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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]

This paper describes the role of sense of place in students' analysis of spatial data toward understanding the distribution of financial services in their city. High school students participated in a 10-session module about their city's two-tiered financial system of banks and alternative financial institutions. The paper analyzes two class sessions organized around the use of ratios, or intensive variables, to understand the distribution: an embodied distribution activity atop a large floor map and individual exploration of scalable, data-rich digital maps. Analysis investigates the role of students' sense of place as they grappled with these ratios. Students drew on their senses of place to interpret data and generate their own sets to associate in ratios. Abstract measures and data visualizations contained in the digital maps were less accessible to students in this iteration of the module. [ABSTRACT FROM AUTHOR]

The body of research examining students' graphing understandings across STEM fields indicates students are not developing productive meanings for graphs. We conjecture such failings may, in part, be explainable by features of students' use of coordinate systems and graphing activity that are under examined. In this theoretical report, we present a conceptual analysis of different ways students may reason about graphs and coordinate systems. Specifically, we describe two different uses of coordinate systems--spatial and quantitative--students might leverage and two ways of reasoning--static and emergent--students might engage in as they construct or interpret graphs. We characterize how a student may engage in each kind of reasoning in each use of coordinate system. We intend this paper to serve as a theoretical lens for future empirical studies examining students' developing graphing understandings. [ABSTRACT FROM AUTHOR]

This paper presents the results of a study that examined how preservice teachers used the Internet to find information that would help them better understand one of the Common Core Standards around the Pythagorean Theorem. A brief review of the research around the participants' search strategies, and content understanding is presented first. The connections between the searching strategies used, and the form and quality of the mathematical connections made is the focus of this paper. [ABSTRACT FROM AUTHOR]

This paper reports two prospective teachers' and a mathematician's ways of identifying similar triangles and hexagons through the analysis of routines, a characteristic of geometric discourse. The findings show that visual recognition was a common approach for the mathematician as well as the two prospective teachers. However, when asked for justification, their routines of identifying similar polygons diverged. The paper also discusses the implication of classroom discourse practices to enhance prospective teachers' communication and reasoning skills while learning geometric concepts such as similarity. [ABSTRACT FROM AUTHOR]

Research has illustrated that prospective teachers struggle to construct word problems for integer operations. This paper extends the current research by examining the ways in which four classes of prospective teachers responded to a child's temperature story for an integer addition number sentence (i.e., -9 + -6 = ☐). One hundred prospective teachers responded to whether they believed that the story matched the number sentence and provided justification for their reasoning. The child's story, an actual story posed by a Grade 5 student, did not utilize temperature realistically (realism), nor was it consistent with the given number sentence (consistency). The results indicated that when prospective teachers were asked to evaluate the child's story, they tended to either focus on realism or consistency, but not both. Benefits of using children's thinking during instruction as well as implications for research are discussed. [ABSTRACT FROM AUTHOR]

In this paper we examine the application of Tzur's (2007) fine-grained assessment to the design of an assessment measure of a particular multiplicative scheme so that non-interview, good enough data can be obtained (on a large scale) to infer into elementary students' reasoning. We outline three design principles that surfaced through our recent effort to devise a sequence of items to assess at which stage--participatory or anticipatory--a child might have constructed the multiplicative double counting (mDC) scheme (Tzur et al., 2013). These principles include the nature and sequencing of prompts, number choice, and authenticity of responses. [ABSTRACT FROM AUTHOR]

Computer scientists have reported on "computational thinking," which Aho (2012) defines as thought process involved in formulating problems so their solutions can be represented as computational steps and algorithms" (p. 832). We wanted to investigate such thinking in mathematics. To study this, we interviewed five mathematicians about the role of computation their work, and the notion of "algorithmic thinking" developed from these interviews. We foun mathematicians value this thinking, and in this paper we define algorithmic thinking and prese number of applications and affordances of such thinking. [ABSTRACT FROM AUTHOR]

This paper proposes to use communalism, an asset-driven strategy effective with Black learners, with racially mixed study groups. This study found that students who studied in racially mixed groups performed better when told that individuals would receive a reward for scoring the highest in their group. Also, the White students' perceptions of how much they mattered differed based on the racial composition of the study groups where those studying with only White students felt they mattered more than those who studied with White and Black participants. Implications of the importance to have culturally relevant and responsive resources for racially diverse classrooms are discussed. [ABSTRACT FROM AUTHOR]

