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).     

A Framework for Analysis of Teachers’ Geometric Content Knowledge and Geometric Knowledge for Teaching

Current reform-driven mathematics documents stress the need for teachers to provide learning environments in which students will be challenged to engage with mathematics concepts and extend their understandings in meaningful ways (e.g., National Council of Teachers of Mathematics, 2000, Curriculum and evaluation standards for school mathematics. Reston, VA: The Council). The type of rich learning contexts that are envisaged by such reforms are predicated on a number of factors, not the least of which is the quality of teachers’ experience and knowledge in the domain of mathematics. Although the study of teacher knowledge has received considerable attention, there is less information about the teachers’ content knowledge that impacts on classroom practice. Ball (2000, Journal of Teacher Education, 51(3), 241–247) suggested that teachers’ need to ‘deconstruct’ their content knowledge into more visible forms that would help children make connections with their previous understandings and experiences. The documenting of teachers’ content knowledge for teaching has received little attention in debates about teacher knowledge. In particular, there is limited information about how we might go about systematically characterising the key dimensions of quality of teachers’ mathematics knowledge for teaching and connections among these dimensions. In this paper we describe a framework for describing and analysing the quality of teachers’ content knowledge for teaching in one area within the domain of geometry. An example of use of this framework is then developed for the case of two teachers’ knowledge of the concept ‘square’.     

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.     

Prospective primary mathematics teachers’ learning from on-line discussions in a virtual video-based environment

The aim of this study was to investigate how participation and reification of ideas about mathematics teaching are constituted in on-line discussions when prospective primary mathematics teachers analysed video-cases about mathematics teaching. Prospective teachers enrolled in a mathematics methodology course participated for 4 weeks in two virtual learning environments that integrated the analysis of video-clips, on-line discussions and writing essays about key aspects of mathematics teaching. Three aspects were considered relevant to explain the prospective teachers’ learning: the way in which the theoretical information was used to frame and to interpret the events from mathematics teaching; the characteristics of engagement with others participating in the on-line discussions and the role played by prospective teachers’ beliefs. Possible reasons for the importance of these features include the specific questions posed in on-line discussions and the use of video-clips of mathematics teaching. These findings are considered useful in designing virtual learning environments and the kinds of tasks through which the understanding of mathematics teaching and learning-to-notice skills can be developed.     

The relationship among elementary teachers’ content knowledge,attitudes, beliefs,and practices

This study investigated 481 in-service elementary teachers’ level of mathematical content knowledge, attitudes toward mathematics, beliefs about the effectiveness of inquiry-based instruction, use of inquiry-based instruction and modeled the relationship among these variables. Upper elementary teachers (grades 3–5) were found to have greater content knowledge and more positive attitudes toward mathematics than primary teachers (grades K-2). There was no difference in teachers’ beliefs about effective instruction, but primary level teachers were found to use inquiry-based instruction more frequently than upper elementary teachers. Consistent with Ernest’s [Ernest (1989). The knowledge, beliefs and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15(1), 13–33] model of mathematics teaching, content knowledge, attitudes, and beliefs were all found to be related to teachers’ instructional practice. Furthermore, beliefs were found to partially mediate the effects of content knowledge and attitudes on instructional practice. Content knowledge was found to be negatively related to beliefs in the effectiveness of inquiry-based instruction and teachers’ use of inquiry-based instruction in their classrooms. However, overall, teachers with more positive attitudes toward mathematics were more likely to believe in the effectiveness of inquiry-based instruction and use it more frequently in their classroom. Teacher beliefs were found to have the strongest effect on teachers’ practice. Implications for the goals and objectives of elementary mathematics methods courses and professional development are discussed.

