School of Education

The University of Queensland

St Lucia Qld 4072

Paper presented at the Annual Conference of

the Australian Association for Research in Education

Brisbane, 1-5 December 2002

Paper code GOO02058

Although research on mathematics teacher education has grown rapidly in the past 10 to 20 years, there are those who argue that teacher education remains an under-theorised field of inquiry, lacking coherent conceptual frameworks that address the complexity of individuals acting in social situations (Cooney, 1994; Lerman, 2001a). For example, much of the literature on teacher socialisation identifies and analyses the beliefs that prospective teachers bring to the pre-service course - beliefs which arise largely from their experiences as students in classrooms and frequently are reinforced by their practicum observations and initial teaching experiences (Brown & Borko, 1992). Functionalist perspectives on teacher socialisation tend to view teachers as being passively moulded by external forces to fit the existing culture of schools, and hence unable to implement innovative approaches (e.g. those involving the use of educational technologies) they may have experienced during their pre-service courses (Loughran, Mitchell, Neale, & Toussaint, 2001). However, Lerman (2001a) claims that the study of teacher beliefs is too static and decontextualised to adequately describe how teachers acquire the attitudes, values, and knowledge structures of their professional culture. Instead he proposes that teachers' learning is better understood as increasing participation in sociocultural practices that develop their identities as teachers. Sociocultural theories view learning as a collective process of enculturation into the practices institutionalised by specific knowledge communities. While recent research in this theoretical tradition has investigated school students' learning in classroom communities (e.g. see Boaler & Greeno, 2000; Lerman, 2000; Lerman 2001b), the study described here develops the concept of community of practice in the context of pre-service teacher education and its interface with beginning teaching. This paper reports on the first year of a three year study investigating how pre-service and beginning teachers are initiated into a collaborative professional community featuring both web-based and face to face interaction, and how participation in such a community shapes their pedagogical beliefs and practices.

Learning in Communities

Few studies have applied sociocultural theories to teacher learning in general, and to pre-service teacher education in particular. Yet the community of practice model seems to hold promise for understanding how novices (i.e. pre-service and beginning teachers) gain access to professional knowledge through collaboration with more experienced members (i.e. teacher educators and supervising teachers), and gradually adopt the practices of the community. A community of practice is a sustained social network of individuals who share common beliefs, values, and practices in the pursuit of a mutual enterprise that is connected to the larger social system in which the community is nested. Such communities have a common cultural and historical heritage, and it is through the sharing and re-construction of this collective knowledge base that individuals come to define their identities in relationship to the community. Because communities of practice evolve over time they also have mechanisms for reproduction through which the community can maintain itself (Barab & Duffy, 2000; Lave & Wenger, 1991; Wenger, 1998).

Although communities of practice are generally constituted through face to face interaction, communication technologies such as the internet can provide an alternative medium for participation. Within mathematics education, there have been attempts to design such online communities so as to challenge prospective teachers' beliefs about learning, teaching, and the nature of mathematics (e.g. Schuck & Foley, 1999). Nevertheless, when student teachers move out of this community into practicum placements, they frequently find it difficult to sustain any shift in beliefs towards the less procedural, more investigative view of mathematics typically promoted by the university program. Enabling pre-service students to continue their participation in a technology-mediated community during the practicum may assist them to critically reflect on the dilemmas of practice and persist with innovative teaching approaches (see Brett, Woodruff & Nason, 1997).

Two additional difficulties identified in research into online communities concern the tension between designed and emergent communities, and the question of sustainability of such communities. For example, Derry, Lee, Kim and Seymour (2001) identified various forms of resistance to an online community designed for a cohort of secondary teacher education students. In imposing a structure on the operation of the community, the researchers incorrectly assumed that students and instructional staff shared their research goals. They concluded that any attempt to design a community should respect the context and follow from a careful analysis of local conditions and the needs of members. Likewise, Barab (2001) writes of the inherent tension in managing the interplay between the designed and the emergent in building online (or face to face) communities of in-service and pre-service mathematics and science teachers. He maintains that it is preferable to create a framework and then facilitate the growth of the community (i.e. adopt an emergent design) so that participants build the space, rather than imposing a design completely formulated in advance. The sustainability of a community of practice is related to the designed/emergent duality in that an emergent community is more likely to meet the needs of its members because they have played a part in its development and thus identify with its goals and values. Also, particularly in online communities, participation involves costs (investment of time, and effort in using the electronic tools) that may outweigh any benefits to members.

