International Teacher Education Conference 2014 1

International Teacher Education Conference 2014
fields of study. I also encouraged students to form their own small groups consisting of two or three people. By
encouraging students to make those decisions regarding their learning and classroom environment, I aimed to
break away from the deeply entrenched modernist assumption that the teacher should be in control of the
classroom, learning, and the outcomes. Doll (1993) views self-organization to be the central concept around
which a post-modern education emerges. Following up on this idea Fleener (2002) points out that “selforganization
occurs not when there is a control…” The modernist assumption that the teacher should be in
control is so entrenched among us as individuals and in society that it took some time for a good number of my
students to get comfortable with the “unconventional” approach. Three weeks into the semester, a couple of my
students were disturbed enough by this emerging classroom environment to ask me, on two different occasions;
one privately and one publicly in class, if “this was going to be how the class would be taught the entire semester
and if I was going to start teaching them.” Upon sharing my response, as well as my philosophy, which was, in
essence, that we were all learners and teachers in a collective community of learners, the two students
unfortunately switched over to another section of the course. A third student, about six or seven weeks into the
semester, indicated being misadvised into enrolling into calculus without indeed needing the course, and
dropped the course.
Breaking away from the deeply entrenched modernist assumption that the teacher should be in control of the
classroom required me to be flexible, and also stay responsive to the needs and the contributions of my students.
For example, in investigating one of the student selected problems we all embarked on a grand tour of ideas
from critical points and inflection points to implementation of rational zeros theorem and onto synthetic division
and Descartes’ rule of sign change, and finally to Newton’s Method all in only one problem and over a period of
several days of immersion. I thought of this particular segment of our classroom experience to be a vivid
illustration and a strong evidence of a widely accepted learning theory based on social constructivism, which
views learning as building schemas, picking up concepts, procedures, skills, and making sense of these by
establishing links among them over time (Begg, 2004). The collective sense of accomplishment generated, as a
result of such an experience of teaching and learning, among the members of the community of learners would
not have been possible if the students were not to have a say in selecting the problems which they would want to
investigate. I certainly would not have picked such a “messy” problem had I been following a linear lesson plan
that was teacher controlled and objective-driven. As Davis & Sumara (2007) indicated, “the lesson plan had to
be non-linear to allow for unexpected changes.” The predictability and certainty we have grown accustom to
expecting from a mechanistic system is not a valid expectation or assumption of the behavior of a complex
learning system, as was the case for this particular community of learners in a calculus classroom. Hence, it is
only natural to have neither reasonable predictability nor any certainty in the classroom, unless the classroom as
a learning system is forced to be a closed system and made to yield to control and prediction. Davis & Sumara
(2007) indicated that the living systems, as well as human systems, “resist prediction because they constantly
interact with one another and actively learn and adapt.” If a system such as a community of learners in a calculus
class is forced to yield to control and prediction, the life support for the system is significantly severed and the
collective learning process and growth as a community of learners gradually diminish.
I would like to offer another brief segment of my own calculus classroom experiences, which I think may be
a parallel to the example of “multiplying understandings” offered by Davis & Sumara (2007) who posed the
question “What is multiplication?” As a short essay assignment, I posed the question “What is a derivative?” to
my twenty or so calculus students, all engineering majors. They were given about 15-20 minutes to generate
their responses. Generally speaking, I found that most of my students struggled not with a response to the
question, but rather with the question itself, as was the case for students with whom Davis & Sumara were
working. Several of my students were insistent on giving an example of a particular function, expression, etc.
and were wanting to illustrate the derivative numerically, graphically or symbolically (i.e., process of finding the
derivative-differentiation). They were having difficulties writing about what a derivative was, because they were
focusing on the how aspect, on the skills, rather than the what aspect of the derivative as a concept.
The diverse educational backgrounds of the students in our classroom ranged from two or three students
already with B.S. degrees to a few sophomores, and a few freshman students just out of high school. Davis and
Sumara (2007) described internal redundancy, the complement of diversity, as a common ground of participants
in subject matter as well as culture, language, history and expectation. At times in our calculus classroom some
level of the internal redundancy, especially in subject matter, was necessary in order for this diverse community
of learners to be able to contribute to the collective learning system individually, as well as a group, collectively.
The development of some common vocabulary, concepts, meaning-makings and experiences are necessary for
meaningful interactions to take place in the classroom environment. For example, a certain redundancy in
factoring of algebraic expressions and the function notation are required prior to discussing the essential ideas of
calculus. More often than not, this redundancy is to be (re)-established at the beginning of many calculus classes.
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