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15 NOVEMBER 2009

Learning algebra in a computer algebra environment : design research on the understanding of the concept of parameter

"Design research – also known as developmental research or development research – is a research methodology that aims at developing theories, instructional materials and an empirically grounded understanding of 'how the learning works' (Research Advisory Committee, 1996). The main objective of design research is understanding and not explaining (Bruner, 1996). This objective implies different norms of justification than would be the case in comparative empirical research. One important feature of design research is the adaptation of the learning trajectory throughout the research; based on previous experience, the instructional sequences and teaching experiment conditions are adjusted. Therefore, design research is particularly suitable in situations where a full theoretical framework is not yet available and where hypotheses are still to be developed. The methodology of design research is addressed in many recent publications (e.g. van den Akker, 1999; Brown, 1992; Edelson, 2002; Freudenthal, 1991; Gravemeijer, 1993, 1994, 1998; Gravemeijer & Cobb, 2001; Leijnse, 1995; Treffers, 1993). In spite of varying interpretations of the notion of design research, there is agreement on the identification of two key aspects: the cyclic character of design research and the central position of the design of instructional activities. We now address these two issues.

The cyclic character of design research

Design research has a cyclic character: a design research study consists of research cycles in which thought experiments and teaching experiments alternate. We distinguish macro–cycles that concern the global level of the teaching experiments, and micro–cycles that concern the level of subsequent lessons. Gravemeijer argued that the cycles lead to a cumulative effect of small steps, in which teaching experiments provide 'feed–forward' for the next thought experiments and teaching experiments (Gravemeijer, 1993, 1994).

A macro–cycle of design research consists of three phases: the preliminary design phase, the teaching experiment phase, and the phase of retrospective analysis. In the last–mentioned phase, the reflection captures the development of the insights of the researcher. Following Goffree (1986) and Schön (1983), Gravemeijer called this 'reflection–in–action' (Gravemeijer, 1993, 1994). As a result, new theories, new hypotheses and new instructional activities emerge, that form the feed–forward for the next research cycle that may have a different character, according to new insights and hypotheses. The process of the researcher's thinking should be reported, to ensure the trackability of this development for others (Freudenthal, 1991; Gravemeijer, 1994).

As far as the role of theory in design research is concerned, the term 'theory–guided bricolage' is used (Gravemeijer, 1994). The researcher is like a tinkerer, who tries to combine and integrate global and local theories, which may be issued from other domains, to develop a learning trajectory and a local instruction theory for a specific topic. This local instruction theory contributes to the development of the domain–specific instruction theory.

In our study, three full macro–cycles – indicated as G9–I, G9–II and G10–II – and one intermediate cycle were carried out. The first phase of preliminary design includes two related parts, the development of a hypothetical learning trajectory (HLT) and the design of instructional activities. This phase is followed by the teaching experiment and the retrospective analysis. Fig. 2.1 shows the three full research cycles. Cycle 1 started with a conceptual analysis that is described in Chapter 4. Each of the phases is elaborated on in Sections 2.3 – 2.6, whereas specific information on each of the cycles is presented in 2.7.

The role of design

A second characteristic of design research is the importance of the development of a learning trajectory that is made tangible in instructional activities (Gravemeijer, 1994). The design of instructional activities is more than a necessity for carrying out teaching experiments. The design process forces the researcher to make explicit choices, hypotheses and expectations that otherwise might have remained implicit. The development of the design also indicates how the emphasis within the theoretical development may shift and how the researcher's insights and hypotheses develop. We agree with Edelson, who argued that design of student texts is a meaningful part of the research methodology:

(...) design research explicitly exploits the design process as an opportunity to advance the researchers' understanding of teaching, learning, and educational systems. Design research may still incorporate the same types of outcome–based evaluation that characterize traditional theory testing, however, it recognizes design as an important approach to research in its own right. (Edelson, 2002, p.107)

This is particularly the case when the theoretical framework involved is under construction:

(...) it [the research] started with only a partial theory and has proceeded with the explicit goal of elaborating that theory before attempting any summary evaluation. The lessons that are emerging from this effort are being shaped by the concrete, practical work of design. (Edelson, 2002, p. 112)"

(Paulus Drijvers, Maria Hendrikus, 2003)

Fig.1 David Coghlan & Teresa Brannick (2001).



2003action researchalgebra • CAS • computer algebra • cyclicdesign-based research • development research • developmental research • how learning works • instructional materials • instructional sequences • learning trajectorymathematicsmathematics educationmathsmethodology • realistic mathematics education • reflection-in-actionresearchresearch methodologyresearch methods • retrospective analysis • technologytheoretical framework • theory-guided bricolage • tinkerer


Simon Perkins

Types of research in the creative arts and design

"scholarly research – creates and sustains the intellectual infrastructure within which pure, developmental and applied research can be conducted. it aims to map the fields in which issues, problems, or questions are located (what is known or understood in the general area of the proposed research already, and how addressing or answering the issues, problems or questions specified will enhance the generally–available knowledge, and, understanding of the area in question). it documents/compiles the knowledge, resources, methods, tools and models evolved through pure, developmental and applied research along with the subsequent results. pure research – asks fundamental questions in the field and explores hypotheses experimentally. it searches for pure knowledge that may uncover issues, theories, laws or metaphors that may help explain why things operate as they do, why they are as they are, or, why they appear to look the ways they do. it generates significant new facts, general theories or reflective models where immediate practical application or long–term economic, social or cultural benefits are not a direct objective. the results may be unexpected and yield original theories, discoveries or models that are unrelated to the disciplines in which the research has been conducted – they may be applied in another research context. developmental research – serves two purposes (a) it identifies the limitations of existing knowledge as evolved through pure research by creating alternative models, experiences and/or thought–systems so to generate useful metaphors for organising insight and expanding/reframing the base of existing knowledge (b) it harnesses, tests and reworks existing knowledge so to evolve special methods, tools and resources in preparation for the solving of specific problems, in specific contexts, through applied research. applied research – involves a process of systematic investigation within a specific context in order to solve an identified problem in that context. it aims to create new or improved systems (of thought or production), artefacts, products, processes, materials, devices, or services for long–term economic, social and/or cultural benefit. it is informed by the intellectual infrastructure of scholarly research in the field; it applies and/or transfers enhanced knowledge, methods, tools and resources from pure and developmental research; it also contributes to scholarship in the field through systematic dissemination of the results. the outcomes cannot usually be directly applied to other contexts because of the specificity of the situation in which the research has been applied although the methods/tools evolved are often transferable."
(Bruce Brown, Paul Gough, Jim Roddis, March 2004)

1). Brown, B., Gough, P. and Roddis, J. (2004) Types of Research in the Creative Arts and Design [online]. Bristol, UK: E–Papers, University of Brighton.


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