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The purpose of this study was to examine patterns of mathematical practices evident during children's interactions with touchscreen mathematics virtual manipulatives. Researchers analyzed 33 Kindergarten children's interactions during activities involving apps featuring mathematical content of early number sense or quantity in base ten, recorded during one-to-one task-based interviews. Iterative analysis involved applying learning progression rubrics to video data, using hierarchical clustering to visualize the progressions via heatmaps with dendrograms, and returning to video data to investigate emergent patterns. Results indicated that overall, children's mathematical practices aligned with research on development of mathematical understandings, but that individual children's mathematical practices changed little within a given activity. The study extends existing research by shedding light on children's interactions with touchscreen mathematics virtual manipulatives and supports the use of these analysis and visualization techniques.

The purpose of this study was to explore the relationship between instructional modality used for teaching fractions and third- and fourth-grade students' responses and strategies to open-response fraction items. The participants were 155 third-grade and 200 fourth-grade students from 17 public school classrooms. Students within each class were randomly assigned to two instructional treatment groups: a virtual manipulatives representations (VMR) instruction group and a physical manipulatives and textbook representations (PMTR) instruction group. A conversion mixed methods analysis was used to examine quantitative and qualitative data. The quantitative analysis showed achievement outcomes were the same for both groups. The qualitative analysis revealed shifts in learning that were otherwise hidden with solely quantitative achievement results. Specifically, the results indicated VMR group success in understanding fractions as relationships and PMTR group success in maintaining conceptualization of the whole. Overall, the results of this study corroborate previous research indicating the importance of both types of instructional modalities, showing that virtual manipulatives and physical manipulatives are effective instructional tools with positive effects on student learning. The study expands existing research by offering an opportunity to explore the nuances of students' fractions understanding and provide a window into students' shifts in fraction learning.

Recently, over 40 states in the United States adopted the Common Core State Standards for Mathematics (CCSSM) which include standards for content and eight standards for mathematical practices. The purpose of this study was to better understand the nature of young children's mathematical practices through an exploratory examination of the practices of a group of second-grade students engaged in several mathematical tasks focused on rational number concepts. Twenty-five second-grade students completed three fraction tasks in structured clinical interviews. The interviews and student work were analyzed using an interpretational analysis to examine the data for constructs, themes, and patterns that were useful in explaining children's mathematical practices. The results reveal that children used a variety of mathematical practices during the interviews to respond to the mathematical problems presented. Children's mathematical practices were both a product that they used to solve the mathematical situations, and a process that was developing during the interactions of the interview. The findings lead to new insights about how mathematical practices develop and what promotes their development.

The purpose of this study was to explore relationships between app affordances and user abilities in second graders' interactions with mathematics virtual manipulative touchscreen tablet apps. The research questions focused on varying manifestations of affordance-ability relationships during children's interactions with mathematics virtual manipulative touchscreen tablet apps. Researchers qualitatively analyzed video recordings and ethograms from clinical interviews of 33 second-grade children. Each 45-min clinical interview involved one child interacting with two sequences of mathematics virtual manipulative touchscreen tablet apps: one sequence focusing on place value concepts and the other sequence focusing on skip counting concepts. Results provided evidence of Moyer-Packenham and Westenskow's ("Int J Virtual Pers Learn Environ" 4(3):35-50, 2013) five affordance categories of virtual manipulatives. Approach to and degree of affordance access varied depending on a child's corresponding ability, and some children modified their affordance access as their ability changed. Results also indicated that outcomes of accessing an affordance also related to a child's ability. Context also influenced affordance access. These results imply that affordance-ability relationships are multifaceted. Overall, these results imply that is important to consider affordance-ability relationships in relation to mathematics education technology.

This paper discusses changes in young children's learning performance and efficiency during clinical interviews in which each child interacted with a variety of virtual manipulative mathematics apps on iPads. Researchers interviewed 100 children ages 3 to 8 using a protocol format with two pre-assessment apps, four learning apps, and two post-assessment apps. Data were gathered quantitatively and qualitatively, using wall and screen-capture videos, pre and post assessments of performance, and time-stamping to record efficiency. Following 30-40 minute interviews where children interacted individually with the mathematics apps, results showed that children in the Preschool group increased efficiency while maintaining performance, children in the Kindergarten group increased performance while maintaining efficiency, and children in the Grade 2 group increased their performance and efficiency in skip counting, but not in place value. Overall, children in different age groups responded in different ways to the apps and some apps had a greater influence on children's learning performance and efficiency than others. To conduct this study, researchers created a variety of research tools that were not readily available because research on children's interactions with touch-screen-based virtual manipulative mathematics apps is still in its infancy. The paper describes the processes our research team used to develop new tools for novel and unique research situations. [ABSTRACT FROM AUTHOR]