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Many scholars have argued that mathematics classrooms often offer narrow conceptions of mathematical excellence, recognizing only some kinds of thinking and some kinds of people as valuable, and conflating mathematical aptitude with overall intelligence. Play offers the potential to disrupt such classroom mathematical practices, by offering new and broader ways to exercise agency, and, relatedly, more expansive visions of who is seen as mathematically capable. Offering an in-depth analysis of the participation of two students as they engage in mathematics in their Kindergarten class, whole group rug time and small group play centers, we investigate how different activity structures create space for students to exercise agency in ways that demonstrate multiple forms of competence, creating liberating mathematical spaces. [For the complete proceedings, see ED658295.]

Dramatic differences in children's math knowledge at school entry are thought to originate in the Home Math Environment (HME), where parents and caregivers are the primary provider of experiences that influence children's early math knowledge development. Little is known about what informs parent decision-making around "mathematical parenting" (i.e., parents' cognitions, motivations, and behaviors that impact and influence child math development in the HME). This study uses the RESET Framework and survey instrument to investigate parents' mathematical parenting perceptions (n = 847) across the domains of Role, Expectations, Skills, Efficacy, and Time. Parent self-reports of early childhood math knowledge and of shared math activity are also examined to shed light on the factors that influence mathematical parenting of 4-5-year-old children in the home. [For the complete proceedings, see ED658295.]

Bridging the gap between mathematical learning at home and school has been an issue for education research for decades (Galindo & Sheldon, 2012). Expectations for mathematics do not often align for teachers and parents (Posey-Maddox & Hayley-Lock, 2016) and a limited view of what counts as mathematics persists. What needs more attention is the meaningful mathematical learning that happens at home but is rarely seen as mathematics. Parents frequently struggle in supporting their children's mathematical learning, but that struggle becomes productive when parents are recognized as mathematically capable. This paper shows how two mothers shift their perspectives of what counts as mathematics and recognize the rich content in their current interactions with young children. Making such connections between mathematics and parent action can strengthen the relationship between at-home and school learning. [For the complete proceedings, see ED630060.]

In this paper we address the question, how do quantitative reasoning and covariational reasoning present as students build structural conceptions of real-world situations. We use data from an exploratory teaching experiment with an undergraduate STEM major to illustrate the explanatory roles quantitative reasoning and covariational reasoning play in, (a) coordinating more than two interdependent quantities, (b) conceiving of real-world situations in more than one way, (c) constructing networks of quantitative relationships, and (d) creating a mathematical expression. We make the case that looking at mathematical model construction through the lens of quantitative reasoning and covariational reasoning may provide insights into students' mathematical decisions as they structure complex real-world scenarios. [For the complete proceedings, see ED630060.]

Several researchers have promoted reimagining functions and graphs more quantitatively. One part of this research has examined graphing "conventions" that can at times conflict with quantitative reasoning about graphs. In this theoretical paper, we build on this work by considering a widespread convention in mathematics teaching: putting related, derived graphical objects (e.g., the graphs of a function and its inverse or the graphs of a function and its derivative) on the same set of axes. We show problems that arise from this convention in different mathematical content areas when considering contextualized functions and graphs. We discuss teaching implications about introducing such related graphical objects through context on separate axes, and eventually building the convention of placing them on the same axis in a way that this convention and its purposes become more transparent to students. [For the complete proceedings, see ED630060.]

In the mathematical community, two notions of "function" are used: the set-theoretic definition as a univalent set of ordered pairs, and the Bourbaki triple. These definitions entail different interpretations and answers to mathematical questions that even a secondary student might be prompted to answer. However, mathematicians and mathematics educators are often not explicit about which definition they are using. This paper discusses these parallel usages and the related implications for the field of mathematics education. [For the complete proceedings, see ED629884.]

Noticing children's mathematical thinking is foundational to teaching that is responsive to children's thinking. To better understand the range of noticing expertise for teachers engaged in multiyear professional development, we assessed the noticing of 72 upper elementary school teachers using three instructional scenarios involving fraction problem solving. Through a latent class analysis, we identified three subgroups of teachers that reflected different profiles of noticing expertise. Consideration was given to the noticing component skills of attending to children's strategy details, interpreting children's understandings, and deciding how to respond on the basis of children's understandings. We share theoretical and practical implications for not only the three profiles but also our choice to explore separately two versions of deciding how to respond (deciding on follow-up questions and deciding on next problems). [For the complete proceedings, see ED630060.]

