Taking up space : children's opportunities to learn in one-on-one interactions involving fraction story problems
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Mathematics instruction can work to open or constrain children's opportunities to learn, and the busy, interactive nature of instruction makes it challenging for researchers to identify patterns in high-quality opportunities to learn. While researchers have documented how whole-group discussions and small-group work can contribute to such opportunities, there has been little attention to other parts of instruction. This study focused on the part of instruction in which the teacher circulates while children work to solve a problem and have one-on-one conversations with individual children. I used the construct taking up space, adapted from Hand (2012) and Johnson (2017), to examine the quality of children's opportunities to learn fractions during these conversations. I chose taking up space because it connotes an expansive view of how children make sense of fractions, including through their spoken responses, gestures, and written work. Taking up space with fractions captured children's mathematical activity--in the form of problem-solving and articulating that problem-solving--and agency as they worked to solve an equal sharing story problem with a fractional answer. An additional exploratory analysis showed how children also bring who they are into how they take up space, which I summarized as revealing my mathematical self. This study drew on data from the Responsive Teaching in Elementary Mathematics project (Jacobs et al., 2019). Lessons were chosen from classrooms in grades 3, 4, and 5 with teachers who demonstrated some skill at questioning in ways that were responsive to children's mathematical thinking (Empson et al., 2022; Jacobs and Empson, 2016). A total of 20 teachers were included, drawn from schools that represented socio-economically and racially diverse populations of children. All interactions during the circulating phase (N=185) were scored for the extent to which children took up space with fractions, which included to a great extent, a limited extent, or little to no extent. Analyses revealed that children took up space to a great or limited extent in a majority of interactions, suggesting that one-on-one conversations about children's strategies for story problems contribute to children's opportunities to learn. Furthermore, a majority of children used direct modeling strategies to solve fraction story problems, which tended to support children in articulating their problem-solving decisions. Finally, revealing my mathematical self highlighted children's use of informal language and emotion as ways to bring more of themselves in addition to their strategies into these interactions.
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Ph. D.