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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
05 JUNE 2007

Ten Canoes: an allegory about respect and community

"Bout time to tell you a story, eh? Then I'll tell you one of ours...

It is longtime ago. It is our time, before you other mob came from cross the ocean...longtime before then. The rains been good and ten of the men go on the swamp, to hunt the eggs of gumang, the magpie goose. One of the men, the young fella, has a wrong love, so the old man tell him a story...a story of the ancient ones, them wild and crazy ancestors who come after the spirit time, after the flood that covered the whole land...

It's a good story, this story I'm gonna be tellin' you 'bout the ancient ones. There's more wrong love in this story, and plenty spears too, and plenty wives...too many wives if you ask me...a beautiful young one and a bit of a jealous one and the older wise one and even more wives than that.

And there's plenty happens in this story...there's a stranger comes and one of them wives goes missin' and there's a man with a belly big as a mountain. There's sorcery and magic too, and a wrong spear in the wrong body, and more spears and bad spirits. And even that's not everything that's happening in this story.

Ahh, you gotta see this story of mine cause it'll make you laugh, even if you're not a blackfella. Might cry a bit too eh? But then you laugh some more...cause this story is a big true story of my people. True thing.

The film Ten Canoes employs a cyclical structure to represent its Indigenous Australian world view. This is done through using a story within a story that works an allegory about respect and community."
Rolf De Heer (Director) & Peter Djigirr (Co–Director)



AboriginalallegoryAustraliaAustralian Aboriginecanoecommunitycycliccyclic narrativecyclical narrative • De Heer • Djigirr • dramafilmIndigenousIndigenous Australiansmobnarrativeparablerespect • Ten Canoes

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