Multiple-solution tasks: from a teacher education course to teacher practice

While development of a teacher’s expertise includes continuous incorporation of innovations throughout his/her career, teachers are often reluctant to adopt and implement new practices when challenged by innovative teaching approaches. This paper presents an analysis of the development of teachers’ expertise in relation to the implementation of novel (for them) instructional material. The study examines the ways in which teachers implement multiple-solution tasks (MSTs) (as an example of instructional tools new to the teacher) in their classes, following a professional development course in which they participated. The analysis focuses on the nature of MSTs implemented by the teachers and of the subsequent class discussion. The nature of MSTs is analyzed focusing on the goals with which MSTs were implemented, mathematical connections embedded in the MSTs, scaffolding provided to the learners and the learning settings. This analysis has led to the identification of four main implementation styles: straightforward, simple, adaptive and inventive. Concluding discussions are examined with respect to elevating and framing elements. Two lessons by mathematics teachers are described in the paper to explain how lessons were analyzed, and to exemplify adaptive and inventive implementation styles.     

Combining theories in research in mathematics teacher education

In this paper, we describe how the combination of two theories, each embedded in a different realm, may contribute to evaluating teachers’ knowledge. One is Shulman’s theory, embedded in general, teacher education, and the other is Fischbein’s theory, addressing learners’ mathematical conceptions and misconceptions. We first briefly describe each of the two theories and our suggestions for combining them, formulating the Shulman–Fischbein framework. Then, we present two research segments that illustrate the potential of the implementation of the Shulman–Fischbein framework to the study of mathematics teachers’ ways of thinking. We conclude with general comments on possible contributions of combining theories that were developed in mathematics education and in other domains to mathematics teacher education.     

Research into teacher knowledge: a stimulus for development in mathematics teacher education practice

In this paper, we document some developments in teacher education practice at one university, brought about by reflection on research into mathematics teacher knowledge. The authors are three members of the Cambridge-based research team who developed the Knowledge Quartet (KQ), a theory of mathematics teacher knowledge, with a focus on classroom situations in which this knowledge is applied. At the same time as being researchers, the authors were elementary mathematics teacher education instructors. They found that the KQ research brought about new awareness of the importance of some components of mathematics didactics, as well as providing new tools for undertaking some aspects of their teacher educator role. The paper explores some of these awarenesses and tools in detail.     

Unpacking Online Asynchronous Collaboration in mathematics teacher education

We report on one aspect of an extended research and development project that was conducted to support teachers?? development of mathematical knowledge for teaching (MKT) algebra through participation and authentic engagement in online collaborative mathematical problem solving. This article expands on our recent work, which has succeeded in developing a model for supporting teachers?? mathematical development at a distance that has shown great promise for supporting significant gains in teachers?? MKT. Specifically, this ex-post-facto analysis consisted of qualitative, micro-level analysis of the content of teachers?? activity and generated artifacts and helps us understand how the various collaborative activities (specific combinations of interaction, instructor support and feedback, and technology) supported and/or constrained the development of MKT algebra in an online environment.     

Epistemological beliefs concerning the nature of mathematics among teacher educators and teacher education students in mathematics

Beliefs constitute a central part of a person’s professional competences as beliefs are crucial to the perception of situations and as they influence our choice of actions. The present article focuses on epistemological beliefs about the nature of mathematics among future teachers and their educators at university and post-university teacher-training institutions in Germany. The data reported are part of a larger sample originating from the MT21 study [supported by the National Science Foundation through a grant to W. S. Schmidt and M. T. Tatto (REC-0231886). MT21 started in 2003] which explores and compares mathematics teacher education in Bulgaria, Germany, Mexico, South Korea, Taiwan, and the United States. In this article, we examine the structure and level of beliefs concerning the nature of mathematics for teacher education students in Germany both at the beginning (n = 368) and the end of their education (n = 286) as well as their educators (n = 77) in three academic disciplines (mathematics, mathematics pedagogy and general pedagogy). In the first part of the article, the literature on epistemological beliefs and their structure will be reviewed. In the empirical part, analyses on the level and the structure of beliefs for our samples and subsamples will be presented. Relations between educators’ and students’ beliefs will be explored.     

