Michael De Villiers

Stellenbosch University | SUN · Applied Curriculum Studies: Mathematics Education

In this chapter, a brief outline of the traditional approach to the teaching of proof in geometry is critiqued from a philosophical as well as a psychological point of view, and in its place an alternative approach to the teaching of proof (in a dynamic geometry environment) is proposed. In the alternative approach it is suggested that students are...

Traditionally the function of proof has been seen almost exclusively in terms of the verification of the correctness of mathematical statements. This paper strongly criticizes this view as one-sided, and instead proposes a model which distinguishes between five different functions of proof within mathematics. This analyses is based on epistemelogic...

This article provided an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by a Grade 11 student. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). Different proofs are given, ea...

This article examines the role and function of so‐called quasi‐empirical methods in mathematics, with reference to some historical examples and some examples from my own personal mathematical experience, in order to provide a conceptual frame of reference for educational practice. The following functions are identified, illustrated, and discussed:...

The article generalizes a theorem about tangential quadrilaterals & the incentres of some triangles subdividing it. The given proof gives insight into why it is true allowing generalization to higher tangential polygons. An analogue result for cyclic quadrilaterals is also given, and generalized further. A link to a dynamic geometry sketch online...

This paper describes a dynamic geometry investigation of the sum of alternate angles of crossed cyclic hexagons using directed angles. It is an easily accessible high school extension of the result that the sum of the alternate angles of a convex cyclic hexagon is 360 degrees.

This paper argues that any similarity proof of the theorem of Pythagoras can be translated into an equivalent trigonometric (using the basic trig ratio definitions) since trigonometry is based on similarity. It also gives a brief synopsis of a recent trigonometric proof by two high school students from the USA.

This article presents a 'side-divider' theorem for a triangle that is accessible to high school learners. It is then shown that Conway's Circle Theorem is a special case of this theorem. A dual 'angle-divider' theorem is also presented. A dynamic geometry link is also available.

An interesting geometric conservation problem is presented & generalized. Here proof is presented in a ‘proof without words’ style, with the aim of developing the reader's visual proof ability. The study of the task and its expansion is accompanied by a dynamic sketch to highlight the conservation property.

We understand experimental mathematics as the systematic experimental investigation of concrete examples of a mathematical structure in the search for conjectures about its properties. Experiments might be done by pencil-and-paper work, building physical models, and, of course, by using available computer programs for doing time-consuming calculati...

In this paper we present two purely geometric proofs for Dao Tanh Oai's generalization of Napoleon's theorem to a hexagon, and some further generalizations.

This paper follows on a previous paper about a particular hexagon and proves additional properties. For example, proving that the hexagon in question is tangential, i.e. has an incircle, formulating & proving a converse, as well as exploring the conditions under which the hexagon becomes cyclic. Generalizations to particular 2n-gons are included.

This paper explores some circle concurrencies that should be accessible to high school geometry students. It has extensions to Napoleon & Miquel's theorems.

This paper presents a little known theorem of similarity that is a nice exercise for high school students in geometry.

We deal with an extension of the six-point circle theorem for the quadrilateral [1] when the Van Aubel configuration is generalised as in [2] and [3]: similar parallelograms are constructed, all internally or all externally, on the sides of a given quadrilateral.

In this paper we present a new generalisation of Napoleon's theorem to a hexagon with equilateral triangles constructed on the sides as well as a purely geometric proof of the result.

This paper briefly describes a heuristic investigation using dynamic geometry of some concurrency, collinearity and other properties of a particular hexagon that would be of interest to mathematics olympiad enthusiasts. After managing to prove the initial result, further reflection on the proof led to an immediate generalisation, illustrating the s...

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 dyn...

This paper generalizes IMO 2014, problem 4, to cyclic 2n-gons. It also gives a proof Cartensen's concurrency theorem for a cyclic hexagon in the appendix.

The short paper describe a simple way to construct a bi-centric quadrilateral with dynamic geometry.

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...

Revisits a challenging geometry problem providing several different proofs.

This paper briefly discusses and proves some mathematical results related to a tangential (or circumscribed) quadrilateral. These could be interesting investigations or extensions of high school geometry. Links to some interactive (dynamic) web pages are also provided.

The value of used signed quantities in geometry is briefly discussed by way of two instructive examples. The first example deals with the use of signed angles for the familiar angle-at-centre-of-circle theorem (twice the angle on the circumference). The second example relates to using signed distances when distances fall outside a geometric figure....

Some interesting area, perimeter and other properties of a bicentric isosceles trapezium, as well as for a right kite, are derived. These could provide nice investigation with dynamic geometry for high school learners.

Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new results are unaffected by the digital era. The reality is quite d...

This book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Provi...

102.30 A surprising 3-D result involving a hexagon - Volume 102 Issue 554 - Michael de Villiers, Heinz Schumann

A not so familiar card-trick is mathematically explained. Reflection on the proof leads to further generalizations and variations, nicely illustrating the so-called 'discovery' function of proof. It provides a suitable example for mathematics classroom use as the proof only involves elementary school algebra.

The paper describes and illustrates an example of problem-posing starting with the median triangle and asking some 'what-if' questions. This leads to an interesting result about a closed (crossed) hexagon inscribed on a conic. The result is proved using Ceva's and Pascal's theorems and is a suitable challenge for talented mathematical learners at h...

This paper extends the concept of a Kepler triangle to that of a Cyclic Kepler Quadrilateral with sides in a geometric progression of phi. Some interesting properties of this quadrilateral is then proved. An external link to a dynamic geometry sketch is also provided.

This paper gives a review of research on the Van Hiele Theory over the past 30 years, and highlights some important issues regarding theoretical implications for specifically designing learning activities in dynamic geometry contexts, as well as issues for further research such as the role of proof and hierarchical class inclusion.

This is a summary report of the ICME-13 survey on the theme of recent research in geometry education. Based on an analysis of the research literature published since 2008, the survey focuses on seven major research threads. These are the use of theories in geometry education research, the nature of visuospatial reasoning, the role of diagrams and g...

Mathematical knowledge and the interplay of practices, by Ferreirós José , pp. 337, £31.00 (hard), ISBN 978-0-691-16751-0, Princeton University Press (2016). - Volume 101 Issue 551 - Michael De Villiers

The paper presents multiple solutions to a SA Math Olympiad problem.

Constructive defining occurs when one defines new mathematical in terms of existing definitions but modifying, extending, generalizing or specializing them. Starting with the standard definition for a 'golden rectangle', various possibilities for constructively defining certain golden quadrilaterals that involve the golden ratio areexplored. Exampl...

This paper discusses an interesting, classic problem that provides a nice classroom investigation for dynamic geometry, and which can easily be explained (proved) with transformation geometry. The deductive explanation (proof) provides insight into why it is true, leading to an immediate generalization, thus illustrating the discovery function of l...

This paper briefly discusses the importance of generalization in mathematics, and then presents two possible examples of appropriate generalizations to engage gifted secondary school learners in. The one is the generalization of a familiar theorem for cyclic quadrilaterals to cyclic polygons while the other is the generalization, through the proces...

This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span fro...

This paper presents multiple solutions for a problem solving task used in the First Round of the 2016 SA Mathematics Olympiad for high school learners in South Africa.

In this paper, we present, analyse and critique an episode from a secondary school lesson involving an introduction to the definition of the scalar product. Although the teacher attempted to be explicit about the difference between a definition and a theorem, emphasizing that a definition was just an arbitrary assumption, a student rejected the tea...

This paper explores a 1949 geometry exam question, generalizing it by 'looking back' at the solution in Polya style, as well as considering some interesting variations of it, including an optimization problem that is solved purely geometrically, without recourse to calculus.

This published letter argues that a well-defined arithmetic order of operations (BODMAS/PEMDAS/PEDMAS) is needed to avoid ambiguity and confusion when given a string of mathematical operators such as +, x, -, /. Attempts to justify the order of operations in terms of real world contexts are grossly misguided and circular since they are based on the...

This paper starts with a poem lamenting that the study of crossed quadrilaterals is largely ignored in high school geometry as well as university level. It then briefly discusses some of the Lakatosian opportunities offered by using crossed quadrilaterals in a geometry class. Such a learning experience of modifying, adapting and reformulating defin...

Two convincing geometry conjectures of points appearing to be collinear are given that easily pass a normal 'drag-test' with dynamic geometry software. Counterexamples only became evident after dilating the figures by a very large scale factor of about 100:1. One conjecture is proved. A URL link to dynamic geometry sketches are given for readers to...

This paper explores and proves some generalizations of a result by Gregory about a triangle circumscribed around an ellipse.

1. Introduction In the article [1], a follow-up to my article [2], Christian and Trustrum cite empirical evidence that the probability of a family giving birth to a boy or to a girl may vary from family to family. Letting b i and g i = 1 − b i respectively denote the probability of the i th family giving birth to a boy or to a girl, the suggestion...

