Summary

foundation;

awareness of purpose
identifying errors
overt subject knowledge
theoretical underpinning of pedagogy
use of terminology
use of textbook

This category consists of trainees’ knowledge, beliefs and
understanding acquired in the academy, in preparation
(intentionally or otherwise) for their role in the classroom.
Such knowledge and beliefs inform pedagogical choices and
strategies in a fundamental way. The key components of this
theoretical background are: knowledge and understanding of
mathematics per se and knowledge of significanttracts of the
literature and thinking which has resulted from systematic
enquiry into the teaching and learning of mathematics. The
beliefs component relates to convictions held, and values
espoused, by prospective teachers. Such beliefs typically
concern different philosophical positions regarding the
nature of mathematical knowledge, the purposes of mathematics
education, and the conditions under which pupils will best
learn mathematics.

transformation;

teacher demonstration
use of instructional materials
choice of representation
choice of examples

This category concerns knowledge-in-action as demonstrated
both in planning to teach and in the act of teaching itself.
At the heart of this category, is Shulman’s observation that
the knowledge base for teaching is distinguished by “ … the
capacity of a teacher to transform the content knowledge he
or she possesses into forms that are pedagogically powerful”
(1987, p. 15). As Shulman indicates, the presentation of
ideas to learners entails their re-presentation (our hyphen)
in the form of analogies, illustrations, examples,
explanations and demonstrations (Shulman, 1986, p. 9). This
second category picks out behaviour that is directed towards
a pupil (or a group of pupils) which follows from
deliberation and judgement. Of particular importance is the
trainees’ choice and use of examples presented to pupils to
assist their concept formation, language acquisition and to
demonstrate procedures

connection;

making connections between procedures
making connections between concepts
anticipation of complexity
decisions about sequencing
recognition of conceptual appropriateness

This category binds together certain choices and decisions
that are made for the more or less discrete parts of
mathematical content. It concerns the coherence of the
planning or teaching displayed across an episode, lesson or
series of lessons. Our conception of coherence includes the
sequencing of topics of instruction within and between
lessons, including the ordering of tasks and exercises which
reflect deliberations and choices entailing both knowledge of
structural connections within mathematics and an awareness of
the relative cognitive demands of different topics and task
contingency

respond to children’s ideas
use of opportunities
deviation from agenda
teacher insight

This category concerns classroom events that are almost
impossible to plan for. In commonplace language it is the
ability to ‘think on one’s feet’. In particular, the
readiness to respond to children’s ideas and a consequent
preparedness, when appropriate, to deviate from an agenda set
out when the lesson was prepared. A constructivist view of
learning provides a valuable perspective on children’s
contributions within lessons. To put aside such indications,
or simply to ignore them or dismiss them as ‘wrong’, can be
construed as a lack of interest in what it is that that child
(and possibly others) have come to know as a consequence, in
part, of the teacher's teaching. However, Brown and Wragg
(1993) observe that “our capacity to listen diminishes with
anxiety” (p. 20). Uncertainty about the sufficiency of one’s
subject matter knowledge may well induce such anxiety,
although this is just one of many possible causes.

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