A teacher education experiment to challenge conceptions and practices

This study describes a teacher education experience with grade 5–6 teachers, based on a calculator module within a national program for mathematics in-service teacher education. The aim was to challenge the teachers’ conceptions about the role of the calculator in mathematics teaching and to promote their reflection about professional practices. The research methodology was qualitative and interpretive, with data collection through interviews and observation of teacher education and classroom supervision sessions, as well as analysis of teachers’ portfolios. The results indicate that some teachers are clearly against the use of the calculator in the mathematics classroom, while others allow students to use it in a passive way and some others are very affirmative about its use. The teachers who argue against the use of the calculator seem to predominate, suggesting a great distance between the curriculum orientations and classroom practice. The methodology of the course, combining collective sessions and individual classroom supervision, proved to be fruitful, providing new information, practice and discussion that allowed teachers to analyze different kinds of tasks in which the calculator might be useful, experiment using them in the classroom and reflect about the students’ work. The no imposing and questioning approach used in collective discussions encouraged teachers to assume their own positions; sharing and discussing in the collective reflections during the course stimulated a deeper reflection of their practice. Therefore, in this course, in-service teacher education focused on practice contributed to teachers to reflect on their conceptions and practices.     

Constructs of engagement emerging in an ethnomathematically-based teacher education course

This paper presents a case study, in which we apply and develop theoretical constructs to analyze motivating desires observed in an unconventional, culturally contextualized teacher education course. Participants, Israeli students from several different cultures and backgrounds (pre-service and in-service teachers, Arabs and Jews, religious and secular) together studied geometry through inquiry into geometric ornaments drawn from diverse cultures, and acquired knowledge and skills in multicultural education. To analyze affective behaviors in the course we applied the methodology of engagement structures recently proposed by Goldin and his colleagues. Our study showed that engagement structures were a powerful tool for examining motivating desires of the students. We found that the constructivist ethnomathematical approach highlighted the structures that matched our instructional goals and diminished those related to students’ feelings of dissatisfaction and inequity. We propose a new engagement structure “Acknowledge my culture” to embody motivating desires, arising from multicultural interactions, that foster mathematical learning.     

Theories in practice: mathematics teaching and mathematics teacher education

Whilst research on the teaching of mathematics and the preparation of teachers of mathematics has been of major concern in our field for some decades, one can see a proliferation of such studies and of theories in relation to that work in recent years. This article is a reaction to the other papers in this special issue but I attempt, at the same time, to offer a different perspective. I examine first the theories of learning that are either explicitly or implicitly presented, noting the need for such theories in relation to teacher learning, separating them into: socio-cultural theories; Piagetian theory; and learning from practice. I go on to discuss the role of social and individual perspectives in authors’ approach. In the final section I consider the nature of the knowledge labelled as mathematical knowledge for teaching (MKT). I suggest that there is an implied telos about ‘good teaching’ in much of our research and that perhaps the challenge is to study what happens in practice and offer multiple stories of that practice in the spirit of “wild profusion” (Lather in Getting lost: Feminist efforts towards a double(d) science. SUNY Press, New York, 2007).     

Connecting social perspectives on mathematics teacher education in online environments

This article explores theoretical issues underpinning the design and use of online learning environments in mathematics teacher education. It considers the contribution of social theories of learning to conceptualising technology-mediated interaction, focusing specifically on community of practice models and the notion of digital mathematics performance. The article begins by introducing social perspectives on collaboration. Because of the diversity of theories within this broad research paradigm, the next section outlines networking strategies that have been proposed for connecting theoretical approaches. There follows a discussion of studies that illustrate the community of practice and performance-based approaches to research into online mathematics teacher education. The main purpose of the article is to show how these approaches could be connected by examining the same teaching and learning scenarios through different theoretical lenses. The final section identifies implications of this exploration for the design of online learning environments in mathematics teacher education to capitalise on the affordances of Web-based technologies.     

