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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. [For the complete proceedings, see ED606556.]

This paper explores the nature of students' quantitative reasoning and conceptions of functions supporting their ability to symbolize quadratic function rules, and the meanings students make of these rules. We analyzed middle school students' problem solving activity during a small group teaching experiment (n = 6) emphasizing quadratic growth through covarying quantities. Results indicate four modes of reasoning supportive of students' symbolization of quadratic function rules: (a) correspondence, (b) variation and correspondence, (c) covariation, (d) flexible covariation and correspondence. We discuss implications for research on learning vis-à-vis students' representational fluency, as well as design principles to support quantitative reasoning. [For the complete proceedings, see ED583608.]

In this paper, we describe our learning progressions approach to early algebra research that involves the coordination of a curricular framework, an instructional sequence, written assessments, and levels of sophistication describing the development of students' thinking. We focus in particular on what we have learning through this approach about the development of students' abilities to generalize and represent functional relationships in a grades 3-5 early algebra intervention by sharing the different levels of responses we observed in students' written work and the percent of students situated at each level across different tasks.

We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional sequence of tasks based on objectives concerning content and algebraic thinking practices; assessments; and posited levels of sophistication in children's reasoning about algebraic concepts within big ideas of early algebra. This research balances the goals of longitudinal research on supporting students' preparedness for algebra while attending to the practical goals of establishing connections among curriculum, instruction, and student learning. [This paper was published in: Bartell, T. G., Bieda, K. N., Putnam, R. T., Bradfield, K., & Dominguez, H. (Eds.) (2015). "Proceedings of the 37th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education" (p201-204). East Lansing, MI: Michigan State University.]

This research targets children's informal strategies and knowledge of fractions by examining their ability to create, interpret, and connect representations in doing and communicating mathematics when solving fractions tasks. Our research group followed a constant comparative method to analyze clinical interviews of children in grades 2-6 solving fraction tasks. Several iterations of coding yielded an emergent coding scheme that helps capture the nuances of children's reasoning with multiple types of fractions representations. Initial results from the interview analyses suggest variation in children's reasoning across four categories: (a) model types, (b) children's use of representations and representational fluency, (c) connectedness of children's informal to formal reasoning type, and (d) meanings of fractions. We discuss the challenges of negotiating first-order versus second-order models of children's meaning of fractions, and the relationship between children's representational fluency and conceptions of fractions.

Partnerships between researchers and teachers are central to stimulating advancements in a linking research and practice agenda. This paper addresses two key aims. First, research on supporting students' representational fluency in technology-rich algebra learning environments is used to illustrate a linking research and practice agenda. Three themes are addressed as outcomes of this researcher-practitioner partnership: (a) the importance of addressing a shared problem of practice, (b) the role of theoretical lenses in supporting both practice and research, and (c) how to design for collaboration using both practice-based and research-based methods. Second, emerging issues in establishing productive researcher-practitioner partnerships are introduced from the perspective of both researchers and practitioners. Viable next steps for bridging the experiences of researchers and practitioners are discussed.

This research study was specifically concerned with the development, testing, and revision to an instructional theory for studying the mathematical concepts of equivalence and equation solving with multiple representations and multiple tools. Following a design research approach, a collaborative teaching experiment was conducted with a ninth-grade algebra teacher in which instruction was guided by a specifically designed sequence of tasks, techniques using paper-and-pencil and computer algebra systems (CAS), and theory on a hypothesized progression of learning. Retrospective analyses of data informed revisions to a resulting progression of learning and activity sequence that are being tested with pre-service secondary mathematics teachers. [For the complete proceedings, see ED584443.]

To better support students' conceptual understanding of algebra in the information age, we need an improved understanding of how students interpret connections between multiple representations using CAS as a representational toolkit. This research centers on the development of an analytic framework for categorizing students' connections between multiple representations, an indicator of students' representational fluency. Videotaped task-based interviews with high school algebra students solving equations using CAS were analyzed to generate the proposed analytic framework, which has a hierarchical structure based on the direction and purpose of students' translations between representations. [For the complete proceedings, see ED585874.]

