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Teachers’ Knowledge and Practice and the Teaching of Algebra

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A presentation by the Working Group on Teachers’ Knowledge for Teaching Algebra at the Twelfth ICMI Study conference on The Future of the Teaching and Learning of Algebra, University of Melbourne, December, 2001 Cecilia Agudelo, Monash University Elizabeth Belfort, Rio de Janeiro Helen Doerr, Syracuse University Mary Enderson, Middle Tennessee State University George Gadanidis, Western Ontario Brigitte Grugeon, University Paris VII Sylvia Johnson, Sheffield Hallam University Vilma Mesa, University of Michigan Sheryl Stump, Ball State University

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What are the problems? After 30 years of research on children’s learning of algebra, there has been little change in how algebra is taught and learned in schools Much of school algebra is learned as procedures that are disconnected from meaning and from purpose and from mathematics This points to a significant disconnect between the research on children’s learning and actual classroom practice

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Problems (cont.) Teachers’ knowledge and practice and its development for the teaching of algebra have been largely unexamined This points to a need for theory building The practical wisdom of teacher educators is fragmented and uncodified and not subject to the scrutiny of research and revision by the community This points to a need for principles and/or cases of practice

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Our strategy Identify what we already know from research –An outcome of our work is an annotated bibliography on teachers’ knowledge and its development Identify key issues related to the nature and development of teachers’ knowledge and practice Discuss and analyze these issues in light of the research and practical wisdom in order to suggest areas in need of future research

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Dilemmas The dilemmas of experience –How to simultaneously build on pre-service teachers’ experience and to break the mold of that experience –Collective experience of teacher educators and professional developers has not been systematically investigated and reported The dilemma of “what algebra?”

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Dilemmas (cont.) The dilemma of a much larger body of research on teacher development and how to use that work in understanding the teaching of algebra The dilemma of “knowledge” versus “knowing”

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Findings from research What do we know and the limitations of some of that work –There is a great deal of research on the learning of algebra, but often the teacher is absent from the picture –The research on algebra learning tends to focus on the mathematics, the task, the learner and (in some cases) the technology, but rarely is the teacher the focus of the study –There is little literature on teachers becoming teachers of algebra!

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Research diagram

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What do we know? Shulman’s (1986) pedagogical content knowledge is widely used as a framework for studying teachers’ knowledge (especially in the US) The conceptions and mis-conceptions that secondary teachers have about functions and variable Experiences that promote reflection with a view towards changing teachers’ views of mathematics are sometimes successful

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What do we know (cont.) Changing teachers’ views of mathematics may lead to changes in practice, but even when it does, it is not clear how or why this happens “I know mathematics differently” does not easily translate into teaching mathematics differently

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What do we know (cont.) There is evidence that when working in depth with small numbers of teachers (n = 1 or 2) over an extended amount of time changes in practice occur There is almost no evidence of changes in practice on a medium or large scale Success in change (or “reform”) can come about when teachers possess strong content connections

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What do we know (cont.) Success in change (or “reform”) can come about when professional development is reflexively embedded in practice There is a body of literature on elementary teachers’ development with respect to number and ratio There is a need for research on elementary teachers’ understanding of the shift from arithmetic reasoning to algebraic reasoning

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Our Questions 1. What is it that teachers need to know about algebra and its teaching and learning? 2. What do we know about appropriate practices? 3. How is it that teachers learn to become teachers of algebra? 4. What will make teaching change for the better? 5. What are appropriate research strategies and methodologies for investigating the practice of teachers?

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What is it that teachers need to know about algebra and its teaching and learning? We propose a framework to help us organize the knowledge we think is necessary to teach algebra. This is meant to be a tool for us, as researchers, to organize our work. It is not meant to suggest that all Teacher Education programs should be structured in this way or that we plan to indoctrinate future teachers in this model.

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This framework currently has three dimensions Epistemological dimension Cognitive dimension Didactic/Pedagogical dimension

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Framework One reason we decided to use this framework was to help future teachers in their growth to make “good” decisions about problems with teaching algebra. The three dimensions are interrelated

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Epistemological Dimension This dimension involves knowing: (a) content and structure of algebra (b) historical evolution of algebra (c) the distinction of algebra as a tool and as an object (d) what a valuable algebraic activity “looks” like

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Cognitive Dimension This dimension might include knowing: (a) different learning theories (b) the development of algebraic thinking (c) specific aspects of learning algebra (d) students’ misconceptions in algebra

