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The purpose of this study was to examine shifts in young children's learning progression levels while they interacted with virtual manipulative mathematics apps on touch-screen devices. A total of 100 children participated in six mathematics learning sequences while using 18 virtual manipulative mathematics touch-screen apps during clinical interviews. Researchers developed a micro-scoring tool to analyze video data from two camera sources (i.e., GoPro camera, wall-mounted camera). Our results showed that it is possible to document evidence of shifts in children's learning progressions while they are interacting with mathematics apps on touch-screen devices. Our results also indicated patterns in the children's interactions that were related to the shifts in their learning progression levels. These results suggest that an open-ended number of tasks with a variety of representations and tasks at varying levels of difficulty led to children refining their understanding and shaping their concept image of mathematical ideas resulting in incremental shifts in learning. The results of this study have important implications about how mathematical tasks in touch-screen apps may prompt children's incremental learning progression shifts to occur, and thereby promote opportunities for learning. We propose that design features in mathematics apps can be created to support and encourage these learning shifts.

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Extensive research has been published on the nature of classroom mathematical discourse and on the impact of technology tools, such as virtual manipulatives (VM), on students' learning, while less research has focused on how technology tools facilitate that mathematical discourse. This paper presents an emerging construct, the Techno-Mathematical Discourse (TMD) framework, as a means for analyzing and interpreting aspects of learning when students use technological representations to mediate mathematical discussions. The framework focuses on three main components: classroom discourse, technology tools, and mathematical tasks. This paper examines each of these components, and then illustrates the framework using examples of students' exchanges while interacting with virtual manipulatives. The TMD Framework has applications relevant to teachers, teacher educators, and researchers concerning how technology tools contribute to discourse in mathematics classrooms. The TMD framework addresses a critical issue in mathematics education, in that classroom teachers and researchers need to understand how technology facilitates classroom interactions and how to best leverage technology tools to enhance students' learning of mathematics.

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