A Mathematics Teachers’ Perspective and its Relationship to Practice

In this paper we try to characterize the pedagogical approaches that mathematics teachers are developing to meet the challenges posed by education reforms. A key aspect is the identification of the perspectives that underlie those pedagogical approaches, using the term perspective to include a broad pedagogical structure composed of multiple conceptions that are related to some aspects of a teacher’s practice. Through the study of the practice of a secondary mathematics teacher, we try to explore how his/her pedagogical approaches on mathematics, mathematics learning, and mathematics teaching are related to the relational architecture that is established in the classroom during the development of an instructional unit of similarity at a secondary school level, and we examine if that relationship can be explained in terms of the underlying perspective. The results of the study have shown the characteristics of that relationship, and the important role that the teacher’s knowledge of the students’ difficulties plays both in making decisions and in developing the teachers’ actions.     

Viewing Mathematics Teachers’ Beliefs as Sensible Systems*

This article discusses theoretical assumptions either explicitly stated or implied in research on teachers’ beliefs. Such research often assumes teachers can easily articulate their beliefs and that there is a one-to-one correspondence between what teachers state and what researchers think those statements mean. Research conducted under this paradigm often reports inconsistencies between teachers’ beliefs and their actions. This article describes an alternative framework for conceptualizing teachers’ beliefs that views teachers as inherently sensible rather than inconsistent beings. Instead of viewing teachers’ beliefs as inconsistent, teachers’ abilities to articulate their beliefs as well as researchers’ interpretations of those beliefs are seen as problematic. Implications of such a view for research on teacher beliefs as well as for the practice of mathematics teacher education are discussed. This article is based on the author’s doctoral dissertation completed at the University of Georgia under the direction of Thomas J. Cooney. Parts of this article were presented at the 2004 meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Toronto, Canada).     

The meaning of mathematics instruction in multilingual classrooms: analyzing the importance of responsibility for learning

In the multilingual mathematics classroom, the assignment for teachers to scaffold students by means of instruction and guidance in order to facilitate language progress and learning for all is often emphasized. In Sweden, where mathematics education is characterized by a low level of teacher responsibility for students’ performance, this responsibility is in part passed on to students. However, research investigating the complexity of relations between mathematics teaching and learning in multilingual classrooms, as well as effect studies of mathematics teaching, often take the existence of teachers’ responsibility for offering specific content activities for granted. This study investigates the relations between different aspects of responsibility in mathematics teaching and students’ performance in the multilingual mathematics classroom. The relationship between different group compositions and how the responsibility is expressed is also investigated. Multilevel structural equation models using TIMSS 2003 data identified a substantial positive influence on mathematics achievement of teachers taking responsibility for students’ learning processes by organizing and offering a learning environment where the teacher actively and openly supports the students in their mathematics learning, and where the students also are active and learn mathematics themselves. A correlation was also revealed between group composition, in terms of students’ social and linguistic background, and how mathematics teaching was performed. This relationship indicates pedagogical segregation in Swedish mathematics education by teachers taking less responsibility for students’ learning processes in classes with a high proportion of students born abroad or a high proportion of students with low socio-economic status.     

Development and validation of a new measure of teacher perceptions of science and mathematics learning environments

This article describes the development and validation of the Inventory of Teaching and Learning (ITAL) as a new measure of teacher perceptions of science and mathematics learning environments. The ITAL was initially developed and administered in 2004 and has subsequently been revised annually. The ITAL is administered using a confidential web-based procedure to large samples of K–12 teachers in Georgia as part of an external evaluation of an NSF-funded effort to improve K–16 science and mathematics education. Results of extensive Principal Components Analyses of the 2006 and 2007 ITAL databases are reported to support the replicated measurement structure of this new measure. Alpha reliabilities for three ITAL dimensions of teaching and learning for the 2007 ITAL administration with a sample of 3,012 teachers were: 0.95 for Inquiry-Based, 0.90 for Standards-Based and 0.85 for Traditional. Implications of the research findings for future research and evaluation studies involving teacher perceptions of science and mathematics and other learning environments are described.     

