A Tri-ple Analysis of thinking multiplicatively around/with TouchTimes

Multiplication is important for future mathematical competency. However, many students have difficulty thinking multiplicatively. Researchers attribute this difficulty to the widespread use of the repeated addition model when introducing learners to multiplication. In this dissertation, I explore how learners' multiplicative thinking emerges around/with a gesture-based, multi-touch, iPad application called TouchTimes (TT), which enables learners to create and manipulate a multiplication model that is different from the repeated addition model. Learning mathematics by using digital tools is a complex phenomenon. Drawing on my diffractive reading of the theory of semiotic mediation through enactivism, my thesis addresses multiple dimensions of this phenomenon, presented in three separate qualitative studies which followed the methods of videography. The first study explores the semiotic potentials of TT and pencil-and-paper to engage students with multiplicative thinking. The analysis was conducted with respect to the same multiplication task which was initially designed for TT and modified for pencil-and-paper. The data were created through the recordings of the critical gestures that were required to solve the task. The findings show that two artefacts share some semiotic potentials and also that each of them has some singular contributions to students' understanding of multiplication. The second study examines how young children make sense of TT when they use a duo of artefacts (pencil-and-paper and TT) back-and-forth. The data were created through a video-recording of a five-year-old child using the duo. The findings were strongly related to those of previous research and showed that back-and-forth use of the duo helped the child bring forth different aspects of multiplicative relationships. The third study attends to how children collaboratively structure quantities in TT. The data were created by a video-recording of two third-graders' interaction around/with TT. The findings showed that the structures of the students' bodies and TT co-evolved through reciprocal interactions. While at the beginning, these structures were aligned with additive relationships, they were multiplicative towards the end. The peer's body contributed to this shift in various ways other than allowing for the verbal exchange of ideas.