Challenges in deepening prospective teachers’ understanding of multiplication through justification

This article reports challenges faced by prospective elementary teachers as they revisited whole number multiplication through a sequence of tasks that required them to develop and justify reasoning strategies for multiplication. Classroom episodes and student work are used both to illustrate these challenges, as well as to demonstrate growth over time. Implications for the design of mathematics courses for prospective teachers’ are discussed. Although the study is situated in the context of multiplication, it has implications for teachers reasoning and justification in other areas of mathematics.

Learning mathematics for teaching in the student teaching experience: two contrasting cases

Student teaching (guided teaching by a prospective teacher under the supervision of an experienced “cooperating” teacher) provides an important opportunity for prospective teachers to increase their understanding of mathematics in and for teaching. The interactions between a student teacher and cooperating teacher provide an obvious mechanism for such learning to occur. We report here on data that is part of a larger study of eight student teacher/cooperating teacher pairs, and the core themes that emerged from their conversations. We focus on two pairs for whom the core conversational themes represent disparate approaches to mathematics in and for teaching. One pair, Blake and Mr. B., focused on controlling student behavior and rarely talked about mathematics for teaching. The other pair, Tara and Mr. T., focused on having students actively participating in the lesson and on mathematics from the students’ point of view. These contrasting experiences suggest that student teaching can have a profound effect on prospective teachers’ understanding of mathematics in and for teaching.

Learning to observe: using video to improve preservice mathematics teachers’ ability to notice

Video has assumed an increasingly prominent role in teacher education, particularly in the form of the viewing of videotaped class lessons by preservice teachers. Yet there is little research that confirms whether preservice teachers attend to the aspects of the video(s) that teacher educators anticipate or desire. This article explores this issue and reports on the impact of video viewing as a means to improve teachers’ ability to be observers of classroom practice. We utilized a pre- and post-test design to measure the quantity and type of classroom events that preservice mathematics teachers noticed before and after a teaching methods course where improving observation skills was an explicit goal. The results of the pre-assessment suggest that preservice teachers generally do not enter teaching methods courses with well-developed observation skills. The post-assessment indicates that the course led to significant increases in preservice teachers’ observation skills, particularly in teachers’ ability to notice features of the classroom environment, mathematical content of a lesson, and teacher and student communication during a lesson.

Socio-emotional orientations and teacher change

In this article we consider how elementary education students’ views of mathematics changed during their mathematics methods course. We focus on four female students: two started the course with mainly positive views of mathematics and a task orientation, two with negative views of the subject and an ego-defensive orientation. The biggest change observed was that the trainees’ views of teaching and learning mathematics became more positive. Moreover, what had been an ego-defensive orientation changed towards a social-dependence orientation. The crucial facilitators of change seemed to be (1) handling of and reflection on one’s experiences of learning and teaching mathematics, (2) exploring content with concrete materials, and (3) collaboration with a partner or working as a tutor of mathematics.

Computer Manipulatives in an Ordinary Differential Equations Course: Development, Implementation, and Assessment

The d’Arbeloff Interactive Mathematics Project or d’AIMP is an initiative that seeks to enhance and ultimately transform the teaching and learning of introductory mathematics at the Massachusetts Institute of Technology. A result of this project is a suite of “mathlets,” a carefully developed set of dynamic computer applets for use in the university’s ordinary differential equations course. In this paper, we present the rationale for such computer innovations, the philosophy behind their design, as well as a discussion of their careful development and implementation. Survey results are reported which yielded positive student feedback and suggestions for improvement.

Mathematical preparation of elementary teachers in China: changes and issues

It is generally perceived that Chinese elementary teachers have a profound understanding of the school mathematics they teach. This perception has led to further interest in understanding teacher education practices in China. As some dramatic changes in elementary teacher preparation have taken place in China over the past decade, this article aims to outline these changes with a focus on curriculum provided in the new 4-year bachelor preparation programs. Sample mathematics teacher educators in China were also surveyed to gather insiders’ views about teacher preparation practices and to identify relevant issues. We believe that elementary teacher preparation and its changes in China can provide an important case for mathematics teacher educators around the world to reflect on teacher education practices in their own systems.

Mathematical belief change in prospective primary teachers

The development and influence of beliefs in teacher education has been a topic of increasing interest for researchers in recent years. This study explores the responses of a group of prospective primary teachers to attempts to facilitate belief change as part of their initial teacher education programme in mathematics. The students’ responses seemed to fall into three categories: non-engagement; building a new set of beliefs and; reforming existing beliefs. In this article the participants’ responses are outlined and illustrated with stories from three individuals. This study suggests that belief reform is complex and fraught with ethical dilemmas. Certainly there is a need for further research in this area, particularly given the pervasive influence of beliefs on teaching practice.

