Foundation
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.
- Awareness of purpose
- Identifying errors
- Overt subject knowledge
- Theoretical underpinning of pedagogy
- Use of terminology
- Use of textbook
Transformation
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.
- Teacher demonstration
- Use of instructional materials
- Choice of representation
- Choice of examples
Connection
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.
- Making connections between procedures
- Making connections between concepts
- Anticipation of complexity
- Decisions about sequencing
- Recognition of conceptual appropriateness
Contingency
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.
- Respond to children’s ideas
- Use of opportunities
- Deviation from agenda
- Teacher insight