Comments on The Co-emergence of Machine Techniques, Paper-and-Pencil Techniques, and Theoretical Reflection
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OMMENTS ON THE CO-EMERGENCE OF MACHINE TECHNIQUES, PAPER-AND-PENCIL TECHNIQUES, AND THEORETICAL REFLECTION
ABSTRACT. We comment on the paper The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra by Carolyn Kieran and Paul Drijvers. We look at aspects of Kieran and Drijvers’ analysis with regard to ‘task-technique-theory’ in the light of a model of abstraction in context and suggest that this re-viewing brings mutual analytic benefits.
KEY WORDS: abstraction, task, technique, theory
We comment on the paper The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra by Carolyn Kieran and Paul Drijvers in collaboration with others, hereafter referred to as KD, in Vol II(Z) of this journal. This paper advances knowledge of the Ôtask-techniquetheoryÕ (TTT) dialectic as espoused by, for example, Artigue (2002) and Lagrange (2005). Apart from informative analysis of studentsÕ work with computer algebra systems, KD advance theory by noting the situated co-emergence of technique and theory and set new research questions concerning the epistemic value of CAS and byhand techniques and concerning the nature of tasks in Ôkick-startingÕ actions in which the epistemic value of techniques come into play. In these comments we address theoretical issues. We look at what KD call TT in the light of Hershkowitz et al. (2001) model of abstraction, hereafter referred to as HSD, and the ideas of Davydov (1990), which inspired the HSD model. We see Davydovian ideas in the influential work of Noss and Hoyles (1996) and also comment on this link. These theories/models/accounts are complex and a description of them would make for a very long paper. These comments are to be short so we merely allude to the main features of these accounts. The main thing we want to say is that aspects of KD can be viewed in terms of HSDÕs model and DavydovÕs ideas and that
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JOHN MONAGHAN AND MEHMET FATIH OZMANTAR
this re-viewing brings mutual analytic benefits. We wish to emphasise that this re-viewing addresses issues of substance and it is not a theoretical game of let us see what happens if we interpret KDÕs data from another theoretical perspective. The remainder of these comments are in five sections. We outline the theories we consider. We then look at selected extracts from KD from the viewpoint of HSD theory and then consider (with regard to DavydovÕs ideas) issues that this re-viewing brings about. In the penultimate section we consider insights that KDÕs work bring to HSD and Davydovian ideas. We end with a note on Ômixing theoriesÕ. 1. THEORETICAL POSITIONS KD clearly describe TTT, which owes a great deal to ChevallardÕs (1999) anthropological approach, so we do not repeat this description. An important ÔplayerÕ in these comments is Davydov (1990) and his Marxist account of mathematical abstraction. We sketch the basic ideas here but return to Davydov later. Abstraction starts
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