Teaching problem-solving in undergraduate mathematics by BadgerM.S., SangwinC.J. and HawkesT.O. with BurnsR.P., MasonJ. and PopeS., pp 142, £5.25 plus postage, available as print on demand at: http://www.lulu.com/shop/product-20405493.html or as free pdf download at: http://mellbreak.lboro.ac.uk/problemsolving/sites/default/files/guide/Guide.pdf (2...

This short note first proves an elementary property of the quasi-circumcenter of a quadrilateral, and then generalizes the quasi-Euler line of a quadrilateral to a hexagon involving its quasi-circumcenter, its quasi-orthocenter and its lamina centroid.

This paper discusses a 'converse-like' variation of Miquel's theorem, and its generalization to higher polygons. Provides a suitable exploration with dynamic geometry software, and intellectual challenge for talented mathematical learners at high school, or for undergraduate students. A dynamic geometry Java applet is also available at: http://dyna...

Reviews - Introduction to differential equations using Sage, by Joyner David and Hampton Marshall . Pp. 264. $60 (USD). 2012. ISBN 978-1-4214-0637-4, e-ISBN 978-1-4214-0724-1 (Baltimore, MD: The John Hopkins University Press). - Volume 98 Issue 542 - Michael De Villiers

Reviews - Iris Runge: A life at the crossroads of mathematics, science and industry, by Tobies Renate . Pp. 442. £99.00 (Hardback). 2012. ISBN: 978-3-0348-0229-1 (Birkhauser). - Volume 98 Issue 541 - Michael De Villiers

“The ambitious learner should carefully study a new fact; he should turn it over and over, consider it under various aspects, scrutinize it from all sides … Moreover, he should try to expand and enlarge any newly acquired knowledge by application, generalization, specialization, analogy, and in all other ways.” [1] Vincenzo Viviani, a 17th century...

The paper describes a classroom episode in which the circumcentre of a triangle was modeled as the 'best' (fairest) solution for placing a water reservoir for three towns (of more or less equal size). However, in the subsequent classroom exploration of the case when the triangle is obtuse, a 'better' solution was argued for at the midpoint of the l...

This paper presents and briefly discusses an algebraic expression that came up in a proof that could not be factorized by current computer algebra systems, but had to be done by hand using high school techniques. It therefore gives an example of why factorizing by hand is still a useful skill for mathematical problem solvers to have.

A heuristic description of the experimental exploration and discovery of a further generalization of a theorem about triangles by Arsalan Wares.

How do you know if an election result is fair? Or if the result truly represents the choice of the people? In Making Democracy Fair students use mathematical methods to explore different kinds of ballots, election decision procedures, and apportionment methods. In the first half of the book, students are introduced to a variety of alternatives to t...

The Euler line of a triangle is mostly valued, not for any practical application, but purely as a beautiful, esoteric example of post-Greek geometry. Much to his surprise, however, the author recently came across the following result and theorem by Sylvester (1814-1897) in [1] that involves an interesting application of forces that relate to the Eu...

This chapter introduces the ICMI Study Volume Proof and Proving in mathematics education. First it states the concept and rationale of the Study Volume. It then describes the development and organisation of the ICMI 19th Study Conference and the genesis of this Volume. Next, it clarifies the range of conceptions of proof with which the chapter auth...

This book is now available for free to download as an Open Access book on SpringerLink: https://link.springer.com/book/10.1007/978-94-007-2129-6

Rethinking Proof harnesses the power of Dynamic Geometry to engage students in proving conjectures and thinking about proof in different ways. Rethinking Proof offers other motivations for proof besides verification, including explanation, discovery, challenge, and systemization. In the activities, students work with SketchpadTM sketches to make co...

The dual results for semi-regular angle-and side-gons, which respectively are generalizations of a rectangle and a rhombus, can be further generalised as follows. Theorem 1: A cyclic 2n-gon has n distinct pairs of adjacent angles equal, if and only if, one set of alternate sides are equal. Theorem 2: A circumscribed 2n-gon has n distinct pairs of a...

This paper describes the generalization of the concepts of a rectangle and rhombus respectively to "cyclic polygons with equal angles" and "circumscribed polygons with equal sides".

Reviews - A primer for mathematics competitions, Zawaira Alexander and Hitchcock Gavin . Pp. 338. £50.00 (Hardback), £22-50 (Paperback). 2009. ISBN 978-0-19-953987-1; 978-0-19-953988-8 (Paperback). (Oxford University Press). - Volume 95 Issue 532 - Michael De Villiers

This article presents a generalization of the concurrency of the maltitudes of a cyclic quadrilateral, as well as a generalization of the Euler line to cyclic n-gons. The role of computer exploration and proof in this discovery is also briefly discussed.

