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Thomas J. Cooney, Barry E. Shealy, and Bridget Arvold

This is a study of the beliefs and belief structures of 4 preservice secondary mathematics teachers as they progressed through a 4-quarter sequence in mathematics education including student teaching. We considered the notions of centrally and peripherally held beliefs and whether beliefs were held dualistically or contextually. The various ways in which the teachers structured their beliefs helped account for the fact that some beliefs were permeable whereas others were not. The nature of the evidence supporting the teachers' beliefs was considered particularly as that evidence related to the voices of significant others or to what the individuals valued. A scheme for conceptualizing the professional development of preservice teachers is posited.

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Jennifer Earles Szydlik

In this study, I investigated 27 university calculus students' mathematical beliefs and connections between those beliefs and their understandings of limit. Participants were selected on the basis of questionnaire and interview responses to real-number, infinity, function, and sourcesof- conviction items. Data obtained in a subsequent limit interview suggest a relationship between sources of conviction and understanding of limit; students with external sources of conviction gave more incoherent or inappropriate definitions of limit, held more misconceptions of limit as bound or unreachable, and were less able to justify limit calculations than those with internal sources of conviction. The influence of content beliefs on understanding of limit is less evident.

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Jane M. Watson and Jonathan B. Mortiz

One hundred eight students in Grades 3, 5, 6, 7, and 9 were asked about their beliefs concerning fairness of dice before being presented with a few dice (at least one of which was “loaded') and asked to determine whether each die was fair. Four levels of beliefs about fairness and four levels of strategies for determining fairness were identified. Although there were structural similarities in the levels of response, the association between beliefs and strategies was not strong. Three or four years later, we interviewed 44 of these students again using the same protocol. Changes and consistencies in levels of response were noted for beliefs and strategies. The association of beliefs and strategies was similar after three or four years. We discuss future research and educational implications in terms of assumptions that are often made about students' understanding of fairness of dice, both prior to and after experimentation.

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James A. Middleton

In this study I examine the structures of 2 teachers' beliefs about what makes mathematics intrinsically motivating and provide instances of the representations of their beliefs at 2 times: before the introduction of middle school mathematics curricula organized around the tenets of Realistic Mathematics Education and after 1 year of implementing a pilot program. Personalconstructs analyses are paired with observations of teachers' classrooms and their beliefs and perceptions as reported in semistructured interviews. Results indicate that the teachers became more attuned to the conceptual complexity and challenge of mathematics activities and placed less emphasis on task ease over their year of involvement in the pilot program. Results are discussed in relation to “job-embedded learning,” a form of staff development that fosters teachers' development of meaning with regard to reforms, and how such learning enables shifts in teacher beliefs and practice.

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Randolph A. Philipp, Rebecca Ambrose, Lisa L.C. Lamb, Judith T. Sowder, Bonnie P. Schappelle, Larry Sowder, Eva Thanheiser, and Jennifer Chauvot

In this experimental study, prospective elementary school teachers enrolled in a mathematics course were randomly assigned to (a) concurrently learn about children's mathematical thinking by watching children on video or working directly with chil-dren, (b) concurrently visit elementary school classrooms of conveniently located or specially selected teachers, or (c) a control group. Those who studied children's mathematical thinking while learning mathematics developed more sophisticated beliefs about mathematics, teaching, and learning and improved their mathematical content knowledge more than those who did not. Furthermore, beliefs of those who observed in conveniently located classrooms underwent less change than the beliefs of those in the other groups, including those in the control group. Implications for assessing teachers' beliefs and for providing appropriate experiences for prospective teachers are discussed.

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AnnaMarie Conner and Laura Marie Singletary

Connecting teachers’ beliefs and their classroom practice has been a long-standing endeavor in the search for how to understand and improve teaching. Previous studies have yielded mixed results, with some claiming that teachers’ beliefs determine

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Carlos Nicolas Gomez and AnnaMarie Conner

There is a rich literature base focusing on the study of beliefs 1 in mathematics education. These studies focus on beliefs about mathematics, beliefs about mathematics teaching, or beliefs about learning mathematics and study populations of

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Bilge Yurekli, Mary Kay Stein, Richard Correnti, and Zahid Kisa

). Recent reviews of research ( Cross Francis, Rapacki, & Eker, 2015 ; Fives & Buehl, 2012 ) have shown that teachers’ enactment of instructional practices aligned with reform initiatives is substantially shaped by their beliefs regarding how important

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Denise A. Spangler

The beliefs that students and teachers hold about mathematics have been well documented in the research literature in recent years (e.g., Cooney 1985; Frank 1988, 1990; Garofalo 1989a, 1989b; Schoenfeld 1987; Thompson 1984, 1985, 1988). The research has shown that some beliefs are quite salient across various populations. These commonly held beliefs include the following (Frank 1988):

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Salvador R. Vazquez, Bradley A. Ermeling, and Gerardo Ramirez

measuring parents’ beliefs about productive struggle and analyzed the relationship between these indices and parents’ math homework help. Defining Productive Struggle For the purposes of this study, we define productive struggle as the act of