In the spirit of questioning, crossing, and blurring disciplinary borders with(in) mathematics education, this paper proposes a theoretical framework that merges conceptions of critical literacy and statistical literacy for high school mathematics curriculum. I begin by discussing the political nature of education and then situate this work within the goals of education for active citizenship. Next, I provide a detailed background of both critical and statistical literacy from multiple educational research perspectives. Lastly, by synthesizing notions of critical and statistical literacy, I present a theoretical framework for a critical statistical literacy in the context of high school mathematics curriculum. [ABSTRACT FROM AUTHOR]

Our study investigates preservice teachers' perceptions of real-world problems; their beliefs about teaching real-world contexts, especially ones sociopolitical in nature; and their ability to pose meaningful real-world problems. In this paper we present cases of three preservice teachers who participated in interviews that probed their thinking about real-world problems, and asked them to create problems they would consider using in their future classrooms. We use the three cases to propose a potential trajectory for preservice teachers from ambivalence to certainty about teaching mathematics through real-world and controversial contexts. [ABSTRACT FROM AUTHOR]

The calculus concepts of concavity and inflection points are often given meaning through the shape or curvature of a graph. However, there appear to be deeper core ideas for these two concepts, though the research literature has yet to give explicit attention to what these core ideas might be or what it might mean to "understand" them. In this paper, I propose a framework for the concavity and inflection point concepts, using the construct of covariation, wherein I propose conceptual (as opposed to mathematical) definitions that can be used for both research and instruction. I demonstrate that the proposed conceptual definitions in this framework contain important implications for the teaching and learning of these concepts, and that they provide more powerful insight into student difficulties than more traditional graphical interpretations. [ABSTRACT FROM AUTHOR]

In this study I investigate Saldanha and Thompson's (1998) claim that conceptualizing a coordinate pair in the Cartesian coordinate system as a multiplicative object, a way to unite two quantities' values, supports students in conceptualizing graphs as emergent representations of how two quantities' values change together. I presented three university precalculus students with an animated task showing varying values of two quantities along the axes and asked each student to sketch a graph of how the two quantities changed together. In this paper I document the difficulty students encountered when they did not conceptualize a coordinate pair as a multiplicative object. I address why the convention of "over x and up y" inhibits students from constructing a coordinate pair as a multiplicative object and I provide recommendations for supporting students in constructing coordinate pairs as multiplicative objects. [ABSTRACT FROM AUTHOR]

This paper considers what students attend to as they first encounter R³ coordinate axes and are asked to graph y = 3. Graphs are critical representations in single and multivariable calculus, yet findings from research indicate that students struggle with graphing functions of more than one variable. We found that some students thought y = 3 in R³ would be a line, while others thought it would be a plane. In creating their graphs, students attended to equidistance, parallelism, specific points, and the role of x and z. Students' use of these ideas was often generalised from thinking about the graphs of y = b equations in R². A key finding is that the students who thought the graph was a plane always attended to the z variable as free. [ABSTRACT FROM AUTHOR]

This exploratory qualitative study describes how teachers in a yearlong professional development project conceptualized and used formative assessment within their instruction. The study focuses on how teachers interpreted formatives and the instructional decisions that teachers made. This paper documents the shifts in the lenses teachers adopted when interpreting student work and the instructional decisions that are possible based on the information teachers gain from looking at student work. [ABSTRACT FROM AUTHOR]

This paper reports results from investigating the role of facilitators in professional learning communities of mathematics and science teachers. Facilitators who made efforts to understand the reasoning patterns of PLC members were more effective in promoting quality discourse among the members of the PLC. The paper describes and contrasts various manifestations of decentering that were observed among PLC facilitators in four different PLC's over the course of one year. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

The following paper examines the cognitive processes that occur in the course of student interactions during collaborative learning activities. We find that conceptual blending and distributed cognition are useful models for interpreting these interactions using the cooperative group itself as the cognitive unit of analysis. Conceptual blending is rarely characterized in the literature as a group phenomenon. This paper advances that notion and describes how these theories can be integrated to provide a framework for understanding the cognitive processes of small groups in a classroom setting. ..PAT.-Conference Proceeding [ABSTRACT FROM AUTHOR]

