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As part of a longitudinal study focused on mathematical play, we (Melissa, Amy, and Anita) are often faced with questions about what counts as play and what mathematics (and other learning) we see in play, and whose play is most likely to be seen or dismissed. Rather than discuss our findings from classroom videos of kindergarten children engaged in mathematical play, we asked scholars who bring different lenses to research on play, young children, and teaching and learning mathematics to look at some of our data and provide their perspectives. In this session, we will share video and discuss with our panel (Nathaniel, Naomi, and Tran) various ways to interpret that video. This paper provides background on the potential of mathematical play and the details of the study that generated data for analysis. We conclude with a copy of a transcript that is associated with a video we will watch during the plenary with hopes that participants will watch prior to the session and come with their own questions/perspectives. [ABSTRACT FROM AUTHOR]

Many studies use instructional designs that include two or more artifacts (digital manipulatives, tables, graphs) to support students' development of reasoning about covarying quantities. While students' forms of covariational reasoning and the designs are often the focus of these studies, the way students' interactions and transitions between artifacts shape their actions and thinking is often neglected. By examining the transitions that students make between artifacts as they construct and reorganize their reasoning, our study aimed to justify claims made by various studies about the nature of the synergy of artifacts. In this paper, we present data from a design experiment with a pair of sixth-grade students to discuss how their transitions between artifacts provided a constructive space for them to reason about covarying quantities in graphs. [ABSTRACT FROM AUTHOR]

In this paper, I discuss an Archeological analysis of two high school mathematics textbook series in an effort to investigate how actions for the doing of statistics are formed by statements from the statistics lessons the texts. The objective of this paper is not to present all of the findings related to the posed question, but to focus on discussing the methodology employed (i.e. Foucauldian Discourse Analysis), which is relatively novel for analyzing textbooks in mathematics education. One finding will be presented to highlight how the methodology employed can contribute to the field helping to uncover patterns not found using well established deductive frameworks. This methodological discussion is meant to provide an example of how using different methodologies can help to investigate issues in the field and problematize taken for granted norms against a new horizon to embrace the sociopolitical turn. [ABSTRACT FROM AUTHOR]

In this paper, we explore the ways in which learning more about research on students' experiences in mathematics classrooms has the potential to transform the work we do with teachers in teacher preparation, professional development, and research settings. We focus in particular on questions of student access to and participation in mathematics and highlight studies of the racialized and gendered experiences of students and the connections between these experiences and broader narratives about race, gender, and ability/disability. We conclude with questions and possibilities raised by these studies for our individual and collective efforts to support and understand teacher learning and changes in teacher practice. [ABSTRACT FROM AUTHOR]

The phenomenon of the sea level rise is a pressing environmental and social issue of the present age. Starting with the assumption that mathematics can be utilized to help students explore this phenomenon, we designed a simulation in NetLogo, in which students investigated the relationships between the quantities of temperature rise, height of future sea level, and total land area. In this paper, we present the analysis of a whole-class design experiment in a sixth-grade classroom and discuss how our design helped students to examine sea level rise as both an environmental and a social issue. [ABSTRACT FROM AUTHOR]

The increased use of poststructural theories in mathematics education provides a pathway for novel methodological approaches. In this paper, I introduce feminist poststructural discourse analysis (FPDA) as a methodology to study the intersection of gender and racial inequities in mathematics education. Specifically, I explore one example of a conceptualization of such a study to understand gendered, racialized, mathematical discourses in a middle school classroom. [ABSTRACT FROM AUTHOR]

We studied an online course designed to provide rural mathematics teachers access to high quality professional development. The online course emphasized productive discussions related to high-leverage discourse practices. We applied a community of inquiry framework, which emphasizes deep intellectual work, and its three tenets: cognitive presence, social presence, and teaching presence. In this paper we focus on cognitive presence. A major contribution of our study is the use of mediating processes to characterize cognitive presence; this allowed us to conceptualize the interaction between the three presences (social, teaching, and cognitive) and their connections to our design conjectures. [ABSTRACT FROM AUTHOR]

