International Teacher Education Conference 2014 1

International Teacher Education Conference 2014
commitment to an open an inclusive pedagogical approach which values students’ opinions, seeks and welcomes
students’ contributions to the classroom community of learners. This open and inclusive pedagogy was
described by Davis & Sumara (2007) as being “oriented toward unimagined and not-yet-imaginable
possibilities.” Davis & Sumara were not simply interested in finding a neat encapsulation. Their interest was to
integrate some of the elements of complexity theory into the teacher’s role in learning, and outline “some of the
specific guidelines that complexity thinking offers on how to enhance the possibilities of collectives by ensuring
that the conditions for complex-self organization are in place”. Next, I will proceed to offer several segments of
my own calculus classroom experiences, and try to tie them to the framework offered by Davis & Sumara
(2007).
Complexity of Teaching and Learning Calculus
The classroom environment, which I envision, is based on small group learning strategies and activities
specifically designed to foster the main traits of an integrated calculus curriculum in order to develop students’
understandings through sustained interaction, conversation, and discussion. This vision, in essence, aligns nicely
also with Doll’s (1993) post-modernist and process oriented ideas of curriculum built from the base of a
constructivist and experiential epistemology. The commitment on my part to an open and inclusive pedagogical
approach, which values students’ opinions, seeks, and welcomes students’ contributions to the classroom
community of learners, needed to be communicated to my students during the first class meeting. Not being used
to any other type of an environment except the traditional lecture-and-listen mode, it took some time for my
students to begin to realize that an unconventional approach and a classroom environment were shaping up. I
also paid specific attention to carefully fostering, and developing a sense of belonging, mutual respect and
responsibility by designing collective experiences such as collaborative activities and group presentations based
on those activities. According to Bowsfield, Breckenridge, Davis, et al. (2004), the aims of schooling, from a
point of view of complexity research, shift our mindset to thinking in terms of being parts of larger social,
cultural, technological, and ecological systems. I was committed to have a classroom environment toward
helping individuals contribute and, maybe more importantly, have a sense of belonging to a community of
learners. Breaking away from the role of teacher as a figure with centralized control and increasing neighbor and
group interactions were some of the critical ideas on which I needed to focus. I believe that a genuine sense of
belonging, on the part of students and teacher, and being responsible to and for one another are the necessary
conditions for more ambitious learning in a classroom environment.
Based on what Whitehead (1929) suggested on avoiding an education filled with inert ideas, I decided that I
should be introducing the few and important main ideas of calculus into my students’ education, and as a
community of learners, including myself as the teacher and a part of that community, investigate these main
ideas as they are “thrown into every imaginable possible combination”, as well as the relations among these
main ideas. I decided not to sweat the small stuff or have us bogged down in the proofs or in the rote
memorization of formulae. I took up Whitehead (1929) on his suggestion and I did abandon “the fatal habit of
cramming the students with theorems, which they do not understand and will never use.” For example, the
concept of rate of change and the differentiation of a few fundamental polynomials, sin x and cos x were
achieved easily with the aid of geometry. This approach, to which I affectionately and colloquially refer as
“don’t sweat the small stuff”, resulted in creating spaces to discuss concepts which really influence thought.
Palmer (2007) argued for a pedagogical circle to be neither teacher-centered nor student-centered, but
subject-centered. I considered this subject-centered pedagogical circle to be very suitable to our classroom
environment, and I shared this idea with my students. It should be noted, however, that a subject-centered
pedagogical circle does not by any means imply that students are ignored. On the contrary, a subject-centered
approach, as Palmer (2007) envisions it, has strong implications for opening spaces where students can have an
ongoing conversation with the subject and with each others. In order to open such spaces, I, as a teacher, realized
that I needed to break away from the old habit of “covering the field” via lecture-and-listen mode and to strive to
teach more with less. Hence, I aim to create spaces and simultaneously investigate the subject in a much deeper
level than just at a glance.
Following some of the practices suggested by Wheatley (1991) on establishing learning environments
conducive for students to construct their own mathematics in social settings, I used, in particular, problem
centered learning with the intention of providing opportunities for learners to share their ideas for solutions both
within small groups and within the whole community of learners in the classroom by way of group
presentations. I encouraged students to select the problems they would like to investigate from a variety of
sources including, but not limited to, the pool of problems I have collected and created over the years. Several of
the students were also in a variety of engineering courses and chose problems from their respective engineering
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