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An investigation of teachers' beliefs of students' algebra development
 Cognition and Instruction
, 2000
"... Elementary, middle, and high school mathematics teachers (N = 105) ranked a set of mathematics problems based on expectations of their relative problemsolving difficulty. Teachers also rated their levels of agreement to a variety of reformbased statements on teaching and learning mathematics. Anal ..."
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Elementary, middle, and high school mathematics teachers (N = 105) ranked a set of mathematics problems based on expectations of their relative problemsolving difficulty. Teachers also rated their levels of agreement to a variety of reformbased statements on teaching and learning mathematics. Analyses suggest that teachers hold a symbolprecedence view of student mathematical development, wherein arithmetic reasoning strictly precedes algebraic reasoning, and symbolic problemsolving develops prior to verbal reasoning. High school teachers were most likely to hold the symbolprecedence view and made the poorest predictions of students ’ performances, whereas middle school teachers ’ predictions were most accurate. The discord between teachers ’ reformbased beliefs and their instructional decisions appears to be influenced by textbook organization, which institutionalizes the symbolprecedence view. Because of their extensive content training, high school teachers may be particularly susceptible to an expert blindspot, whereby they overestimate the accessibility of symbolbased representations and procedures for students ’ learning introductory algebra. The study of people engaged in cognitively demanding tasks must consider the relation between people’s judgments and actions and the beliefs they hold. Several aspects of people’s decision making are well established. People do not strictly follow the laws of logic and probability when weighing information or following im
Calculational and conceptual orientations in teaching mathematics
 In D. B. Aichele & A. F. Coxford (Eds.), Professional Development for Teachers of Mathematics: 1994 Yearbook. Reston, VA: National Council of Teachers of Mathematics
, 1994
"... and do not necessarily reflect official positions of NSF or Apple Computer. ..."
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and do not necessarily reflect official positions of NSF or Apple Computer.
The beginning science teacher: Classroom narratives of convictions and constraints
 Journal of Research in Science Teaching
, 1992
"... This article is a case study of a secondyear middle school science teacher’s beliefs about science and science teaching and how these beliefs influencedor failed to influenceclassroom instruction. It illustrates how beginning teachers struggle to reconcile (a) conflicting beliefs about what is de ..."
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This article is a case study of a secondyear middle school science teacher’s beliefs about science and science teaching and how these beliefs influencedor failed to influenceclassroom instruction. It illustrates how beginning teachers struggle to reconcile (a) conflicting beliefs about what is desirable, and (b) conflicts between what they believe is desirable and what is possible within the constraints of their preparation and the institutions in which they work. This teacher, for example, struggled to reconcile his view of science as a creative endeavor with his belief that students need to be provided with a high degree of structure in order to learn within the context of formal schooling. He also had difficulty resolving the conflict between the informal (“messing about”) type of science learning that he believed was desirable and the personal and institutional constraints he faced in the classroom. The problem of teacher retention is a national concern. The first few years of teaching are particularly difficult (Marso & Pigge, 1987; Veenman, 1984); some districts report losing 40 % of their beginning teachers within the first two years (Wise, DarlingHammond, & Berry, 1987). Case studies of firstyear teachers have provided some understanding of how particular classroom events, institutional characteristics, and the personal qualities of
The Symbol Precedence View of Mathematical Development: A Corpus Analysis of the Rhetorical Structure of Textbooks
, 2000
"... This study examined a corpus of 10 widely used prealgebra and algebra textbooks, with the goal of investigating whether they exhibited a symbol precedence view of mathematical development as is found among high school teachers. The textbook analysis focused on the sequence in which problemsolvi ..."
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This study examined a corpus of 10 widely used prealgebra and algebra textbooks, with the goal of investigating whether they exhibited a symbol precedence view of mathematical development as is found among high school teachers. The textbook analysis focused on the sequence in which problemsolving activities were presented to students. As predicted, textbooks showed the symbol precedence view, presenting symbolic problems prior to verbal problems.
Issues of Methods and Theory in the Study of Mathematics Teachers' Professed and Attributed Beliefs
 Educational Studies in Mathematics
, 2005
"... ABSTRACT. In research on teachers ’ beliefs, a distinction is often made between what teachers state (“professed beliefs”) and what is reflected in teachers ’ practices (“attributed beliefs”). Researchers claim to have found both consistencies and inconsistencies between professed and attributed bel ..."
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ABSTRACT. In research on teachers ’ beliefs, a distinction is often made between what teachers state (“professed beliefs”) and what is reflected in teachers ’ practices (“attributed beliefs”). Researchers claim to have found both consistencies and inconsistencies between professed and attributed beliefs. In this paper, methods and research designs typically used in studies of teachers ’ beliefs are examined. It is asserted that, in some cases, the perceived discrepancy between professed and attributed beliefs may actually be an artifact of the methods used to collect and analyze relevant data and the particular conceptualizations of beliefs implicit in the research designs. In particular, the apparent dichotomy can be the result of a lack of shared understanding between teachers and researchers of the meaning of terms used to describe beliefs and practices. In addition, it is asserted that it is inappropriate to classify any belief as entirely professed since researchers make various attributions to teachers through choices about data collection, theory, analysis of data, and presentation of findings. Moreover, the emphasis on classifying beliefs in this manner may be inhibiting researchers from developing a more comprehensive understanding of teachers ’ beliefs. Traditional and alternative methods are described, a data example is provided to illustrate the claims, and implications for future research are discussed.
