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extended abstract

The ability to learn and the passion to do so may be innate human characteristics. Education, on the other hand, concerns identifying specific structures of knowledge and directing learning towards them by assembling resources and activities under cultural, institutional, social, and psychological constraints. In other words, education is designed learning, and as such, an incredibly complex and inherently multidisciplinary endeavour. Any framework attempting to address this domain needs to identify methodological tools which allow it to confront these complexities. Such tools need to balance the need for a crisp directive for action with a rich representation of context, intentions and possible solutions. The characterisation of education as designed learning establishes a multi-faceted link between design and epistemology, or the creation of knowledge. This link is a motif that weaves through this thesis. Design emerges as a method of study, an object of study and an outcome of study, leading to an overarching theme of this thesis:

To consider the study of technology-enhanced mathematics education as a design science; highlight the implications of such a paradigm, and propose ways to theorize design in a manner which draws both on educational research and computer science.

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. The pragmatist nature of a design stance in educational research entails a tight dependency between the three levels of epistemology: the method by which we study education (its normative epistemology) links our understanding of how people learn (genetic epistemology) to how we design artefacts for learning (generative epistemology). The specific aims of this thesis are derived from this realisation, namely, to:

·         Identify potential elements of an epistemic infrastructure for a design science of TEME.

·         Combine and elaborate the elements identified into a coherent methodological framework in a given research TEME context.

·         Apply the methodology in a problem domain and demonstrate its potential by producing a contribution towards a language of pedagogical patterns for TEME.

Three characteristics of design science, derived from Herbert Simon’s work (1969), were found to be highly relevant to the study of education: a value-driven agenda, a functional axis of decomposition and the role of representation. The value dimension is reflected in education’s remit to improve the life of individuals and communities. A functional axis of decomposition means that the objects of educational research should be categorized and analysed according to their effect rather than their structure. The importance of role of representation is generally accepted in the learning sciences. Iterative methodologies are gaining ground, in particular where technology is involved.

A design approach was found to be better adapted to the complex and dynamic nature of the circumstances and questions studied by educational science. This approach has the potential to offer a cohesive paradigm, bridging across practice and multiple theories.  These advantages are even more salient in the face of the rapid pace of change induced by technological developments, calling for agile, responsive and proactive approaches to educational research.

Simon’s argument for an inherently iterative method of inquiry resonates well with the growing tradition of design-based research (DBR) in TEME. DBR was characterised as pragmatic, committed to theoretical as well as practical innovation, highly interventionist, iterative. DBR produces modest theories, constrained to well-defined domains of learning and contexts of design. A design science of TEME produces design knowledge characterised as: problem driven, solution oriented, value laden; Situated in context; and holistic, i.e. inherently inter-disciplinary.

A review of the field identified the need for a clearly articulated consensual epistemic infrastructure; the explicit and implicit rules and assumptions which bound the discourse of the scientific community, and a logical system by which claims are presented and justified, independently of their content. Some desirable features of such an infrastructure emerged from the discussion: arguments should be accessible to the scientific community, as well as by practitioners and by policy makers; the processes of design study should be transparent and traceable; the forms used for communicating results should be sufficiently expressive to articulate all that is needed to support the above requirements. These forms should promote cumulativity and aggregation of knowledge, and its organisation should follow a functional-pragmatist orientation aligned with the functional axis of design knowledge.

A review of existing design approaches to educational research identified common methodological characteristics, including a dual focus on practical and theoretical contributions, a highly interventionist and agile attitude, and a cycle of iterative research. This cycle includes phases of theory, design, implementation, execution (experiment / practice), articulation of experience, interpretation, evaluation and analysis, and feedback to both theory and design. The products of this cycle are validation or critique of existing theory, evidence regarding the effectiveness of artefacts and practices in well-defined settings, and innovations in practice and theory. A frequent by-product of research is the synthesis of multiple frameworks. This cycle is embedded in a meta-cycle, which includes a framing phase, an empirical phase and a retrospective analysis phase.