This paper articulates a stance on the study of students’ meaningful mathematics understanding with multiple representations. We integrate Thompson’s theory of quantitative reasoning and Dreyfus’ theory of multiple representations in our approach to frame and conduct empirical investigations of the study of meaningful understanding of function. We provide empirical data to support our approach to examining representational fluency and functional thinking from this networked stance. Our research articulates how the coordination of theories can be productive in informing the design, conduct, and analysis of contexts aimed to understand students’ meaningful math learning with a focus on functional thinking. [ABSTRACT FROM AUTHOR]

This study investigates the student-created representations in a synchronous online mathematics course. Over a 15-week semester, a select group of mathematics specialists candidates took the course, MATH 613 - Algebra for K-8 Math Specialists using a 100% synchronous online format. Data collection included interactive slides and reflection paper assignments. The findings indicate that the mathematical representations included abstract, pictorial, concrete and dynamic-pictorial. Dynamic-pictorial is defined in this research as pictorial models that use the advantages of technology to move representations on the screen in a way that could not be reasonably replicated using hand held manipulatives. Implications for this research include supporting instructors of math content courses with planning and delivery of online coursework with the expectation of student-created dynamic-pictorial representations. Further research could address the variety of dynamic-pictorial representations to analyze the complexity of these representations. [ABSTRACT FROM AUTHOR]

Just as the learning of mathematics requires that students have opportunities to learn, it also requires that children have opportunities to see themselves and be seen by others as mathematical doers and knowers. This paper proposes a framework for identifying teacherintroduced opportunities to identify through the examination of 1) instructional tasks, 2) participation structures and 3) storylines. The framework was developed and is illuminated through a year-long ethnographic case study of Ms. Wong’s classroom, a place where students had opportunities to identify (a) as people with dignity and competence, (b) who have mathematical ideas worthy of consideration, (c) and mathematical lives worthy of exploration, and (d) whose participation is vital to their communities. [ABSTRACT FROM AUTHOR]

In mathematics classrooms, teachers are often the authority as a knower of mathematics. To support students to develop their identity as doers of mathematics, teachers need to decenter the authority and to place students in a more powerful position as autonomous thinkers. Building on the work of Herbel-Eisenmann and colleagues (2013), this paper aims to expand the notion of powerful discourse from a conversation analytic approach. By examining the fine-grain epistemic characteristics of interactional sequences, I argue for the significance of the interactional knowledge domain, in which positioning of students as more or less knowledgeable takes place. Further, I discuss the need for examining the discursive dimension of knowledge and discursive practices that can redistribute the epistemic authority in mathematics classrooms. [ABSTRACT FROM AUTHOR]

Examining the Trajectories of Black Mathematics Teachers: Learning From the Past, Drawing on the Present, and Defining Goals for the Future is a three year-long research study funded by the National Science Foundation. Our study was designed to reflect on the dearth of Black mathematics teachers, including problems with recruiting and retaining Black mathematics teachers. The research study mitigates some of the oversights in the history of mathematics education. The first phase of the study was a large-scale survey of more than 500 currently practicing Black mathematics teachers. Simultaneously, the research team began conducting oral histories of retired Black mathematics teachers. In the second phase of the project, there will be focus group with currently practicing Black mathematics teachers. Ultimately, the objective of the research is to leverage the epistemological wealth of Black teachers, particularly through an historical lens, to guide contemporaneous teacher education policy and practice. This paper focuses on preliminary findings from the oral histories part of this multifaceted research study. [ABSTRACT FROM AUTHOR]

In this paper we describe bonusing as a strategy to continuously extend students' mathematical understanding from tasks initially offered in the classroom. While this strategy resembles enrichment activities often described in approaches based on Mastery Learning, there are fundamental differences in our use of assessment and the nature of the tasks we propose. After two years of implementing this strategy as part of the Math Minds initiative, we have found that teachers perceived bonusing as relevant and engaging for students. However, they experienced difficulties in the implementation of this strategy. In this paper, we draw from interviews with 14 elementary teachers to identify difficulties they perceive while attempting to implement bonusing. We propose the use of Variation Theory of Learning to inform the creation of bonus material and discuss connections to knowledge for teaching. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to examine the link between students’ mathematical experiences and the epistemological orientations that they develop. In keeping with the theme of this conference, building connections between communities, this paper seeks to make an important connection highlighted by Schoenfeld in his AERA presidential address (1999), between research that focuses on the social perspective and research that focuses on the individual perspective. We will illustrate how it is that individuals develop orientations to mathematics through social practice and map out the different orientations that students are likely to develop. Our research draws from a large data set of approximately 1000 students who experienced different mathematics approaches over four years of high school. [ABSTRACT FROM AUTHOR]

