Showing papers in "The Mathematical Gazette in 1990"

An introduction to chaotic dynamical systems (2nd edition), by Robert L. Devaney. Pp 336. £34·95. 1989. ISBN 0-201-13046-7 (Addison-Wesley)

Abstract: Part One: One-Dimensional Dynamics Examples of Dynamical Systems Preliminaries from Calculus Elementary Definitions Hyperbolicity An example: the quadratic family An Example: the Quadratic Family Symbolic Dynamics Topological Conjugacy Chaos Structural Stability Sarlovskiis Theorem The Schwarzian Derivative Bifurcation Theory Another View of Period Three Maps of the Circle Morse-Smale Diffeomorphisms Homoclinic Points and Bifurcations The Period-Doubling Route to Chaos The Kneeding Theory Geneaology of Periodic Units Part Two: Higher Dimensional Dynamics Preliminaries from Linear Algebra and Advanced Calculus The Dynamics of Linear Maps: Two and Three Dimensions The Horseshoe Map Hyperbolic Toral Automorphisms Hyperbolicm Toral Automorphisms Attractors The Stable and Unstable Manifold Theorem Global Results and Hyperbolic Sets The Hopf Bifurcation The Hnon Map Part Three: Complex Analytic Dynamics Preliminaries from Complex Analysis Quadratic Maps Revisited Normal Families and Exceptional Points Periodic Points The Julia Set The Geometry of Julia Sets Neutral Periodic Points The Mandelbrot Set An Example: the Exponential Function

Mathematical enculturation: a cultural perspective on mathematics education, by Alan J. Bishop. Pp 195. £39·50. 1988. ISBN 90-277-2646-9 (Kluwer Academic Publishers)

Abstract: 1/Towards a Way of Knowing.- 1.1. The conflict.- 1.2. My task.- 1.3. Preliminary thoughts on Mathematics education and culture.- 1.4. Technique-oriented curriculum.- 1.5. Impersonal learning.- 1.6. Text teaching.- 1.7. False assumptions.- 1.8. Mathematical education, a social process.- 1.9. What is mathematical about a mathematical education?.- 1.10. Overview.- 2/Environmental Activities and Mathematical Culture.- 2.1. Perspectives from cross-cultural studies.- 2.2. The search for mathematical similarities.- 2.3. Counting.- 2.4. Locating.- 2.5. Measuring.- 2.6. Designing.- 2.7. Playing.- 2.8. Explaining.- 2.9. From 'universals' to 'particulars'.- 2.10. Summary.- 3/The Values of Mathematical Culture.- 3.1. Values, ideals and theories of knowledge.- 3.2. Ideology - rationalism.- 3.3. Ideology - objectism.- 3.4. Sentiment - control.- 3.5. Sentiment - progress.- 3.6. Sociology - openness.- 3.7. Sociology - mystery.- 4/Mathematical Culture and the Child.- 4.1. Mathematical culture - symbolic technology and values.- 4.2. The culture of a people.- 4.3. The child in relation to the cultural group.- 4.4. Mathematical enculturation.- 5/Mathematical Enculturation - The Curriculum.- 5.1. The curriculum project.- 5.2. The cultural approach to the Mathematics curriculum - five principles.- 5.2.1. Representativeness.- 5.2.2. Formality.- 5.2.3. Accessibility.- 5.2.4. Explanatory power.- 5.2.5. Broad and elementary.- 5.3. The three components of the enculturation curriculum.- 5.4. The symbolic component: concept-based.- 5.4.1. Counting.- 5.4.2. Locating.- 5.4.3. Measuring.- 5.4.4. Designing.- 5.4.5. Playing.- 5.4.6. Explaining.- 5.4.7. Concepts through activities.- 5.4.8. Connections between concepts.- 5.5. The societal component: project-based.- 5.5.1. Society in the past.- 5.5.2. Society at present.- 5.5.3. Society in the future.- 5.6. The cultural component: investigation-based.- 5.6.1. Investigations in mathematical culture.- 5.6.2. Investigations in Mathematical culture.- 5.6.3. Investigations and values.- 5.7. Balance in this curriculum.- 5.8. Progress through this curriculum.- 6/Mathematical Enculturation - The Process.- 6.1. Conceptualising the enculturation process in action.- 6.1.1. What should it involve?.- 6.1.2. Towards a humanistic conception of the process.- 6.2. An asymmetrical process.- 6.2.1. The role of power and influence.- 6.2.2. Legitimate use of power.- 6.2.3. Constructive and collaborative engagement.- 6.2.4. Facilitative influence.- 6.2.5. Metaknowledge and the teacher.- 6.3. An intentional process.- 6.3.1. The choice of activities.- 6.3.2. The concept-environment.- 6.3.3. The project-environment.- 6.3.4. The investigation-environment.- 6.4. An ideational process.- 6.4.1. Social construction of meanings.- 6.4.2. Sharing and contrasting Mathematical ideas.- 6.4.3. The shaping of explanations.- 6.4.4. Explaining and values.- 7/The Mathematical Enculturators.- 7.1. People are responsible for the process.- 7.2. The preparation of Mathematical enculturators - preliminary thoughts.- 7.3. The criteria for the selection of Mathematical enculturators.- 7.3.1. Ability to personify Mathematical culture.- 7.3.2. Commitment to the Mathematical enculturation process.- 7.3.3. Ability to communicate Mathematical ideas and values.- 7.3.4. Acceptance of accountability to the Mathematical culture.- 7.3.5. Summary of criteria.- 7.4. The principles of the education of Mathematical enculturators.- 7.4.1. Mathematics as a cultural phenomenon.- 7.4.2. The values of Mathematical culture.- 7.4.3. The symbolic technology of Mathematics.- 7.4.4. The technical level of Mathematical culture.- 7.4.5. The meta-concept of Mathematical enculturation.- 7.4.6. Summary of principles.- 7.5. Socialising the future enculturator into the Mathematics Education community.- 7.5.1. The developing Mathematics Education community.- 7.5.2. The critical Mathematics Education community.- Notes.- Index of Names.