Narrative was identified as a powerful epistemic form, leading to a proposal for a formalisation of design narratives as a form of scientific discourse. Design patterns, organised into pattern languages, emerged as a promising form for encoding design knowledge in educational research. The constructs identified at the epistemological level were operationalised as a methodological framework by projecting them into a specific research setting of the demonstrator study.

Once the research setting was described, it was used to illustrate how these constructs interact in practice. Special attention was given to the trajectory from experience to structured theory, focusing on the processes by which data was collected and how design narratives and design patterns were produced from the data and validated. The result is a description of a methodological framework and a set of instruments for the demonstrator study, from data collection and management, through interpretation and systemisation of observations as design narratives and on to the formalisation of research outcomes as design patterns. The latter provided a full specification for implementation of the analytical hemicycle of the design experiment cycle and of the retrospective analysis phase of the design research meta-cycle.

A structured process of selection and construction of design narratives was identified, using Bruner’s ten principles (1991) as guidelines. These principles, adapted to the needs of scientific form, were expressed in the design narrative template. Design patterns were extracted from design narratives through a six-step process devised to capture, systemise and substantiate the key design elements. This was followed by a phase of refactoring: structural manipulations which gave the pattern language as a whole greater coherence.

The methodological framework was applied in the demonstrator domain to the question of learning about number sequences. The demonstrator study followed the design research cycle and meta-cycle noted above. The framing phase produced a pragmatic review of the literature, whose aim was to identify key challenges in the domain and raise conjectures as to ways in which they may be addressed. These conjectures were translated into the design of activities and the technology to support them. The iterative process of design, implementation and evaluation was captured as a set of design narratives that were analysed to produce the collection of design patterns. 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.

The 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. Number patterns and sequences are broadly accepted as a viable gateway to advanced mathematical subjects which is intuitively appealing to learners. Yet the review exposed several pedagogical and epistemic issues concerning this topic. Foremost among these are the difficulties of formulating a structural view of sequences, specifically with respect of the process-product duality and recursive vs. closed form. These difficulties were linked to a dissonance between intuitive perceptions and the prevalent school representations of mathematical objects. This observation was inspected in the broader context of the relationship between communication, representation and meaning in mathematical learning. Combining the communicational approach with the concept of situated abstraction led to considering narrative as a fundamental social and cognitive epistemic force. Narrative form would appear to be at odds with the nature of mathematics, raising the question of their reconciliation – which has been a central question of the demonstrator study. Bringing narrative into the domain of constructionist learning and educational programming raised questions regarding its manifestation in computer-based representations, specifically, the representatiuon of number sequences. The Streams design pattern was proposed as a promising candidate. The union of the various approaches calls for educational designs which weave construction, communication and collaboration.

The observations and conjectures emerging from the review of the literature were translated into design of activities and tools. These were tested and refined in two sites over three years. The process of design and the outcomes of testing were captured as series of nine design narratives which form the empirical content of this thesis. They are the first tier of interpretation, not the data itself. Each narrative recounts a particular incident, defined by a single problem to be solved or task to be accomplished.

Several generalisable observations stood out: the potential of the Streams design pattern, the roles and characterisations of narrative, the importance of combining construction, communication and collaboration, and the recursive intuition of sequences. Among the more general themes that surfaced were the co-evolution of technology and pedagogy, the interdependence of interface and substance, and consequently the fluidity of design and the need for flexibility and malleability.

Applying the methodological framework to a demonstrator study yielded a contribution towards a pedagogical pattern language of construction, communication and collaboration in TEME. This contribution includes thirteen design patterns, supported by nine design narratives. Seven of these patterns have been fully specified, while six others are provided as thumbnails. The design patterns unpack the general themes emerging from the design narratives, translating them into theoretically and empirically substantiated context-aware guidelines for educational design. This collection of design narratives and patterns demonstrates the validity of the proposed approach in dealing with the immense complexity of designing for learning, and calls for further work in this vein.

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 a design science of TEME which combines design narratives and design patterns into structured cycles of enquiry. This framework is supported both theoretically and empirically. It is, inter alia, used to derive a contribution towards a pedagogical pattern language of construction, communication and collaboration in TEME.

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