Our study differs from much previous research in that the community it is creating will expand longitudinally, over the three years of the project, to include not only pre-service students but also newly graduated beginning teachers and their professional mentors in schools. Thus the focus is not on a single cohort of pre-service students or the professional development of a particular group of teachers, but on inter-generational interaction emerging over time and how this shapes participants' beliefs about teaching and learning, and their own teaching practice. The aim of this paper is to analyse processes through which the emerging community was established and sustained during the first year of the study.

Technology and Teacher Education

In keeping with current developments in mathematics education, the function (or mutual enterprise) of this community of practice centres on integrating technology resources into the teaching of secondary school mathematics. One of the major themes in current debates on educational reform identifies the need for teachers to become more effective, confident, and creative users of technology in their teaching (e.g. Web-Based Education Commission, 2000). Similarly, there is growing recognition that pre-service teacher education programs must integrate technology into their own curricula to ensure beginning teachers are adequately prepared (e.g. McCoy, 1999). Within mathematics education, research studies have yielded descriptions of pre-service methods courses that help student teachers design technology intensive lessons using, for example, spreadsheets (e.g. Connell & Abramovich, 1999), multimedia (e.g. Bell, 1996; Kim, Sharp & Thompson, 1998), and the World Wide Web (e.g. Halpin & Kossegi, 1996). As graphics calculators are a relatively new technology their impact on such courses has yet received little attention, and this is a major focus of our study.

Research Design

We aim to create a learning environment in the pre-service course that is consistent with recent mathematics education research and curriculum reforms relating to learning mathematics in secondary schools (e.g. Australian Education Council 1991; Goos, Galbraith & Renshaw, 1999) in emphasising mathematical thinking and problem solving, communication and real world applications. There is growing agreement in the mathematics education community that learning to think mathematically involves not only acquiring knowledge, skills, and strategies, but also habits and dispositions of interpretation and meaning construction (Schoenfeld, 1994), that is, a mathematical point of view. Thus mathematics education is a process of enculturation into the practices of the discipline, and students' understanding of what mathematics is about is shaped by their participation in the classroom mathematical community of practice. Applying these ideas to teacher education, we involve our pre-service students in activities that immerse them in a culture of collaboration and mathematical sense-making. Our goal is to create a community of practice where interdependence can be nurtured, and shared purposes and values can be enacted. At the same time we want to challenge our pre-service students to adopt ways of working with students in their own classrooms that may differ substantially from their own experiences of learning mathematics at school.

The pre-service mathematics curriculum course is also enriched with the use of educational technologies, and students experience regular and intensive use of graphics calculators, computer software, and internet applications (see Goos, 1999). Integration of technology into mathematics education is emphasised through continuous personal access to a graphics calculator for the duration of the course (including the practicum), and assignment tasks that require preparation and presentation of technology-based classroom activities.

Quantitative and qualitative measures are combined in the four components of the study:

(i) a large scale survey of secondary mathematics teachers' experience with and attitudes towards educational uses of technology (currently in progress);

(ii) cohort studies of pre-service students' beliefs about mathematics teaching and learning, their practicum experiences in technology integration, and the ways in which their shared experiences become part of the common culture and history of the pre-service community of practice;

(iii) individual case studies of selected students to gain in-depth understanding of issues in technology integration in specific school and classroom contexts;

(iv) a community-centred professional development intervention (in the third year of the study).

This paper draws only on data from component (ii), the cohort study of 2002.

As the conceptual framework for the study is derived from sociocultural theories of learning, particularly the concept of learning in communities of practice, the focus of the cohort studies is on identifying features of the emerging professional learning community in both its face to face and online forms. A mathematics community website has been established via Yahoo Groups, with membership restricted to current students (and graduates from 2003). The website offers access to bulletin board, email, and file sharing facilities. The advantage of such a community over web-based course tools used for flexible learning in university programs lies in its continued accessibility to members after graduation, when student email accounts are cancelled.

Messages posted to the bulletin board are automatically archived and thus available for analysis of interaction patterns. Bearing in mind the lessons from previous research on designed versus emergent online communities, we decided to impose minimal structure on bulletin board communication. This involved posting four questions to which students were asked to respond during the first practicum session, on the availability and accessibility of technology resources in their practicum schools, and the frequency and mode of use they observed during lessons and assessment tasks. (This contrasts with the approach imposed by most other courses in the Bachelor of Education, which require students to make a specified number of posts to a Web CT bulletin board as part of the course assessment program.) Students were also free to use the mathematics curriculum bulletin board (or its email equivalent) for any other purposes they chose.

