Despina Potari

Tackling Complexity In Mathematics Teaching Development: Using the Teaching Triad as a Tool for Reflection and Analysis

This paper reports research that attempts tomake sense of the complexity of mathematicsteaching and its development at secondaryschool level. The research was conducted inpartnership between two teachers and twoeducator/researchers over one school term intwo U.K. schools. A theoretical construct, theteaching triad, was used as an analyticaldevice (by the researchers) and as a reflectiveagent for teaching development (by theteachers). The focus of analysis was theinteractions between teacher and students atwhole class and small group level. Both micro-and macro-analyses were undertaken. We presentdetails of the processes involved in examplesfrom the teaching of one teacher as shetranslated theoretical aims into classroompractice. The use of the triad allowed accessto complexity, involving both psychological andsociological elements, and to the position of asincere teacher with respect to competingforces in the educational system. Thepotential of the triad for teacher and teachingdevelopment is discussed.

Children's approaches to the concept of volume

The research reported is an attempt to explore 11-year-old childrens approaches to the concept of volume. Six tasks have been devised to identify any commonalities in childrens responses. Results suggest that children hold and use different conceptions in their effort to explain and compare aspects of volume. However, these conceptions fall in certain categories which are met in almost all the tasks: volume is the space occupied, the capacity, related to the material substance, related to the geometrical properties, related to the weight and indefinable. Childrens comparing actions (justifications and strategies) have also been analyzed and characterized as tautological, phenomenological, experimental, or as referred to the properties of the object. Common patterns in childrens conceptions and justifications across the different tasks showed a variety of possible approaches to volume which will probably be met in other contexts and are presented in the form of a systemic network. As our study indicates, the concept of volume is related to a number of factors like the shape and geometrical properties, the nature of the matter, the mass and the weight, the hollowness, and the openness of the object. This complexity is not appreciated in teaching where the concept of volume is approached in a fragmented way and without paying attention to the referent. It is suggested that appropriate integrated activities could facilitate children to construct and unify a plurality of ideas in order to form a “mature concept” of volume.

Response to Part II: Emerging Issues from Lesson Study Approaches in Prospective Mathematics Teacher Education

This chapter is a response to the three chapters of Fermandez & Zilliox, Murata & Pothen, and Yu based on Lesson Study at pre-service teacher education. The chapter is structured around three dimensions: the way that Lesson Study has been implemented in the three studies; the prospective teachers’ learning about teaching mathematics; and the main factors that framed learning. Common approaches but also different perspectives and ways of implementation were identified. Moreover, a number of issues emerged related to the relation of theory and practice, the role of teacher educator and group collaboration. Finally, some ideas about future research and development in the context of Lesson study are discussed.

Creating a learning environment for 16+ mathematics

The reorganization and reform of mathematics courses in Scottish Colleges of Further Education is described. The teaching and learning approaches for the new courses are discussed and an evaluation is made of their implementation in one such course.

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

Prospective Mathematics Teacher Argumentation While Interpreting Classroom Incidents

This paper aims to analyze the structure and quality of prospective mathematics teachers’ (PMTs)’ argumentation when identifying and interpreting critical incidents from their initial field experiences. We use Toulmin’s model and recent elaborations of it to analyze the discussions that took place at the university where PMTs reflected on their recent classroom experiences. Our aim is to identify the structure of the argumentation and characterize the emerging warrants, backings, and rebuttals. Results indicate different argumentation structures and types of warrants, backings, and rebuttals in the process of PMTs’ interpretations of students’ mathematical activity. We discuss these findings from the perspective of noticing to identify shifts at the level of PMTs’ interpretations.

Summary and Looking Ahead

Throughout the survey one can see that there is a steady growth in the research around prospective secondary mathematics teacher education. Also several of the studies were repeated in at least two of the sections due to the nature of the factors that were studied. It is difficult to write about teacher knowledge without some acknowledgment of how knowledge or lack of knowledge can impact teachers’ mathematics identities. Furthermore, field experiences are contexts in which PSMTs’ knowledge and identities are impacted and shaped. In addition, the survey revealed that mathematics education researchers are thinking more deeply about how to foster the growth of effective mathematics teachers in a myriad of ways.

Current Research on Prospective Secondary Mathematics Teachers’ Knowledge

This chapter focuses on research on pre-service secondary school mathematics teachers’ (PSMT) knowledge. Fifty-nine papers were reviewed and each paper was coded according to its research focus, theoretical and methodological framework, main findings and the contribution of the study. We identified three main thematic areas to frame the discussion of the papers: the exploration of knowledge, the impact of teacher education on its development, and the process that this knowledge develops. The review shows that: both small-scale and large-scale studies continue to recognize different aspects of teacher knowledge and indicate certain difficulties of PSMTs; a large number of teacher education practices have a positive impact on PSMTs’ knowledge; research has started to focus on the process of how PSMTs’ knowledge develops either in mathematical or in teaching discourses.

Mathematics Teachers’ Learning at the Boundaries of Teaching and Workplace

This chapter describes how novice and experienced mathematics teachers integrate authentic workplace contexts into mathematics teaching. This goal was inspired by the European MaSciL project and introduced to the teachers in the context of a masters programme in mathematics education. Under an Activity Theory perspective, we use the notions of activity system and boundary crossing to study the process of teachers’ professional learning. In particular, we analyse teachers’ boundary crossings between two activity systems: mathematics teaching and workplace. Results indicate that collaborative task design and reflection made teachers combine elements from the workplace into mathematics teaching. Different ways of linking reality and mathematics teaching were identified in the modelling process in which the students were asked to be engaged.

Becoming a mathematician: an international perspective

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