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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.

Background and Context: We explored how learning trajectories (LTs) might be used to design variables instruction. Objective: We aimed to develop an LT for variables and use it to guide curriculum development for fourth graders working in Scratch in an integrated mathematics+CS curriculum. Method: We synthesized learning goals (LGs) and levels of thinking about variables from existing research, then designed instruction to support progress along the levels of thinking while addressing a subset of the LGs. Findings: Our literature synthesis articulated four levels of thinking and eight LGs about variables. Curriculum development allowed us to support movement through the levels of thinking while addressing five LGs in an integrated math+CS course for fourth grade using Scratch. Implications: Our model of curriculum development around levels of thinking can support the creation of customized instructional sequences.

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.

Culturally responsive computing (CRC) frames the localized knowledges and practices of Black, Brown, and Indigenous communities as assets for working toward racial justice in science, technology, engineering, and mathematics (STEM). A key part of CRC is the role that local communities play in designing and/or implementing curricula and technologies. Yet, there is a dearth of research on collaborating with local knowledge experts and what they think about CRC. In response, this paper details a two-year long research project on the design and implementation of one CRC program called pH Empowered. pH Empowered uses computing to bridge Black hairstyling, chemistry, and entrepreneurship. Through a mixed-methods study of one pH Empowered professional development workshop, we show how cosmetologists, urban farmers, and librarians had diverse perspectives about how to be culturally responsive with STEM and the racial justice goal of broadening participation in STEM 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.

The aim of this article is to inspire more teachers to explore the use of codable robots for teaching mathematics. The authors begin by discussing four benefits experienced by incorporating robots into middle school mathematics classes: (1) reasonable time frames; (2) authentic purposes for mathematics; (3) visual and nonevaluative feedback; and (4) introduction to computational thinking. Next, the authors provide some ideas about how to get started with using robots in the classroom, focusing on Sphero, a small, rolling robot that uses a simple programming language available in a tablet app.

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.

A key debate in computer science education is whether and how computational thinking (CT) is used within disciplines other than computer science. Broad definitions provide many avenues for developing integrated instruction, as practices within existing activities can simply be reframed in terms of CT. But such general use of the term CT may confuse its meaning and dilute its power as a tool for bringing CS to all. In this paper, we take the viewpoint that mathematical thinking and computational thinking share common practices, such as using repetition to accomplish tasks, but these shared ideas may develop differently in mathematics and computer science. We use document analysis to analyze the K-5 Common Core State Standards for Mathematics for the presence of these common practices and describe the specific ways they develop in mathematics. We compare these elements of mathematical thinking to related elements of CT to identify synergies and differences between CT and mathematical thinking. We argue that these synergies and differences have implications for the development of integrated instruction designed to bring CS to all.

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.

In order to create professional development experiences, curriculum materials, and policies that support elementary school teachers to embed computational thinking (CT) in their teaching, researchers and teacher educators must understand ways teachers see CT as connecting to their classroom practices. We interviewed 12 elementary school teachers, probing their understanding six components of CT (abstraction, algorithmic thinking, automation, debugging, decomposition, and generalization) and how those components relate to their math and science teaching. Results suggested that teachers saw stronger connections between CT and their mathematics instruction than between CT and their science instruction. We also found that teachers draw upon their existing knowledge of CT-related terminology to make connections to their math and science instruction that could be leveraged in professional development. Teachers were, however, concerned about bringing CT into teaching due to limited class time and the difficulties of addressing high level CT thinking in developmentally appropriate ways. We discuss these results and their implications future research and the design of curriculum and professional development.

Culturally responsive computing (CRC) frames the localized knowledges and practices of Black, Brown, and Indigenous communities as assets for working toward racial justice in science, technology, engineering, and mathematics (STEM). A key part of CRC is the role that local communities play in designing and/or implementing curricula and technologies. Yet, there is a dearth of research on collaborating with local knowledge experts and what they think about CRC. In response, this paper details a two-year long research project on the design and implementation of one CRC program called pH Empowered. pH Empowered uses computing to bridge Black hairstyling, chemistry, and entrepreneurship. Through a mixed-methods study of one pH Empowered professional development workshop, we show how cosmetologists, urban farmers, and librarians had diverse perspectives about how to be culturally responsive with STEM and the racial justice goal of broadening participation in STEM education. [ABSTRACT FROM AUTHOR]

A key debate in computer science education is whether and how computational thinking (CT) is used within disciplines other than computer science. Broad definitions provide many avenues for developing integrated instruction, as practices within existing activities can simply be reframed in terms of CT. But such general use of the term CT may confuse its meaning and dilute its power as a tool for bringing CS to all. In this paper, we take the viewpoint that mathematical thinking and computational thinking share common practices, such as using repetition to accomplish tasks, but these shared ideas may develop differently in mathematics and computer science. We use document analysis to analyze the K-5 Common Core State Standards for Mathematics for the presence of these common practices and describe the specific ways they develop in mathematics. We compare these elements of mathematical thinking to related elements of CT to identify synergies and differences between CT and mathematical thinking. We argue that these synergies and differences have implications for the development of integrated instruction designed to bring CS to all. [ABSTRACT FROM AUTHOR]

The article offers information on create professional development experiences, curriculum materials, and policies that support elementary school teachers to embed computational thinking (CT) in their teaching. It mentions knowledge of CT-related terminology to make connections to their math and science instruction that could be leveraged in professional development.

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 "realworld" 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. [ABSTRACT FROM AUTHOR]