Small groups (SG) is a space where three aspects, such as mathematical thinking, status, and funds of knowledge, come into play. How teachers draw upon these three aspects in facilitating small group work is shaped by what teachers notice. Researchers have used the professional noticing framework (Jacobs, Lamb, & Philipp, 2010) to understand teachers' noticing of students' mathematical thinking. This paper proposes a teacher noticing framework based upon Louie (2017) and Jacobs and her colleagues (2010) to expand teacher noticing for facilitating SG. [ABSTRACT FROM AUTHOR]

As schools turn towards co-teaching models to implement inclusion practices (US Department of Education, 2010), teachers are increasingly being asked to operate with a co-teacher (Scruggs et al., 2007). However, most research on co-teaching remains at the survey level; it does not examine factors that may influence co-teaching relationships over time (Scruggs et al., 2007). This paper seeks to address that gap by examining the feedback a first and second year coteacher give one another as they teach a high school mathematics course. The question under consideration is: what kinds of feedback may beginning co-teachers offer one another and how may it influence their classroom practice over time? The results of this work contribute to the broader body of co-teaching research by offering insights into a specific case of teacher collaboration. [ABSTRACT FROM AUTHOR]

This study on the mental solving of systems of linear equations is part of a larger research program, aimed to gain a better understanding of the potential of mental mathematics activities with topics/objects other than numbers. Through outlining the study details and the activities engaged with, the paper reports on the variety of strategies developed for mentally solving systems of linear equations. The analysis pays attention to specificities and particularities of these strategies, focusing on their economical, tailored, and spontaneous nature to solve the tasks. In reflecting on the context of mental mathematics, the paper closes with discussions of the potential and richness of these strategies for exploring systems of linear equations. [ABSTRACT FROM AUTHOR]

This paper focuses on research that can inform the improvement of mathematics teaching and learning at scale. We first argue that such research should view mathematics teachers' instructional practices as situated in the institutional settings of the schools and districts in which they work. We then discuss a series of hypotheses about school and district structures that might support teachers' ongoing improvement of their classroom practices. In the latter part of the paper, we outline an analytic approach for documenting the institutional settings of mathematics teaching established in particular schools and districts that can feed back to inform the instructional improvement effort. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

The purpose of this paper is to describe a model for analyzing the quality of mathematical discourse in online asynchronous discussions. Using the constant comparative methodology for developing theory, the analysis revealed five hierarchical stages for coding mathematical discourse: participation, contribution, deliberation, justification, and expansion. The paper also discusses implications of the findings and recommendations for research. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

The paper will be a discussion of the implications and conclusions of my dissertation reseach on cooperative mathematical problem solving. The following is a summary of that study: [ABSTRACT FROM AUTHOR]

Recent mathematics standards (NCTM, 2000) place central importance on the development of mathematical reasoning. Reasoning involves more than dealing with individual concepts and focusing on conventions, skills, and techniques that produce answers to "type problems." Reasoning involves creating a rich cognitive map of content along with habits of mind that help connect and make sense of mathematics. Many areas of mathematics have recently been framed by describing the nature of applicable mathematical reasoning; for example, geometric reasoning, proportional reasoning, algebraic reasoning, and statistical reasoning. While there are similarities across all types of reasoning (e.g., deductive, inductive, and intuitive reasoning), there are also content-specific reasoning processes. As an example, statistical reasoning begins with a process of data analysis - an iterative process that includes asking a question, collecting data, analyzing the data, and forming and communicating conclusions. Learning statistics therefore includes two central components: the concepts and conventions of the discipline itself (Konold, 2002) and the ability to apply this knowledge in a broader, more flexible, and skeptical manner when analyzing, interpreting and/or communicating results obtained by statistical analysis. Garfield (2002) concurs when she defines statistical reasoning as "the way people reason with statistical ideas and make sense of statistical information. This involves making interpretations based on sets of data, graphical representations, and statistical summaries" (p. 1). In this paper, we build on the research literature and our current work to argue that this kind of statistical reasoning is facilitated by using tasks that are designed to engage students in comparing distributions of two or more data sets (e.g., Russell, Schifter & Bastable, 2002; Author, in press, 2002; Author, in press). [ABSTRACT FROM AUTHOR]

