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.     

Building coherence in research on mathematics teacher characteristics by developing practice-based approaches

Scholars have debated which teacher characteristics are primary in determining teachers’ practice. Some claim that identity is at the core of teachers’ ways of being and acting; others argue that teachers’ actions depend principally on their knowledge or beliefs. We argue that, whichever is examined, it is important to study how teachers use specific characteristics in their work, and how the work of teaching is shaped by that use. We claim that this can be done by addressing research questions about teacher characteristics in ways that provide insight into how they contribute to shaping interactions in classrooms—what we call a practice-based approach. To develop and illustrate our argument, we discuss studies that exemplify what we mean by a practice-based approach to the study of a teacher characteristic and we unpack ways in which these studies distinctively contribute to understanding and improving practice. Further, we explore ways in which the development of practice-based approaches might support coherence across efforts to study different characteristics and innovation in studies that consider the interplay of different teacher characteristics in teaching.     

Beliefs and beyond: hows and whys in the teaching of proof

In this paper, we report on a study aimed at describing the way secondary school teachers treat proof and at understanding which factors may influence such a treatment. This study is part of a wider project on proof carried out for many years. In our theoretical framework, we combine references from research on proof with those from research on teachers in relation to their beliefs. The study was carried out through interviews with secondary school teachers aimed at learning how they describe their work with proof in the classroom, and to elicit beliefs and other factors that shape this work. Through the interviews we were able to detect reasons behind teachers’ choices in planning their work in the classroom. In the present paper, we concentrate on four cases that, among other factors, offer elements suitable to unravel the problem of inconsistencies using the construct of leading beliefs, i.e., beliefs (whose nature may vary from teacher to teacher) that seem to drive the way each teacher treats proof.     

Using Sociocultural Theory to Teach Mathematics: A Vygotskian Perspective

This study describes an elementary teacher's implementation of sociocultural theory in practice. Communication is central to teaching with a sociocultural approach and to the understanding of students; teachers who use this theory involve students in explaining and justifying their thinking. In this study ethnographic research methods were used to collect data for 4 1/2 months in order to understand the mathematical culture of this fourth‐grade class and to portray how the teacher used a sociocultural approach to teach mathematics. To portray this teaching approach, teaching episodes from the teacher's mathematics lessons are described, and these episodes are analyzed to demonstrate how students created taken‐as‐shared meanings of mathematics. Excerpts from interviews with the teacher are also used to describe this teacher's thinking about her teaching.     

A teacher’s learning process in dual design research: learning to scaffold language in a multilingual mathematics classroom

In this paper, we argue that dual design research (DDR) is a fruitful way to promote and trace the development of a mathematics teacher’s expertise. We address the question of how a teacher participating in dual design research can learn to scaffold students’ development of the language required for mathematical learning in multilingual classrooms. Empirical data were collected from two teaching experiments (each with 8 lessons, and 21 and 22 students, aged 11–12 years), for which lesson series about line graphs were co-designed by the researchers and the teacher. The teacher’s learning process was promoted (e.g. by conducting stimulated recall interviews and providing feedback) and traced (e.g. by carrying out 5 pre- and post-interviews before and after the teaching experiments). An analytic framework for teachers’ reported and derived learning outcomes was used to analyse pre- and post-interviews. The teacher’s learning process was analysed in terms of changes in knowledge and beliefs, changes in practice and intentions for practice. Further analysis showed that this learning process could be attributed to the characteristics of dual design research, for instance the cyclic and interventionist character, the continuous process of prediction and reflection that lies at its heart, and the process of co-designing complemented with stimulated recall interviews.     

Multi‐Year Professional Development Grounded in Educative Curriculum Focused on Integrating Technology With Reformed Science Teaching Principles

Visions of science teaching and learning in the newest U.S. standards documents are dramatically different than those found in most classrooms. This research addresses these differences through closely examining one professional development (PD) project that connects teacher learning and teacher practice with student learning/achievement. This study examines the effects on eighth grade science teachers and their students in the context of a PD focused on the integration of information communication technologies and reformed science teaching practices. Findings from this investigation suggest that teachers who participated in PD for two years learned more about technology, improved their practice, and their students’ achievement was significantly higher compared to teachers who participated in one year of the PD or non‐participating peers. Science educators face multiple challenges as they attempt to deliver instruction in fundamentally different ways than what they experienced as learners. The delivery of this professional learning suggest that PD for science teachers should include educative learning experiences if understandings of reforms supported by research are to be realized.     

Teachers' beliefs on teacher training contents and related characteristics of implementation—the example of introducing the topic study method in mathematics classrooms

So-called “bottom-up” strategies for implementation based on mathematics teachers' own developmental activities are considered to be a powerful approach when encouraging teachers to introduce alternative instructional practices. For evaluational research of in-service teacher training programs using “bottom-up” implementation strategies, the way how teachers implement contents of the teacher training is at the centre of interest. As the teachers' active role in the implementation process is necessary, their individual beliefs on the contents of the teacher training and their expectancies might influence the teachers' implementational activities. These beliefs can be considered as components of professional knowledge and pedagogical contents knowledge (Shulman, 1986) in particular. For this reason, the study focuses on the development of beliefs on contents of a teacher training program throughout the training on the one hand and relationships with characteristics of implementation on the other hand. We consider the example of introducing a student-centred learning environment, the so-called topic study method, in the teachers' classrooms. The results indicate that there are interdependencies between beliefs on the teacher training contents and characteristics of implementation.     