Impact of a content and methods course sequence on prospective secondary mathematics teachers’ beliefs

In this article, we report on a study of beliefs about mathematics, teaching, and proof conducted with six prospective secondary mathematics teachers as they completed a two-semester sequence of a content course and a methods course. The initial beliefs of the participants were identified using interview and survey data, and potential shifts in beliefs were examined through further interview and survey data combined with classroom observations and written work. While their beliefs about mathematics and proof appeared to be relatively stable, their beliefs about teaching shifted from a more teacher-centered view to beliefs that foreground the activities and understandings of the students. These shifts are analyzed using the construct of belief structures, and activities and events from the courses that may have facilitated the shifts are identified. The results are consistent with the literature in some respects, such as the stability of the participants’ beliefs about mathematics. On the other hand, our results present new information about how prospective secondary mathematics teachers’ beliefs about teaching may be impacted.     

Using Self-Efficacy as a Construct for Evaluating Science and Mathematics Methods Courses

The focus of this study was elementary preservice teachers’ development as effective teachers of science and mathematics as influenced by their participation in elementary science and mathematics methods courses. Preservice teachers’ reports of factors that influenced their perception of their teaching abilities were analyzed according to Bandura’s (1994) 4 sources of efficacy: mastery experiences, vicarious experiences, social persuasion, and stress reduction. This investigation allowed the researchers to evaluate the courses based on these sources. The analysis indicated all 4 sources influenced preservice teachers’ teaching self-efficacy beliefs, with mastery experiences considered the most influential. Embedded within discussions of mastery experiences were references to the other sources of efficacy, which suggest an interrelationship between mastery experiences and the other sources.     

Do Pre-service Chemistry Teachers Reflect their Beliefs about Constructivism in their Teaching Practices?

This study aimed to explore pre-service chemistry teachers’ beliefs about constructivism and the influence of their beliefs in their teaching practice. For this purpose, semi-structured interviews were carried out with eight pre-service teachers in order to understand their belief structures. Pre-service teachers’ beliefs about constructivism were classified in three categories which are weak, moderate, and strong conceptions of constructivism. For detailed exploration, three cases of pre-service teachers representing these three categories were selected. The findings of this study showed that most pre-service teachers in this study did not have a strong conception of constructivism and the relationship between the pre-service teachers’ beliefs and their practice was not clear-cut.     

Teachers’ conceptions of representation in middle school mathematics

Current reform efforts call for an emphasis on the use of representation in the mathematics classroom across levels and topics. The aim of the study was to examine teachers’ conceptions of representation as a process in doing mathematics, and their perspectives on the role of representations in the teaching and learning of mathematics at the middle-school level. Interviews with middle school mathematics teachers suggest that teachers use representations in varied ways in their own mathematical work and have developed working definitions of the term primarily as a product in problem solving. However, teachers’ conception of representation as a process and a mathematical practice appears to be less developed, and, as a result, representations may have a peripheral role in their instruction as well. Further, the data suggested that representation is viewed as a topic of study rather than as a general process, and as a goal for the learning of only a minority of the students—the high-performing ones. Implications for mathematics teacher education, prospective and practicing, are discussed.     

Taiwanese primary school teachers’ perceived enablers for and barriers to the integration of children’s literature in mathematics teaching and learning

Educational Studies in Mathematics - This study is part of the international survey studies on teachers’ beliefs concerning the integration of children’s literature in mathematics...     

Perceptions that may affect teachers’ intention to use technology in secondary mathematics classes

Technology is available and accessible in many mathematics classrooms. Adopting technology to support teaching and learning requires teachers to change their teaching practices. This paper reports the responses of a diverse cohort of 92 secondary mathematics teachers who chose to respond to an Australian state-wide survey (Mathematics with Technology Perceptions Survey) developed using a Theory of Planned Behaviour framework. The items discussed in this paper targeted mathematics teachers’ perceptions of possible barriers and enablers to their intention to use technology in their teaching. The responses are varied but, overall, strength of agreement with enablers outweighed agreement with perceived barriers. However, it is clear that despite an overall positive attitude towards the use of technology for teaching mathematics, some perceived barriers to change are notable. It is, therefore, helpful if those responsible for professional development, promoting the use of technology, recognise and address these barriers as well as working to strengthening enablers.     