Since shared beliefs are a feature of a community of practice, changes in students' mathematical and pedagogical beliefs over the first year of the study have been investigated by administering a questionnaire at the beginning and end of the one year course that focused on their beliefs about the nature of mathematics, mathematics teaching, and mathematics learning. The questionnaire (based on that designed by Frid, 2000) invites both open ended and structured responses. Students are first asked to describe what mathematics is to them, and then to explain why they think mathematics is taught in schools. The remainder of the questionnaire consists of 40 statements to which students respond using a Likert-type scale based on scores from 1 (Strongly Disagree) to 5 (Strongly Agree), with a score of 3 corresponding to Undecided. The statements are underpinned by current research on mathematics teaching and learning and are representative of the values enacted in the pre-service course.

Section 1 includes 14 statements about the nature of mathematics, Section 2 has 12 statements about mathematics teaching in secondary schools, and Section 3 has 14 statements on mathematics learning in secondary schools. Within each section, statements are paired to create positive and negative poles of a particular idea; for example, items 4 and 11 represent extreme positive and negative positions with respect to solution methods for mathematics problems:

In the literature these poles have been variously referred to as representing beliefs about mathematics being rule-based versus non-rule-based (Tharp, Fitzsimons & Ayers, 1997), tramsmissive versus child-centred (Perry, Howard & Tracey, 1999), or closed versus open (Boaler, 1998).

Features of the Emerging Community of Practice

Feature of the community of practice as it took shape in the pre-service course during 2002 are analysed in terms of students' developing identities as mathematics teachers, shifts in their beliefs about mathematics and mathematics teaching and learning, and interaction patterns observed in bulletin board discussions.

How did participation in the pre-service community of practice shape the students' developing identities? This question is considered in terms of two community features, interdependence - members' sense that they are part of a larger social system, or professional network- and a reproductive cycle - the way in which newcomers engage with the practices of 'old timers' to generate overlapping histories.

A typical approach within pre-service teacher education is to invite practising teachers with expertise in a specific field to present guest workshops as part of the regular on-campus program. While this demonstrates to students that there is indeed a broader community of practising teachers beyond the university, the activities of the workshop do not necessarily allow them to make a contribution to the community of practitioners - the social system in which the pre-service community is nested. Thus, while we did occasionally call on the expertise of school-based teaching colleagues to offer workshops on graphics calculators, data logging equipment, and newly released mathematical software packages, we also explicitly encouraged students to transform materials they created in course assignments into professional development workshops and journal articles on technology-enriched mathematics learning. For example, in 2002 two pairs of pre-service students presented workshops in a technology seminar organised by the Queensland Association of Mathematics Teachers, and they will offer these workshops again at the forthcoming biennial conference of the Australian Association of Mathematics Teachers in January 2003. Thus there is potential for pre-service students' identities-in-practice to develop through legitimate participation in the greater community - where their interdependence is demonstrated through producing, rather than consuming, professional knowledge.

A community of practice reproduces itself through the enculturation of new members who appropriate the community's common heritage through interaction with more mature members. Such interaction is possible if the pre-service community maintains itself through inter-generational communication, rather than being created anew with each successive cohort. Although this is the first year of the study in the formal sense, the pre-service community has evolved over several years and members already have multiple overlapping histories. One feature of the reproductive cycle involves recent graduates now teaching mathematics in schools sending us technology-enriched materials they have developed for their own classrooms to share with current BEd pre-service students via the mathematics curriculum website. This year we have uploaded materials created by graduates of the classes of 2000 (an interactive spreadsheet for introducing the trapezoidal rule in integral calculus) and 2001 (an assignment on three-dimensional vectors that investigates the question of who shot John F. Kennedy). Face to face engagement of newcomers and 'old timers' has occurred when recent graduates were invited to give pre-practicum and end-of-year advice to the current cohort of BEd students.

