Introduction to Geometry. By H. S. M. Coxeter. Pp. xiv, 443. 1961. (John Wiley)

Concrete materials have a long history in the mathematics classroom, although they have not always been readily accepted or used appropriately. They disappeared when written computational methods arose and little premium was placed on understanding the algorithms being learned. Comenius and Pestalozzi began the process of reintroduction, with Montessori and many others in the present century providing new materials and new rationales for their use, so that today one finds hundreds of ‘manipulatives’ available. Arguments have persisted, however, as to whether common tools from daily life might be better than specially constructed educational materials and whether, in fact, all such materials might do more harm than good. Educational materials are not miracle drugs; their productive use requires planning and foresight.

Concrete Materials in the Classroom

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.

Experimentation and Proof in Mathematics

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.

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The role and function of quasi‐empirical methods in mathematics

The idea of personal computational technology in mathematics education is explored, and different portable devices, from the arithmetic calculator to the laptop computer, are described and appraised in terms of their potential as individual resources for classroom use. The relatively modest impact of calculators on mathematics education is documented, and factors which inhibit and support development are discussed: resourcing; curricular influence; public perceptions; and teacher confidence. Two examples of ways in which calculators have been integrated into classroom mathematics are sketched: one involving arithmetic calculators in the lower primary school; the other, graphic calculators at upper secondary level. Findings from research into the effects of calculator use are summarised and evaluated. Various aspects of the engraining of calculator use in processes of mathematical thinking and of teaching mathematics are identified and discussed, and emerging issues highlighted.

Calculators in the Mathematics Curriculum: the Scope of Personal Computational Technology

This article considers the use of a range of computer tools for aiding geometric exploration, together with a suitable algebraic representation for objects connected with the triangle. The techniques are applied to produce new results in the geometry of the triangle. This is an extended version of part of the contribution: How do computers change the way we do mathematics? given at the Association’s 1994 Easter Conference.

Computer aided research into triangle geometry

An analysis of an algorithm

On Twin Peaks and Flat Functions

This is the second of two articles presenting new results which extend the Soddy circles of a triangle to Soddy spheres of a special class of tetrahedra. In Part 1 [1], after a brief resume of the Soddy theorems in two dimensions, we stated the corresponding theorems in three dimensions, discovered partly as the result of computer investigations. These threedimensional Soddy theorems are valid only for a special type of tetrahedron - the four-ball tetrahedron - and we went on to prove some basic results about such tetrahedra.

The Soddy spheres of a 4-ball tetrahedron: Part 2

New discoveries about the Soddy circles of a triangle were published in the Gazette [1] in 1995; they are summarised below. Extensions of wellknown results in the geometry of the triangle to that of the tetrahedron were presented by Zeeman in his 2004 Presidential Address at the Mathematical Association Conference in York [2]. Using the computer software package Cabri 3D, we have discovered new results which extend the Soddy circles of a triangle to Soddy spheres of a special class of tetrahedra. This article is the first of two which present our discoveries, as well as relevant aspects of the established geometry of tetrahedra, together with their proofs.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/D07C85365437427C659BDE2B9B718769/S0025557200183032a.pdf/div-class-title-the-soddy-spheres-of-a-4-ball-tetrahedron-part-1-div.pdf

Adrian Oldknow

Papers

Computer aided research into triangle geometry

An analysis of an algorithm

On Twin Peaks and Flat Functions

The Soddy spheres of a 4-ball tetrahedron: Part 2

The Soddy spheres of a 4-ball tetrahedron: Part 1