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.     

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM—The International Journal on Mathematics Education—by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers’ and didacticians’ engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice.     

Being a mathematics teacher in times of reform

Within research on mathematics teachers and/or their professional development, the concept of identity emerges as a critique of views of how teaching practice is related to teachers’ ‘internal states’ of knowledge and beliefs. Identity relates teachers’ professional lives to teaching practices and to the contexts in which the teaching and/or professional development occurs. However, what might count as the context still needs in-depth discussion. In order to contribute to the development of a theoretical framework for understanding mathematics teachers’ professional lives, we will draw on one remarkable teacher’s identity as a primary mathematics teacher in relation to one political, sociocultural, and pedagogical context. We use this teacher’s experience to discuss how education policies that create what Ball (2003) called ‘terrors of performativity’ tend to impede the formation of a balanced teacher identity.     

Making mathematics and science integration happen: key aspects of practice

The integration of mathematics and science teaching and learning facilitates student learning, engagement, motivation, problem-solving, criticality and real-life application. However, the actual implementation of an integrative approach to the teaching and learning of both subjects at classroom level, with in-service teachers working collaboratively, at second-level education, is under-researched due to the complexities of school-based research. This study reports on a year-long case study on the implementation of an integrated unit of learning on distance, speed and time, within three second-level schools in Ireland. This study employed a qualitative approach and examined the key aspects of practice that impact on the integration of mathematics and science teaching and learning. We argue that teacher perspective, teacher knowledge of the ‘other subject’ and of technological pedagogical content knowledge (TPACK), and teacher collaboration and support all impact on the implementation of an integrative approach to mathematics and science education.     

Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one assessment interviews

In this paper, we outline the benefits to teachers’ expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers’ knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge.     

Promoting fundamental change in mathematics teaching: a theoretical, methodological, and empirical approach to the problem

Mathematics education reform, as conceptualized in the United States and a number of other countries, represents a fundamental change in the teaching of mathematics and the results it would produce for students. Whereas there are data that suggest some progress is being made in the direction of reform, teacher education and professional development during the last two decades have been largely unsuccessful in preparing teachers to enact the reform vision. In this article, I present a theoretical construct, major assimilatory structures, that can contribute to explaining the difficulty of promoting change in mathematics teaching. I describe a methodology—accounts of practice—for identifying major assimilatory structures of teachers and present an example of a major assimilatory structure, perception-based perspective, that emerged from our empirical work.     

Conception of expert mathematics teacher: a comparative study between Hong Kong and Chongqing

This paper reports the similarities and differences in how “expert mathematics teacher” is conceptualized by mathematics educators in Hong Kong and Chongqing, two cities in China which share similar but different cultural and social backgrounds. Thirty-seven mathematics education researchers, school principals with mathematics education background, and mathematics teachers were interviewed on their perceptions of expert mathematics teacher. It is found that in both cities an expert mathematics teacher should have a profound knowledge base in mathematics, teaching, and students; strong ability in teaching; and a noble personality and a spirit of life-long learning. As for differences, an expert mathematics teacher should have the ability to conduct research, mentor other teachers, and have profound knowledge of examination and educational theories in Chongqing. These attributes were not found in Hong Kong. These similarities and differences are discussed, and relevant social and cultural factors in the two contexts are examined.     

Knowledge and confidence of pre-service mathematics teachers: the case of fraction division

To make teacher preparation and professional development effective, it is important to find out possible deficiencies in teachers’ knowledge as well as teachers’ own perceptions about their needs. By focusing on pre-service teachers’ knowledge of fraction division in this article, we conceptualize the notion of pre-service teachers’ knowledge in mathematics and pedagogy for teaching as containing both teachers’ perceptions of their preparation and their mathematics knowledge needed for teaching. With specific assessment instruments developed for pre-service middle school teachers, we focus on both pre-service teachers’ own perceptions about their knowledge preparation and the extent of their mathematics knowledge on the topic of fraction division. The results reveal a wide gap between sampled pre-service middle school teachers’ general perceptions/confidence and their limited mathematics knowledge needed for teaching fraction division conceptually. The results suggest that these pre-service teachers need to develop a sound and deep understanding of mathematics knowledge for teaching in order to build their confidence for classroom instruction. The study’s findings indicate the feasibility and importance of conceptualizing the notion of teachers’ knowledge in mathematics and pedagogy for teaching to include teachers’ perceptions. The applicability and implications of this expanded notion of teachers’ knowledge is then discussed.     