Mathematics, computers, and young children as a research-oriented learning environment for a teacher candidate

The advent of computer technology in the classroom raised the issue of its appropriate use by teachers and their students alike. It has been recommended that teacher education programs provide more opportunities for teacher candidates’ use of technology including teaching their own technology-enhanced lessons. With a goal of integrating scholarship into student teaching, a teacher candidate enrolled in a graduate program in childhood education carried out a technology-enhanced research project within a professional development school. Examining the impact of the project on the teacher candidate, this article describes how one’s pedagogical content knowledge and technological competence can be developed through a research-oriented teaching experience. The article also demonstrates the emergence of a community of practice that shares the goal of providing learning spaces for the teacher candidate and young children in the context of mathematics enrichment with computers.     

Enhancing part-time in-service teacher training programmes to facilitate rural teacher development in China

Despite the frequent criticisms directed toward them, in-service teacher training (INSET) programmes remain one of the most commonly used approaches for enhancing teachers’ professional development. This article reports on a study that examined the effects of a part-time masters of education INSET programme based in Central China. The main source of data was reflective papers written by 38 teacher participants; classroom observations and informal interviews also helped reveal the teachers’ perspectives on the programme. Whilst the programme was very popular, and had a variety of positive impacts on the teachers’ professional development, the programme paradigm involved several drawbacks. The results suggest that part-time INSET programmes for rural teachers should be supplemented by various other forms of teacher development activities to meet the dual goals of developing such programmes and providing professional development for teachers.     

Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices

This collective case study reports on an investigation into the relationship between mathematics teachers’ beliefs and their classroom practices, namely, how they organized their classroom activities, interacted with their students, and assessed their students’ learning. Additionally, the study examined the pervasiveness of their beliefs in the face of efforts to incorporate reform-oriented classroom materials and instructional strategies. The participants were five high school teachers of ninth-grade algebra at different stages in their teaching career. The qualitative analysis of the data revealed that in general beliefs were very influential on the teachers’ daily pedagogical decisions and that their beliefs about the nature of mathematics served as a primary source of their beliefs about pedagogy and student learning. Findings from the analysis concur with previous studies in this area that reveal a clear relationship between these constructs. In addition, the results provide useful insights for the mathematics education community as it shows the diversity among the inservice teachers’ beliefs (presented as hypothesized belief models), the role and influence of beliefs about the nature of mathematics on the belief structure and how the teachers designed their instructional practices to reflect these beliefs. The article concludes with a discussion of implications of teacher education.     

Promoting equity in mathematics teacher preparation: a framework for advancing teacher learning of children’s multiple mathematics knowledge bases

Research repeatedly documents that teachers are underprepared to teach mathematics effectively in diverse classrooms. A critical aspect of learning to be an effective mathematics teacher for diverse learners is developing knowledge, dispositions, and practices that support building on children’s mathematical thinking, as well as their cultural, linguistic, and community-based knowledge. This article presents a conjectured learning trajectory for prospective teachers’ (PSTs’) development related to integrating children’s multiple mathematical knowledge bases (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge), in mathematics instruction. Data were collected from 200 PSTs enrolled in mathematics methods courses at six United States universities. Data sources included beginning and end-of-semester surveys, interviews, and PSTs’ written work. Our conjectured learning trajectory can serve as a tool for mathematics teacher educators and researchers as they focus on PSTs’ development of equitable mathematics instruction.     

Teacher education through the history of mathematics

In this paper I consider the problem of designing strategies for teacher education programs that may promote an aware style of teaching. Among the various elements to be considered I focus on the need to address prospective teachers’ belief that they must reproduce the style of mathematics teaching seen in their school days. Towards this aim, I argue that the prospective teachers need a context allowing them to look at the topics they will teach in a different manner. This context may be provided by the history of mathematics. In this paper I shall discuss how history affected the construction of teaching sequences on algebra during the activities of the ‘laboratory of mathematics education’ carried out in a 2 year education program for prospective teachers. The conditions of the experiment, notably the fact that our prospective teachers had not had specific preparation in the history of mathematics, allow us to outline opportunities and caveats of the use of history in teacher education.     

