Theoretical Perspectives on Teacher Knowledge and their ...



Theoretical Perspectives on Teacher Knowledge and their Implications for Cross-National Research on Mathematics Teacher Education Impact

Submitted by Teresa Tatto, Jack Schwille and Eduardo Rodrigues

Abstract. This paper discusses salient theoretical issues associated with the definition and conceptualization of mathematics knowledge and beliefs, the situated character of mathematics teacher knowledge, and the process of knowledge acquisition by novices. It then discusses the implications of these theories for the on-going development of the conceptual framework and instrumentation for the IEA first cross-national study of teacher education in mathematics (TEDS-M).

Introduction

The Teacher Education Study in Mathematics (TEDS-M) is a cross-national research study sponsored by the International Association for the Evaluation of Educational Achievement (IEA) and coordinated by Michigan State University (MSU) and the Australian Council for Educational Research (ACER). TEDS-M builds on TIMSS and other similar large-scale cross-national studies to examine the nature and impact of mathematics teacher education for primary and lower secondary school teachers.

The overall study consists of three overlapping sub-studies:

1. Studies of policy and context in the recruitment, preparation, induction and retention of competent primary and secondary mathematics teachers. This sub-study will collect data using qualitative analysis of existing documentation, and interviews with key informants and focus groups.

2. Studies of the standards, expectations and opportunities to learn embodied in the routes, approaches and programs to prepare and induct competent primary and secondary mathematics teachers. This sub-study will collect data through surveys, curriculum analysis, and focused field studies.

3. Studies of the impact of teacher preparation and induction in producing knowledgeable primary and secondary mathematics teachers. This sub-study will collect data through surveys and focused field studies.

A research project with these aims has to consider the complexities of teacher education and teacher learning, such as lack of agreement among experts, policymakers, and reformers about what model of education for mathematics teachers is important; competing views concerning the importance of subject-matter and pedagogy; and disagreement over what teachers learn from experience; among others. TEDS-M focuses on knowledge as a crucial issue in the quality of mathematics teachers (Ball, 2001; Gess-Newsome and Lederman, 1999); it documents those features in teacher education expected to influence the mathematics knowledge teachers acquire; and it takes into account how teacher knowledge develops along the continuum of teacher learning (Feiman-Nemser, 2001; Britton et al, 2003; Grossman 1990).

Theoretical issues in mapping the outcomes of learning mathematics for teaching

In our study meetings we have identified five areas of theoretical import to our study framework and instrumentation.

Mapping the content and boundaries of the knowledge domains.

Since the 1980s under the influence of Shulman (1987), teachers’ professional knowledge has been conceptualized as subject-matter content knowledge, pedagogical content knowledge and general pedagogical knowledge. Although there is much agreement on pedagogical content knowledge (PCK) as a shared construct of teacher knowledge, in our project meetings it has become evident that there is no consensus on how to conceptualize and measure it. For example, in certain countries (e.g. Germany) teacher educators appear to treat PCK as a domain of theoretical knowledge; while in the U.S., PCK is typically understood to have both theoretical and practical knowledge components.

This hypothesized difference about the nature of PCK, if upheld, has important implications for how the education of future teachers of mathematics is organized and what its impact is likely to be on the knowledge of such teachers. It also raises a number of conceptual challenges implicated in strong differences in positions taken by experts. First, what is “practical knowledge”? This question is important as this term has been defined and applied in various and somewhat inconsistent ways. Is it idiosyncratic knowledge acquired in teaching a particular classroom which does not generalize or transfer in a way that can readily be assessed? Is it justifiable to assume that this knowledge transfers to new or changing situations in useful ways? Different epistemological and disciplinary traditions of educational research give different answers to these questions (see Ball, 2001; De Corte & Verschaffel, 1996; Hamilton & McWilliams, 2001; Munby et al., 2001). Moreover, research by Ma (1999), Britton, Paine and colleagues (2003) demonstrates that there can likewise be important differences from country to country in the nature of knowledge for teaching mathematics.

Conceptualizing and mapping the depth/levels of understanding within knowledge domains

Although there is agreement on the importance of considering and classifying gradations in cognitive understanding, differences in what it means to understand, however, can be formulated in a subject-matter specific or a generic fashion. A classification commonly used in mathematics education is the distinction between procedural and conceptual knowledge. But Brodie (2004) argues that this dichotomy is too limited and limiting and that a more elaborated and interactive classification is needed.

A more elaborate generic and psychometrically oriented view is represented in the recent revision of Bloom’s classic taxonomy (Anderson et al, 2001), now with two dimensions, one representing gradation in terms of increasing cognitive complexity. The original Bloom taxonomy was criticized, however, by scholars in mathematics education (e.g. Freudenthal) for its atomistic, engineering character. In contrast, mathematicians and mathematics educators have used other distinctions to characterize the understanding of mathematics (either of the content of mathematics in general or of mathematics for teaching). For example the TIMSS curriculum analysis used a classification of performance expectations that is specific to mathematics. Likewise, a specific classification was proposed in Kilpatrick et al (2001) with five interrelated strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. Ma (1999) analyzes the attributes of teacher understanding without reducing this to any underlying continuum. Likewise, the current Knowledge of Algebra for Teaching (KAT) project at Michigan State University in the U.S. (MSU) has used a framework in which school knowledge, related mathematical knowledge and knowledge for teaching are all broken down into four not mutually exclusive major categories: core concepts/procedures, representations, applications, and reasoning/proof (Ferrini-Mundy et al., 2004).

