Dreams of Calculus, Perspectives on Mathematics Education, J. Hoffman C. Johnson A.Logg, Springer
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The Mathematics Delegation was created by the Minister of Education Thomas Ostros at a Swedish government cabinet meeting on 23rd January 2003 with the following main task.
“Analyse the current situation in terms of the teaching of mathematics in Sweden and assess the need for changing current syllabuses and other steering documents.”
In public presentations of the work of the Mathematics Delegation by its chaiman, the following two statements have been expressed as a basis for the analysis:
> A1: There is -no- crisis in the mathematics education today. > A2: There is -no change od paradigm- in mathematics education now going on because of the computer.
Our motivation to write this book comes from our conviction that
> B1: There is -a crisis- in the mathematics education today. > B2: There is -a change of paradigm- in mathematics education now going on because of the computer.
Of course, the problems are not solved by just identidying B1-2 as more true than A1-2; if B1-2 indeed describe the realities, it remains to come up with a reformed mathematics program reflecting the change od paradigm and which may help to resolve the crisis. We briefly present our work in this direction within our Body & Soul mathematics education reform project (www.phi.chalmers.se/bodysoul/).
The Body and Soul project has grown out of a 30 year activity of the senior author in international research with several influential articles and books. This book may be viewed as a kind of summary of this work.
Mathematics education is of course not a Swedish affair; the issues are similar in all countries, and in each country the different points of view with A1-2 and B1-2 as clearly expressed alternatives, have their supporters.
“I admit that each and every thing remains in its state until there is reason for change” –Leibniz–
“The mathematician's pattern's, like those of the painter's or the poet's, must be beautiful, the ideas, like the colours or de words, must fit together in a harmonious way. There is no permanent place in the world for ugly mathematics.” –Hardy–
Mathematics is an important part of our culture and has a central role in education from elementary pre-school, through primary, secondary schools and high schools to many university programs.
The basic questions of mathematics education are: What to teach? How to teach? Whom to teach? and Why to teach? The answers to these questions are changing over time, as mathematics as a science and our entire society are changing. Education in general, and in particular education in Mathematics and Science, is supposed to have a scientific basis. The scientific basis of the standard mathematics education presented today was formed during the 19th century, well before the computer was invented starting in the mid 20th century. Therefore the current answers to the basic questions formulated above reflect a view of mathematics without the computer.
Today computers are changing our lives and our society. The purpuse of this book is to stimulate a much needed debate on mathematics education. We seek to reach a wide public of teachers and students of mathematics on all levels, and we therefore seek to present some basic ideas as simply and clearly as possible, with a minimu of mathematical notation.
The terms pure mathematics and applied mathematics are used to identify different areas of mathematics as a science, with different focus. In applied mathematics the main topics of investigation would come from areas such as mechanics and physics, while in pure mathematics one could pursue mathematical questions without any coupling to applications. The distintion between pure and applied mathematics is quite recent and gradually developed during the 20th century. But even today, there is no clear distintion between pure and applied mathematics; a mathematical technique once developed within pure mathematics may later find applications and thus become a part of applied mathematics. Another distintion is now developing: mathematics withour computer and mathematics with computer. Applied mathematics today can largely be described as mathematics with computers, or computational mathematics. Most of the activity of pure mathematics today can correspondingly be described as mathematics without computer, although computers have been used to solve some problems posed within pure mathematics. One example is the famous 4-color problem asking for a mathematical proof that 4 colors are enough to color any map so that neighboring countries do not get the same color.
In pure mathematics a question may receive attention just because it represents an intellectual challenge, not because it has a scientific relevance as a question of some importance to mankind. The famous mathematician G. Hardy (1877-1947) expressed this attitude very clearly in his book “A Mathematician's Apology”, although the title indicates some doubts about public acceptance. The prime example of this form is the proof of the famous Fermat's Last Theorem in number theory, which was the most famous open problem in pure mathematics for 300 hundred years until Andrew Wiles completed his 130 page proof in 1994 after 8 years of heroic lonely constant struggle. For this achievement Wiles effectively received a Field Medal in 1998, the Novel Prize of Mathematics, at the International Congress of Mathematicians in Berlin 1998, although technically the prize was awarded in the form of a special tribute connected to the Fields Medal awards. (because Wiles had passed the limit of 40 years of age to receive the medal).