In this paper, a model is proposed to characterize some of the ways mathematical proofs are generated and understood. The model uses the notion of ‘Key Idea’, which is defined and illustrated in the context of a task from a pre-service course for secondary school teachers. [ABSTRACT FROM AUTHOR]

This paper describes a tool we developed for analyzing small-group discussions in mathematics classrooms. We applied this tool to videotaped sessions of thirty groups of four students, each working together on an open-ended problem. Our analysis shows that small group discussions offer teachers and researchers important knowledge about student thinking and understanding. These insights are often invisible in the written work of the students. We also found that such student discussions frequently emulate discussions among professional mathematicians, thus creating authentic engagement experiences for the students. These findings may help bring traditionally disparate groups of thinkers together. Namely, teachers and researchers are presented with a common set of features to look for in mathematical discussions, and mathematical researchers and mathematicians are presented with striking parallels between student discussions and professional mathematical work. [ABSTRACT FROM AUTHOR]

Copyright of Conference Papers -- Psychology of Mathematics & Education of North America is the property of Psychology of Mathematics & Education of North America and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Conference Papers -- Psychology of Mathematics & Education of North America is the property of Psychology of Mathematics & Education of North America and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In order to make sense of mathematics and to communicate mathematical ideas, it is essential to be able to assess and produce mathematical arguments, including formal proofs. Although student difficulty with proof has been well established in the literature (Chazan, 1993; Hart, 1994; Martin & Harel, 1989; Senk, 1985), research connecting teachers’ actions in the classroom to students’ understanding of proof is limited (Herbst, 2002). In this paper, we describe a classroom-based interpretive study in which we explore the relationship between teaching and understanding of proof in a proof-based geometry class. In particular, we address the following question: [ABSTRACT FROM AUTHOR]

Research suggests a strong relationship between teachers’ beliefs and their teaching practices. However, relatively little is known about the relationship between teachers’ beliefs and their knowledge. For the past two years, we have been conducting research concerning the mathematics beliefs and conceptions of preservice elementary teachers. In this paper, we describe how we have been doing this research and report on results of this on-going study. Specifically, we report on some of the general beliefs and conceptions of preservice teachers and those of mathematics specialists and non-traditional preservice teachers. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to share some of the results of a qualitative study, which indicate that in an appropriately supportive environment students are capable of constructing and justifying mathematical ideas. The students in this study solved challenging combinatorics problems and developed mathematical proofs as convincing arguments to justify their ideas to themselves and others. [ABSTRACT FROM AUTHOR]

This paper reports findings from a study that explored the effect of a secondary mathematics teacher's level of attention to quantitative reasoning on the quality and coherence of his instruction of angle measure. I analyzed 37 videos of an experienced teacher's instruction of trigonometric functions to characterize the extent to which the teacher attended to supporting students in reasoning quantitatively, and to examine the consequences of this attention (or lack thereof) on the quality and coherence of the meanings the teacher's instruction supported. My analysis revealed that the incoherencies in the teacher's instruction were occasioned by his inattention to quantitative reasoning. [ABSTRACT FROM AUTHOR]

This paper builds on the cannon of literature on the concept of slope to introduce a framework for describing conceptions of slope. The framework will allow us to consider how slope is viewed and used. It also provides researchers a way to classify student understanding of slope and educators a guide to help design slope tasks. The framework brings together ideas from APOS theory with research on slope conceptualizations to describe how students may develop an understanding of slope. [ABSTRACT FROM AUTHOR]

In this paper we describe a study involving middle school mathematics teachers who are participating in a large-scale professional development program designed to improve their conceptual understanding of mathematics. The design and implementation of this study were guided by the research question: In what ways do teachers reason through tasks designed to elicit proportional reasoning? Our analysis of the data was conducted through the lens of quantitative reasoning as described by Thompson. We highlight the issues that arose when some teachers performed numerical operations on values that they did not clearly connect to quantities. [ABSTRACT FROM AUTHOR]