Online mathematics teacher education: overview of an emergent field of research

Online mathematics teacher education is characterized as an emergent area of research in mathematics education. We identify some key topics that require further research: communities and networks of teachers in online environments; sustainability of these communities and kinds of organizational structures; knowledge-building practices in technology-mediated work group interactions; and online interactions among teachers. The emergence of new research issues also gives rise to new theoretical approaches or the adaptation of existing theoretical perspectives that are presented in this special issue. We summarize some of these theoretical perspectives and attempt to show how online environments have changed them, as well as some theoretical problems that remain to be solved.     

Mathematics teacher education advanced methods: an example in dynamic geometry

We present a research work about an innovative national teacher training program in France: the Pairform@nce program, designed to sustain ICT integration. We study here training for secondary school teachers, whose objective is to foster the development of an inquiry-based approach in the teaching of mathematics, using investigative potentialities of dynamic geometry environments. We adopt the theoretical background of the documentational approach to didactics. We focus on the interactions between teachers and resources: teachers’ professional knowledge influences these interactions, which at the same time yield knowledge evolutions, a twofold process that we conceptualise as a documentational genesis. We followed in particular the work of a team of trainees; drawing on the data collected, we analyse their professional development, related with the training. We observe intertwined evolutions and stabilities, consistent with ongoing geneses.     

On music production in mathematics teacher education as an aesthetic experience

ZDM – Mathematics Education - This paper addresses the integrated use of the arts and digital technology in mathematics education—specifically involving aspects of preservice...     

Representing context in video records of practice for urban mathematics teacher education

Mathematics education research has not sufficiently theorized about mathematics teacher knowledge and practice, teacher learning, and teacher education in ways that are reflective of the specificities of the sociopolitical contexts of schooling. In the USA, this is particularly important for urban mathematics education. This paper examines the affordances and challenges of representing context in video records of practice, particularly in the urban context, for use in the preparation of mathematics teachers for urban settings. The discussion, grounded in current research and theory relevant to representations of teaching, urban education, and mathematics teacher education, takes up three key issues: how is a focus on the urban context relevant to the design of video records of practice for mathematics teacher education? How can video records support prospective teachers’ understandings of the sociopolitical contexts of mathematics teaching? How does a focus on the urban context impact the meaning teachers make of video records?     

The conceptualisation of mathematics competencies in the international teacher education study TEDS-M

The main aim of the international teacher education study Teacher Education and Development Study in Mathematics (TEDS-M), carried out under the auspices of the International Association for the Evaluation of Educational Achievement (IEA), was to understand how national policies and institutional practices influence the outcomes of mathematics teacher education. This paper reports on the definition of effective mathematics teacher education in TEDS-M, distinguishing between mathematics content knowledge and mathematics pedagogical content knowledge as essential cognitive components of mathematics teachers’ professional competencies. These competence facets were implemented as proficiency tests based on extensive coordination and validation processes by experts from all participating countries. International acceptance of the tests was accomplished whereas, by necessity, national specifications had to be left out, as is common in comparative large-scale assessments. In this paper, the nature of the TEDS-M tests for the primary study is analysed and commented on detail. The aims are to increase our understanding of mathematics content knowledge and mathematics pedagogical content knowledge, which are still fuzzy domains, to provide a substantive background for interpretations of the test results and to examine whether some educational traditions may be more accurately reflected in the test items than others. For this purpose, several items that have been released by the IEA are presented and elaborately analysed in order to substantiate the test design of TEDS-M. Our main conclusion is that the overall validity of the TEDS-M tests can be regarded as a given, but that readers have to be aware of limitations, amongst others from a continental European point of view.     

