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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.

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]