The topic of rural mathematics education and the experiences of teachers working in these schools is one that has not been fully discussed within the literature. The relative isolation imposed by their circumstances create specific challenges, such as lack of resources, which is especially true for secondary teachers. This paper reports the results of a yearlong professional development project based on teacher noticing of student thinking involving the use of video club meetings at one rural school district. Findings show that while the project teachers developed deeper understanding of student thinking and task development/implementation they also changed their view of collaboration with their fellow teachers. [ABSTRACT FROM AUTHOR]

Research on teachers frequently illustrates the persistence of beliefs about students and mathematics that reproduce inequity. The prevalence of these beliefs suggests that they not only belong to individual teachers but also are part and parcel of culturally dominant discourses concerning the nature of mathematical ability. In this paper, I show how these discourses interfere with equity-oriented reforms, focusing on a teacher who was deeply engaged in redefining what counts as "smart in math," dismantling the borders between "smart" and "not smart." In spite of her efforts, the teacher continued to see students' deficits and, more significantly, she constantly doubted the legitimacy of the "smartnesses" she worked hard to see. I call this phenomenon the Smartness Dilemma. Understanding how it functions is important for understanding how to support teachers to disrupt hierarchies of perceived ability. [ABSTRACT FROM AUTHOR]

In this paper, we show how one teacher used software, curriculum and new teaching moves to advance student argumentation and promote equity in discourse. The professional development (PD) she engaged in, Bridging PD, focuses on assisting teachers in diverse urban school communities to infuse argumentation into their practices so that all students actively engage in mathematical argumentation. Bridging PD integrates technology-based curriculum and dynamically-linked representations in the PD and then teachers use these in their classrooms. [ABSTRACT FROM AUTHOR]

In this paper we explore the relationship between certification path and teaching self-efficacy for teachers of K-12 mathematics. We focused on teaching self-efficacy for a content area that spans the K-12 curriculum: fractions, ratios, and proportions. Initial findings indicated that alternate certification is significantly correlated to teaching self-efficacy. However, in multiple regression analysis of certification after including other variables that predicted teaching self-efficacy, the relationship between certification path and teaching self-efficacy was no longer statistically significant. Findings suggest that the apparent relationship between alternate certification and teaching self-efficacy is likely due to other factors (age, highest grade taught, MKT) that differentiate alternately certified and traditional teachers. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present preliminary findings from my doctoral research. The study focuses on the interrelationship between teachers’ mathematics histories, confidence and personalities, social context, and the development of mathematics and pedagogic beliefs. The interrelationships are investigated as factors impacting the process of changing mathematics teaching practices. The paper discusses the purpose for combining two data collection instruments: mathematics life story interviews and concept mapping activities. I suggest that through utilizing both instruments, in combination, it may be possible to develop a more comprehensive understanding of teachers’ past experiences and the impact of those experiences on the development of beliefs. [ABSTRACT FROM AUTHOR]

In the past decade there is growing interest in mathematics education research that has emerged from complexity sensibilities that focuses on the collective (group) as a learning system. With this growing emphasis there is a need for methodological tools to study those systems. I am concerned with ways in which we can observe for the emergence of mathematics and mathematical thinking in the collective (rather than the individual). The purpose of this paper is to explore means and methods that can be used to anticipate, describe and account for the mathematics that emerges in collective activity in its own right. What are the markers of collective learning and understanding? How can we observe those markers of collective learning and understanding? [ABSTRACT FROM AUTHOR]

Digital curricula (DC) are gaining ubiquity in elementary mathematics classrooms and dramatically impacting the nature of mathematics teaching. Teacher educators and professional development providers are not keeping pace with the changing learning environments. This Working Group will examine video case studies of teachers implementing DC in varied ways, identify unique features of instruction with DC, brainstorm innovations to teacher education and professional development, and determine a research agenda. Pepin et al.'s (2017) theoretical framework of Learning Spaces will guide our discussion as well as four thematic questions addressing students' procedural fluency versus conceptual understanding, Common Core Mathematical Practices, affordances and constraints of DC, and teachers' use of DC generated data to engage all learners. [ABSTRACT FROM AUTHOR]