Fermat's Last Theorem states that there are no integers x, y and z which satisfy the equation x^n+y^n=z^n, where 'n' is an integer larger than 2. It was stated in the notes of Fermat (1601-1665) in the margin of a copy of 'Arithmetica by Diofantes of Alexandria (around 250 AD). Fermat himself indicated a proof for n=4 and Euler developed a similar proof for n=3. The French Academy of Science offered in 1853 its big prize for a full proof and drew contributions to famous mathematicians like Cauchy, but none of the submitted proofs at the dead-line 1857 was correct (and none after that until Wiles proof in 1994). The great mathematician Gauss (1777-1855), called the king of mathematics, decided not to participate, because he viewed the problem to be of little interest. Gauss believed in a synthesis of pure and applied mathematics, with mathematics being the queen of the science.
The format of Fermat's Last Theorem makes it particularly difficult to prove, since it concerns 'non-existence' of integers x, y and z and a integer n>2 such that x^n+y^n=z^n. The proof has to go by contradiction, by proving that the assumption of existence of a solution leads to a contradiction (reduccion al absurdo?). It took Wiles 130 pages to contruct a contradiction, in a proof which can be followed in detail by only a few true experts.
Fermat's Last Theorem may seem appealing to a pure mathematician because it is (very) easy to state, but (very) hard to prove. Thus, it can be posed to a large audience, but the secret of the solution is kept to a small group of specialists, as in the Pythagorean society build on number theory.
But what is the scientific meaning of a proof of Fermat's Last Theorem? Some mathematicians may advocate that (apart from aesthetics) it is not the result itself that is of interest, but rather the methods developed to give a proof. Gauss would probably not be too convinced by this type of argument, unless some striking application was presented, while Hardy would be.
The incredible interest and attention that Wiles proof did draw within trend-setting circles of mathematics, shows that the point of view of Hardy, as opposed to that of Gauss, today is dominating large parts of the scene of mathematics, but the criticism of Gauss may still be relevant.
To sum up: Today there is a dividing line between pure mathematics and computational mathematics. We will come back to this split, which originated with the bird of computer in the 1930's in a great clash between the formalist (mathematics without computer) and the contructivis (mathematics with computer).
Body & Soul contains books (Applied Mathematics: Body & Soul, Vol I-II, Vol IV to appear), software and educational material and builds on a modern synthesis of Body (computation) and Soul (mathematical analysis). Body & Soul presents a synthesis of 'analytical mathematics and computational mathematics', where analytical mathematics is used to capture basic laws of science in mathematical notation (mathematical modeling) and to investigate qualitative aspects of such laws, and computational mathematics is needed for simulation and quantitative prediction.
Body & Soul offers a basic studies in science and engineering and also for further studies in mathematics, and includes modern tools of computational mathematics. Body & Soul is a unique project in scope and content and is attracting quickly increasing interest.
Mathematics is perceived as a difficult subject by most people and feelings of insufficiency are very common, among both laymen and professionals. This is not strange, mathematics is difficult and demanding, just like classical music or athletics may be very difficult and demanding, which may creat a lot of negative feelings for students pushed to perform. There is no way to eliminate all the difficulties met in these areas, except by trivialization. Following Einstein, one should always try to make Science and educational based on Science as simple as possible, but not simplier.
In music and athletics the way out in our days is clear: the student who does not want to spend years on practicing inventions by Bacj on the piano, or to become a master of high-jump, does not have to do so, but can choose some alternative activity. In mathematics this option is not available for anyone in elementary education, and not even an arts student at an American college may get away without a calculus course, not to speak of the engineering student who will have to pass several mathematics courses.
Mathematics education is thus compulsory for large gruops of student and since mathematics is difficult, for student on all levels, problems are bound to arise.
To come to grips with this inherent difficulty of mathematics education, on all levels. we propose to offer a differentiated mathematics education on all levels, according to the interest, ability and need of the different students. Although mathematics is important in our society of today, it's very possible to both survive and have a successful professional career, with very little formal training in mathematics. It is thus important to identify the real need of mathematics for different groups of students and the shape educational programs to fill these needs.
Classical Greek or Latin formed an important part of secondary education only 50 years ago, with motivations similar to those currently used for mathematics: studies in these subjects would help develop logical thinking and problem solving skills. Today, very few students take Greek and Latin with the motivation that such studies are both difficult and of questionable usefulness for the effort invested.
One may ask if mathematics education is bound to follow the same evolution? The answer is likely to directly couple to the sucess of efforts to make mathematics both more understandable (less difficult) and more useful today (up to day), in contrast to the traditional education where many fail and even those who succeed may get inadequate training on inadequate topics.