From Peasant to Princess: A Standards Conversion Story with a Twist

The first author, a student in a graduate children’s literature class, designed a project to locate “good” mathematics-based children’s literature selections. However, the reference tools usually consulted (e.g., Books in Print) to locate books by topic were of little help, and those she located under individual mathematics topics were mostly traditional mathematics books rather than good read-aloud selections. Consequently, she perused the university library’s sizeable juvenile collection to find books that would meet her selection criteria. This article describes the influence of two landmark documents for mathematics teaching and learning—Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics [NCTM], 1989) and Principles and Standards for School Mathematics (NCTM, 2000)—as she engaged in the process.

Teachers’ Perspectives on a Professional Development Intervention to Improve Science Instruction Among English Language Learners

Our 5-year professional development intervention is designed to promote elementary teachers’ knowledge, beliefs, and practices in teaching science, along with English language and mathematics for English Language Learning (ELL) students in urban schools. In this study, we used an end-of-year questionnaire as a primary data source to seek teachers’ perspectives on our intervention during the first year of implementation. Teachers believed that the intervention, including curriculum materials and teacher workshops, effectively promoted students’ science learning, along with English language development and mathematics learning. Teachers highlighted strengths and areas needing improvement in the intervention. Teachers’ perspectives have been incorporated into our on-going intervention efforts and offer insights into features of effective professional development initiatives in improving science achievement for all students.

Using the history of mathematics to induce changes in preservice teachers’ beliefs and attitudes: insights from evaluating a teacher education program

Scholars and teacher educators alike agree that teachers’ beliefs and attitudes toward mathematics are key informants of teachers’ instructional approaches. Therefore, it has become clear that, in addition to enriching preservice teachers’ (PSTs) knowledge, teacher education programs should also create opportunities for prospective teachers to develop productive beliefs and attitudes toward teaching and learning mathematics. This study explored the effectiveness of a mathematics preparatory program based on the history of mathematics that aimed at enhancing PSTs’ epistemological and efficacy beliefs and their attitudes toward mathematics. Using data from a questionnaire administered four times, the study traced the development of 94 PSTs’ beliefs and attitudes over a period of 2 years. The analysis of these data showed changes in certain dimensions of the PSTs’ beliefs and attitudes; however, other dimensions were found to change in the opposite direction to that expected. Differences were also found in the development of the PSTs’ beliefs and attitudes according to their mathematical background. The data yielded from semi-structured follow-up interviews conducted with a convenience sample of PSTs largely corroborated the quantitative data and helped explain some of these changes. We discuss the effectiveness of the program considered herein and draw implications for the design of teacher education programs grounded in the history of mathematics.

Tracing the development of preservice teachers’ efficacy beliefs in teaching mathematics during fieldwork

This paper focuses on efficacy beliefs with respect to teaching mathematics; it aims at testing and validating an existing scale for measuring efficacy beliefs in the domain of mathematics, examining the development of preservice teachers’ efficacy beliefs in mathematics during fieldwork, and identifying factors that contribute to the development of these beliefs. Exploratory factor analysis applied to longitudinal data collected through a questionnaire administered to almost 90 preservice teachers elicited a two-factor model, reflecting efficacy beliefs in mathematics instruction and classroom management. Using cluster analysis, we found four patterns in the development of preservice teachers’ efficacy beliefs regarding these two factors. Our analysis of semi-structured interviews with eight participants representing these four patterns suggested that preservice teachers’ efficacy beliefs were mainly informed by experimentation with teaching and interaction with mentors, tutors, peers, and pupils. Based on our findings, we draw implications for mathematics preservice teacher education and provide suggestions for future research.

Lacan,subjectivity and the task of mathematics education research

This paper addresses the issue of subjectivity in the context of mathematics education research. It introduces the psychoanalyst and theorist Jacques Lacan whose work on subjectivity combined Freud’s psychoanalytic theory with processes of signification as developed in the work of de Saussure and Peirce. The paper positions Lacan’s subjectivity initially in relation to the work of Piaget and Vygotsky who have been widely cited within mathematics education research, but more extensively it is shown how Lacan’s conception of subjectivity provides a development of Peircian semiotics that has been influential for some recent work in the area. Through this route Lacan’s work enables a conception of subjectivity that combines yet transcends Piaget’s psychology and Peirce’s semiotics and in so doing provides a bridge from mathematics education research to contemporary theories of subjectivity more prevalent in the cultural sciences. It is argued that these broader conceptions of subjectivity enable mathematics education research to support more effective engagement by teachers, teacher educators, researchers and students in the wider social domain.

