Laurinda C Brown

Responses to ‘The Core of Algebra’

The crucial point is the power of algebra to gain insight into something that you could not do without it: Algebraic symbolism ⋯ introduced from the very beginning in situations in which students can appreciate how empowering symbols can be in expressing generalities and justifications of arithmetical phenomena ⋯ in tasks of this nature manipulations are at the service of structure and meanings. (Arcavi, 1994, p. 33) Arcavi’s point applies not only with arithmetical phenomena, but with any phenomena of objects and relationships between them. The different frameworks proposed by various authors to define the core of what algebra is, can each be used to give perspectives on algebraic activity. Sometimes these activities will be generational, sometimes transformational, sometimes functional, and sometimes involving generalised arithmetic, but all are encompassed by the global/meta-level activities that give purpose for algebra.

CERME7 Working group 17: From a study of teaching practices to issues in teacher education

We report on a pilot project that has investigated the hypothesis that, in addition to subject and pedagogical knowledge, much of what experienced teachers know is what we call attention-dependent knowledge, and that it is this knowledge that enables them to respond effectively to what happens during lessons. A study of mathematics lessons taught by six teachers has led to some further conjectures about the role of attention-dependent knowledge in teaching, and about the interplay between different knowledge sources in planning and teaching.From a study of teaching practices to issues in teacher education : Introduction to the the papers and posters of WG17

SEEING PATTERNS: SOMATIC MARKERS IN TEACHERS’ DECISION-MAKING AND STUDENTS’ REASONING IN MATHEMATICS CLASSROOMS

Ideas of categorisation and pattern have been used in the past research of all three authors; on teachers’ complex decision-making (Brown and Coles, 2000) and students’ reasoning in mathematics classrooms (Reid, 1999, 2002). We illustrate these ideas with an analysis of two transcripts. As we reflect on how categories are formed it becomes clear that decision-making in the complex world of classrooms often takes place without time for reflection. Damasio (1996) develops what he calls his ‘somatic marker hypothesis’ to account for how people manage such decision-making. By ‘somatic marker’ Damasio means a bodily predisposition that informs decision-making. We then provide a second analysis of the two transcripts, illustrating how we currently observe the development of somatic markers through the language used and decisions made in classrooms.

Emotional orientations and somatic markers: expertise and decision-making in the mathematics classroom

Abstract Learning to teach is learning to make the kinds of decisions in classroom situations that teachers make. In this chapter, we explore a theoretical basis for describing this process of making decisions as a teacher does. We analyze the teaching and reflections of two experienced teachers using theoretical constructs at three levels. At the most general level, we adapt Maturanas phrase “emotional orientation” to refer to a “teacherly emotional orientation,” which is the set of decision-making criteria appropriate to teaching. Having a teacherly emotion orientation simply means a person behaves as one expects a teacher to behave. At the most specific level, we name the criteria themselves “somatic markers,” after Damasio. Individuals have many emotional orientations related to different communities, characterized by different (probably overlapping) constellations of somatic markers. For a teacher one emotional orientation is the teacherly emotional orientation, which contains somatic markers leading the teacher to make specific decisions in classrooms consistent with that identity. At the middle level, “purposes” group somatic markers into patterns that make sense at a conscious level. They are linked to actions and collections of purposes that, in turn, form emotional orientations. Purposes are significant for researchers, teachers, and teacher educators in that they provide a level of description that allows an individual to see whether they are acting effectively or not, and they can lead to changing a persons behaviors.

Commentary for Section 3: From Diversity to Practices: Addressing, Redressing and Taking Action

In Peter Sullivan’s opening chapter, he sets out the intentions of this book and also presents me with a perspective through which to discuss issues arising after reading the three chapters in this section. In the introduction to Chapter 1 (p. 3), Peter asks, “whether the goal of any recommendations for change is to improve the education of all students, without addressing the differences, or to find ways to reduce the differences between groups of students”. In the conclusion to the chapter, as researchers addressing inclusivity, we are asked to report on “what redressing disadvantage might look like” (p. 13). I will comment on each chapter in these terms, particularly highlighting what any advice for implementation might be, in order to discuss, from a UK perspective, different levels of advice and their relation to actions.

Working within the tower of Babel: the learner's perspective study

APPA group 2004) of the chapter contributed by that group to the study volume: ‘‘The Working Group maintains that such differences often remain implicit within metalevel discussions of the mathematics education community’’ (72). What we were concerned to do was work on developing a ‘toolkit’ to support the analysis of practice within a country that might uncover these unquestioned assumptions and also what has not even been considered in that cultural practice, what I call the ‘nots’ of that practice. How do these researchers handle these tensions? The LPS sought to address ‘‘the interactive and mutually dependent character of teaching and learning [requiring] the simultaneous documentation of the practices of both teacher and learners and the identification of the meanings each constructs for (and from) the practices of the other’’ (LPS-1 6). Following this statement there are the common set of seven questions underpinning the research, worth reproducing here because they give some insight into the organisation and structure of the two books from the study: Research in Mathematics Education 217