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Covariational reasoning and the creation and interpretation of graphs of covariational situations are important skills in math and science. Unfortunately, research shows that students often struggle to make meaningful connections between graphs and the covariational situations they represent. Educational activities designed to help students overcome this struggle tend to use either student-generated or automatically-generated graphs, and have students either act out covariational situations or more passively observe them. In this paper, we present the design of a tool and task that enabled two students to simultaneously embody both the creation of a graph and the covariational actions that the graph represents. Through a process of collaborative instrumentation, the students made meaningful connections between their motions and the embodied traces they created as they reasoned about the covarying quantities of height and time. [For the complete proceedings, see ED630210.]

Despite significant research exploring students' quantitative reasoning, few studies have explored the semiotic processes that mediate its development. In this report, we present a case study to show how one student constructed a semiotic chain for a quantity as he worked with a mathematical task. Importantly, we connect frameworks for quantitative reasoning and semiotics to make sense of this process. Our findings show how our case student constructed a sign for a chunk of change in a triangle to support his later construction of the quantity of amount of change of area. We also describe how the case student leveraged these signs to bolster his development of the quantity of total area. We emphasize the role of artifacts, such as physical manipulatives, a digital applet, and a diagram, in this process. Finally, we discuss the implications of this analysis for future studies that explore students' constructions of quantity. [For the complete proceedings, see ED630210.]

In this proposal, we share research that explores the potential benefits of a novel Making experience within mathematics teacher preparation that we hypothesized would inform the pedagogical and curricular thinking of prospective teachers of elementary mathematics (PMTs). That experience had PMTs exploring at the intersection of content, pedagogy, and design to digitally design, 3D print, and share an original manipulative with a child to promote their mathematical thinking. We share several vignettes of our research that aim to discern some of the potential benefits the experience might offer PMTs. These take a variety of theoretical and methodological approaches at the intersections of teacher knowledge, identity, pedagogy, and design. Implications of our findings for teacher preparation and professional learning are provided throughout the paper and in its conclusion. [For the complete proceedings, see ED630210.]

Positioning teachers as designers of curricular resources invites opportunities for exploration at the intersection of content, pedagogy, and design. As researchers accepting greater responsibility for preparing teachers to maintain a commitment to their pedagogical vision in practice, this work seeks to cultivate the imagination of humanistic forms of mathematics teaching and learning by supporting these explorations. Toward that end, this paper reports on research that examines connections between the pedagogical/conceptual knowledge that prospective teachers embed in the designs of original manipulatives and how those designs mediate the pedagogical moves they make in teaching situations. The promise of this work is that these connections may reveal a viable means to support bolder connections between teacher preparation and practice. Implications of our findings for teacher preparation are considered. [For the complete proceedings, see ED630060.]

This work seeks to understand the emergent nature of mathematical activity mediated by learners' engagement with multiple artifacts. We explored the problem solving of two learners as they aimed to make sense of fraction division by coordinating meanings across two artifacts, one being a physical manipulative and the other a written expression of the standard algorithm. In addressing the question, "How do learners make sense of and coordinate meanings across multiple representations of mathematical ideas?" we took an enactivist perspective and used tools of semiotics to analyze the ways they navigated the dissonance that arose as they sought to achieve harmony in meanings across multiple representations of ideas. Our findings reveal the value of such tool-mediated engagement as well as the complexity of problem solving more broadly. Implications for learning mathematics with multiple artifacts are discussed. [For the complete proceedings, see ED630060.]

We describe how we supported middle-school students developing meanings for quadratic growth by bringing together three theoretical framings and designing a task sequence grounded in that framework. Specifically, this framework was assembled from the research on students' quantitative and covariational reasoning, Realistic Mathematics Education, and the theory of designing for mathematical abstraction. We present data from a teaching experiment to highlight how this task sequence supported students as they imagined constant increases in the amounts of change in one quantity, a kind of change they later understood to be a defining characteristic of quadratic growth. We conclude with a discussion of our findings and how our integration of the three frameworks proved useful to design a productive task sequence. [For the complete proceedings, see ED629884.]

In this paper, we describe an experience within mathematics teacher preparation that engages preservice teachers (PSTs) in Making and design practices that we hypothesized would inform their conceptual and pedagogical thinking. With a focus on the design of new tools to support mathematics teaching and learning, this Learning by Design experience has PSTs exploring at the crossroads of content, pedagogy, and Making. We report our findings of the variety of forms of knowledge that PSTs brought to bear on their design work. As the engagement and advancement of these forms of knowledge is essential to effective mathematics teaching, these findings suggest the promise of a making-oriented experience within mathematics teacher preparation coursework. [For complete proceedings, see ED581294.]