Student and early career mobility patterns among highly educated people in Germany,Finland, Italy,and the United Kingdom

This article addresses the question of how great are higher education students’ incentives to change study programs or institutions to improve one’s personal employability in the course of the higher education-to-work transition process. The posed question is addressed at a system level. Students’ mobility between programs and institutions is referred to as ‘student mobility’ and graduates’ mobility between jobs is referred to as ‘early career mobility.’ The relationship between these two separate components of mobility is discussed in three different institutional frames: German/Finnish, Italian, and British. When depicting the relationship between the two components of mobility, the article also considers parallel phenomena such as prolongation of the degree-earning process and participation in work-life with student status. Indicator level analysis concerning graduates of the year 2000 reveals important differences between the three institutional frames: in Germany and Finland, there is a high level of student mobility at the basic degree level combined with a low level of career mobility after graduation; in the UK the opposite transition logic than that of Germany and Finland occurs, and in Italy students demonstrate prolonged transitions with little student mobility and early career mobility.

The relationship between performance on mathematical word problems and language proficiency for students learning through the medium of Irish

Ireland has two official languages—Gaeilge (Irish) and English. Similarly, primary- and second-level education can be mediated through the medium of Gaeilge or through the medium of English. This research is primarily focused on students (Gaeilgeoirí) in the transition from Gaeilge-medium mathematics education to English-medium mathematics education. Language is an essential element of learning, of thinking, of understanding and of communicating and is essential for mathematics learning. The content of mathematics is not taught without language and educational objectives advocate the development of fluency in the mathematics register. The theoretical framework underpinning the research design is Cummins’ (1976). Thresholds Hypothesis. This hypothesis infers that there might be a threshold level of language proficiency that bilingual students must achieve both in order to avoid cognitive deficits and to allow the potential benefits of being bilingual to come to the fore. The findings emerging from this study provide strong support for Cummins’ Thresholds Hypothesis at the key transitions—primary- to second-level and second-level to third-level mathematics education—in Ireland. Some implications and applications for mathematics teaching and learning are presented.

Developing a communal identity as beginning teachers of mathematics: Emergence of an online community of practice

The aim of this study is to investigate how a community of practice focused on becoming a teacher of secondary school mathematics emerged during a pre-service teacher education programme and was sustained after students graduated and began their first year of full-time teaching in schools. Bulletin board discussions of one pre-service cohort are analysed in terms of Wenger’s (1998) three defining features of a community of practice: mutual engagement of participants, negotiation of a joint enterprise, and development of a shared repertoire for creating meaning. Emergence of the online community was associated with our own role in facilitating professional dialogue, the voluntary and unstructured nature of participation, initial face-to-face interaction that created familiarity and trust, and the convenience of using email rather than logging on to a website. The study shows that the emergent design of the community contributed to its sustainability in allowing the pre-service and beginning teachers to define their own professional goals and values.

Exploring pedagogical strategies to integrate a holistic approach to education into ESL classes

Holistic education is a popular trend in the midst of various educational reforms, yet little systematic research has been done to consider its application in the teaching of English as a Foreign Language (EFL) and English as a Second Language (ESL) for Asian Chinese learners. According to the New Zealand Ministry of Education, holistic education is defined as an approach to teaching to include the physical, social, emotional, cultural, and cognitive dimensions of a person’s growth. Education reform in Taiwan has adopted a Western constructivist approach. However, in Confucian Heritage Culture (CHC), open teacher–student and student–student interactions are not as common as they are in the social constructivist model found in the Anglo-Saxon culture. CHC values social harmony, teachers’ authority, and silence to avoid confrontation. Without understanding the affective and social environment in the Western context, holistic education is unlikely to be successfully implemented in Taiwan. The purpose of this research is to understand the contextualization of the holistic environment and pedagogical strategies that nurture student–teacher and student–student interactions in a holistic approach to EFL. This study utilized qualitative research methods of observation and in-depth interviews of New Zealand English teachers and observed the pedagogical practices in New Zealand English classes. This research provided effective pedagogical strategies for implementing the social and affective support of a holistic educational approach for Asian EFL students.

Toward a framework for the development of mathematical knowledge for teaching

Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident—that content knowledge itself was not sufficient for teachers to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics. One such conceptualization that shows promise is mathematical knowledge for teaching—mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has started to gain attention as an important concept in the mathematics teacher education research community, there is limited understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article, we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in both mathematics education and the learning sciences.

Jong-Hsiang Yang: a major promoter of Taiwanese science education

This article reviews the work of Jong-Hsiang Yang in science education and his efforts in creating a research culture in Taiwan. Following in Yang’s footprints, the rebuilding of science education, implementing a new science curriculum, and gaining the academic status of science education, we go through the important years of the development of science education in Taiwan. His leadership in introducing interpretive research methods and expanding international studies catalyzed profound changes to science education research in Taiwan.