Previous work has examined how new theoretical, methodological, and design frameworks for engaging classroom learning are provoked and supported by the highly interactive and group-centered capabilities of a new generation of classroom-based networks. This paper extends this ‘mathematics structuring the social sphere’ analysis and presents a taxonomy of kinds of generative activity useful both in clarifying the relations between kinds of generative activity and in clarifying internal aspects of generative design, especially as supported by next-generation classroom networks. Examples of pre-service and in-service mathematics teachers’ generative classroom activities are to be presented and discussed. [ABSTRACT FROM AUTHOR]

Many jurisdictions are implementing a concentrated professional development program accompanied by focused goal-setting as a method for improving mathematics achievement. This paper presents a research project designed to study the implementation of one such initiative. [ABSTRACT FROM AUTHOR]

Despite over a decade of active research on the organization of collaborative learning environments, these remain more the exception than the rule. Additional research that focuses closely on teachers’ successful enactment of reform pedagogies is needed (Boaler, 2003). This paper presents results from an in-depth longitudinal case study of an accomplished teacher who regularly organizes collaborative mathematical discussions in her classrooms. The analyses suggest that the teacher guides the development of mathematical ideas through collaborative by attending to students’ thinking, high level task implementation, and her conception of how students develop proficiency with the subject matter. [ABSTRACT FROM AUTHOR]

Currently, the precise characterization of advanced mathematical thinking (AMT) is a subject of some controversy in the mathematics education community. On both sides of the Atlantic, researchers have used the term to describe a goal we would like our undergraduates to achieve -- the ability to do mathematics as mathematicians do. However, defining AMT for purposes of research has been problematic. In this paper, I present a framework that both characterizes AMT and provides a means of analyzing undergraduatesf use of AMT. [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 on a study that investigated an approach for preservice elementary teachers to develop flexibility in their thinking of and use of word problems in their teaching of mathematics. [ABSTRACT FROM AUTHOR]

In a group professional development project, teachers’ participation is an important structure of the professional development process because their participation influences each other’s learning experiences in the group. Wenger (1998) defines learning as follows: ‘For individuals,…learning is an issue of engaging in and contributing to the practices of their communities’ (p. 7). Therefore, the way the teachers participate and interact with others can be considered as an important factor of their own learning as well as the development of their group’s understanding. In fact, some teachers play a significant role in helping the group move forward and make a pivotal contribution to the group. These teachers serve as key members by creating learning opportunities not only for themselves but also for the whole group. This paper describes two key teachers’ cases while they are participating in a one-year professional development project: Kathy, a beginning teacher, and Jenny, an experienced teacher. While the two teachers had different backgrounds in terms of teaching and professional development experiences, they revealed common characteristics in the way they participated in the project. The purpose of the paper is to identify these key teachers’ characteristics and role in the group teacher learning context. [ABSTRACT FROM AUTHOR]

Number sense is a key component in being able to make sense of problem situations that involve fraction addition. Positioning students to invent algorithms for adding fractions involves their ability to use number sense-based reasoning. This paper offers five forms of mathematical interaction that teachers engaged in during classroom discussions to support student engagement in number sensebased reasoning when estimating fraction sums. [ABSTRACT FROM AUTHOR]

In this paper we report the interview questions used in a study of middle school students' mathematical problem solving behaviors which were chosen based on Vygotsky's concept formation theory and Berger's appropriation theory. We discuss the task design/selection process along with the findings associated with the use of these tasks so to provide new direction for gauging research on mathematical problem solving. [ABSTRACT FROM AUTHOR]

