Michael De Villiers

Using dynamic geometry to expand mathematics teachers’ understanding of proof

This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers’ understanding of other functions of proof than just the traditional function of ‘verification’. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called ‘reconstructive’ approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry.This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers’ understanding of other functions of proof than just the traditional function of ‘verification’. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called ‘reconstructive’ approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry.

The Role and Function of Quasi-empirical Methods in Mathematics 1

Abstract 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: conjecturing, verification, global refutation, heuristic refutation, and understanding. After some fundamental limitations of quasi‐empirical methods have been pointed out, it is argued that, in genuine mathematical practice, quasi‐empirical methods and more logically rigorous methods complement each other. The challenge for curriculum designers is, therefore, to develop meaningful activities that not only illustrate the above functions of quasi‐empirical methods but also accurately reflect an authentic view of the complex, interrelated nature of quasi‐empiricism and deductive reasoning.

Proof and Proving in Mathematics Education

One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.

Experimentation and Proof in Mathematics

This paper examines the role and function of experimentation in mathematics with reference to some historical examples and some of my own, in order to provide a conceptual frame of reference for educational practise. I identify, illustrate, and discuss the following functions: conjecturing, verification, global refutation, heuristic refutation, and understanding. After pointing out some fundamental limitations of experimentation, I argue that in genuine mathematical practise experimentation and more logically rigorous methods complement each other. The challenge for curriculum designers is therefore to develop meaningful activities that not only illustrate the above functions of experimentation but also accurately reflect the complex, interrelated nature of experimentation and deductive reasoning.

Generalizations involving maltitudes

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.

From nested Miquel triangles to Miquel distances

Neubergs theorem can, however, be generalised by starting with a point P and constructing lines to the sides of a triangle ABC so that these lines all form equal angles with the sides as shown in Figure 1 (i.e. ZPA1B = ZPB1C = ZPC1A). Following [2], we shall call the triangle A1B1CI formed by these lines, a Miquel triangle. From the same point P, construct a second Miquel triangle in the first Miquel triangle, and then another Miquel triangle in the second one. Then the third Miquel triangle is similar to the original triangle ABC. 390

A Fibonacci Generalization and its Dual

An interesting dual sequence for the Fibonacci sequence is presented in which the consecutive terms are constructed via multiplication of the preceding terms, instead of addition. Well-known results illustrating this duality are also generalized, showing how these relate to generalizations of the golden ratio.

A generalized dual of Napoleon's theorem and some further extensions

Utilizing a duality between the concepts incentre and circumcentre, a dual to a well‐known generalization of Napoleons theorem is conjectured, experimentally confirmed and eventually proved. The proof then also shows that the result is merely a special case of a more general result. As a further consequence, two interesting related results are also derived.

Vertical line and point symmetries of differentiable functions

A heuristic account is given of the authors personal investigation of some aspects of the vertical line and point symmetries of differentiable functions. Starting from examples of third degree polynomials there is first generalized to polynomials in general, and later even more generally to differentiable functions. The role of quasi‐empirical testing is demonstrated throughout, not only in the gaining of confidence in conjectures, but also in improving them through the production of counter‐examples. In some examples the role of deductive proof is also clearly shown to be far less that of verification than that of explanation, systematization and/or discovery. Symmetry, as wide or as narrow as you may define it, is one idea by which man through the ages has tried to comprehend, and create order, beauty and perfection. (Hermann Weyl).

The ‘stop after k girls or N children’ policy

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 is that b i may be distinct from b j for distinct i and j . As they point out, this has implications for the expected society-wide number of boys and girls. Following my discussion of the ‘stop after k girls’ policy, Christian and Trustrum introduce the related policy ‘stop after k girls or N children’. They argue that the expected number of children under this latter policy is an increasing function of b i (when 0 k N ). The present article complements their discussion by examining this alternative stopping policy in more detail.