A Design Approach to Research in Technology Enhanced Mathematics Education
A Thesis Submitted for the Degree of Doctor of Philosophy
Institute of Education
University of London
winner of the second TELEARC award for PhD excellence, 2011.
This thesis explores the prospect of a design science of technology enhanced mathematics education (TEME), on three levels: epistemological, methodological and pedagogical. Its primary domain is the identification of scientific tools for design research in TEME. The outputs of this enquiry are evaluated by a demonstrator study in the domain of secondary school mathematics. A review of existing literature establishes a need for a design perspective in TEME research, but at the same time suggests a need for a consensual epistemic infrastructure for the field: a shared set of rules, processes and representations which bound and support its scientific discourse. Three constructs are proposed towards such an infrastructure: design narratives, design patterns, and the cycles of design research in which they are embedded. The first two are representations of domain design knowledge, the latter is a description of a design-centred scientific process.
The three constructs identified at the epistemological level are operationalised as a methodological framework by projecting them into a specific research setting of the demonstrator study. Appropriate methods and procedures are identified for collecting data, organising and interpreting them as design narratives, and extracting design patterns from these narratives.
The methodological framework is applied in the demonstrator domain to the question of learning about number sequences. A review of the educational research on number sequences identifies challenges in this area related to the tension between learners’ intuitive concept of sequences and the dominant curricular form. The former appears to be recursive in nature and narrative in form, whereas the latter is a function of index expressed in algebraic notation. The chosen design approach combines construction, collaboration and communication. It highlights the need for representations and activities which lead learners from intuitive concepts to formal mathematical structures.
Three interleaved themes connect the primary and the demonstrator domains: narrative, systematisation and representation. Narrative emerges as a key element in the process of deriving knowledge from experience. Systemisation concerns the structured organisation of knowledge. The tension between the two calls for representations which support a trajectory from the intuitive to the structural.
The main outcome of this study is a methodological framework for design science of TEME which combines design narratives and design patterns into structured cycles of enquiry. This framework is supported both theoretically and empirically. Inter alia, it is used to derive a contribution towards a pedagogical pattern language of construction, communication and collaboration in TEME.