Isomorphism and homomorphism are topics central to abstract algebra, but research on mathematicians' views of these topics, especially with respect to sameness, remains limited. This study examines 197 mathematicians' views of how sameness could be helpful or harmful when studying isomorphism and homomorphism. Instructors saw benefits to connecting isomorphism and sameness but expressed reservations about homomorphism. Pedagogical considerations and the dual function-structure nature of isomorphism and homomorphism are also explored. [For the complete proceedings, see ED630060.]

This study aims to describe how preservice secondary mathematics teachers (PSMTs) reason about different function representations. The study focuses on two PSMTs' reasonings across static and dynamic representations of functions. Sfard's (2008) Theory of Commognition guided our analysis. Findings indicate that while static representations restrict attention given to covariation, dynamic representations support PSMTs' reasoning about covariation including making connections to how covariation is represented in static graphs. [For the complete proceedings, see ED630060.]

This article describes the results of an investigation based on a Models and Modeling Perspective [MMP]. We present the evolution of the models built by university students when solving a model development sequence designed to promote their learning of the exponential function. As a result, we observed that students' thinking was modified, expanded, and refined, as they developed different iterations of their models. Students' models evolved by creating, first, linear models that required direction; second, models where there was no dissociation between linear and exponential behavior; then, situated exponential models; and finally, sharable, and reusable exponential models. [For the complete proceedings, see ED630060.]

People often view mathematics as abstract, cold, and irrelevant to real-life, and their school experiences influence such views. In this case study, we investigated the mathematics learning experiences of two women who participated in an afterschool girls STEM club 26 years ago. We explored their experiences in and out of school and how such experiences informed their images of mathematics. Data were collected from a survey, focus group interviews, and individual interviews. Using qualitative analysis, we learned that their school mathematics experiences influenced the participants' images of mathematics. The findings also revealed the participants' continuous and discontinuous learning experiences between school and out-of-school mathematics. This study suggests creating spaces to develop curricula that bridge the gap between school and out-of-school learning experiences. [For the complete proceedings, see ED630060.]

As mathematics educators, we teach and research a particular form of knowledge. However, in reacting to Platonic views of mathematics, we often overlook its unique characteristics. This paper presents a Kantian and Piagetian perspective that defines mathematics as a product of psychology. This perspective, based in human activity, unites mathematical objects, such as shape and number, while explaining what makes mathematics unique. In so doing, it not only privileges mathematics as a powerful form of knowledge but also empowers students to own its objects as their own constructions. Examples and interdisciplinary research findings (e.g., neuroscience) are provided to elucidate and support the perspective. [For the complete proceedings, see ED606531.]

In this study, we investigated two students', ages ten and eleven, emotions while they engaged in mathematical problem solving. During three task-based interviews, the students explored parts of the unsolved problem the Graceful Tree Conjecture. While they were engaged in the interviews, they self-identified the emotions of frustration and joy they were feeling using the Wong-Baker Scale. The students displayed the interplay of the emotions of frustration and joy or which we consider to be productive struggle. A descripted case of Georgia is included to describe her emotions while problem solving. [For the complete proceedings, see ED629884.]

This study examines pre-service teachers' (PSTs') views of tasks that engage students in mathematical argumentation. Data were collected in two different mathematics courses for elementary school education majors (n = 51 total PSTs). Analyzed were (a) written journals in which PSTs defined tasks that promote student engagement in argumentation, (b) tasks PSTs posed to engage students in mathematical argumentation, and (c) accompanying explanations in which PSTs motivated tasks they posed. The analysis revealed that PSTs interpret tasks that foster argumentation in terms of activities of argumentation that a task elicits and space for argumentation that the task provides. Several features that PSTs associated with each of the two major task characteristics were identified. While posing tasks to engage students in argumentation, PSTs did not place equal emphasis on all of the identified features. [For the complete proceedings, see ED629884.]

This report proposes a framework for describing student analogical reasoning activities in abstract algebra that moves beyond the traditional literature-based treatment of analogical mapping. The Analogical Reasoning in Mathematics (ARM) framework captures the activities that students engage in when anticipating, creating, and reasoning from mathematical analogies. This considers activities along several dimensions including: inter/intra domain activity, foregrounded/backgrounded domain, and attention to similarity/difference. These dimensions are integrated with Gentner's (1983) analogical mapping framework to characterize student activity when they are presented with tasks where reasoning by analogy can productively support their mathematical investigations. By characterizing these activities, we can better develop tasks to support students in productively analogizing between mathematical domains. [For the complete proceedings, see ED629884.]

