### Synopses & Reviews

Dreams of Calculus presents evidence that mathematics education today is in a process of change of paradigm, caused by the revolutionary new possibilities offered by the computer. The authors complement the physicist Eugene Wigner's famous statement concerning "the unreasonable effectiveness of mathematics in the natural sciences", by presenting evidence of "the reasonable effectiveness of computational mathematics". The book may also serve as an introduction to the Body&Soul mathematics education reform project reflecting the new paradigm. Dreams of Calculus is directed to a large audience of teachers, students and users of mathematics. In a first part the authors present a brief history leading into applications of computational mathematics today. In a second part, they present key applications of computational mathematics for simulation of the motion of the planets in our solar system by solving Newton's equation, and turbulence by solving the Navier--Stokes equations. This book is stimulated by the work of the Mathematics Delegation created by the Swedish Minister of Education in 2003 with the task of analyzing the current crisis in mathematics education on all levels. The authors are leading researchers in computational mathematics. Further information can be found at http://www.phi.chalmers.se/bodysoul/

#### Review

Aus den Rezensionen:

"... Das vorliegende Buch kann als eine Einführung in die Philosophie dieses Projekts bezeichnet werden. ... Das Buch ist eher unkonventionell verfasst: nicht linear, sondern vernetzt und redundant, abwechslungsreich. Es ist kurzweilig zu lesen und liefert allen an einer Reform des Mathematikunterrichts Interessierten Stoff für Diskussionen."

(M. Kronfellner, in: IMN - Internationale Mathematische Nachrichten, 2006, Issue 202, S. 55 f.)

#### Review

From the reviews: "[...] this book fills a gap in the literature of mathematics education although the analysis and conclusions in it are not constraining (but very interesting!)." St.Götz, Monatshefte für Mathematik 148, Issue 3, 2006 "The object of this book is to make a case for computational mathematics to be treated as a subject in its own right. ... The text is copiously illustrated with diagrams associated with the topic discussed and portraits of some of the mathematicians mentioned. ... The printing and format are pleasing and the book will be of interest to those in university departments contemplating revision of syllabi." Ll. G. Chambers, Zentralblatt MATH, Vol. 1079, 2006

#### Synopsis

What is the relationship between modern mathematics - more precisely computational mathematics - and mathematical education? It is this controversal topic that the authors address with an in-depth analysis. In fact, what they present is an extremely well-reasoned account of the development of mathematics and its culture giving concrete recommendation for a much-needed reform of the teaching of mathematics. The book is essential reading for everybody involved in mathematics and science, and mathematics teaching.

#### Synopsis

The purpose of this book is to stimulate a much needed debate on ma- ematics education reform. We do not claim that we present anything near the whole truth, and that there are not many other points of view. We thus invite to debate and urge other people to express their views. As scientists we believe that it is our duty to present our own standpoints and conc- sions as clearly as possible, to open for scrutiny and discussion, rather than ambiguous politically correct views, which are di?cult to question. This book may be viewed as an introduction to our mathematics e- cation reform project Body&Soul based on a synthesis of Body (com- tational mathematics) and Soul (analytical mathematics). We do make a case for computational mathematics, which is the new world of mathem- ics opened by the computer, now waiting to enter mathematics education. But we do also propose a synthesis of this new world, with the kernel being computational algorithms, and the traditional world of analytical ma- ematics, with analytical formulas as the kernel. In fact, algorithms are expressed using analytical formulas before being translated into computer code, and good analytical formulas are necessary for understanding and insight.

#### Synopsis

A first-class debate book on the crucial issues of current mathematics teaching The authors offer startling evidence that computers are changing mathematics in a profound way Raises the question of how to alter teaching in mathermatics as a result of the computer's influence on the field

### Table of Contents

I. Perspectives.- 1 Introduction.- 2 What? How? For Whom? Why?- 3 A Brief History of Mathematics Education.- 4 What is Mathematics?- 5 Mountain Climbing.- 6 Jazz/Pop/Folk vs Classical Music.- 7 Virtual Reality and the Matrix.- 8 Scientific Revolutions.- 9 Education is based on Science.- 10 The Unreasonable Effectiveness of Mathematics in the Natural Sciences?- 11 Do we live in "The Best of Worlds"?- 12 The Reasonable Effectiveness of Computational Mathematics!- 13 The Need of Mathematics according to the Mathematics Delegation.- 14 An Agenda.- II. Essence.- 15 A Very Short Calculus Course.- 16 The Solar System.- 17 Turbulence and the Clay Prize.- III. Appendix.- 18 Appendix: The Mathematics Delegation and its Task.- 19 Preface to Body&Soul.- 20 Do Mathematicians Quarrel?