strong and powerful it can be to name and claim one’s identity and integrity. I have once
come across a phrase which read “there is nothing more powerful than an idea whose time has come.” Although
I have often pondered, and time to time reflected on this phrase, I realize that only now I can truly relate to the
feelings of the person who might have articulated these feelings so eloquently.
According to Catalano (1995), any transformation from a teacher-centered classroom is considered to be a
fundamental change, and that it will meet with resistance in several ways. Resistance from some colleagues can
be expressed as a judgment that any classroom other than a teacher-centered classroom must be without
sufficient rigor. Since most of the teachers have received their formal training in the teacher-center model, many
teachers tend to believe that they should certainly teach as they were taught in order to have rigor. Catalano
(1995) asserts that far too often rigor is the last refuge for ineffective teaching. One of the aims of sharing my
experience of the calculus classroom environment in this paper is based on the Green’s (1988) concept of going
from a personal sphere to a public sphere in the hopes of reaching other colleagues, teachers, and individuals to
form a community of congruence that offer mutual support and opportunity to develop a shared vision.
Literature Review and Background
Generally, the content in calculus courses is linearly and hierarchically ordered as to start with limits, then in
detail to discuss derivatives, and to end the course with integral calculus. I propose to dispute the conventional
curricular approach and its implied assumptions regarding teaching calculus via lecture-and-listen mode as a
sequence of disjoint topics padded with a series of techniques, and offer a different vantage point. The
conventional approach of lecture-and-listen mode of teaching calculus might lead students to view calculus as a
fragmented list of formulae and procedures taught in isolation without any interconnectedness established
among the main ideas and concepts. This particular view of calculus created and promoted by the current
lecture-and-listen classroom practices impresses upon our students, while they are in the classroom and still long
after, an image of calculus as a disconnected collection of formulae. Needless to say, this distorted image of
calculus also may lead to the common sentiment about calculus among general populace that can be summarized
by the following passage taken from Peterson (1998), who states that “[t]here is something about calculus that
can evoke a mixture of both wonder and dread.”
Many teaching methods used in calculus, especially prior to the calculus reform movement of the 1990s, can
be qualified to be structuralistic in nature (Tall, 1996). The structuralistic approach is based on the principle of
transmission, especially via lecture-and-listen mode, of a logical structure within a scientific system. The
popularity of this approach is attributed by Doorman (2005) to the fact that the theories themselves seem elegant
and compact description of what is to be learned. Even though a structuralistic approach to the teaching of
calculus combined with lecture-and-listen classroom practices and deficit-based evaluations seem to be “less
messy” in comparison to what I propose here, the problems associated with a structuralistic approach to teaching
of calculus are well documented in the literature (Doorman, 2005). The “perfect” lesson plans created,
implemented according to a script, and transmitted in a lecture-and-listen classroom environment tend to be
mostly at the expense of the potential contingencies of the classroom as a collective learning community. It is
important to qualify the last sentence, however, by indicating that I am not taking a dualistic stance and imply
that lesson plans are useless and should be discarded for good. Organizing learning in a classroom environment
is just as important and requires a considerable amount of advance preparation on the part of the teacher. The
concept of intermittence as outlined by Briggs & Peat (1999) in their Seven Lessons of Chaos to intertwine the
two approaches as needed seems to be what we may consider here in lieu of a dualistic either or approach.
Theoretical Framework
It is very difficult for researchers to communicate the nature of new ideas without a framework on which to
base new classroom practices. A framework can also be a useful tool in helping teachers understand the types of
thinking that need to be fostered in students. Even though it was originally developed in the fields of
mathematics and physical sciences, complex systems theory has been also widely proposed as a new theoretical
framework in social and cognitive sciences, including language learning (Harshbarger, 2007). For the most part,
the classroom practices presented here are grounded in the complexity theory of teaching and learning as
outlined by Davis & Sumara (2007), Bowsfield, Breckenridge, Davis, et al (2004), and Davis, Sumara and Luce-
Kapler (2000).
Complexity science points out that hardly any event or activity can be reduced to being just a “thing” in
isolation. Teaching, being not an exception, is not reducible to what a teacher does and/both does not do in a
classroom. As a result of this realization about the nature of teaching, there seems to be a need to make a
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