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Dreams of Calculus: Perspectives on Mathematics Education

Publisher: 
Springer Verlag
Number of Pages: 
158
Price: 
29.95
ISBN: 
3-540-21976-5

In January 2003, Sweden's minister of education created a Mathematics Delegation to review mathematics education in Sweden and decide if changes were in order. At the time Dreams of Calculus was published, the Mathematics Delegation had not yet submitted its report, but public statements given by the delegation's chairman expressed the view that

  • There is no crisis in mathematics education today.
  • There is no change in paradigm in mathematics education now going on because of the computer.

Hoffman, Johnson, and Logg are experts in computational mathematics who believe there is a crisis and there is a change in paradigm going on because of the computer. In Dreams of Calculus, they argue their case and ask readers to take a stand on this issue. The authors are not newcomers in this discussion. They are involved in a reform project at Chalmers University of Technology in Göteborg, Sweden that has produced the three-volume text Applied Mathematics: Body and Soul.

In the first section of the book, entitled Perspectives, the authors discuss a variety of topics including the differences between pure mathematics (mathematics without a computer) and computational mathematics (mathematics with a computer), the history of mathematics education, and several philosophical issues regarding the nature of mathematics and science in general. Throughout this section, they make the case for the study of computational mathematics and its use in solving deep questions in science. Their Body and Soul text approaches mathematics from this viewpoint.

The authors point out that in much of pure mathematics, there is a set of known tricks that students must master so they can use them when the need arises. Certainly, computational mathematics has its share of tricks, but there is more opportunity for experimentation and improvisation. Also, since pure analytical solutions to many (most?) important questions in science are extremely difficult to find, it makes sense to look for computational methods that can produce approximate solutions.

In the second section, Essence, the authors give a very brief description of calculus, discuss computational approaches to some famous problems from science, and then return to more philosophical matters. In this section, the material is more technical and is intended to demonstrate how mathematics can be approached in a way that combines analytical and computational techniques. Some of the chapters in this section are reproductions of material from the Body and Soul text.

I recommend this book for anyone interested in computers and the teaching of calculus. Although I have not seen the Body and Soul text, I like Dreams of Calculus and the approach it suggests. I believe that significant problems in physical science are most likely to be solved by people who are well versed in both analytical and computational mathematics. Furthermore, I like the Body and Soul project because I feel that many teachers of calculus overemphasize the use of the computer in demonstrating concepts and underemphasize the use of the computer in solving problems. The recent trend of offshoring computer programming jobs to Asia has affected our students' desire to study programming and computing in general. I hope that projects such as Body and Soul will help remind students and teachers that professionals who are employed as scientists (not programmers) may need to write programs to solve problems.

At times, however, the authors seem a bit out of touch with the current state of affairs — at least in the United States. On page 12 the authors say that "not even an arts student at an American college may get away without a calculus course," and on page 16 they claim that "both calculus and linear algebra are still presented as if the computer does not exist."

There are some grammatical/typographical errors and awkward sentences that detract from the writing. For example, on pages 12 and 30 there are sentences that begin with lower case letters, and on pages 103 and 132 there are sentences where verb and subject do not agree. The book is easy to read, but I think Springer could have done a better job of editing the manuscript.

The Mathematics Delegation's final report (including a summary in English) can be found at http://www.regeringen.se/content/1/c6/03/03/48/6a32d1c0.pdf.


Keith Brandt is Associate Professor of Mathematics at Rockhurst University in Kansas City, Missouri.

