Mathematics

• The use of repetition in some papers;
• Counting the number of references for and against a particular position
to identify bias;
• Noticing and commenting on extensive self-referencing in some papers;
and
• Commenting on the uncritical use of strings of references as a tool to
suggest widespread agreement with a position.
It was clear that the participants engaged critically with the papers, and were
able to identify and describe strengths and weaknesses. We maintain that the
capacity to critically analyse literature is crucial in coming to be a teacher
educator, as it enables the teacher educator to evaluate the relative merits of
the large volume of research in the field as well as to be critical of initiatives
introduced, often with little basis in research, by schools or systems.
In introducing key ideas relating to the nature of mathematics and science we
presented participants with an initial activity that engaged them in collaborative
investigation or problem solving. For example, in mathematics we presented a
problem that asked participants to change five coins, initially all showing
heads, by flipping any two at a time until they all showed tails. After
considerable investigation and discussion, the participants decided that it was
impossible. What ensued was a lively discussion on why it was impossible,
and more particularly on what would constitute a convincing proof. This
introduced some key aspects of mathematics as a dynamic activity, but one in
which the dynamic nature of mathematical discovery is often obscured by
carefully prepared and well-synthesised results. The idea of mathematics as
simultaneously dynamic and static was one that continued throughout the
discussions and readings in the first unit of the course.
5 Future developments and research
There is a steep learning curve for us, as course developers and leaders.
Although we communicate regularly via Skype and email, as one of us
(Thornton) is based in Australia and the other (Childs) in Oxford, we miss the
sense of collegiality generated by being onsite. The course development
process is ongoing, with much of the work being done “on-the-run”. We are
also continuing to find effective ways to communicate with the course
participants and to encourage regular collaboration and discussion. Each of
the participants is employed full-time, and each is very busy in his or her own
context. This places great demands on their time and capacity to engage fully
with all aspects of the course design. We have wrestled with identifying what
are reasonable expectations with regard to professional reading, and whether
to expect tasks and reflections to be completed on a weekly basis or at less
regular intervals. However we have been encouraged by the feedback of the
participants, their insistence that as adults they do not need to be “spoon-fed”
and the way in which they have embraced the variety of readings presented.
As a design principle for future units we therefore seek to find an appropriate
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