This paper aims to advance understanding of instructional phenomena by focusing on the nature of an instructional situation. We argue that the decisions in managing a situation require a teacher to respond to norms by negotiating dispositions activated in the situation that could contradict each other. We illustrate this point examining the situation of installing theorems in high school geometry. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper reports a study of prospective bilingual teachers and the intersectionality of language and mathematics in a non-traditional field experience. Qualitative data were analyzed in order to map Spanish and English language practices to teachers' assistance strategies. We define assistance strategies as those actions that are intended to help students understand content. Findings demonstrate the complexity of prospective teachers' use of language in mathematics teaching and suggest implications for teacher preparation programs. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper explores the emphasizing of mathematical relationships through socially negotiated links as a way to ground important concepts in sociomathematically meaningful activity. Findings are presented from a detailed analysis of a classroom episode in which a succession of student contributions from different representational perspectives supported the development of a collective understanding of the relationships among multiple function representations. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper examines narratives one teacher told during participation in a Professional Study Group. Through various retellings, one sees her shift the responsibility for underachievement in mathematics from the child to herself. Specific plans the teacher made to adapt her teaching practice are noted. Through her retelling of events we gain access to what supported her in forming these plans. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper describes elementary pre-service teacher's models of connecting algebraic generalizations with a related contextual problem. After a pre-test, subjects engaged in model-eliciting activities by finding objects to represent a generalization of the form y=ax+b, and an accompanying survey sheet regarding the intervention (Doerr & Lesh, 2003). Results indicated refinement and expansion of models despite initial difficulty with authentic applications. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Teachers must have an understanding of students' mathematical thinking in order to create meaningful learning opportunities. Literature on learning trajectories and geometric reasoning provide a framework for investigating the ways that teachers make sense of their students' thinking. This paper will report on two case studies that examined the experiential models of students' geometric thinking created by secondary teachers. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Students' solutions to the "student-professor problem" have provided a source for investigating students' algebraic mental models. This paper seeks to expand on previous investigations by approaching "student-professor type" situations from a new perspective. It uses ideas of equality and covariation while analyzing students' successful strategies to suggest new pedagogical strategies for helping students develop algebraic reasoning. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Recent calls for research on the impact of the Principles and Standards for School Mathematics (NCTM) indicate a need to better understand the relationship between teachers and curricula. This paper examines 2nd grade teachers' implementation of Standards-based whole number lessons in terms of an enactment's fidelity to two curricular forms: the literal and the intended curricula, and then in terms of teachers' instructional moves. In so doing, we explore interactions between teachers' instructional moves and students' opportunities to learn mathematics. We conclude by considering how curricula support or fail to support teachers' instructional moves and choices. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

In this paper, we examine theory and practice related to creating spaces for pre-service teachers to learn about incorporating family and community resources into elementary mathematics and literacy. Using interview data, as well as excerpts from pre-service teachers' reflections, we first define overlaps between mathematics and literacy, and between family and community funds of knowledge and school-based content. We then describe how pre-service teachers learned about, experienced, and enacted these overlaps in three distinct, but intersecting spaces - the elementary classroom, the university methods class, and the community. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

In this paper, we examine elementary pre-service teachers' understanding of the phrase "is a special kind of" as in "a square is a special kind of rectangle." Data collected from 109 students indicated that they possess varying interpretations of the phrase. Our findings suggest that difficulty with shape categorization may be related to the meanings students attach to such phrases. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

In this paper we characterise the growth of collective mathematical understanding as an improvisational process. Drawing on elements of improvisational theory, in particular the notion of etiquette, to analyse extracts of classroom video data, we demonstrate how this theoretical framework can illuminate and inform collaborative group processes in a mathematics classroom. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

In this paper we share our professional development effort within one urban high school. Throughout intervention teachers' emotional needs and practical problems impeded efforts to facilitate change in mathematical understandings and practices. Results revealed that the complex relationship between teachers' mathematical anxiety and professional identity, including fear of public recognition of a lack of content knowledge, was crucial to learning. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]