Teaching and Coherent Science: An Investigation of Teachers’ Beliefs About and Practice of Teaching Science Coherently

This article is about an investigation of six middle school science teachers’ beliefs and instructional practice about the coherence of the science they teach as articulated by National Science Education Standards ( NRC, 1996 ). Many well intentioned reform efforts focus on improving content knowledge of teachers, but many classroom teachers regularly miss opportunities to provide conceptual connections within the science ideas building the sense of coherence in science. This investigation involved a quasi experimental study to examine the efficacy of a method for collecting data about middle school science teachers’ thinking about science and to determine if they teach science coherently. The teachers were surveyed, interviewed, provided concept maps about their thinking of the science they taught, and observed to investigate whether their practice reflects their beliefs. An examination of the teachers’ beliefs, stated and unstated curriculum, the connections among topics and the nature of science revealed that one, the observation tool may have merit for identifying the content and connections among science topics, and two, that teachers ‘stated beliefs consistent with the National Science Education Standards’ vision for coherent science, did not match their demonstrated practice. The content taught could be characterized in three ways; coherent content and few connections, coherent content and connections, and not coherent content. This indicates for this group of middle school science teachers that knowing how they think about science and how those beliefs are reflected in their teaching is complex. This study can inform teacher education and professional development efforts about the need to move beyond just content enhancement to examine prior beliefs about the connections of concepts within science.     

Facilitating parental engagement in school mathematics and science through inquiry-based learning: an examination of teachers’ and parents’ beliefs

This study examined teachers’ and parents’ beliefs on the implementation of inquiry-based modeling activities as a means to facilitate parental engagement in school mathematics and science. The study had three objectives: (a) to describe teachers’ beliefs about inquiry-based mathematics and science and parental engagement; (b) to describe parents’ beliefs about inquiry-based mathematics and science and their engagement in inquiry-based problem solving; and (c) to explore the impact of an inquiry-based learning environment comprising a model-eliciting activity and Twitter. The research involved three sixth-grade teachers and 32 parents from one elementary school. Teachers and parents participated in workshops, followed by the implementation of a model-eliciting activity in two classrooms. Three teachers and six parents participated in semi-structured interviews. Teachers reported positive beliefs on parental engagement in the mathematics and science classrooms and the potential positive role of parents in implementing innovative problem-solving activities. Parents expressed strong beliefs on their engagement and welcomed the inquiry-based modeling approach. Based on the results of this aspect of a four-year longitudinal design, implications for parental engagement in inquiry-based mathematics and science teaching and learning and further research are discussed.     

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.     

Enhancing elementary-school mathematics teachers' efficacy beliefs: a qualitative action research

Individuals and societies that can use mathematics effectively in this period of rapid changes will have a voice on increasing the opportunities and potentials which can shape their future. This has brought affective characteristics, such as self-efficacy, that affect mathematics achievement into focus of the research. Teacher efficacy refers to the extent to which a teacher feels capable to help students learn, influence students’ performance and commitment, and thus plays a crucial role in developing the student in all aspects. In this study, we used two sources of efficacy beliefs, mastery experiences and physiological and emotional states, in an interesting and challenging seven month workshop, as tools to foster teacher efficacy for six elementary-school teachers who were frustrated and wanted to leave their job. Our aim was to study the nature of these teachers’ efficacy in order to change it. In this qualitative action research, we used open interviews, non-participant observations and field notes. Results show that these teachers became efficacious, their students’ achievements and motivation were enhanced, and the school climate was changed. Qualitative inquiry of this construct sheds light on efficacy beliefs of mathematics teachers. Nurturing teacher efficacy has borne much fruit in the field of mathematics in school.     

Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers

This mixed-methods study describes classroom characteristics and student outcomes from university mathematics courses that are based in mathematics departments, targeted to future pre-tertiary teachers, and taught with inquiry-based learning (IBL) approaches. The study focused on three two-term sequences taught at two research universities, separately targeting elementary and secondary pre-service teachers. Classroom observation established that the courses were taught with student-centred methods that were comparable to those used in IBL courses for students in mathematics-intensive fields at the same institutions. To measure pre-service teachers' gains in mathematical knowledge for teaching, we administered the Learning Mathematics for Teaching (LMT) instrument developed by Hill, Ball and Schilling for in-service teacher professional development. Results from the LMT show that pre-service teachers made significant score gains from beginning to end of their course, while data from interviews and from surveys of learning gains show that pre-service teachers viewed their gains as relevant to their future teaching work. Measured changes on pre-/post-surveys of attitudes and beliefs were generally supportive of learning mathematics but modest in magnitude. The study is distinctive in applying the LMT to document pre-service teachers' growth in mathematical knowledge for teaching. The study also suggests IBL is an approach well suited to mathematics departments seeking to strengthen their pre-service teacher preparation offerings in ways consistent with research-based recommendations.     