The Penguin dictionary of mathematics , edited by John Daintith and R. D. Nelson. Pp 350. £4·99. 1989. ISBN 0-14-051119-9 (Penguin)

Abstract: From algebra to number theory and from statistics to mechanics, this versatile dictionary takes in all branches of pure and applied mathematics up to first-year university level. Invaluable for mathematicians, it is also a useful source book for economists, business people, engineers, technicians and scientists of all kinds who need a knowledge of mathematics in the course of their work.

The World Cup draw's flaws

Abstract: I don’t know how many readers of this journal would have been watching the draw for the opening matches of the 1990 football World Cup, made in Rome on Saturday, 9 December 1989, and televised worldwide. But I do hope that most of you that did—and who were paying attention!—at least thought “Hey, wait a minute, that can’t be right!” Sure enough, the draw was flawed: it was not as fair to all concerned as it might have been. This was a great pity given that the world football authority, FIFA, were at pains to stress how fair these proceedings would be, and the agonies of FIFA trying to be seen to live up to their “fair play” motto were indeed prolonged. The errors lay not in the process of drawing balls from urns itself, but in the algorithm used to define the drawing procedure, as I shall explain below. The problem was caused by FIFA’s decision to keep the four South American countries involved apart and, in particular, by their inability to implement this decision in the most fair manner. There is no allegation of anything being “fixed” here, just one of incompetence on behalf of football’s governing body. That there should be errors of this kind in such an overtly public affair is, however, yet another sad indictment of the general lack of understanding of matters numerical and probabilistic in the public at large.

Sophie Germain: or Was Gauss a feminist?

Abstract: This article is focussed on a letter that perhaps the greatest mathematician ever, Carl Frederich Gauss, sent to a female mathematician, Sophie Germain. The letter is interesting because it is the first that Gauss wrote to Germain in the knowledge that she was a woman, and it contains a piece of mathematics that reveals better than the text of the letter what Gauss really thought of Germain. To put the letter in context I shall give some of Germain’s biography and also briefly describe female mathematicians that pre-date Germain.