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Activity theory may be used as a language of description and of analysis (see Figure 1 below). In terms of the construction of this framework the central components are the Learner as Subject, Mathematical or Numerate Activity as the Object, and Technology as the Mediating Artefact or medium for the delivery of instruction. Technology also supports a range of mathematical tools for exploring new possibilities, developing understanding, maintaining skills, communicating with others, and acting as a labour-saving device. These are all set in the socio-cultural context of Rules, Community, and Division of Labour. Engeström (1987) notes that these vertices form part of a dynamic system, with each vertex in a continual state of contradiction or tension with all others — indicated by the double arrows. He also proposes that there are multiple levels of contradictions so that each vertex may be expanded to form a new activity system with its own tensions and contradictions. He also underlines the fact the activity systems of learners and teachers necessarily have different objects — a point which is often overlooked in the pedagogical design process. In a later work the tensions and contradictions between designers and users of technological artefacts are highlighted (Hasu & Engeström, 2000).
Figure 1. The basic mediational triangle expanded (after Engeström, 1987)
In my original proposal I had intended to have three major dimensions to the framework: content, technology, and pedagogy. Now that the research project is complete, it seems more appropriate to redesign the framework to incorporate the finer nuances highlighted by the research, but in as simple a manner as possible. The critical feature of mathematics or numeracy instruction is that, as noted above, technology has dual aspects — as a medium of delivery and as an integral tool — and it is this dual focus which sets the field apart from most other discipline areas. In particular, there is a need to focus on the use of technology as a tool, but also as an object of learning in its own right. In order to keep the diagrammatic form as simple as possible, I have included work on Technology as Object under the Rules vertex, rather than form another sub-triangle. At the very least there are rules to be learned or invented so that the artefact may become useful and transparent as an instrument (Trouche, 2004). There are also compelling arguments for the importance of collaboration in adult mathematics/numeracy education, particularly where it is, or can be, mediated by information and communication technologies (ICTs). Following a similar line of reasoning, I have included collaborative learning under the Community vertex. In this context, collaboration can go well beyond the realms of formally enrolled classmates. Finally, the Division of Labour vertex has been expanded to include learning design, interactivity, and general pedagogical principles (see Figure 2).
Figure 2. Revised mediational triangle
In order to refine these down to three major categories as originally envisaged, Community could be subsumed under Learner, Division of Labour under Technology as Tool, and Rules under Mathematics/Numeracy.
In a project such as this, involving boundary crossings between academic disciplines, different contexts of learning and work, and the complicated profiles of adult learners with their diversity of education, life and work experiences, there are necessarily areas of overlap; discrete and binding divisions would be meaningless. Ultimately, this project will remain a work-in-progress, not least because the world of technology is in a constant state of evolution, but also because the project is largely the work of only one person, sustained by rich personal interactions with local and international colleagues as well as by the wealth of literature currently available online.
References
Engeström, Y. (1987). Learning by expanding: An activity-theoretical approach to developmental research. Helsinki : Orienta-Konsultit. Retrieved February 20, 2003 , from the World Wide Web: http://lchc.ucsd.edu/MCA/Paper/Engeström/expanding/toc.htm
Hasu, M. & Engeström, Y. (2000). Measurement in action: An activity-theoretical perspective on producer-user interaction. International Journal of Human Computer Studies, 53, 61-89. (Special issue on “Understanding Work and Designing Artifacts”). [Retrieved August 23, 2002 , from the World Wide Web: http://www.edu.helsinki.fi/activity/people/yrjo.htm]
Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematics Learning, 9(3), 281-307.
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