The teacher–tool–mathematics ensemble: Before, during and after implementing TouchTimes

Using the theoretical constructs of double instrumental genesis, instrumental distance, didactical landmarks and instrumental orchestration, I examined case studies of four elementary school teachers in British Columbia, Canada, who implemented TouchTimes (hereafter, TT) as a pedagogical tool for teaching mathematics. TT comprises a pair of novel, multi-touch iPad applications that provide embodied and relational experiences of multiplication through two different dynamic multiplicative models. In interviews, these teachers shared their personal experiences learning about this relatively new digital application themselves, the obstacles they encountered and their professional experiences integrating TT into their instructional repertoires as a tool for student learning. The aim of this research was to study the teacher, the digital tool – in this case, TT – and the mathematical concept of multiplication as an ensemble, as it played out in the classroom. Rather than examining each of the parts separately, the ensemble views each piece in relation to the whole, rather than individually, and became my way of honouring the complexity and the nuances that emerged from the information and experiences shared. I examined how TT, and its way of materialising the mathematical concept of multiplication, affected the personal and professional instrumental geneses of the teachers and, conversely, how the teachers' accommodated TT within their previous ways of thinking about, and teaching, multiplication. Furthermore, I studied how the teachers' experiences of double instrumental genesis evolved during their use of TT as a pedagogical instrument. My analysis indicates that the implementation of TT, and its way of presenting multiplication, was multi-faceted and complex, that the personal and professional instrumental geneses that teachers undergo may be closely intertwined and that, when speaking of TT, they clearly differentiate between ways of teaching with it and how students may learn using this technology. In keeping the teacher–TT–mathematics ensemble in mind, a few noteworthy implications of the research include: (1) the mutually interactive nature of personal and professional instrumental genesis; (2) an acknowledgement of the agency of the tool; (3) the importance of sociocultural influences on the teachers' processes of double instrumental genesis; (4) the identification of three new instrumental orchestrations that emerged from this research.