Designing for and enacting social justice mathematics tasks is one approach to supporting students' meaningful mathematics learning. We share our design conjectures that were developed out of a theoretically grounded, empirically based investigation of undergraduate students' experiences of a social justice mathematics lab. We engaged an interdisciplinary lens of historically responsive literacy and identity in both the lab design and analyses of student responses. We have conducted four cycles of action research that contribute to a broader program of design-based research. Based on thematic analyses of qualitative data, we introduce the construct of locality-identity as a main theme in the data. Attending to locality-identity in the design of this lab increased student engagement with mathematics by making mathematics relevant and authentic. [ABSTRACT FROM AUTHOR]

In this exploratory action research study we investigated teaching mathematics for social justice in undergraduate mathematics. We asked: How if at all do undergraduate students consider the importance of local social justice issues as embedded in the context of a precalculus course assignment?; and As instructors, how might feedback from students' experiences of a social justice mathematics lab inform our future teaching and research practices? We discuss emerging findings from our initial analyses of student surveys. We also elaborate how these findings will inform our efforts to link research and practice at the intersection of supporting students' math cognition and exploring social justice issues through teaching mathematics for social justice. [ABSTRACT FROM AUTHOR]

In this theoretical report we focus on the issue of communicating learning trajectories (LTs) to researchers. There is great variation in the body of work on LTs including how researchers communicate what a LT entails, and the kinds of metaphors employed for making meaning of LTs. We elaborate possible affordances and limitations of different metaphors for LTs including "a garden path" and "growing flowers." This work has implications for how LTs are taken up by researchers, and also how LTs are leveraged to inform student-centered teaching practices. [ABSTRACT FROM AUTHOR]

The establishment of learning trajectories has become a key area of research in mathematics education, but there remains a need to characterize the role of instruction on students’ developing concepts. This paper offers a way to establish a connection between student concepts, tasks, and pedagogical supports. We draw on two constructs, the epistemic student and DNRbased instruction, and articulate how these constructs can work in concert to inform the three phases of design, enactment, and analysis of learning trajectories. We close with an example from a learning trajectory of middle-school students’ understanding of quadratic growth. [ABSTRACT FROM AUTHOR]

In this study we explored how preservice teachers’ (PTs) demonstrated representational fluency and functional thinking in approaching quadratic function tasks within a realizable task context. Participants were secondary school mathematics PTs in their final semester. We employed a multi-case study method with task-based interviews to deeply explore PTs problem-solving activity. The findings showed representational fluency pushed learners to switch back and forward between covariational and correspondence reasoning. [ABSTRACT FROM AUTHOR]

Representational fluency is key to understanding mathematical ideas is deeper ways. This research advances a framework for characterizing sophistication in representational fluency that builds on a structure of observed learning outcome (SOLO) taxonomy. Drawing on an analysis of interviews with 9th-grade algebra students solving equations using a computer algebra system and paper-andpencil, I discuss four levels of sophistication: prestructural, multistructural, unistructural, and relational. This characterization is sensitive to the nature of students' creations, interpretations, and meanings. Future directions for research are noted. [ABSTRACT FROM AUTHOR]

This paper explores the nature of students' quantitative reasoning and conceptions of functions supporting their ability to symbolize quadratic function rules, and the meanings students make of these rules. We analyzed middle school students' problem solving activity during a small group teaching experiment (n=6) emphasizing quadratic growth through covarying quantities. Results indicate four modes of reasoning supportive of students' symbolization of quadratic function rules: (a) correspondence, (b) variation and correspondence, (c) covariation, (d) flexible covariation and correspondence. We discuss implications for research on learning vis-à-vis students' representational fluency, as well as design principles to support quantitative reasoning. [ABSTRACT FROM AUTHOR]

We detail a learning progressions approach to early algebra research and how existing work around learning progressions and trajectories in mathematics and science education has informed our development of a four-component theoretical framework consisting of: a curricular progression of learning goals across big algebraic ideas; an instructional sequence of tasks based on objectives concerning content and algebraic thinking practices; assessments; and posited levels of sophistication in children's reasoning about algebraic concepts within big ideas of early algebra. This research balances the goals of longitudinal research on supporting students' preparedness for algebra while attending to the practical goals of establishing connections among curriculum, instruction, and student learning. [ABSTRACT FROM AUTHOR]