Learning to learn to teach: An "experiment" model for teaching and teacher preparation in mathematics
 Journal of Mathematics Teacher Education
, 2003
"... ABSTRACT. This paper describes a model for generating and accumulating knowledge for both teaching and teacher education. The model is applied first to prepare prospective teachers to learn to teach mathematics when they enter the classroom. The concept of treating lessons as experiments is used to ..."
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ABSTRACT. This paper describes a model for generating and accumulating knowledge for both teaching and teacher education. The model is applied first to prepare prospective teachers to learn to teach mathematics when they enter the classroom. The concept of treating lessons as experiments is used to explicate the intentional, rigorous, and systematic process of learning to teach through studying one’s own practice. The concept of planning teaching experiences so that others can learn from one’s experience is used to put into practice the notion of contributing to a shared professional knowledge base for teaching mathematics. The same model is then applied to the work of improving teacher preparation programs in mathematics. Parallels are drawn between the concepts emphasized for prospective teachers and those that are employed by instructors who study and improve teacher preparation experiences. In this way, parallels also are seen in the processes used to generate an accumulating knowledge base for teaching and for teacher education. KEY WORDS: knowledge for mathematics teaching, knowledge for mathematics teacher education, learning to teach, lesson study An enduring problem in mathematics education is how to design prepara
Viewing mathematics teachers' beliefs as sensible systems
 Journal of Mathematics Teacher Education
, 2006
"... ABSTRACT. 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 onetoone correspondence between what teachers state and what researchers thin ..."
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ABSTRACT. 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 onetoone 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.
Teachers' mathematical beliefs: A review
 The Mathematics Educator
, 2003
"... This paper examines the nature and role of teachers ’ mathematical beliefs in instruction. It is argued that teachers ’ mathematical beliefs can be categorised in multiple dimensions. These beliefs are said to originate from previous traditional learning experiences mainly during schooling. Once acq ..."
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This paper examines the nature and role of teachers ’ mathematical beliefs in instruction. It is argued that teachers ’ mathematical beliefs can be categorised in multiple dimensions. These beliefs are said to originate from previous traditional learning experiences mainly during schooling. Once acquired, teachers ’ beliefs are eventually reproduced in classroom instruction. It is also argued that, due to their conservative nature, educational environments foster and reinforce the development of traditional instructional beliefs. Although there is evidence that teachers ’ beliefs influence their instructional behaviour, the nature of the relationship is complex and mediated by external factors. For the purpose of this paper, teachers’ mathematical beliefs refers to those belief systems held by teachers on the teaching and learning of mathematics. Educationalists have attempted to systematize a framework for teachers ’ mathematical belief systems into smaller sub–systems. Most authors agree with a system mainly consisting of beliefs about (a) what mathematics is, (b) how mathematics teaching and learning actually occurs, and (c) how mathematics teaching and learning should occur ideally (Ernest, 1989a, 1989b; Thompson, 1991). Certainly, the range of teachers ’ mathematical beliefs is vast since such a list would include all teachers ’ thoughts on personal efficacy, computers, calculators, assessment, group work, perceptions of school culture, particular instructional strategies, textbooks, students’ characteristics, and attributional theory, among others. In this paper, the concept of progressive instruction is associated with a socioconstructivist view of teaching and learning mathematics. Socioconstructivism, which for the sake of brevity will be called just constructivism, gives recognition and value to new instructional strategies in which students are able to learn mathematics by personally and socially constructing mathematical knowledge. Constructivist strategies advocate instruction that emphasises problemsolving and generative learning, as well as reflective processes and exploratory learning. These strategies also recommend group learning, plenty of discussion, informal and lateral thinking, and situated
Assessing students’ beliefs about mathematics, The Mathematics Educator (on line
 Handbook of Research on Mathematics Teaching and Learning
, 1992
"... The beliefs that students and teachers hold about mathematics ..."
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The beliefs that students and teachers hold about mathematics
Preservice secondary mathematics teachers’ beliefs about the nature of technology in the classroom
 Canadian Journal of Science, Mathematics, and Technology Education
, 2007
"... This study investigated preservice secondary mathematics teachers ’ (PSTs) beliefs about teaching mathematics with technology, the experiences in which those beliefs were grounded, and how those beliefs were held. Beliefs were defined as dispositions to act. Coherentism and the metaphor of a belief ..."
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Cited by 9 (2 self)
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This study investigated preservice secondary mathematics teachers ’ (PSTs) beliefs about teaching mathematics with technology, the experiences in which those beliefs were grounded, and how those beliefs were held. Beliefs were defined as dispositions to act. Coherentism and the metaphor of a belief system provided a conceptual framework through which the PSTs ’ beliefs were seen as sensible systems. Coherentism was posited as an alternative way of interpreting apparent inconsistencies between teacher’s beliefs and their practice. Through the qualitative research methodology called ground theory, four PSTs were purposefully selected and studied. Data stories were written that demonstrated the organization and structure of the PSTs ’ belief systems. From an analysis of the PSTs ’ experiences with technology, a theory was posited that focused on the PSTs ’ ownership of learning mathematics with technology. Experience, knowledge, and confidence were the primary factors that constituted ownership. The primary dimensions of the PSTs ’ core beliefs with respect to technology, referred to as their beliefs about the nature of technology in the classroom, were the availability of