Euclid's Elementa, with its axioms, theorems and the rules and compasses as tools, was the canon og mathematics education for many centuries into the mid 20th century, until it quite suddenly disappeared from the curriculum along with Greek and Latin. Not because geometry ceased to be of importance, but because Euclid's geometry was replaced by “computational geometry” with the tools being Descartes analyticl geometry in modern computational form.
The difficult of mathematics presents serious obstacles to discussions on mathematics and mathematics education. A pure mathematician of today would usually state that it is impossible to convey any true picture of contemporary research to anybody outside a small circle of experts. As a consequence, there is today little interaction between pure mathematicians and mathematics didactics.
On the other hand, presenting essential aspects of contemporary research in computational mathematics may be possible for large audiences, including broad groups of students of mathematics. Typically, research in computational mathematics concerns the design of a computational algorithm for solving some mathematical equatio, for example the Navier-Stokes equations for fluid flow. The result of the algorithm may be visualized as a movie describing some particular fluid flow, such as the flow of air around a car or airplane, and the objective of the computationn could be to compute the 'drag force' of a particular design, which directly couples to fuel consumption and economy. Although the details of the computation would be difficult to explain to a layman, the general stuctures of the computation and its meaning could be conveyed.
To sum up: Discussions on mathematics education have to struggle with some key difficulties: (i) contemporary research in pure and computational mathematics seem to live largely separated lives, and (ii) mathematics didactics largely lives a life separated from both contemporary pure and computational mathematics.
'Life is good for twoo things; to study mathematics and to theach it' –Poison (1781-1840)
The Babylonians formed a rich culture in Mesopotamia 2000-1000 BC based on advanced irrigation systems and developed mathematics for varios practical purposes related to building and maintaing such systems. Using a numer system with base 60. The Babylonians could do arithmetics, and could also solve quadratic equations. The development continued in ancient Greece 500 BC-100 AD, where the schools of Pythagoras and Euclid created eternal foundations of arithmetics and geometry. The next leap came with the 'calculus' of Leibniz and Newton preceded by the analytic geometry by Descartes, which opened for the scientific revolution starting in the 18th century and leading into our time, calculus (or differential and integral calculus) is the mathematical theory of functions, derivatives, integrals, and differential equations.
Calculus (also referred to as mathematical analysis) developed over the centuries with important contributions from many great mathematicians such as Cauchy and Weirstrass and found its in the present curriculum of science and engineering education in the beginning of the 20th century. Together with 'linear algebra' including vector and matrix calculus introduced in the 1950's, calculus today forms the core of mathematics education at the university level, and simplified forms thereof fill the mathematics curricula in secondary schools. The introduction of the 'linear algebra' was probably stimulated by the development of the computer, but both calculus and linear algebra are still presented as if the computer does not exist. The foreword of a new standard calculus text usually pays tribute to an idea that calculus found its form in the early 20th century and that new book can only polish on a forever given picture.
We all know that the modern computer is fundamentally changing our society. One may describe this development alternatively as a mathematics revolution reflecting that the computer may be though os as a mathematical machine. More precisely, the modern computer is referred as the 'von Neumann machine' after the famous mathematician John von Neumann who first formulated the mathematical principles of modern computer design and computer programming in the early 1940's.
Von Neumann followed up on a long tradition within mathematics of contructing machines for automated computation. from the mechanical calculators of Pascal and Leibniz doing elementary arithmetics, over Babage's Difference Engine and Analytical Computing Engine designed to solve differential equations, to the theorical Turing machine capable of mimicing the action of any conceivable computer. The motivation of all these machines was to enable automated computation for various purposes, typically connected to calculus.
We observed above that mathematics education in its current canonical form of calculus, is basically the same as before the computer revolution. We argue that this state of affairs is not motivated from either scientific or applications point of view. We belive that the computer is radically changing calculus and that this change has to be seen in the mathematics curriculum.
The multiplication table may be viewe as the corner stone of elementary mathematics education. The educated man of the 17th century did not necessarily master the multiplication table, as this knowledge was something for practitioners like merchants and carpents, not for men of learning. This is illustrated in the famous diary by Samuel Pepys 1660-69 during his studies at the University of Cambridge, where he describes his difficulties of learning to master the table: “The most difficult subject he had ever encountered”. This was the same Pepys that created the modern English fleet and became the chairman of the ROyal SOciety and publisher of Newton's monumental 'Principia Mathematica“.