We report on findings from two one-on-one teaching experiments with prospective middle school teachers (PTs). The focus of each teaching experiment was on identifying and explicating the mental processes and types of intermediate, supporting reasoning that each PT used in their development of combinatorial reasoning. The teaching experiments were designed and facilitated to guide each PT toward reinventing multiple combinatorial formulas. Drawing on a subset of this data, we describe the development of the PTs' mental processes and reasoning as they came to construct formulas for counting permutations and arrangements without repetition, and we analyze our findings through a psychological constructivist framework. [For the complete proceedings, see ED629884.]

One important aspect of teaching is reviewing tasks in preparation for instruction. The goal of the multicase study of four secondary teachers was to examine the interplay between their mathematical knowledge for teaching [MKT] and what they attend to when reviewing a mathematics task. We engaged secondary mathematics teachers in a semi-structured, clinical interview focused on a nonroutine mathematical task involving exponential growth. The results suggest experienced teachers may not explicitly attend to learning opportunities in their review of a task, and their own mathematical work contributes to their anticipation of student work and thinking. This work highlights how researchers focused on MKT can use clinical interviews as a tool for extracting and describing a teacher's MKT. [For the complete proceedings, see ED629884.]

We report our findings and perspective to document the knowledge exhibited by three experienced high school teachers in their instructional decisions for lessons on the equation of an exponential function. We describe the nature of the mathematical ideas and connections teachers promoted in discourse and the decisions that supported the emergence and connections of the mathematics. Despite similarities in the structure of the mathematical activities, differences existed in the ideas that emerged in the three teachers' discussions regarding the relationship between the exponent value and the independent variable. We describe links between collections of teacher decisions to their influences on the mathematics discourse. [For the complete proceedings, see ED629884.]

Using neurodiversity as our theoretical framework, rather than a deficit or medical model, we analyze the narratives of five dyslexic research mathematicians to find common strengths and challenges for dyslexic thinkers at the highest level of mathematics. We report on 3 themes: (1) highly visual and intuitive ways of mathematical thinking; (2) pronounced issues with memorization of mathematical facts and procedures; and (3) resilience as a strength of dyslexia that matters in mathematics. We introduce the idea of Neurodiversity for Mathematics, a research agenda to better understand the strengths (as well as challenges) of neurodiverse individuals and to use that knowledge to design better mathematical learning experiences for all. [For the complete proceedings, see ED629884.]

Generalization is a critical aspect of doing mathematics, with policy makers recommending that it be a central component of mathematics instruction at all levels. This recommendation poses serious challenges, however, given researchers consistently identifying students' difficulties in creating and expressing normative mathematical generalizations. We address these challenges by introducing a comprehensive framework characterizing students' generalizing, the Relating-Forming-Extending framework. Based on individual interviews with 90 students, we identify three major forms of generalizing and address relationships between forms of abstraction and forms of generalization. This paper presents the generalization framework and discusses the ways in which different forms of generalizing can play out in activity. [For complete proceedings, see ED581294.]

This paper reports from a broad investigation of mathematics knowledge as dependent on interconnected concepts. The paper focuses specifically on illustrating how network analysis may be used in examining spatiotemporal relationships between learned mathematics concepts, or curriculum outcomes, and concepts inherent in assessment items. Connections both within and between year levels are shown, based on primary years' multiple-choice assessment items related to measurement. Network analysis provides a potentially powerful tool that may offer educators greater specificity in approaches to the design of revision and intervention through a view of complex rather than linear conceptual connectivity in mathematics learning. [For the complete proceedings, see ED597799.]

We explore the epistemological issues that arise when considering STEM as a curricular and instructional construct. Our approach is somewhat unique in that we are not focused on the curricular or instructional boundaries of STEM education, but consider the nature of the cognitive activity at play during STEM-focused activity, with an emphasis on mathematical thinking. We focus specifically on the epistemological underpinnings of mathematics and other STEM disciplines, and the possibility of an epistemology of STEM as a curricular construct. The implications on students' STEM ways of thinking (SWoT) are discussed in detail from a theoretical and empirical lens. Future research directions are identified. [For the complete proceedings, see ED606556.]