Inductive and Deductive Justification of Knowledge: Flexible Judgments Underneath Stable Beliefs in Teacher Education

Personal epistemological beliefs are considered to play an important role for processes of learning and teaching. However, research on personal epistemology is confronted with theoretical issues as there is conflicting evidence regarding the structure, stability, and context-dependence of epistemological beliefs. We give evidence how theoretical and methodological issues can partly be resolved by distinguishing between relatively stable “epistemological beliefs” and situation-specific “epistemological judgments.” A qualitative content analysis of a series of semistructured interviews (study 1) with pre-service teachers, teachers, and teacher educators as well as a statistical analysis of pre-service teachers’ extensive answers in questionnaires (study 2), both on the topic of “mathematical discovery,” reveal not only beliefs of the participants but also different qualities of judgments. Therefore, in further research both aspects of beliefs should be considered in a more differentiated manner when categorizing belief structures.     

Collective work with resources: an essential dimension for teacher documentation

In this paper we study the collective dimensions of teachers’ work in their ordinary daily practice. We argue that teachers’ ordinary work comprises many collaborative aspects, and that the interactions with colleagues, often through resources, are crucial for teacher professional development. Using a theoretical framework based on teacher documentation work, we focused on interactions between teachers and resources to study two teachers’ work: one teacher in France and one teacher in Norway. For both teachers we identified collective dimensions in their work with resources. Contrasting these two cases enabled us to identify particular features, but also possible invariants, of teachers’ work with resources. Our results indicate that collective dimensions are always present in teachers’ work, but that true communities of practice, which could be considered as an achievement of collective teachers’ work, are scarce. We discuss the conditions for the emergence of such communities.     

Standards for professionalization of mathematics teachers: policy, curricula, and national teacher employment test in Korea

International comparative studies such as the Trend in International Mathematics and Science Study (TIMSS) and the OECD Programme for International Student Assessment (PISA) indicate that Korean students have consistently performed well. In addition, a recent study titled Mathematics Teaching in the 21st Century (MT21) compared prospective teachers’ knowledge and beliefs about teaching and learning in six participant countries, reporting that Korean prospective secondary mathematics teachers were better prepared than those in other countries. In this context, this study has examined the curricula for mathematics teacher education and teacher employment tests in order to investigate the social expectation for teacher professionalization in Korea, particularly focusing on teacher knowledge. The analysis shows that while elementary mathematics teacher education emphasizes pedagogical knowledge, the secondary mathematics education curricula are highly content knowledge oriented. However, the secondary mathematics teacher education includes various aspects of pedagogical content knowledge in its curricula and teacher employment test. This research also identifies the discourse concerning mathematics instruction for diversity and equity and the emphasis of reflective practice as the significant development of the current Korean teacher education.     

Using Metaphors as a Tool for Examining Preservice Elementary Teachers' Beliefs About Mathematics Teaching and Learning

This study used metaphors as a tool to gain insight about preservice teachers' conceptualizations of the role of the teacher and the learner and held the view that the examination of these metaphors might provide an opportunity for teacher educators to reflectively and critically examine those beliefs. Thus, this research examined possible differences in the reflected beliefs of elementary preservice teachers as depicted in their metaphors about mathematics teaching and learning at three different points throughout their mathematics education methods courses. The results of this study indicated that elementary preservice teachers' beliefs primarily remained static throughout their mathematics methods courses despite ongoing experiences designed to challenge and extend those beliefs.     

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.     

MATHEMATICS STUDENT TEACHERS’ RESPONSES TO INFLUENCES AND BELIEFS

This article forms part of an ongoing study of student teachers of secondary mathematics. The aspect reported on in this article is an analysis of the effects of the influences brought to bear upon four individual student teachers of secondary mathematics as they progress through a one-year postgraduate course of teacher training (PGCE) based at a British University. The students have differing initial beliefs about teaching, learning and mathematics. As anticipated in the literature, the student's initial beliefs survive virtually intact throughout the year. However, the study suggests that the link between initial beliefs and teaching approach is not deterministic. The study suggests ways of encouraging student teachers to employ a range of pupil activities in their teaching.     

United States teachers’ views of effective mathematics teaching and learning

This study investigates US teachers’ cultural beliefs concerning effective mathematics teaching using semi-structured interviews with 11 experienced teachers. For US teachers, effective teaching is student-centered. Cognitively appropriate mathematical content should be understood through many hands-on activities that allow students to explore by themselves the relationship between mathematical knowledge and their life experiences. Correspondingly, the US teachers view an effective teacher as a facilitator who is sensitive to student social and cognitive needs and is skillful at organizing collaborative learning. The result of this study helps researchers and educators understand the student-centered learning model in US classrooms.