A recent review of research on virtual manipulatives (VMs), defined as technology-based interactive representations of mathematical ideas (Moyer-Packenham, & Bolyard, 2016), suggests strengths relative to other instructional tools in supporting student learning of mathematics (Moyer-Packenham & Westenskow, 2013). Researchers have identified affordances of VMs that may help to explain their value and role in learning. Three such affordances are efficient precision, or support for creating precise representations quickly and easily, focused constraints, or limits placed on how VMs can be manipulated in order to focus attention on mathematical ideas or processes, and linked representations, or sets of representations that are dynamically connected such that one changes in response to the other (Sarama & Clements, 2009). These affordances hold particular promise for supporting student learning of fraction concepts, as VMs can help students overcome whole-number biases when operating with fractions (Hansen, Mavrikis, & Geraniou, 2016) and support accurate estimation of magnitudes of fractions and fraction sums (Braithwaite & Siegler, 2021). Although the evidence of the relationship between VMs and student learning is promising, the impact of all educational resources is shaped by how teachers use them (Cohen, Raudenbush, & Ball, 2003). As such, it is critical to understand how teachers think about VM affordances and how they plan to use them in their instruction. In this dissertation, I explored the thinking of six fourth- and fifth-grade teachers about fractions VMs three contexts: exploration, problem solving, and lesson planning. I employed the professional noticing framework (Jacobs, Lamb, & Philipp, 2010) to analyze how teachers thought about VM features in each context. I coded the VM features to which they attended, synthesized their interpretations of features reflecting each of the three affordances, and analyzed how they used these features in problem solving and lesson planning. Teachers attended to features reflecting all three affordances across contexts, but how they interpreted them varied by context and by affordance. Teachers had positive interpretations of features reflecting efficient precision across contexts because they relieved burdens on students for creating equal parts or carefully aligning models. Only in lesson planning, however, did some teachers reflect on how features reflecting efficient precision might support students in reaching more ambitious learning goals. Teachers’ interpretations of features reflecting focused constraints were negative in exploration and problem solving because they interfered with their familiar problem-solving strategies. By contrast, teachers noted both positive and negative potential effects of focused constraints on student thinking during lesson planning. Teachers attended to linked representations in exploration but showed minimal interest in these features in the other contexts. These results provided helpful guidance for teacher professional learning about VMs in teaching mathematics. Teacher thinking about VMs was more flexible and included more consideration of student thinking in the lesson planning context, establishing the importance of connecting professional learning about VMs to teachers’ day-to-day practice. Additionally, since teachers often connected VM affordances to their existing practices rather than thinking about how to use them to introduce new strategies or learning goals, they will likely need additional support for conceptualizing why the VM affordances are important and how VMs might be used to their best advantage.
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