Browsing by Author "De Villiers, Michael"

Now showing 1 - 5 of 5
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    An associated result of the Van Aubel configuration and its generalization
    (Taylor and Francis Group, 2022 ) De Villiers, Michael; Curriculum Studies
    This note presents some novel generalizations to similar quadrilaterals, similar parallelograms, and similar triangles of a result associated with Van Aubel’s theorem about squares constructed on the sides of a quadrilateral. These results provide problem posing opportunities for interesting, challenging explorations for talented students using dynamic geometry at high school or for university students.
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    Generalising some geometrical theorems and objects
    (Association for Mathematics Education of South Africa, 2016) De Villiers, Michael
    This article presents two possible examples of appropriate generalisation activities. The first is the generalisation of a familiar theorem for cyclic quadrilaterals to cyclic polygons, while the second is the generalisation of the concept of a rectangle to a higher order polygons.
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    An instrumental approach to modelling the derivative in Sketchpad
    (AOSIS Publishing, 2011-11) Ndlovu, Mdutshekelwa; Wessels, Dirk; De Villiers, Michael
    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.
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    Some more properties of the bisect-diagonal quadrilateral
    (The Mathematical Association, 2021) De Villiers, Michael
    Martin Josefsson [1] has coined the term ‘bisect-diagonal quadrilateral’ for a quadrilateral with at least one diagonal bisected by the other diagonal, and extensively explored some of its properties. This quadrilateral has also been called a ‘bisecting quadrilateral’ [2], a ‘sloping-kite’ or ‘sliding-kite’ [3], or ‘slant kite’ [4]. The purpose of this paper is to explore some more properties of this quadrilateral.
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    The vertex centroid of a Van Aubel result involving similar quadrilaterals and its further generalisation
    (Taylor & Francis Group, 2024 ) De Villiers, Michael; Humenberger, Hans
    This paper explores the position of the vertex centroid for a generalisation of Van Aubel’s theorem: specifically we look at what happens to the vertex centroid when directly similar quadrilaterals are placed on the sides of an arbitrary quadrilateral. After giving a simple proof that the position of the vertex centroid remains unchanged, the result is further generalised to directly similar triangles (or other directly similar shapes) on the sides of polygons using vectors. Not only are the results mathematically interesting, but can also provide an appropriate classroom opportunity for dynamic geometry exploration, and to build 2D models with clay and drinking straws (or thin wire) to illustrate and check the theoretical solutions. [ABSTRACT FROM AUTHOR]