The challenges facing initial teacher education: Irish prospective elementary teachers’ mathematics subject matter knowledge

Given the acknowledged relationship between teachers’ knowledge, their teaching and pupil learning, teachers’ mathematics subject matter knowledge (MSMK) has received increased attention internationally. As children's early mathematics experiences have been recognized as a critical stage, elementary teachers’ MSMK has become a focal point among researchers and policy makers alike. International research findings have uncovered that in many cases, there is a mismatch between what is perceived to be an appropriate MSMK for teaching elementary mathematics and that demonstrated by many qualified and prospective elementary teachers. Following repeated incidences of weak MSMK during interactions with prospective elementary teachers in one Irish College of Education (provider of initial teacher education programme for elementary teachers), this study sought to examine and address the issue purposefully through two cycles of action research. This article focuses on the data collected prospective teachers’ MSMK in the initial stage (reconnaissance) of these cycles, i.e. pre-test findings. While considerable differences were evident among the pre-test population, the findings suggest that prior to the intervention stage many participating prospective teachers; regardless of previous mathematics achievements or the level of mathematics study; demonstrate weaknesses and gaps in their ‘common’ MSMK. Particular difficulties were evident in relation to pre-test items requiring knowledge of rational numbers, conceptual understanding or problem solving. These findings highlight the inadequacy of previous mathematics achievements and indeed minimum entry requirements as predictors of MSMK for teaching. As well as its contribution at a local and national level, the findings provide an Irish perspective on this international issue.     

Online education for in-service secondary teachers and the incorporation of mathematics technology in the classroom

This paper reviews existing research on how in-service high school teachers have learned about, worked on or thought about the incorporation of mathematics technology into their teaching practices. The paper reviews different scenarios of instruction issuing from important research related to teacher professional development. Specifically, we will deal with contributions to online in-service mathematics teacher education that refer to the use of digital technologies in classroom teaching practices. The different articles reviewed belong to a range of teams of researchers from several universities and countries, and who have implemented distinct online education approaches. That work has allowed the gaining of knowledge on the specificities of using Web 2.0 tools for mathematics professional development (MPD), the function that online teacher interaction has in teacher learning, and the actual classroom conditions in which mathematics technology is incorporated into instructional practice. This paper describes and discusses the design features of those approaches emphasizing the main concepts and their underpinning theoretical frames, noting important design elements, and specific results. Finally, the paper discusses how some of these research findings are connected with emergent issues in the field of MPD.     

Teaching and learning of mathematics: what really matters to teachers and students?

The learner’s perspective study, motivated by a strong belief that the characterization of the practices of mathematics classrooms must attend to learner practice with at least the same priority as that accorded to teacher practice, is a comprehensive study that adopts a complementary accounts methodology to negotiate meanings in classrooms. In Singapore, three mathematics teachers recognized for their locally defined ‘teaching competence’ participated in the study. The comprehensive sets of data from the three classrooms have been used to explore several premises related to the teaching and learning of mathematics. In this paper the student interview data and the teacher interview data were examined to ascertain what do students attach importance to and what do teachers attach importance to in a mathematics lesson? The findings of the student interview data showed that they attached importance to several sub-aspects of the three main aspects, i.e., exposition, seatwork and review and feedback of their teachers’ pedagogical practices. The findings of the teacher interview data showed that they attached importance to student’s self assessment, teacher’s demonstration of procedures, review of prior knowledge and close monitoring of their student’s progress in learning and detailed feedback of their work. It was also found that teachers and students did attach importance to some common lesson events.     

Re-sourcing teachers’ work and interactions: a collective perspective on resources, their use and transformation

This paper reviews the literature on the theme of mathematics teachers’ work and interactions with resources, taking a particular perspective, the so-called ‘collective perspective’ on resources, their use and transformation. The review is presented under three headings: (1) theoretical frameworks commonly used in this area of research; (2) teachers’ interactions with resources in terms of their design and use; and (3) teachers’ interactions with resources in terms of teacher learning and professional development. From the literature, and the collection of papers in this issue, we argue that the collective dimensions play an important role in mathematics teachers’ work with resources and in their professional learning/development. Further empirical investigations are likely to be needed on: how teachers may work in collectives and with resources, and in which ways ‘productive’ collectives may form and work together; which roles particular resources can play in these delicate constellations and how particular resources may support teachers in their work and learning; and which kinds of resources offer opportunities for community building.     

A Teacher’s Conception of Definition and Use of Examples When Doing and Teaching Mathematics

To contribute to an understanding of the nature of teachers’ mathematical knowledge and its role in teaching, the case study reported in this article investigated a teacher’s conception of a metamathematical concept, definition, and her use of examples in doing and teaching mathematics. Using an enactivist perspective on mathematical knowledge, the authors give an account of the case of Lily, a prospective, then beginning, teacher who conceived of mathematical definition as an object with particular form and function and engaged in purposeful, specialized use of examples when doing and teaching mathematics. Lily’s case illustrates how a teacher’s interpretation of examples (as exemplifications or single instances) and conception of the form and function of definitions can influence her doing and teaching mathematics. An implication is that teacher preparation should foster teachers’ abilities to use examples purposefully to provide students with rich opportunities to engage in mathematical processes such as defining.     