Contributions from the Philosophy of Science to the Education of Science Teachers

One of the most important topics on the international agenda in educational research is to gain an understanding of the processes of educational change in teachers and of the factors that favour or hinder it. Such understanding is, for instance, an essential element in planning and putting into practice initial and ongoing teacher education programs. This article reviews the research on science teachers’ educational change. To organize the information, an analogy is made with the process of scientific change, analyzing and evaluating the contributions of the different models taken from the philosophy of science – positivism, Popper’s principle of falsifiability, Lakatos’ scientific research programs, Laudan’s research traditions, Toulmin’s evolutionism, and Kuhn’s relativism. We conclude the article with the implications for science teacher education.     

Challenges of critical colleagueship: examining and reflecting on mathematics teacher study group interactions

This article examines mathematics teacher collegiality by focusing on both the ways in which teachers interacted as critical colleagues in a long-term professional development project and the evolving role of the teacher–educator–researcher as the facilitator of this project. The professional development collaboration comprised two phases: one focused on reading classroom discourse literature and one focused on supporting each other through cycles of action research related to mathematics classroom discourse. Lord’s (1994) critical colleagueship framework is used to examine how a study group of middle-grades (ages 11–16) mathematics teacher–researchers took (or did not take) a more critical stance toward their own teaching practice and that of their colleagues. We found that challenging interactions were related to instances in which the teachers interacted as critical colleagues and were marked by particular features including the use of particular words and the use of personal experience as a form of evidence. We present the ways in which we came to understand what it might look like to scrutinize one’s practice and findings related to the development of this type of collegiality across the two different phases of this project. We end with a section in which the teacher–educator–researcher who facilitated the professional development project reflects on the ways in which the analysis caused her to reconsider both the nature of argumentation in mathematics study group settings and what implications this has with respect to her own practice as a facilitator.     

Teachers’ innovative change within countrywide reform: a case study in Rwanda

This article presents practical perspectives on mathematics teacher change through results of collaborative research with two mathematics secondary school teachers in order to improve the teaching and learning of mathematics in Rwanda. The 2006 national mathematics curriculum reform stresses pedagogies that enhance problem-solving, critical thinking and argumentation. Teachers need to use new teaching strategies. This article is a case study looking at issues around developing teachers’ use of interactions in mathematics classrooms independently of the national programme. Outputs of the study include teachers’ awareness of the need for change and their increased flexibility to accept learners’ autonomy in shifting from teacher-centred to learner-centred pedagogy. Geometer’s Sketchpad challenged teachers’ practice and then provoked reflection to improve student learning.     

Expanding the instructional triangle: conceptualizing mathematics teacher development

As mathematics educators think about teaching that promotes students’ opportunities to learn, attention must be given to the conceptualization of the professional development of teachers and those who teach teachers. In this article, we generalize and expand the instructional triangle to consider different interactions in a variety of teacher development contexts. We have done so by addressing issues of language for models of teachers’ professional development at different levels and by providing examples of situations in which these models can be applied. Through the expansion of our understanding and use of the instructional triangle we can further develop the concept of mathematics teacher development.     

Explanations of attitudes to change: Colombian mathematics teachers’ conceptions of the crucial determinants of their teaching practices of beginning algebra

This article arises from a study whose overall purpose was to investigate the relationship between Colombian mathematics teachers’ conceptions of beginning algebra and their conceptions of their own teaching practices. The teachers’ understandings of their teaching practices were explored with a view to unravelling their conceptions of change in their teaching. Focusing on the perspectives of teachers afforded opportunities that exposed the powerful role that the teachers’ conceptions of social/institutional factors of teaching played in their conceptions of their practices. The degree to which the teachers attributed these (external) factors as crucial reasons for what they do in their teaching was the basis of a categorisation of their conceptions of the crucial determinants of their teaching practices into three types. The findings are particularly relevant to our understanding of the stability of mathematics teaching approaches in the Colombian context but have likely implications for a range of international education contexts. Specific implications for the development of the research into teachers’ conceptions of mathematics and its teaching, and for teacher education programmes are presented.