This study was conducted to gain understanding about potential influences that learning about quadratic functions has on high school algebra students' action versus process views of linear functions. Pre/post linear functions tests were given to two classrooms of Algebra II students (N=57) immediately before and immediately after they participated in a multi-day unit on quadratic functions. The purpose was to identify ways that their views of linear functions had changed. Results showed that on some measures, students across both classes shifted their views of linear functions similarly. However, on other measures, the results were different across the classes. These findings suggest that learning about quadratic functions can influence students' action or process views of linear. Furthermore, the instructional differences between classes provide insights into how to promote those influences that are productive for students' views. [For the complete proceedings, see ED606556.]

In this paper we propose a refinement of Michener's (1978) well-known example classification based on data from university mathematicians. The refinement takes into account the mathematician's perspective on the role of examples in doing mathematics. More specifically, our work provides insight into the ways in which mathematicians talk about using examples in their scholarly work and their work with students. The proposed classification has the potential to inform our work as teachers as we strive to create opportunities to engage students in authentic mathematical work. [For the complete proceedings, see ED583989.]

This paper explores the role examples play in mathematicians' conjecturing activity. While previous research has examined example-related activity during the act of proving, little is known about how examples arise during the formulation of conjectures. Thirteen mathematicians were interviewed as they explored tasks that required the development of conjectures. During the interviews, mathematicians productively used examples as they formulated conjectures, particularly by creating systematic lists of examples that they examined for patterns. The results suggest pedagogical implications for explicitly targeting examples in conjecturing, and the study contributes to a body of literature that points to the benefits of exploring, identifying, and leveraging examples in proof-related activity. [For the complete proceedings, see ED597799.]

In this paper I present a case study to illustrate conflicts between different 'realizations' of the concept of 'continuous function' held by a university first year student. Sfard's commognitve framework is used in the analysis of a student's work on continuity. I point out how these conflicting realizations have arisen from the inconsistent definitions presented in text books and other mathematical resources. The study also points to the need of extending the notion of "commognitive conflict" in the framework. [For the complete proceedings, see ED597799.]

In this paper, we describe the process of students' self-assessment of their creativity and its development in the context of posing mathematical problems, presuming that such a process would support the development of their creativity. Examination of two case studies reveals that self-assessment of creativity may support its development provided that one possesses specific personal recourses; however, this process might suppress the creativity of those lacking the needed resources. Therefore, we suggest that self-assessment of creativity cannot stand on its own, and should be supplemented by teachers' feedback or other environmental 'scaffolding'. [For the complete proceedings, see ED597799.]

In this paper we examine a commonly suggested proof construction strategy from the mathematics education literature--that students first produce an informal argument and then use this as a basis for constructing a formal proof. The work of students who produce such informal arguments during proving activities was analyzed to distill three activities that contribute to students' successful translation of informal arguments into formal proofs. These are elaboration, syntactification, and rewarranting. We analyze how attempting to engage in these activities relates to success with proof construction. Additionally, we discuss how each individual activity contributes to the translation of an informal argument into a formal proof. [For the complete proceedings, see ED597799.]

Quantitative reasoning plays a crucial role in students' and teachers' successful modeling activities. In a semester-long teaching experiment with an undergraduate student, we explore how her conception of a graph plays a role in her ability to quantify and maintain quantitative structures. We characterize here Lydia's conception of a graph as one in which the graph entails several quantities she identified in a given dynamic situation, contradicting the conception of a graph as a representation of a multiplicative object consisting of only two quantities. We also discuss her thinking about her graph in terms of figurative and operative thought during a session in which we support her in disembedding and graphically representing quantities. [For complete proceedings, see ED581294.]

Examples play a critical role in the exploration and proving of conjectures. Although proof has been studied extensively, the precise ways in which examples might facilitate successful proofs are not well documented or understood. Working within a larger set of studies that argue for the value of examples in proof-related activity, in this paper we present a case study of one mathematician's work on a conjecture in which his strategic, intentional use of examples led to a proof of that conjecture. By examining his work in detail, we highlight specific mechanisms by which the mathematician's examples led to successful proof production. These mechanisms shed light on precise ways in which examples can directly lead to proof and inform our understanding of the conceptual landscape of the interplay between examples and proof. [For the complete proceedings, see ED584443.]