Teachers’ views on creativity in mathematics education: an international survey

The survey described in this paper was developed in order to gain an understanding of culturally-based aspects of creativity associated with secondary school mathematics across six participating countries. All participating countries acknowledge the importance of creativity in mathematics, yet the data show that they take very different approaches to teaching creatively and enhancing students’ creativity. Approximately 1,100 teachers from six countries (Cyprus, India, Israel, Latvia, Mexico, and Romania) participated in a 100-item questionnaire addressing teachers’ conceptions about: (1) Who is a creative student in mathematics, (2) Who is a creative mathematics teacher, (3) In what way is creativity in mathematics related to culture, and (4) Who is a creative person. We present responses to each conception focusing on differences between teachers from different countries. We also analyze relationships among teachers’ conceptions of creativity and their experience, and educational level. Based on factor analysis of the collected data we discuss relevant relationships among different components of teachers’ conceptions of creativity as they emerge in countries with different cultures.     

Mathematics professional development as design for boundary encounters

This theoretical paper examines a process for researchers and teachers to exchange knowledge. We use the concepts of communities of practice, boundary encounters, and boundary objects to conceptualize this process within mathematics professional development (MPD). We also use the ideas from design research to discuss how mathematics professional development researchers can make professional development the focus of their research. In particular, we examine the question: How can MPD be conceptualized and designed around research-based knowledge in ways that promote knowledge exchange about students’ mathematics and mathematics learning among researchers and teachers to improve the practices of both the research and the teaching communities? We propose that MPD is a premier space for researchers and teachers to exchange knowledge from their communities, impacting both researchers’ and teachers’ practices without reducing the importance of either.     

Improving teacher expertise through master teacher work stations: a case study

Professional development is emphasized in China not only for junior teachers, but also for experienced teachers. In this study, we investigated a master teacher??s work station that has been established at a provincial level for training experienced ??key?? teachers across the whole province. Three types of data were collected to examine the focuses and approaches used in this master teacher??s work station, teacher participants?? learning experiences and feedback, and selected teachers?? expertise improvement. The results show that the master teacher emphasized teachers?? development of a deeper understanding of elementary mathematics content and its structure through intensive studies of a textbook series. His training led to very positive feedback from teacher participants regarding what they have learned, especially in elementary mathematics content and content connections as presented in the textbooks. The detailed analysis of two teachers?? knowledge learning revealed dramatic changes in their thinking and instruction about specific content topics. Teacher expertise improvement through this master teacher??s work station is then summarized and discussed.     

Assessment of teacher knowledge across countries: a review of the state of research

This review presents an overview of research on the assessment of mathematics teachers’ knowledge as one of the most important parameters of the quality of mathematics teaching in school. Its focus is on comparative and international studies that allow for analyzing the cultural dimensions of teacher knowledge. First, important conceptual frameworks underlying comparative studies of mathematics teachers’ knowledge are summarized. Then, key instruments designed to assess the content knowledge and pedagogical content knowledge of future and practicing mathematics teachers in different countries are described. Core results from comparative and international studies are documented, including what we know about factors influencing the development of teacher knowledge and how the knowledge is related to teacher performance and student achievement. Finally, we discuss the challenges connected to cross-country assessments of teacher knowledge and we point to future research prospects.     

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.     

Addressing prospective elementary teachers’ mathematics subject matter knowledge through action research

There is international dissatisfaction regarding the standard of mathematics subject matter knowledge (MSMK) evident among both qualified and prospective elementary teachers. Ireland is no exception. Following increasing anecdotal evidence of prospective elementary teachers in one Irish College of Education (provider of initial teacher education programme) demonstrating weaknesses in this regard, this study sought to examine and address the issue through two cycles of action research. The examination of the nature of prospective teachers’ MSMK (as well as related beliefs in the main study) informed the design and implementation of an intervention to address the issue. A mixed method approach was taken throughout. In both cycles, Shapiro's criteria were used as a conceptual framework for the evaluation of the initiative. This paper focuses on the perceived and actual effects of the intervention on participants’ MSMK. As well as its contribution at a local and national level, the study provides an Irish perspective on approaches taken to address the phenomenon internationally.     

Teachers’ views on effective mathematics teaching: commentaries from a European perspective

The previous papers of this issue discuss the views teachers from the United States, Australia, Hong Kong SAR, and Mainland China have on effective mathematics teaching and learning. Similarities and differences are found and a differentiation from West to East can be worked out in the order of the regions as listed above. The picture of teachers’ views can, however, be differentiated when they are looked at from a European perspective. On the basis of the analysis of two comparative studies on teaching cultures in three European countries and a questioning of teachers, this commentary therefore locates France, Germany, and England within this framework so that the East–West-contrast is distinguished in more detail.