Date Received: 
Sunday, May 1, 2005
Reviewable: 
Include In BLL Rating: 
J. Hoffman, C. Johnson, and A. Logg
Publication Date: 
2004
Format: 
Paperback
Category: 
General
Keith Brandt
10/15/2005
I Perspectives 1
1 Introduction 3
1.1 The Mathematics Delegation and its Main Task . . . . 3
1.2 Crisis and Change of Paradigm, or Not? . . . . . . . . 4
1.3 The Body&Soul Project . . . . . . . . . . . . . . . . . 4
1.4 Same Questions in All Countries . . . . . . . . . . . . 5
1.5 WhyWeWrote this Book . . . . . . . . . . . . . . . . 5
2 What? How? For Whom? Why? 7
2.1 Mathematics and the Computer . . . . . . . . . . . . . 7
2.2 Pure and Computational Mathematics . . . . . . . . . 8
2.3 The Body&Soul Reform Project . . . . . . . . . . . . . 11
2.4 Difficulties of Learning . . . . . . . . . . . . . . . . . . 12
2.5 Difficulties of Discussion . . . . . . . . . . . . . . . . . 13
2.6 Summing up the Difficulties . . . . . . . . . . . . . . . 14
3 A Brief History of Mathematics Education 15
3.1 From Pythagoras to Calculus and Linear Algebra . . . 15
3.2 Fromvon Neumann intoModern Society . . . . . . . . 16
3.3 Mathematics Education and the Computer . . . . . . . 17
3.4 The Multiplication Table . . . . . . . . . . . . . . . . . 17
3.5 Again: What? . . . . . . . . . . . . . . . . . . . . . . . 18
x Contents
4 What is Mathematics? 19
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 TheModernWorld . . . . . . . . . . . . . . . . . . . . 19
4.3 The Role of Mathematics . . . . . . . . . . . . . . . . 23
4.4 Design and Production of Cars . . . . . . . . . . . . . 26
4.5 Navigation: From Stars to GPS . . . . . . . . . . . . . 26
4.6 Medical Tomography . . . . . . . . . . . . . . . . . . . 26
4.7 Molecular Dynamics and Medical Drug Design . . . . 27
4.8 Weather Prediction and Global Warming . . . . . . . . 28
4.9 Economy: Stocks and Options . . . . . . . . . . . . . . 28
4.10 The World of Digital Image, Word and Sound . . . . . 29
4.11 Languages . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.12 Mathematics as the Language of Science . . . . . . . . 30
4.13 The Basic Areas of Mathematics . . . . . . . . . . . . 30
4.14 What is Science? . . . . . . . . . . . . . . . . . . . . . 31
4.15 Mathematics is Difficult: Choose Your Own Level
of Ambition . . . . . . . . . . . . . . . . . . . . . . . . 32
4.16 Some Parts of Mathematics are Easy . . . . . . . . . . 32
4.17 Increased/Decreased Importance of Mathematics . . . 33
5 Virtual Reality and the Matrix 35
5.1 Virtual Reality . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Digital Cameras . . . . . . . . . . . . . . . . . . . . . . 36
5.3 MP3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 Matrix andMarilynMonroe . . . . . . . . . . . . . . . 36
6 Mountain Climbing 39
7 Scientific Revolutions 41
7.1 Galileo and the Market . . . . . . . . . . . . . . . . . . 41
7.2 Thomas Kuhn and Scientific Revolutions . . . . . . . . 42
7.3 Shift of Paradigm in Mathematics? . . . . . . . . . . . 42
7.4 Quarrelling Mathematicians . . . . . . . . . . . . . . . 43
7.5 Change of Paradigm? or Not? . . . . . . . . . . . . . . 43
7.6 To Prove or Prove Not . . . . . . . . . . . . . . . . . . 44
7.7 The Role of Text Books . . . . . . . . . . . . . . . . . 44
8 Education is Based on Science 45
8.1 The Scientific Basis of Standard Calculus . . . . . . . 45
8.2 A New Scientific Basis . . . . . . . . . . . . . . . . . . 46
8.3 The Scientific Basis of Body&Soul . . . . . . . . . . . 47
Contents xi
9 The Unreasonable Effectiveness of Mathematics
in the Natural Sciences? 49
9.1 Newton's Model of Gravitation . . . . . . . . . . . . . 50
9.2 Laplace's Model of Gravitation . . . . . . . . . . . . . 51
9.3 Molecular Dynamics . . . . . . . . . . . . . . . . . . . 51
9.4 Einstein's Law of Gravitation . . . . . . . . . . . . . . 52
9.5 The Navier-Stokes Equations for Fluid Dynamics . . . 52
9.6 Schrùodinger's Equation . . . . . . . . . . . . . . . . . . 52
9.7 Discussion of Effectiveness . . . . . . . . . . . . . . . . 53
9.8 Weather Prediction . . . . . . . . . . . . . . . . . . . . 54
10 Do We live in The Best of Worlds"? 57