This paper describes research that fosters, documents, and analyzes a collaborative project between a university researcher and local middle and high school teachers. An operational framework of centering the teaching of mathematics is provided. The paper provides a theoretical analysis of how mathematics teachers, as individuals and as a professional community, learn to center their mathematics teaching on students in an urban setting. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This study reports on changes in preservice teachers' knowledge of content, pedagogy, and school cultures developed during field experiences in their mathematics courses. Analysis of data points to positive outcomes and reactions. This paper will address the motivating factors, design issues, and results of implementing field experiences for prospective K-8 teachers in the context of their mathematics coursework. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper reports on the first year of a multi-year project on preservice teachers' learning about posing problems, interpreting students' thinking, and responding to students mathematical ideas. We focus on insights encountered while refining definitions of the practices, constructing an Opportunities to Learn map across teacher preparation experiences, and exploring research instruments for studying teacher candidates' learning over time. ..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 presents results of an intervention study of preservice secondary mathematics and science teachers and their learning about both traditional (print) literacies and new literacies of the Internet as they relate to middle and high school pupil learning of subject matter. This investigation was part of a two-year project funded by the Carnegie Corporation's Adolescent Literacy Preservice Initiative. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Recently has been an interest in the significance of complexity theories for educational research. Complexity theory allows researchers to experience a shift in thinking to a "middle way" mentality of proscriptive studies. Common principles of researching in this middle way, however, have not yet been articulated. The paper reports on an inquiry about methodological commonalities, the principles that appear to guide complexity-based studies. The inquiry involved content analysis of seven complexity based theses. The commonalities amongst others include: task oriented research, researchers as close participant-observer, multiple research observers and diverse data interpretations. We argue that to the extent that studies framed by complexity theory have common modes of inquiry that are unique, complexity theory is a methodology. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper examines the various compromises teachers make between district mandates and meeting student needs. Using classroom observations analyzed with the QUASAR framework, (Stein, Grover, & Henningsen,1996) we found that, even in a high-accountability climate, teachers in the same school made strikingly different mathematics instructional decisions. We found none of the teachers using the mandated text to teach for understanding. ..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 discusses the learning experiences of pre-service teachers in a project called Tech-knowledgy for Building Mathematical Knowledge, where they explored mathematics concepts using various models and representations via virtual manipulatives. As teachers learned how to select and evaluate effective virtual manipulatives and incorporate them meaningfully in teaching mathematics, they gained confidence in their ability to model mathematics using technology. ..PAT.-Unpublished Manuscript [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]

Teachers must be prepared to teach mathematics to all students. In this paper, we describe an on-going study and the preliminary results of preservice elementary teachers' beliefs about teaching diverse learners in mathematics. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Preservice elementary teachers' learning of mathematical content, teachers' instruction in mathematics, and parents' work with their children on mathematics outside of school can all be enhanced through a focus on how children learn and understand mathematics. This paper will describe our efforts with all three audiences to improve the mathematics education of children by focusing on children's mathematical thinking. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This research synthesizes studies that used a Digitalized Interactive Component (DIC) to assess K-12 student mathematics performance during Computer-based-Assessments (CBAs) in mathematics. A systematic search identified ten studies that categorized existing DICs according to the tools that provided language assistance to students and tools that supported students problem solving. We report on the one study that involved students with learning disabilities and three studies involved English Language Learners. One study focused on assessing geometry content and four studies targeted on number and operations understanding. For other studies included a mixture of mathematics domains. Mixed results were reported as to the effectiveness of the availability of DICs. The research suggests that older children were more likely to benefit from availability of the DIC than younger children, and that DICs have greater impact on students with special needs. [ABSTRACT FROM AUTHOR]

This paper is based on a project that investigated teacher thinking in implementing an application -oriented high school mathematics curriculum with a focus on projects. Applications are central to mathematics education as a basis of demonstrating and validating the utility aspect of mathematics. Applications can take different forms in school mathematics, e.g., word problems [WP], projects, investigations. Traditional school curricula have generally focused on WP. Verschaffel (2002) described the goal of WP as "to bring reality into the mathematics classroom, to create occasions for learning and practising the different aspects of applied problem solving, without the practical … inconveniences of direct contact with the real world situation [p. 65]." The same could be said for projects with a real-world orientation. However, traditional WP and traditional instruction have been critiqued for not accomplishing these goals in a meaningful or effective way (e.g., Lave, 1992; Nesher, 1980). In particular, there is concern expressed about the lack of sense making or suspension of sense making (Verchaffel et al., 2000) when working with contextual problems, particularly in connecting school mathematics and reality. In the current movement to reform the teaching of mathematics, e.g., the NCTM standards 1989, applications in a variety of forms are advocated as worthwhile mathematical tasks to enhance the learning of mathematics. An application-oriented, project-based curriculum, then, has the potential of enhancing school mathematics. However, the curriculum by itself is unlikely to make a significant difference in the classroom. How it is realized will likely depend on the teacher. This paper reports on a study that investigated mathematics teachers’ perspective of what is necessary to teach this curriculum in a non-traditional way. [ABSTRACT FROM AUTHOR]

This paper is a case study report from the initial stage of a project to develop an approach to lesson study in courses for elementary teacher candidates. The initial stage involved the development and piloting of an assignment to introduce students to the process of lesson study. There is a growing body of literature on lesson study and its use in professional development (cf., Fernandez, 2002; Murata & Takahashi, 2002; Presmeg & Berrett, 2003; Stigler & Hiebert, 1999). Hiebert, Morris, and Glass (2003) suggest that lesson study is valuable, not only to teachers in the field, but to those in education faculties. The model discussed in this paper involves teacher candidates participating in mini lesson studies of their own lessons. The paper provides a discussion of some results and an analysis of the work of four participants. [ABSTRACT FROM AUTHOR]

