Foundation

Dimensions, Foundation
This category consists of 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
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Transformation

Dimensions, 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…
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Connection

Dimensions, 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 contingency
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Contingency

Dimensions, 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…
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Theoretical underpinning of pedagogy

Theoretical underpinning of pedagogy
One aspect of Mathematical Knowledge for Teaching is a teacher’s use of a theoretical foundation to guide instructional decisions, rather than relying on imitation of another teacher or trial and error. Therefore, it is important for teachers to know factors that are significant in the teaching and learning of mathematics (Rowland et al., 2009; Ball, Lubienski & Mewborn, 2001). Descriptors/Aspects of TUP (Theoretical Underpinning of Pedagogy) "Strong" examples: Draws on knowledge of well-established results in mathematics education research regarding the way pupils learn the topic in hand to underpin the planning and/or delivery of the lesson (whether this is implicit through lesson observation or revealed explicitly later through post-observation interview). Draws on knowledge of well-established results in mathematics education research regarding pupils’ misconceptions to underpin the planning and/or delivery of…
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Use of terminology

Use of mathematical terminology
‘Good’ examples Demonstrate knowledge of the correct mathematical terms & their precise meanings Correct use of mathematical terms and evidence of efforts to teach these terms Use alternative ways & more precise words to describe shapes for example, if child says ‘round’ then suggest ‘curved’ while linking new word to child’s word or words Link use of alternative ways of saying things, e.g.,1/4  can be called  ‘one quarter’ and ‘one fourth’ Add the correct term to help children express what they mean, e.g. Child says threes when describing thirds; child says10 teacher adds ‘centimetres’ Clarity around use of correct terminology when discussing different forms of all four operations on number Precise use of mathematical symbols ‘Bad’ examples Calling a parallelogram a ‘rectangle pulled out of shape’ Calling a sphere a…
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