Leonhard Euler: a man to be reckoned with, by Heyne Andreas and Heyne Alice (illustrations by Pini Elena ). pp. 45. £14.50 (hardback). 2007. ISBN: 978-3-7643-8332-9. (Basel: Birkhäuser Verlag). - Volume 94 Issue 531 - Michael De Villiers

This paper gives a review of research on the Van Hiele Theory over the past 30 years, and highlights some important issues regarding theoretical implications for designing learning activities in dynamic geometry contexts, as well as issues of further research such as hierarchical class inclusion. The Van Hiele theory originated in the respective do...

Este artigo apresenta uma retrospectiva ds pesquisas sobre a Teoria de Van Hiele nos últimos 30 anos. Destaca e ilustra alguns aspectos importantes sobre as implicações teóricas para a concepção de atividades de aprendizagem em contextos de geometria dinâmica. Problemas e questões para novas pesquisas são sugeridos tais como a inclusão hierárquica...

Reviews - Leonhard Euler, by Fellman Emil . Pp. 179. £23.00 (Hardback). 2007. ISBN-13: 978-3-7643-7538-6 (Birkhäuser Verlag). - Volume 94 Issue 529 - Michael De Villiers

Hierdie artikel beskryf die ontdekking van die sogenaamde De Villiers-punte van ’n driehoek die bewyse wat betrokke is, en trek die historiese oorsprong terug na die Fermat-punte van ’n driehoek, die swaartepunt van ’n driehoek, en ’n nuttige veralgemening van die Fermat-punte van ’n driehoek.

Describes the discovery and proof of an interesting area ratio result for a parallelo-hexagon with reference to a related result for quadrilaterals (Varignon's theorem).

This paper describes a generalization of the Fermat point of a triangle, and its application to the discovery and proof of two new triangle points, the so-called De Villiers points.

The purpose of this book is to actively involve the reader in the heuristic processes of conjecturing, discovering, formulating, classifying, defining, refuting, proving, etc. within the context of Euclidean geometry. The book deals with many interesting and beautiful geometric results, which have only been discovered during the past 300 years such...

This paper first discusses the genetic approach and the relevance of the history of mathematics for teaching, reasoning by analogy, and the role of constructive defining in the creation of new mathematical content. It then uses constructive defining to generate a new generalization of the Nagel line of a triangle to polygons circumscribed around a...

This paper argues from a theoretical standpoint that students should be actively engaged in the defining of geometric concepts like the quadrilaterals, and presents some data relating to a teaching experiment aimed at developing students' ability to define.

Conference Proceedings with papers of ICMI Study 19 Conference.

Contains al the papers of ICMI Study 19 Conference: Proof and Proving in Mathematics Education.

This paper presents the discovery of a hexagon result on Geometer's Sketchpad and its generalization via proof for any 2n-gon. The result is : If ABCDEF is a hexagon with opposite sides parallel (not necessarily equal), then the respective centroids G, H, I, J, K and L of triangles ABC, BCD, CDE, DEF, EFA and FAB, form a hexagon with opposite sides...

This paper generalizes Cross' theorem about the areas of the formed triangular 'flanks' (resulting from Vecten's configuration of constructing squares on the sides of an arbitrary triangle) to a convex quadrilateral with two different arrangements of similar parallelograms on the sides as well as to similar cyclic quadrilaterals. Pedagogically, it...

This article discusses the experimental discovery of a generalisation of the Spieker circle and the associated Nagel line using dynamic geometry software, which is possibly new or relatively unknown. It also discusses an interesting analogy (or duality) between the Spieker conic and corresponding Nagel line with that of the Nine-point conic and cor...

A heuristic description is given of the rediscovery with Sketchpad of a less-well-known, but beautiful, generalization of the nine-point circle to a nine-point conic, as well as an associated generalization of the Euler line. The author's initial analytic geometry proofs, which made use of the symbolic algebra facility of the TI-92 calculator, are...

Every year the SA Mathematics Olympiad produces a valuable resource of new, innovative mathematical problems, and manages to stimulate interest in problem solving among learners, teachers and parents. Apart from being an essential resource for preparation for SAMO, a number of problems can also be used as further investigations by teachers. Past pa...

Analiseer die rol en funksie van bewysvoering met spesifieke betrekking tot dinamiese meetkunde.

To most people, including some mathematics teachers, geometry is synonymous with ancient Greek geometry, especially as epitomised in Euclid's Elements of 300 BC. Sadly, many are not even aware of the significant extensions and investigations of Apollonius, Ptolemy, Pappus, and many others until about 320 AD. Even more people are completely unaware...