Using tasks to explore teacher knowledge in situation-specific contexts: An example from the limits teaching

Research often reports an overt discrepancy between theoretically / out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (preand inservice) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of 1 0 x x + − = ) of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore. USING TASKS TO EXPLORE TEACHER KNOWLEDGE IN SITUATION-SPECIFIC CONTEXTS Irene Biza Department of Mathematics, University of Athens (Greece) Elena Nardi School of Education, University of East Anglia (Norwich, UK) Theodossios Zachariades Department of Mathematics, University of Athens (Greece) Explorations of teachers' beliefs and their relation to practice (see, for example, (Thompson 1992) for a review) acknowledge the overt discrepancy between theoretically and out-of context expressed teacher beliefs about mathematics and pedagogy (e.g. in interview-based studies) and actual practice. Therefore teacher knowledge is potentially better explored in situation-specific contexts. Within professional training courses focusing on situation-specificity, or what Shulman (1986, 1987) calls ‘case knowledge’, is far from novel: trainee lawyers are typically required to engage with problems that concern the application of the law in specific cases (thus exploring gaps in their understanding of the law as well as exploring the complexities of applying the law in real cases). Similar requirements feature also in the training of doctors and other professionals. In all of these cases the emphasis is on transforming theoretical knowledge into theoretically-informed practice. In the context of mathematics education, ‘a domain of professional work that makes fundamental use of highly specialized kinds of mathematical knowledge, [...] a kind of applied mathematics’ (Bass 2005), this transformation – see (Watson & Mason, this volume) – has been described by concepts such as Chevallard’s (1985) transposition didactique, Hill and Ball’s (2004) mathematical knowledge for teaching and Shulman’s (ibid) pedagogical content knowledge. In the work we discuss in this paper we engaged mathematics teachers with classroom scenarios which are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal. We thus see these scenarios as likely to occur in actual practice. We perceive the type of task we present here as having both purpose and utility (in the words of Ainley and Pratt (2002) applied to teachers as learners) and we see the potential of these tasks both in terms of research and teacher education. We see these tasks as suitable for engaging both preand in-service mathematics teachers. We also believe that – in contrast to posing questions at a theoretical, decontextualised level – inviting teachers to respond to highly focused mathematically and pedagogically specific situations that are likely to occur in the mathematics classrooms they are (or will be) operating in can generate significant access to teachers’ views and intended practices (Dawson 1999). The mathematically / pedagogically specific situations that we invite teachers to engage with in our work are in the form of tasks which have the following structure: Reflecting upon the learning objectives within a mathematical problem (and solving it) Examining a flawed (fictional) student solution Describing, in writing, feedback to the student