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Incorporating visual representations, rather than strictly relying on symbolic representations, is a research-based strategy for supporting fraction learning. However, students must also make transitions between visual and symbolic fraction representations to apply the conceptual understanding they gained from visual representations to symbolic fraction computation. Virtual manipulatives (VMs) offer opportunities for supporting students in making these transitions, as many fraction VMs integrate visual and symbolic representations into one manipulative. Some VMs also dynamically link the representations, so learners can observe how changes to one representation impact the other. For these features to support students in transitioning among representations, teachers must orchestrate opportunities for students to use and reflect on their use of the features. This study examined how six fourth- and fifth-grade teachers orchestrated opportunities in lesson plans for students to make transitions among the visual and symbolic representations within and beyond fraction VMs. Results showed that teachers used two strategies for orchestrating these transitions: VM choice and direct teacher intervention. Implications of teachers' uses of these strategies are discussed in terms of what kinds of transition opportunities were made available to students and what professional learning experiences could be needed to support teachers in orchestrating transitions.

This paper reviews 10 years of research on virtual manipulatives (VMs) in K-5 mathematics education with a focus on moving beyond generalized conclusions that VMs can support learning to better understanding how and when they support learning. Synthesis of fifteen articles that met the inclusion criteria revealed researchers are beginning to unpack how particular features of VMs and context variables interact with student learning outcomes. The influence of context variables, such as whether VMs are used by individual students or used collaboratively, suggests a need for greater attention to the role of the teacher in implementation of VMs. Attention to the teacher's role in defining how VMs are used could inform future directions for teacher education.

Incorporating computational thinking (CT) ideas into core subjects, such as mathematics and science, is one way of bringing early computer science (CS) education into elementary school. Minimal research has explored how teachers can translate their knowledge of CT into practice to create opportunities for their students to engage in CT during their math and science lessons. Such information can support the creation of quality professional development experiences for teachers. We analyzed how eight elementary teachers created opportunities for their students to engage in four CT practices (abstraction, decomposition, debugging, and patterns) during unplugged mathematics and science activities. We identified three strategies used by these teachers to create CT opportunities for their students: framing, prompting, and inviting reflection. Further, we grouped teachers into four profiles of implementation according to how they used these three strategies. We call the four profiles (1) presenting CT as general problem-solving strategies, (2) using CT to structure lessons, (3) highlighting CT through prompting, and (4) using CT to guide teacher planning. We discuss the implications of these results for professional development and student experiences.

In many discussions of the ways in which abstraction is applied in computer science (CS), researchers and advocates of CS education argue that CS students should be taught to consciously and explicitly move among levels of abstraction (Armoni "Journal of Computers in Mathematics and Science Teaching," 32(3), 265-284, 2013; Kramer "Communications of the ACM," 50(4), 37-42, 2007; "Wing Communications of the ACM," 49(3), 33-35, 2006). In this paper, we describe one way that attention to levels of abstraction could also support learning in mathematics. Specifically, we propose a framework for using abstraction in elementary mathematics based on Armoni's (2013) framework for teaching computational abstraction. We propose that such a framework could address an enduring challenge in mathematics for helping elementary students solve word problems with attention to context. In a discussion of implications, we propose that future research using the framework for instruction and teacher education could also explore ways that attention to levels of abstraction in elementary school mathematics may support later learning of mathematics and computer science.

Moving among levels of abstraction is an important skill in mathematics and computer science, and students show similar difficulties when applying abstraction in each discipline. While computer science educators have examined ways to explicitly teach students how to consciously navigate levels of abstraction, these ideas have not been explored in mathematics education. In this study, we examined elementary students' solutions to a commonplace mathematics task to determine whether and how students moved among levels of abstraction as they solved the task. Furthermore, we analyzed student errors, categorizing them according to whether they related to moves among levels of abstraction or to purely mathematical steps. Our analysis showed: (1) students implicitly shift among levels of abstraction when solving "real-world" mathematics problems; (2) students make errors when making those implicit shifts in abstraction level; (3) the errors students make in abstraction outnumber the errors they make in purely mathematical skills. We discuss the implications for these findings, arguing they establish that there are opportunities for explicit instruction in abstraction in elementary mathematics, and that students' overall mathematics achievement and problem-solving skills have the potential to benefit from applying these computer-science ideas to mathematics instruction.