Prospective mathematics teachers’ development of noticing in an online teacher education program

In this study we aimed to characterize the development of prospective mathematics teachers’ noticing in an online teacher education program. Prospect     

Learning trajectories in teacher education: Supporting teachers’ understandings of students’ mathematical thinking

Recent work by researchers has focused on synthesizing and elaborating knowledge of students’ thinking on particular concepts as core progressions called learning trajectories. Although useful at the level of curriculum development, assessment design, and the articulation of standards, evidence is only beginning to emerge to suggest how learning trajectories can be utilized in teacher education. Our paper reports on two studies investigating practicing and prospective elementary teachers’ uses of a learning trajectory to make sense of students’ thinking about a foundational idea of rational number reasoning. Findings suggest that a mathematics learning trajectory supports teachers in creating models of students’ thinking and in restructuring teachers’ own understandings of mathematics and students’ reasoning.     

Learning from the Eastern and the Western debate: the case of mathematics teacher education

Comparative studies have gained significant influence in the last decades, and school systems of many countries have been revised referring to better results of other countries in international large-scale assessments. Authors of such studies commonly link their interpretations of the results to distinctions between “Eastern” and “Western” cultures, in particular with respect to the consistent and continuing outstanding performance of East Asian learners compared with their Western counterparts. One question is whether the same achievement pattern holds for future teachers and whether similar cultural differences may cause it. International Association for the Evaluation of Educational Achievement’s “Teacher Education and Development Study in Mathematics” (TEDS-M) was the first comparative study that focused on the outcomes of teacher education with standardised testing. In this paper—based on the TEDS-M results—commonalities and differences in the achievement of future teachers from Eastern and Western countries are explored and related to a cultural perspective. Cultural differences between Eastern and Western approaches concerning mathematics, mathematics education and mathematics teachers are analysed with respect to the achievement pattern. The paper closes with reflections on possible consequences concerning the development of teachers’ knowledge and teachers’ expertise in mathematics education.     

Boundary crossing through in-service online mathematics teacher education: the case of scenarios and half-baked microworlds

We address online mathematics teacher education as a means of immersing teachers into new kinds of collectives where professional development may be enhanced by crossing the boundaries of their habitual communities and the norms within these. We analyse the ways in which newly trained teacher educators interacted around their designs and productions for in-service courses they were giving to colleagues. We focus on their uses of scenarios and half-baked microworlds as two kinds of artefacts we had designed to play the role of improvable boundary objects. We consider these interactions as a forum for challenge and for professional development through frequent and relevant boundary crossing.     

Using comics-based representations of teaching, and technology, to bring practice to teacher education courses

This article situates comic-based representations of teaching in the long history of tensions between theory and practice in teacher education. The article argues that comics can be semiotic resources in learning to teach and suggests how information technologies can support experiences with comics in university mathematics methods courses that (a) help learners see the mathematical work of teaching in lessons they observe, (b) allow candidates to explore tactical decision-making in teaching, and (c) support preservice teachers in rehearsing classroom interactions.     

Mobilizing histories in mathematics teacher education: memories, social practices, and discursive games

In the first part of this paper, we share and elucidate the way we mobilize histories in some disciplines that are part of the undergraduate courses in mathematics teacher education offered by State University of Campinas and Federal University of Rio Grande do Norte in Brazil. This way of mobilization can be featured as a set of collective indisciplinary problematizations occurring in a series of student investigations. Mobilizing practices of mathematics culture are the object of these investigations. These practices are performed by different communities both constituted by and constituent of different human activities. In the second part of this paper, we will discuss our way of mobilizing histories, contrasting it with the theoretical perspective of expansive learning, just as it has been defended by Yrjö Engeström, in his article Non scolae sed vitae discimus—towards overcoming the encapsulation of school learning. We will also attempt to highlight the role which this researcher has attributed to history in his model of expansive learning, a perspective based on the current research on activity theory.