11 The Reasonable Effectiveness
of Computational Mathematics 59
11.1 The Solar System: Newton's Equations . . . . . . . . . 60
11.2 Turbulence: Navier-Stokes Equations . . . . . . . . . . 61
11.3 Schrùodinger's Equation . . . . . . . . . . . . . . . . . . 61
11.4 Einstein's Equation . . . . . . . . . . . . . . . . . . . . 62
11.5 Comparing Analytical and Computational
Solution Techniques . . . . . . . . . . . . . . . . . . . 62
11.6 Algorithmic Information Theory . . . . . . . . . . . . 63
11.7 How Smart is an Electron? . . . . . . . . . . . . . . . . 64
11.8 The Human Genome Project . . . . . . . . . . . . . . 65
12 Jazz/Pop/Folk vs. Classical Music 67
13 The Right to Not Know 69
14 An Agenda 71
14.1 Foundations of Computational Mathematics . . . . . . 71
14.2 New Possibilities for Mathematics Education . . . . . 72
II Essence 73
15 A Very Short Calculus Course 75
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 75
15.2 Algebraic Equations . . . . . . . . . . . . . . . . . . . 76
15.3 Differential Equations . . . . . . . . . . . . . . . . . . 76
15.4 Generalization . . . . . . . . . . . . . . . . . . . . . . . 81
15.5 Leibniz' Teen-Age Dream . . . . . . . . . . . . . . . . 82
15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 85
15.7 Leibniz . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
xii Contents
16 The Solar System 89
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 89
16.2 Newton's Equation . . . . . . . . . . . . . . . . . . . . 90
16.3 Einstein's Equation . . . . . . . . . . . . . . . . . . . . 92
16.4 The Solar System as a System of ODEs . . . . . . . . 94
16.5 Predictability and Computability . . . . . . . . . . . . 95
16.6 Adaptive Time-Stepping . . . . . . . . . . . . . . . . . 98
16.7 Limits of Computability and Predictability . . . . . . 99
17 Turbulence and the Clay Prize 101
17.1 The Clay Institute $1 Million Prize . . . . . . . . . . . 101
17.2 Are Turbulent Solutions Unique and Smooth? . . . . . 103
17.3 Well-Posedness According to Hadamard . . . . . . . . 104
17.4 Is Mathematics a Science? . . . . . . . . . . . . . . . . 105
17.5 A Computational Approach to the Problem . . . . . . 106
17.6 The Navier-Stokes Equations . . . . . . . . . . . . . . 109
17.7 The Basic Energy Estimate for the
Navier-Stokes Equations . . . . . . . . . . . . . . . . . 110
17.8 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . 111
17.9 Computational Solution . . . . . . . . . . . . . . . . . 111
17.10 Output Error Representation . . . . . . . . . . . . . . 112
17.11 The Dual Problem . . . . . . . . . . . . . . . . . . . . 113
17.12 Output Uniqueness of Weak Solutions . . . . . . . . . 114
17.13 Computational Results . . . . . . . . . . . . . . . . . . 115
17.14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 119
18 Do Mathematicians Quarrel? 121
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 121
18.2 The Formalists . . . . . . . . . . . . . . . . . . . . . . 124
18.3 The Logicists and Set Theory . . . . . . . . . . . . . . 124
18.4 The Constructivists . . . . . . . . . . . . . . . . . . . . 127
18.5 The Peano Axiom System for Natural Numbers . . . . 129
18.6 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 130
18.7 Cantor versus Kronecker . . . . . . . . . . . . . . . . . 131
18.8 Deciding Whether a Number is Rational or Irrational . 132
18.9 The Set of All Possible Books . . . . . . . . . . . . . . 133
18.10 Recipes and Good Food . . . . . . . . . . . . . . . . . 134
18.11 The "New Math" in Elementary Education . . . . . . 135
18.12 The Search for Rigor in Mathematics . . . . . . . . . . 135
18.13 A Non-Constructive Proof . . . . . . . . . . . . . . . . 136
18.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 137
Contents xiii
III Appendix 141
A Comments on the Directives of the
Mathematics Delegation 143
A.1 The Directives . . . . . . . . . . . . . . . . . . . . . . . 143
A.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . 144
A.3 A SumUp . . . . . . . . . . . . . . . . . . . . . . . . . 145
B Preface to Body&Soul 147
C Public Debate 155"
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