Where's the Math? Prospective Teachers Visit the Workplace

The integration of academic and vocational subject matter is offered in response to efforts to make the study of mathematics meaningful and engaging for all students,as well as aid in the preparation of a mathematically literate workforce. Yet,teachers often come to mathematics education with more ‘pure’ than ‘applied’ backgrounds making it difficult for them to draw upon their own experiences to make subject matter meaningful. This paper analyses prospective teachers' opportunities to connect subject matter with workplace contexts. It examines the degree of importance prospective teachers place on workplace connections and the ways in which they incorporate these connections in classroom lesson plans. Results suggest that given opportunities to visit workplace sites, it is not a trivial task for prospective teachers to: 1) make the mathematics in work explicit, and 2) keep the mathematics contextualized when designing activities and problems for students. These results have implications for teacher education and the support prospective teachers require in building networks connecting mathematics, pedagogy,and work. This revised version was published online in July 2006 with corrections to the Cover Date.     

A review of ICT related courses in pre-service teacher education programs

This study investigated teacher educators’, prospective teachers’, and K-12 teachers’ opinions about the effectiveness of ICT related courses and the ways to improve the courses in Turkey’s pre-service teacher education programs. The researchers used both quantitative and qualitative research approaches within the data collection and analysis processes. The data were collected from 111 teacher educators, 1,330 prospective teachers, and 1,429 K-12 teachers through questionnaires; and from 6 teacher educators, 6 prospective teachers, and 6 K-12 teachers through interviews. Even though a majority of the participants felt that ICT related courses are effective, most participants recommended that these courses need to be redesigned to be more beneficial in practice.     

Qualified Pedagogical Reflection as a Way to Improve Mathematics Education

All changes of the character of education and improvement of its quality depend, in our view, particularly on the teacher. To be able to do their work efficiently, teachers have to be equipped with professional skills and dispositions, with “teachers’ competence”. For their professional growth, we consider indispensable competence in reflection, which is aimed at the teacher’s activities with regard to aspects of pedagogy and/or subject-didactics. From the point of view of the goals and content of their teaching and the methods of work (and their realisation), the development of teachers’ conscious self-reflection on their own teaching and systematic pursuance of joint reflection with other teachers and/or researchers can promote the teacher’s professional growth. In this paper, we show how a selected group of elementary school teachers reflected on their competence in the course of preparing, carrying out and analysing several instruction experiments. The paper is based on samples of teachers’ reflections and their gradual development from mere simple conversations based on intuitive perceptions, through searching for effective teaching approaches, to the deep assessment of mathematics teaching from the point of view of topics and their didactic elaboration and to suggestions for the teachers’ own experiments. May the teacher know what he teaches. May the teacher know how to teach. May the teacher be sincerely interested in teaching. Comenius     

Reflective Teachers and Teacher Educators in the Pacific Region: Conversations with us not about us

This article reports on a study of Pacific primary school teachers’ and university lecturers’ reflections on their involvement in the in-service Bachelor of Education degree programme offered at the regional University of the South Pacific (USP) in Fiji. Two rich sets of data have emerged from this study. Firstly, there are a number of critical reflections by ourselves as teacher educators concerning levels of equitable student access and participation in our degree as it is reconceptualised for distance and flexible delivery to increase levels of teacher professionalism across the Pacific region. Secondly, there has emerged a set of statements from teachers themselves about: teaching and learning; professional development opportunities; and what it means to be a professional educator in the Pacific region. This later data suggests an alternative set of voices in what has largely been a “conversation between us about them” conducted by Ministries of Education, Curriculum Development Units, USP, other educational bodies and the media in the Pacific, but particularly Fiji, about teachers and teachers’ work. Critical reflection upon our own practice as teacher educators and the voices of experienced teachers are particularly pertinent not only as we seek to reshape a degree programme to suit the needs of the region’s primary school teachers but also as “rethinking” debates about the purposes of education in the Pacific region are on-going yet exclusive.