In this paper, I examine the utility of a mathematics learning trajectory as a tool to support teachers' attention to students' mathematical thinking. I present findings from one second grade teacher's use of a learning trajectory as she planned a sequence of three mathematics lessons. Findings suggest learning trajectories support teachers in choosing appropriate tasks and learning goals, and in anticipating students' likely approaches and difficulties. Learning trajectories, as representations of student thinking, provide teachers with a means of evaluating evidence of student learning of intended goals and afford them with a range of instructional moves based on their students' current conceptions. [For the complete proceedings, see ED584829.]

Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, we present a framework that characterizes ways in which mathematicians utilize examples when investigating conjectures and developing proofs. The data consist of 133 mathematicians' responses to two open-ended survey questions. The framework offers categories for the types of examples, uses of examples, and example strategies that mathematicians discussed in reference to their work with conjectures. In addition to presenting the framework, we also discuss potential educational implications of the results. [For the complete proceedings, see ED584829.]

In this paper, we problematize an ontological characterization of change within a complex system by illustrating how epistemological premises of interactionist, individualist, and collectivist theoretical perspectives reveal only specific aspects of a changing system. Methodological considerations resulting from our recognition that change is characterized subjectively within various theoretical perspectives are made. [For the complete proceedings, see ED584829.]

This paper examines ten K-8 preservice teachers' visual representations of fraction operations using the four main pattern blocks. Data consist of figures made using the pattern blocks, drawn colored representations, and detailed written comments and algebraic formalism. The theoretical framework is drawn from representational theories and analyses of fraction operations, and work on coordination of different levels of units. The main result is that only those teachers meaningfully coordinating the different referent units in the fraction situations, were the ones consistent in their representations and reasoning, and in successfully establishing referential commutativity for multiplication of fractions. [For the complete proceedings, see ED585874.]

In this paper we report on a study examining how teacher perspectives on mathematics content and pedagogy changed after the teachers' schools participated for one year in the Mathematics Coaching Program. Teacher narrative responses to questions about student work samples provided qualitative data for analysis. Questions asked for teacher input on student thinking and instructional decisions. Responses reveal teacher change in the area of the program goals of movement toward an integrated procedural/conceptual perspective on content and a learner responsive perspective on pedagogy. [For the complete proceedings, see ED585874.]

Using an innovative recruitment tool, the social network site Facebook, survey data were gathered from samples of the Australian general public and from around the world. Views on the gendering of mathematics, science, and ICT were gathered. In this paper we report the findings from six of the 15 questions on the survey, and only from respondents in predominantly English-speaking countries. The findings reveal that the majority was not gender-stereotyped about mathematics and related careers. However, if a gendered view was held, it was overwhelming to endorse the male stereotype. Male respondents' views were more strongly gendered than were females'. [For the complete proceedings, see ED585874.]

We studied the linguistic norms of mathematical proof writing at the undergraduate level by asking two mathematicians and five mathematics undergraduate students to read seven partial proofs based on student-generated work and to identify and discuss uses of mathematical language that were out of the ordinary with respect to standard mathematical proof writing. By asking participants to discuss the seriousness of each breach, we not only identify and discuss some of these linguistic norms, but also describe important differences between the ways in which mathematicians and students understand them. [For the complete proceedings, see ED583989.]

In this study, we surveyed 570 Calculus I and Calculus II students at two large public universities in the Northeastern region of the United States. We explored the relationship between these students' career goals in mathematics and other STEM fields, with their mathematics identity and self-identified personality attributes. Our findings suggest that mathematics identity can be used as a way of explaining persistence in mathematics and other STEM fields. We also found certain personality attributes to be correlated with persistence in these fields and these personality attributes varied based on gender. We conclude with a detailed analysis of our findings and some implications. [For the complete proceedings, see ED583989.]