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Cognitive (cont.) (e) students’ difficulties with algebra (f) students’ interpretations of notions in algebra (g) epistemological and cognitive obstacles (h) ways to motivate learners

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Didactic/Pedagogical Dimension This might include knowing: (a) the curriculum – algebra (b) resources (textbooks, curriculum materials, manipulatives) (c) technology (d) other teachers’ practices

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Didactic/Pedagogical (cont.) (e) instructional representations (f) different teaching strategies and activities (g) different learning styles (h) the connections among the various grade levels (i) assessment (j) how to develop and promote classroom discourse

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What do we know about “appropriate” practices for teaching algebra? What are they? What do they look like? A teacher must have a vision of how the three previously mentioned dimensions “fit” together to help influence or guide her/his classroom practices.

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Appropriate teacher practices are by necessity linked to student learning of algebra, and characterized by: reflection knowledge about oneself – as a teacher and as a learner a sense of the role of time in development of concepts and in organizing the instructional materials

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Appropriate practices (cont.) the ability to contextualize and decontextualize an event student involvement decision making the culture in which algebra is taught

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What do we know about how teachers learn to teach algebra? There is a significant shortage of research about how teachers learn to teach algebra, how they understand their own practice and how they form and are formed by their own practice within their own specific cultural contexts. In order to implement curricular change we need to understand more about how the teacher mediates this.

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How teachers learn to teach algebra? There needs to be a focus on such research, but if it is to inform change in practice it MUST be collaborative with teachers as professional partners in the acquisition of knowledge about practice. Evidence to date suggests that this activity needs to be longer term rather than shorter term.

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Goal A long term goal is to develop theoretical frameworks to interpret teachers’ development in the teaching and learning of algebra. In the medium term, we need more research data in order to build such theoretical frameworks. We need to consider what are appropriate ways of investigating the practice of teachers of algebra.

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Needed research areas We outline below some of the areas where research is needed if we are to know more about how people learn to teach algebra. These are selected as representative of issues that research suggests are significant in the development of teachers and their practice. Priorities are likely to vary country to country, culture to culture. They aim to address aspects of how teachers acquire epistemological, cognitive and didactic knowledge.

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Directions for Future Research Investigations for Theory: Some components for teacher knowledge have been identified. Is this list sufficient? If not, what’s missing? What are the principles that guide the selection of issues/elements to investigate? With this framework, what kind of phenomena can we observe, explain and make sense of ?

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Investigations for Theory (cont.) What are the theoretical models that help us explain how teachers become teachers of algebra? What is the link between students’ and teachers’ misconceptions of algebra? How can we influence teacher actions – practices & reasoning – based on their experiences and training – to help them make appropriate teaching decisions?

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Investigations for Theory (cont.) How can we carry out systematic research that will influence practice? We need to find out what we know!!!

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Investigations of Practice How can teachers gain a rich view of algebra and algebraic activity (algebraic thinking)? How can teachers make connections between their own knowledge of algebra and the algebraic activity they do with pupils? What motivates teachers to change their teaching of algebra?

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Investigations of Practice (cont.) What do teachers actually do to promote successful learning of algebra? How do teachers make decisions in the algebra classroom? What do teachers pay attention to in algebra classrooms? What questions from learners challenge teachers in algebra classrooms ?

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Investigations of Practice (cont.) What kinds of incidents are critical for pre-service teachers and experienced teachers? Why? What prevents teachers from learning from their experiences of teaching algebra- when is experiential learning not effective learning? Is our own practice as teacher educators/researchers consistent with the pedagogy we encourage teachers to adopt?

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Investigations of Practice (cont.) What are the ways in which reflection specifically supports the development of improved practice in algebra teaching? What is the nature and role of discourse about algebraic activity? What is the role of professional community in teacher development? What is the relationship of teachers to classroom resources, including technology and how these are used effectively?

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Investigations of Practice (cont.) How do teachers articulate algebra topics across the curriculum with respect to students’ understanding of algebra? How do teachers deal with particular algebraic topics with or without technology?

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Practical tasks to support research activity Is it possible to build/generate a list/resource in which we codify the aspects of learners’ misconceptions and students difficulties? The development of cases of effective algebra teaching practices and their use in professional development activity and research The development of cases of effective professional development

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Conclusions Overall, there has been little research about teachers’ knowledge and practice and its development with respect to the teaching and learning of algebra. Research on student learning (and technology) needs to keep the teacher in focus.

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Conclusions (cont.) There is a critical need to build theory about the nature of teachers’ knowledge and practice and its development We need to find new methodological approaches to effectively investigate the practices of teachers of algebra.

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