Elementary Teachers’ Beliefs About Teaching Science and Classroom Practice: An Examination of Pre/Post NCLB Testing in Science

The impact of No Child Left Behind (NCLB) mandated state science assessment on elementary teachers’ beliefs about teaching science and their classroom practice is relatively unknown. For many years, the teaching of science has been minimized in elementary schools in favor of more emphasis on reading and mathematics. This study examines the dynamics of bringing science to the forefront of assessment in elementary schools and the resulting teacher belief and instructional shifts that take place in response to NCLB. Results indicated that teachers’ beliefs about teaching science remained unchanged despite policy changes mandated in NCLB. Teacher beliefs related to their perceptions of what their administrators and peer groups’ think they should be doing influenced their practice the most. Most teachers reported positive feelings and attitudes about science and reported that their students had positive feelings and attitudes about science; however, teachers reported teaching science less as a result of NCLB. Implications for elementary science education reform and policy are discussed.     

Teachers’ beliefs and intentions concerning teaching in higher education

A questionnaire measuring nine different aspects of teachers’ beliefs and intentions concerning teaching in higher education was distributed to teachers at four institutions in the United Kingdom, yielding 638 complete sets of responses. There was a high degree of overlap between the participants’ scores on the subscales measuring beliefs and intentions, and analyses of both sets of scores yielded two factors reflecting an orientation towards learning facilitation and an orientation towards knowledge transmission. However, teachers’ intentions were more orientated towards knowledge transmission than were their beliefs, and problem solving was associated with beliefs based on learning facilitation but with intentions based on knowledge transmission. Differences in teachers’ intentions across different disciplines and between men and women seemed to result from different conceptions of teaching, whereas differences in teachers’ intentions across different institutions and between teachers with different levels of teaching experience seemed to result from contextual factors. Teaching intentions thus reflect a compromise between teachers’ conceptions of teaching and their academic and social contexts     

A knowledge filter model for observing and facilitating change in teachers’ beliefs

This study describes a theoretical knowledge filter model that explains how teacher knowledge (beliefs and practices) shaped the implementation of a science curriculum in Australia. Over four school terms, four elementary and three secondary teachers participated in the study. Through the methodology of educational criticism (Eisner, 1991, The enlightened eye. New York: Macmillan) the results revealed that, in relation to their teaching, the teachers possessed three sets of beliefs: teachers’ expressed beliefs, teachers’ entrenched beliefs and teachers’ manifested beliefs. The outcome of this study is a model for identifying and observing the impact of teachers’ beliefs that can be utilised in facilitating change in education and in influencing communities of practice.     

Are We Having Fun Yet? How Teachers Use Manipulatives to Teach Mathematics

Teachers often comment that using manipulatives to teach mathematics is ‘fun!’ Embedded in the word ‘fun’ are important notions about how and why teachers use manipulatives in the teaching of mathematics. Over the course of one academic year, this study examined 10 middle grades teachers’ uses of manipulatives for teaching mathematics using interviews and observations to explore how and why the teachers used the manipulatives as they did. An examination of the participants’ statements and behaviors indicated that using manipulatives was little more than a diversion in classrooms where teachers were not able to represent mathematics concepts themselves. The teachers communicated that the manipulatives were fun, but not necessary, for teaching and learning mathematics. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mathematics teachers’ beliefs about a new reform in high school geometry in Iran

The geometry textbooks of the new system of secondary education in Iran differed dramatically from the old ones considering the aims, the visions, the content, the approach, and the educational purposes. Four hundred eighty mathematics teachers participated in a nationwide professional development program conducted by the author to facilitate the implementation of the new changes. Based on this program, a research study was designed to investigate the teachers’ beliefs about a new reform in high school geometry as a result of the professional development program. The data of the study were collected through teachers’ reflective writings, project works, group discussions, reflective questionnaires, and oral communications. The purpose of this paper is to draw upon this study and to document mathematics teachers’ beliefs about new changes.     

Connecting with the horizon: developing teachers’ appreciation of mathematical structure

A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’ reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support their students’ present and future learning of mathematics. Further research is needed about professional learning approaches and tasks that may enable teachers to imbed and develop awareness of structure in their practice.     

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.