An examination of past research, policies, and reforms in mathematics education suggests that there have always been, and remain, tensions in conceptualizing the aims and goals of mathematics teaching and learning. While the disproportionality and conditions of marginalized learners is a cause for concern, it is important to understand that addressing the needs of these learners may not have been the primary goal of prior policies and reforms in mathematics education. Derrick Bell, a former attorney with the National Association for the Advancement of Colored People (NAACP) during the Civil Rights Era, employed his interest-convergence principle to explain how the United States Supreme Court issued the landmark ruling in "Brown v Board of Education of Topeka, Kansas" ("Brown I") in 1954. The Supreme Court's ruling in the "Brown" case revoked the "separate but equal" doctrine, which legally sanctioned segregation in public education and all aspects of daily life. Bell (2004) argued that the "Brown" decision was not the result of America coming to terms with its democratic ideals or moral sensibilities. Rather, the Supreme Court was more interested in providing "immediate credibility to America's struggle with communist countries to win the hearts and minds of emerging third world people" than in doing what was morally right (p. 233). "Brown" provided the impetus for legislation, such as the Elementary and Secondary Education Act (ESEA) of 1965 and it reauthorizations Improving America's School Act of 1994 and No Child Left Behind Act (NCLB) of 2001 (Zion & Blanchett, 2011). These along with other legislations and mandates established requirements that address the need to ensure that all students in the United States are provided equal educational opportunities. Although not fully realized, the requirements of these legislative acts and mandates created pressure to address the historical inequity in educational opportunity, achievement, and outcomes. Zion and Blanchett (2011) argued that the reason why large scale improvement in outcomes for all students have yet to be realized is that the problem has not yet been framed appropriately. The problem must be framed as part of the history and legacy of racism, and as an issue of civil rights and social justice, viewed through a critical lens. This article use a critical lens to apply the interest-convergence principle informed largely by the work of legal scholar, Derrick Bell (1980 & 2004), to examine motivating factors of policies and reform efforts in mathematics education. Specifically, this article makes the case that policies and reforms in mathematics education were not designed to address the needs of marginalized learners; rather these policies and reforms are often designed and enacted to protect the economic, technological, and social interests of the dominant culture. [For the complete proceedings, see ED583989.]

The framework of mathematical knowledge for teaching (MKT) is brought under the discursive framework of Commognition in order to track learning in professional development (PD). I follow MKT in differentiating between subject matter discourse and pedagogical discourse. The framework, which I call Mathematical Discourse for Teaching (MDT) permits a combined view on mathematical and meta-mathematical issues as constituted in discourse. Such meta-issues are found to be a significant part of what is taught and learned in a particular PD, where mathematics Ph.D. students teach elementary school teachers. Through the analysis of a lesson on parity I show how "knowing" has different meanings in mathematical and pedagogical discourses, and find evidence of learning in the evolving ways in which the parties use this term. [For the complete proceedings, see ED597799.]

We present a mathematical analysis that distinguishes two quantitative perspectives on ratios and proportional relationships: Multiple Batches and Variable Parts. We argue that (a) existing research on proportional relationships has addressed Multiple Batches but has largely overlooked Variable Parts, (b) Multiple Batches makes the co-variation aspect of proportional relationships more explicit, while Variable Parts makes the fixed multiplicative relationship between two quantities more explicit, (c) the distinction between Multiple Batches and Variable Parts is orthogonal to the within-measure-space versus between-measure-space ratio distinction, and (d) Variable Parts affords promising new approaches for addressing linear relationships. [For the complete proceedings, see ED597799.]

What is entailed in doing mathematics and becoming a mathematician? Working from an autobiographical sketch and biographies of Sofya Kovalevskaya (1850-1891), the first woman generally thought to have gained a doctorate in mathematics, and using the Lacanian notion of desire, I examine the forces that shape and influence engagement with mathematics. This work has consequences for mathematics education in examining the construction of identity/subjectivity in teachers and students alike. [For the complete proceedings, see ED597799.]

This study explored the problem solving experience of pre-service teachers in finding the greatest common factor and the least common multiple using many different approaches. In particular, it examined the effect of pre-service teachers' mathematics content knowledge on how they chose their preferred approach and how they valued the most efficient approach. The findings indicated that the most efficient approach was appreciated only if such approach was reasonably understood by these pre-service teachers. [For the complete proceedings, see ED597799.]

Third- through fifth-grade students participating in a classroom teaching experiment investigating the impact of an Early Algebra Learning Progression completed pre- and post-assessment items addressing their abilities to engage in functional thinking. We found that after a sustained early algebra intervention, students grew in their abilities to shift from recursive to covariational thinking about linear functions and to represent correspondence rules in both words and variables. [For the complete proceedings, see ED584829.]

This study presents a model of three conceptual advances in understanding quadratic functions based on a teaching experiment with 6 8th-grade students. Using a covariation approach, students investigated quadratic growth as a coordinated change of x- and y-values. Qualitative analysis yielded three major shifts in students' understanding: a) developing an understanding of the first and second differences for y as rates of growth with [triangle x] implicit; b) explicitly coordinating [triangle x] with the constant second differences; and c) coordinating covariation and correspondence views through meanings for the parameter "a" in y = ax[superscript 2]. [For the complete proceedings, see ED585874.]