International Teacher Education Conference 2014 1

International Teacher Education Conference 2014
I have found myself having to do just that a couple of times during the semester in order to break off the limit
cycle in the collective self-organization so that the classroom system could bifurcate into a different level of the
evolution of teaching and learning.
In terms of the problem selection and assignments, it was not certainly a case of no guidelines, but the nature
of the guidelines was not constraining or restrictive. The guidelines were provided for what possibilities were
there to investigate as opposed to what was not acceptable. Bowsfield, Breckenridge, Davis, et al. (2004)
articulated this paradoxically as “enabling constraints.” The students were to prepare their problem solving
presentations to illustrate their understanding and share their findings; they were not to do just anything. And the
students had the freedom within the limitations set out in the guiding rules. When these guiding rules are
organized appropriately, in neither too narrow nor too open ways, they allow for unpredictable and often
imaginative outcomes (Bowsfield, Breckenridge, Davis, et al., 2004). I was not expecting at all, but pleasantly
surprised, that a group of my students did develop their presentations as video clips and loaded them up to
YouTube.
By way of having frequent problem solution presentations in our classroom environment, I aimed to allow
opportunities for the different ideas and thought patterns of my students around the subject matter to interact
with one another in a dynamic and non-linear state. These types of collective dynamics, in which the individual
ideas bump into and bounce off each other, were identified by Bowsfield, Breckenridge, Davis, et al. (2004) as
“neighboring interactions.” They pointed out that these dynamic interactions offered the very real potential for
innovative and insightful knowledge to emerge. Based on these dynamic interactions, several ideas for various
types of formulation and notation were developed collectively. For example, an unconventional formulation for
the Chain rule emerged, as opposed to the conventional function composition (i.e., ( ) !
f ! g = f !!
g ⋅ g !) and
the Leibniz (i.e.,
dy dy du
= ⋅ ) notations. Instead, the Chain rule emerged from the dynamic interaction of
ideas during the classroom dx du dx presentations was based on the notions of inner and outer functions, and was
somewhat unconventionally formulated as O ( I( x
) ⋅ I!
( x)
! . The problem solution presentations were done in a
group format with two students presenting two problems. This particular format of presentations seemed to
encourage students not only to combine several of their ideas together but also to incorporate the ideas of others
as well. I regard these diverse contributions as forming a considerable basis for the generation of knowledge
among the community of learners. I would tend to agree with Bowsfield, Breckenridge, Davis, et al. (2004) that
this particular approach to generating knowledge among the community of learners by the community of
learners shifts the role of teacher from a controller to an integral participator in a recursive process of opening up
spaces for not-yet-imaginable new possibilities while exploring the existent spaces.
Conclusions and Implications for Future Research
In this study, I investigated an alternative approach to a lecture-and-listen mode mathematics classroom. The
alternative approach I explored was based on the complexity theory of teaching and learning. Even though one
can never be sure of one’s influences, I hope, in accord with the butterfly effect concept, that the experiences I
shared here may be taken up by other mathematics classrooms and amplified until it transforms the entire
community of mathematics classrooms into something new.
We need to be mindful of and responsive to the experiences that frame learners’ cognitive developments in a
classroom environment. Teaching mathematics solely based on rules and as purely mechanical and repetitive
drill exercises will neither engage nor interest learners in our efforts to support and sustain a community of
learners in a classroom environment. Mathematical content knowledge for teachers also plays an important role
in helping learners’ cognitive development. Being prepared as a teacher with a substantial amount of knowledge
base in mathematics instills confidence in teachers to move away freely from a scripted lesson plan and venture
into exploring the unknown territories of mathematical topics with the students. The explorations into these
unknown borderland territories are best illustrated for me by the Mandelbrot fractal set metaphor articulated by
Fleener (2002). Treading through the very fine border between the known and unknown territories of the
Mandelbrot set provides a vivid representation of a post-modernist mathematics classroom in which teaching
and learning takes place in the borderland between the known and unknown territories of various mathematical
ideas and concepts.
There are ample amount of teaching and learning activities in the literature with an emphasis on developing
an appreciation for and deeper understanding of various mathematics concepts among students and pre-service
teachers. On the other hand, creation of one’s own material, albeit a challenge, is a much needed process,
especially if one desires to seek further insight. However, it is important to realize that there are no best practices
or an ideal set of activities. A suitable set of activities for one particular group at a given time might be
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