Identifying and measuring teacher beliefs and meta-cognitive perspectives

In addition to content knowledge and pedagogical knowledge as conventionally construed, psychological research has demonstrated the importance of meta-cognitive processes (“thinking about thinking”) for a more profound understanding of how future and beginning teachers think about what they know. As an example, Sternberg’s list of meta-cognitive executive processes is relevant to teachers’ knowledge of mathematics and how they attempt to help children learn: “(a) recognizing the existence of a problem, (b) deciding on the nature of the problem, (c) selecting a set of lower-order processes to solve the problem, (d) developing a strategy to combine these components, (e) selecting a mental representation of the problem, (f) allocating one’s mental resources, (g) monitoring one’s problem solving as it is happening, and (h) evaluating problem solving after it is done” (Sternberg, 1992, cited in Shepard, 2001).

In general, conceptualizing how teachers think about mathematics, teaching and learning is important in ascertaining what they “know about knowing” (cf. Toulmin, 1972, on intellectual ecologies; and Sfard’s, 1998, comparison of two metaphors of learning). For example, instruments have been developed to assess whether teachers have a constructivist or a direct transmission view of teaching (see for example the questionnaire developed by Fenema and colleagues, 1992, and used recently in a Germany study by Staub, 2002). As far as mathematical beliefs are concerned, De Corte et al. (2000) offer a useful categorization of the sub-domains involved: beliefs about mathematics education, about the self in relation to mathematics, and about the social context of mathematical learning and problem-solving. Their chapter focuses on students, but has obvious relevance to teachers as well.

Assessing domains when differently situated

Teaching represents what has come to be called situated knowledge, knowledge of and adapted to particular contexts (Putnam & Borko, 2000). To situate knowledge within preparation for and the practice of teaching mathematics, for example, requires attention not only to the varied classroom settings in which teachers ultimately practice, but also to the teachers’ own prior elementary and secondary schooling, the courses in which university-level content knowledge of mathematics is acquired, the courses in which the pedagogy of teaching mathematics is most emphasized, the classroom contexts for acquiring learning about mathematics in teaching during field experience components of teacher education, and special arrangements for internships and induction experiences of beginning teachers. The knowledge developed or modified in each of these contexts contrasts with what has been called general knowledge (knowledge that is applicable across situations and settings) and still more with theoretical knowledge (general knowledge explicitly rooted in terms of interrelated concepts and basic ideas). A question this conceptualization raises is, is it sufficient to measure general or theoretical knowledge without regard for situated knowledge? (cf. Hammer & Elby, 2002). Indeed, one of major issues in research on teacher education is how much to pay attention to the situated character of this knowledge. Ball (2001) takes a strong position on this: “What matters ultimately is not only what courses teachers have taken or even what they know, but also whether and how teachers are able to use mathematical knowledge in the course of their work.”

According to discussions among the international participants in our study, this position contrasts with the view held in certain European countries (notably Germany), that general or theoretical knowledge is considered most appropriate for pre-service teacher education and that knowledge in practice should be left completely for later acquisition in the classrooms of novices.

Understanding novices’ knowledge acquisition for teaching and its institutionalization

Research comparing experts and novices teachers according to a paradigm of cognitive science has been summarized as follows: “Expert teachers possess richly elaborated knowledge about curriculum, classroom routines, and students that allows them to apply what they know to particular cases. Where novices may focus on surface features or particular objects, experts draw on a store of knowledge that is organized around interpretive concepts or propositions that are tied to the teaching environment. Because the knowledge is tacit, it does not translate easily into direct instruction or formalization” (Munby et al, 2001). However, when viewed cross-nationally from a situated perspective, the difference between expert and novice may vary. Inasmuch as the teaching career is defined differently across countries, the nature, path and pace of advance from novice to expert may vary substantially even within countries.

Programs of teacher education typically embody a developmental logic of how teachers acquire this professional knowledge for the teaching of mathematics and other subjects. For example, a common justification of such programs is that theoretical knowledge prepares teachers to acquire practical knowledge in their initial years of practice, leading eventually to a state of expert professional knowledge. But evidence is lacking to show that this model predicts how novices construct their professional knowledge or even that it represents an ideal model that might be realized under more favorable conditions.

While the process of how teachers can best acquire expert knowledge continues to be contested, the literature contains important insights about the state of novices’ subject-matter knowledge for teaching (see Even, 1993, for a study of future secondary school teachers’ understanding of the function concept and their responses to student solutions and errors; also Schmidt, 1994; and Van Dooren et al., 2002, for Canadian and Belgian studies of the preferences of future teachers for algebraic as opposed to arithmetic solutions to word problems).

Conclusion

In this presentation our goal is to discuss the five theoretical dimensions discussed above, and their implications for our conceptual framework and instrument development in particular, and more generally in meeting the operational needs of the study. To achieve this goal we will bring to the presentation examples from developed instruments to show how we are responding to these theoretical issues, and to discuss strengths and weaknesses. TEDS-M is meant to stimulate national and cross-national dialogue among policy makers, and educators regarding mathematics teacher education policy, programs and curricula to improve preparation and practice in mathematics teaching. We expect the study will also make important contributions to the theoretical and knowledge base of mathematics teacher education.

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