Original Research

An instrumental approach to modelling the derivative in Sketchpad

Mdutshekelwa Ndlovu, Dirk Wessels, Michael de Villiers
Pythagoras | Vol 32, No 2 | a52 | DOI: https://doi.org/10.4102/pythagoras.v32i2.52 | © 2011 Mdutshekelwa Ndlovu, Dirk Wessels, Michael de Villiers | This work is licensed under CC Attribution 4.0
Submitted: 13 September 2011 | Published: 23 November 2011

About the author(s)

Mdutshekelwa Ndlovu, University of Stellenbosch, South Africa
Dirk Wessels, University of Stellenbosch, South Africa
Michael de Villiers, University of KwaZulu-Natal, South Africa

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Encouragement to integrate information and communication technologies into mathematics education curricula is an increasingly universal phenomenon. As a contribution to the discourse, this article discusses the potential use in the classroom of The Geometer’s Sketchpad® (Key Curriculum Press, Emeryville, CA, United States) mathematics software in modelling the derivative and related concepts in introductory calculus. In an empirical study involving first-year non-mathematics major undergraduate science students, a hypothetical learning trajectory (HLT) was conjectured and implemented for students to experience the visualisation and multiple representations of calculus concepts on the Cartesian plane with a computer graphic interface. The utilisation scheme is interpreted through the lens of the instrumental1 approach proposed by Trouche. The HLT was partly informed by the historical development of the derivative as synthesised from the literature on the history of calculus and partly by the affordances, enablements, constraints and potentialities of Sketchpad itself. The findings of the study suggest that when exposed to the capabilities of this software, learners can experience Geometer’s Sketchpad® as an effective visualisation tool or instrument for the representation and learning of the derivative and related concepts in introductory calculus. However, the effectiveness of this tool is not a given or a foregone conclusion − it is a product of the teacher’s instrumental orchestration, gradual learner mastery of the software syntax and careful resolution of theoretical-computational conflicts that can arise during early use of the instrument.


Geometer’s Sketchpad; derivative; calculus; instrumental genesis


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