This paper focuses on Mary Boole's construct of teacher lust and its impact on a collegiate mathematics classroom. Within a course for preservice middle grades school teachers, one mathematical task is analyzed for incidents of experienced as well as enacted teacher lust. The author presents his conception of the construct, provides examples within the discusion of a mathematical task known as the marshmallow problem, and offers antecedents connected to the influences of teacher lust. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

This paper considers the ways that teachers' identities for teaching shape the ways that they engage with and plan to implement new curricula in the context of sustained professional development. Findings from a five-year long intensive professional development suggest that teachers' identities are constructed through their engagement in particular figured worlds, and more importantly, that their experiences within these figured worlds come to shape the ways they think about specific aspects of their teaching. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

There has been a strong resurgence of interest of late regarding the role of neuroscience in informing education and vice versa, and the same is holding true regarding mathematics education. This paper proposes to help move studies in the psychology of education forward by demonstrating results from one such effort in mathematics education research. ..PAT.-Unpublished Manuscript [ABSTRACT FROM AUTHOR]

Students' aesthetic and affective responses are interrelated and both central to mathematics learning. This new working group will bring together researchers as a community to explore the connection between affect and aesthetic, and how this connection can help to understand how students experience mathematics. The goals of this working group are to evaluate the state of the field, build shared terms, and identify research questions for further inquiry. [ABSTRACT FROM AUTHOR]

This ethnographic study of one predominantly Latinx-serving high school in California theorizes around the functioning of re/humanizing pedagogies layered through the school and the mathematics department as they challenged flattened social, academic, and mathematical identities. Findings revealed that math department practiced built from school-wide commitments, offering students deflattened identities. However, challenges remained specifically related to the availability of advanced mathematics courses. [ABSTRACT FROM AUTHOR]

The development of proofs and argumentation is one of the major standards for mathematical practices in K-12 education that researchers and practitioners alike are continuing to improve. Further, the use of discourse is considered essential in the learning of mathematical concepts at these levels. However, K-12 educators continue to confound how best to utilize student interpersonal discourse to advance the development of mathematical arguments. This qualitative study examines the nature of student discourse in small-group interactions as students create collective arguments based on mathematical evidence. In examining the patterns of discourse in small groups, the study concludes the effectiveness of various types of discourse in peer-to-peer interaction as students develop more analytical thoughts through the support of their discourse with one another in creating proofs and arguments. [ABSTRACT FROM AUTHOR]

This working group continues to develop a research program on the practice of launching mathematical tasks and its resulting impact on learners. Our research agenda contains two strands of inquiry exploring (1) theory and framework building concerning effective launches and (2) empirical examination of the link between launching and opportunities for students to engage in “productive struggle” while solving worthwhile mathematical tasks. Participants will (a) examine empirical data collected from K-12 mathematics teachers in order to further develop our emerging Launch Framework and (b) discuss the next steps to be taken in the development of an empirical research agenda examining launching and high demand tasks. Participants will collaborate to shape the development of the emerging research agenda and plan future research and dissemination. [ABSTRACT FROM AUTHOR]

The practice of prototyping is challenging to novice designers as they underutilize insights that prototyping offers to solving design problem. Central to this challenge is the abstract nature of design concepts like idea representation, iteration, and problem solution-space exploration. A unique opportunity from mathematics education presents itself for design educators and facilitators; that is, teaching with manipulatives. We seek to transfer such practices in mathematics education to design education and practice. Challenges exist for design researchers to carefully craft activities in design education mainly because of the open-endedness of problems, the decision-making that takes place while designing, and the inherent uncertainties in the design problems. Ultimately, the goal is to develop students’ ability to flexibly transfer expertise to other contexts and new design challenges. [ABSTRACT FROM AUTHOR]

Analysis of data from cognitive interviews with 43 sixth grade students who are English Learners (ELs) identified students’ strengths with making meaning of mathematics tasks, specific challenges with oral language, and complexity of students’ language preferences. We discuss our use of cognitive interviews, and the findings and implications for supporting teacher learning and designing mathematics instruction for ELs. [ABSTRACT FROM AUTHOR]