While the latter interactions have to some extent been structured by us - either by our use of the website or our invitation for graduates to return - there have also been instances of spontaneous engagement between new and old community members in ways that further create a shared history. A pair of students devised an inventive graphics calculator activity, based on solving the murder of a pre-historic mammoth, which was published in a professional journal (Parente & Wethereld, 2000). The following year, a student in the new BEd cohort chose this as an exemplary article to review for a course assignment, thus legitimising the published material as part of the knowledge base of the broader professional community. Another student from this group adapted the murdered mammoth task for use with his own Year 8 students in 2002, and invited the current BEd mathematics cohort to observe and participate in a lesson in which the school students completed part of the mammoth investigation. Each of these inter-generational interactions involved more than transmission of knowledge; rather, individuals shared and re-constructed a particular experience in the process of developing their own identities as teachers.

It is widely recognised that teachers' beliefs about the nature of mathematics and how it is learned influence features of the classroom environment they create (Thompson, 1992). In particular, general pedagogical beliefs of mathematics teachers seem to be reflected in the ways they use technologies such as graphics calculators in the classroom (Simmt, 1997; Tharp, Fitzsimmons & Ayers, 1997). For example, teachers who view mathematics as more rule-based and procedural, and mathematics learning as involving memorisation and symbol manipulation, are more likely to be unconvinced that graphics calculators can enhance students' learning and to use calculators only as a checking device or graphing tool. On the other hand, teachers with more inquiry-based views about mathematics are more likely to use calculators as a means of developing students' conceptual understanding.

Pre-service students' espoused beliefs were investigated via the Mathematical Beliefs Questionnaire described previously. Here we consider only responses to structured items in Sections 1, 2 and 3. Since the item scores ranged from 1 (Strongly Disagree) to 5 (Strongly Agree), and there were 12 students who completed questionnaire both pre-course and post-course, the total score on any item could vary from 12 to 60. Table 1 displays these totals for item pairs (i.e. negative and positive versions of the same construct) in each Section. The magnitude of scores is important for indicating the degree of support for each statement. We identify scores of 48 or more, and 24 or less, as indicating general agreement and general disagreement respectively with particular statements. These statements, and corresponding pre- and post-course scores, are shown by shaded cells in Table 1. Also, we consider a shift in scores of 6 - equivalent to half the class changing their score by 1 in the same direction - as worth looking into. Statements where there was such a shift, together with pre- and post-course scores, are shown in bold type in Table 1.

Table 1(a). Pre- and post-course responses to Mathematical Beliefs Questionnaire Section 1: Nature of mathematics (n = 12)

Note. Item totals are sums of scores for the 12 students who completed the questionnaire pre-course and post-course. Strongly Agree = 5, Agree = 4, Undecided = 3, Disagree = 2, Strongly Disagree =1.

The results of this analysis suggest that this group of students entered the pre-service course with mathematical beliefs that were relatively non-rule-based, open, and student-centred, and - more importantly - that they tended to maintain or strengthen these beliefs throughout the course. For example, students generally viewed mathematics as an accessible human endeavour involving creative thinking and flexible problem solving approaches. They expressed support for teaching approaches based on cooperative work involving activities that encourage students to share their thinking and explore different ways of tackling problems. Their ideas about mathematics learning centred on the importance of building understanding through the use of real life examples, physical objects, and technology resources.

Table 1(b). Pre- and post-course responses to Mathematical Beliefs Questionnaire Section 2: Mathematics teaching in secondary schools (n = 12)

Note. Item totals are sums of scores for the 12 students who completed the questionnaire pre-course and post-course. Strongly Agree = 5, Agree = 4, Undecided = 3, Disagree = 2, Strongly Disagree =1.

In addition, there were many shifts in beliefs towards a more open and constructive view of mathematics and mathematics teaching and learning. Students moved away from viewing mathematics as something that is either right or wrong, best learned by replication of procedures demonstrated by the teacher and practice on lots of problems, where success is judged by getting the right answers. Similarly, there was a shift towards encouraging multiple interpretations and helping learners determine the validity of their methods and answers. Ideas about what constitutes good mathematics teaching moved away from teacher-centred lessons towards discussion based lessons to facilitate the sharing and negotiation of meanings. Students also became more aware of culturally diverse ways of knowing and doing mathematics.

While these results illustrate one feature of an emergent community of practice - an increasing commitment to shared beliefs about mathematics teaching and learning (i.e. those embedded in the pre-service course), some caution is needed in interpreting the questionnaire responses. Lerman (2001a) points out that beliefs are highly contextualised, so it is important also to analyse relationships between espoused beliefs and the constraints and affordances of the practice teaching environments experienced by the pre-service students, to examine how their identities as teachers developed as involvement in practice increased. Such an analysis is planned for the individual case studies conducted in 2002 (see Goos, 2002, for a similar analysis in an earlier study).

