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.

# Search Results

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

### Beth L. Cory and Joe Garofalo

This study investigates 3 preservice secondary mathematics teachers' understandings of limits of sequences and their changing conceptions of limit during and after instruction involving interactive, dynamic sketches embodying the formal definition of the limit of a sequence. Manipulating a coherent visual representation of the formal definition in the contexts of various sequences, coupled with answering carefully chosen questions and completing interview tasks before, during, and after technology-enhanced instruction, gave the participants opportunities to investigate and reflect on their own concept image as they compared their understandings to the results of the actions they performed on the sketch.

### Shiv Karunakaran, Ben Freeburn, Nursen Konuk and Fran Arbaugh

Preservice mathematics teachers are entrusted with developing their future students' interest in and ability to do mathematics effectively. Various policy documents place an importance on being able to reason about and prove mathematical claims. However, it is not enough for these preservice teachers, and their future students, to have a narrow focus on only one type of proof (demonstration proof), as opposed to other forms of proof, such as generic example proofs or pictorial proofs. This article examines the effectiveness of a course on reasoning and proving on preservice teachers' awareness of and abilities to recognize and construct generic example proofs. The findings support assertions that such a course can and does change preservice teachers' capability with generic example proofs.

### Jennifer N. Lovett, Allison W. McCulloch, Lara K. Dick and Charity Cayton

, drawing on the extant literature related to TPACK, video case instruction, and professional noticing, we propose a set of design principles for the development of technology mediated and video-enhanced modules for preservice secondary mathematics teachers

### Kimberly Corum and Joe Garofalo

Incorporating modeling activities into classroom instruction requires flexibility with pedagogical content knowledge and the ability to understand and interpret students' thinking, skills that teachers often develop through experience. One way to support preservice mathematics teachers' (PSMTs) proficiency with mathematical modeling is by incorporating modeling tasks into mathematics pedagogy courses, allowing PSMTs to engage with mathematical modeling as students and as future teachers. Eight PSMTs participated in a model-eliciting activity (MEA) in which they were asked to develop a model that describes the strength of the magnetic field generated by a solenoid. By engaging in mathematical modeling as students, these PSMTs became aware of their own proficiency with and understanding of mathematical modeling. By engaging in mathematical modeling as future teachers, these PSMTs were able to articulate the importance of incorporating MEAs into their own instruction.

### Fran Arbaugh, Duanne Graysay, Nursen Konuk and Ben Freeburn

In the last decade, mathematics teacher educators have begun to design learning opportunities for preservice mathematics teachers using a pedagogies-of-practice perspective. In particular, learning cycles provide a structure for engaging PSTs in learning to teach through the use of representations, approximations, and decompositions of practice (Grossman et al., 2009). In this article, we provide details of one learning cycle designed to support secondary mathematics preservice teachers' learning to *elicit and use evidence of student thinking* and *pose purposeful questions* (National Council of Teachers of Mathematics, 2014). Through qualitative analyses conducted on learning reflections, we provide evidence of the impact on engagement of this cycle through the lens of the Framework for Learning to Teach (Hammerness et al., 2005).

### Elizabeth A. Burroughs

An assignment that asks preservice secondary mathematics teachers to make connections between the mathematics they know and the mathematics they will teach. It describes how one preservice teacher's project resulted in a physical representation of the statement and proof that the sum of cubes of the first n natural numbers is equal to the square of their sum.

### Melvin R. Wilson

This study examines the evolving knowledge and beliefs of a preservice secondary mathematics teacher as she participated in a mathematics education course that emphasized mathematical and pedagogical connections and applications of the function concept. Her conceptions were revealed during a 10-week period through interviews, observations, and written work. The teacher's initial understanding of functions as computational activities (e.g., function machines, point plotting, vertical line test) was consistent with her larger view of mathematics as a collection of “concrete” procedures. Although her understanding of function grew substantially during the study, her anticipated approach to teaching, which was dominated by her narrow view of mathematics, was less significantly affected by course activities.

### Mary T. McMahon and Ellen Hines

The value of collaboration and reflection with peers to improving instructional practices is well known (e.g., Lieberman 1992; Little 1982; Little and McLaughlin 1993; Romberg 1988). However, practicing mathematics teachers are often challenged to find time in their busy schedules to collaborate with peers. Recently, during the implementation of a lesson study experience with a small group of preservice secondary mathematics teachers, we observed firsthand how lesson study could be used to encourage collaborative reflection among preservice teaching peers and how it potentially could be used to support ongoing professional development of in-service teachers while respecting their time constraints.

### Joe Garofalo and Christine Trinter

In this article, we present 2 technology-involved tasks that we use in our mathematics pedagogy courses to ostensibly give preservice secondary mathematics teachers (PSMTs) sample activities they can use in their teaching or use to assess their own future students' ability to apply trigonometric functions in contextual situations using technology. However, we have two other purposes for posing these tasks. One purpose is to provide occasions for PSMTs to self-assess their mathematical and technology knowledge, and subsequently take action to learn mathematics and technology features. The other purpose is to use such tasks as springboards for substantive discussions about teaching, learning, technology, and assessment. Such simulation tasks have engaged PSMTs and helped them develop their knowledge base for teaching mathematics.