Table 1(c). Pre- and post-course responses to Mathematical Beliefs Questionnaire Section 3: Mathematics learning in secondary schools (n = 12)

Note. Item totals are sums of scores for the 12 students who completed the questionnaire pre-course and post-course. Strongly Agree = 5, Agree = 4, Undecided = 3, Disagree = 2, Strongly Disagree =1.

Table 2. Distribution of bulletin board messages

Students posted 79 of the messages, with the remainder being posted by the two teaching staff (43 and 26 respectively). Ten students posted 1 to 5 messages, five students posted 6 to 10 messages, and one student posted more than 10 messages. Interaction was usually initiated by course lecturers rather than students.

The distribution in Table 2 shows three months when interaction peaked: in April and May, which coincided with the first practicum, and in July, when students were preparing for an assignment that involved a seminar presentation. A content analysis of messages revealed three main themes during these times. Interaction in April consisted mainly of questions from students about practicum requirements and school placements. In May, discussion was triggered by the lecturers posting two sets of questions asking students about technology usage in their practicum schools (on 29 April and 6 May respectively). Virtually all students simply supplied the information requested, with no real interaction between students themselves. Messages in July largely arose from the need to organise presentations of the students' technology seminars at a time when classes were not in session. Students notified lecturers of their seminar partners, inquired about the availability of a data projector and link cables for transferring information between computers and graphics calculators, and negotiated the timetable for their presentations. Eleven of these messages were posted by lecturers and nine by students.

Thus the bulletin board was used mainly for administrative purposes, or when lecturers posed specific questions to which students had to respond. Most interactions consisted of only two elements, an initiation and a response. Clearly, students did not see a need to interact online when the option of face to face communication was also available. In addition, we noticed some resistance to bulletin board communication in general arising from students' dislike of another BEd course in which they were obliged to post a minimum number of messages to satisfy course assessment requirements. That the content of these messages was also assessed was another source of dissatisfaction amongst students. Students may not experience any real benefits from using our mathematics course bulletin board and website until they graduate and begin teaching, when access to resources, ideas, and advice becomes crucial.

In this paper we have reported on the early stages of a three year project investigating the impact of a community-based, technology-enriched pre-service teacher education program on beginning teachers' integration of educational technologies into secondary school mathematics classrooms. The study extends previous research that has applied sociocultural theories of learning to developing models of pedagogy for secondary school mathematics, by applying the notion of a community of practice to teacher education settings. Our aim here has been to analyse processes through which the emerging pre-service community of practice was established during the first year of the study. In particular, we considered how participation in this community shaped pre-service students' pedagogical beliefs and their developing identities as teachers.

A community of practice has a history based on shared experiences, beliefs, and purposes, so that members develop a sense of interdependence within the community as well as with the broader society of which the community is part. We noted that interdependence and mechanisms for sharing the pre-service community's history and practices was sometimes structured by the designers (i.e. course lecturers/researchers), but could also take shape spontaneously when the work of recent graduates was integrated into the community's fund of resources. Integral to this community were pedagogical and disciplinary beliefs that we endeavoured to embed within the pre-service course. We found that the students came to share many of these beliefs and often strengthened their espoused commitment to open, investigative, non-rule-based views of mathematics and mathematics teaching and learning. Such beliefs seem to be vital to productive use of technology in supporting learner's conceptual understanding of mathematics; however, the ways in which these beliefs interact with classroom environments in shaping actual teaching practice are yet to be investigated for this group of pre-service students.

A key question that remains concerns the tension between design and emergence in establishing communities of practice, and conditions that favour sustainability of the community. While we created a framework that attempted to facilitate growth (e.g. in engaging students in activities that exemplified certain pedagogical values and beliefs), the lack of structure imposed on bulletin board interaction meant that students made little use of this mediating resource. Clearly, when there was a choice, students decided that the costs involved in using digital tools outweighed any benefits. The issue of sustainability will come to the fore in 2003 and 2004, when students who have graduated from the course are beginning teaching and coming to terms with environments that may not adequately support technology integration. As no community can truly be 'designed', we anticipate that this project will provide rich data on the emergence and sustainability (or otherwise) of communities of practice situated at the interface of pre-service teacher education and the initial professional experience of secondary school mathematics teachers.