Despitey Pepys's difficulties, the multiplication table ans spelling formed the very core of the public school as it was formed in the mid 19th century. These topics gave intruments for exercising control and selection in the shool system, as they gave objective criteria for sorting students to serve in the industrial society.
In the information society of today, difficulties of spelling can be compensated by using a word processor with spell checker, and would not necessarily be a stumbling block for a career in academics, administration or politics for all the talented and intelligent people with some dyslectic syndrom. Likewise, today the pocket calculator can easily compensate a complicated algorithm for long division. Thus, today the corner stones of traditional education, spelling and the multiplication table, seem to be loosing importance as pillars of elementary education.
So if the multiplication table and long division no longer serve as the canon of elementary mathematics education, what could/should then be taugh? What could/should be the purpose of elementary mathematics education? Or should the old canon be resurrected?
“The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we don't know in what direction it will find its final solution or whether a final objective answer may be expected at all. “Mathematizing” may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization” –Weyl–
“The universal mathematics is, so to speak, the logic of the imagination” –Leibniz–
This is the first chapter of Body & Soul Vol I…
The mass comsumption of the industrial society is made possible by the automated production of material goods such as food, clothes, housing, TV sets, cd players, cars, etc. If these items had to be produced by hand they would be the privileges of only a selected few.
Analogously, the emerging information society is based on mass consumption of automated computation by computers that is creating a new “virtual reality”. The information society offers inmaterial goods in the form of knowledge, information, fiction, movies, music, games, etc. The modern PC or laptop is a powerful computing device for mass production/consumption of information e.g. in the form of words, images, movies, music, etc.
Key steps in the automation or mechanization of production were:
The key steps in the automation of computation were:
Mathematics may be viewed as the language of computation and thus lies at the heart of the modern information society. Mathematics is also the language of science and thus lies at the heart of the society that grew out of the scientific revolution in the 17th century that began when Leibniz and Newton created calculus.
Mathematics is a language. There are many different languages. Our mother tongue, whatever it happens to be, English, Swedish, Greek, is our most important language, which a child master quite well at the age of three. To learn to write in our native language takes longer time and more effort and ocuppies a large part of the early school years. To learn to speak and write a foreign language is an important part of the secondary education.
There are also other language like the language of musical notation with its notes, bars, and scores. A musical score is like a model of the real music. For a trained composer, the model of the written score can be very close to the real music. For amateurs, the musical score may say very little, because the score is like a foreign language which is not understood.
Mathematics has been described as the language of science and technology, the words of the language of mathematics often are taken from our usual language, like pointers, lines, circles, velocity, functions, relations, transformations, sequencies, equality and inequality.
A mathematical word, term or concept is supposed to have a specific meaning defined using other words and concepts that are already defined. This is the same principle as is used in a thesaurus, where relativitely complicated words are described in terms of simpler words. To start the definition process, certain fundamental concepts may be described in certain axioms. Funamental concept of Euclidean geometry are point and line, and a basic Euclidean axion states that trought each pair of distict points there is a unique line passing. A theorem is a statement derived from the axioms or other theorems by using logical reasoning following certain rules of logic. The derivation is called a proof of the theorem.
The basic areas of mathematics are:
The basic areas of mathematics education in engineering or science education are:
The basic concepts of calculus are:
The basic concepts of linear algebra are:
First, we have to admit that mathematics is a difficult subject, and we see no way around this fact. Secondly, one should realize that it's perfectly possible to live a happy life with a career in both academics and industry with only elementary knowledge of mathematics. There are many examples including Novel Prize winners. This means that is advisable to set a level of ambition in mathematics studies which is realistic and fits the interest profile of the individual students. Many students of engineering have other prime interests than mathematics, but there are also students who really like mathematics and theoretical engineering subjects using mathematics. The span of mathematical interest may thus be expected to be quite wide in a group of students following a course based on the Body & Soul series of books, and it seems reasonable that this would be reflected in the choice of level of ambition.
On the other hand, there are many aspects of mathematics which are not so difficult, or even “simple”, once they have been properly understood. To help out we have in Body & Soul collected the most essential nontrivial facts in short summaries in the form of -Calculus tool bag I and II, linear algebra tool bag. differential equations tool bag, Applications tool bag, Fourier Analysis tool bag, and Analytic Functions tool bag and together they're only 15-20 pages. If properly understood, this material carries a long way and is “all” one needs to remember from the math studies for further studies and professional activities in other areas.
Body & Soul reflects both the increased importance of mathematics in the information society of today, and the decreased importance of much of the analytical mathematics filling the traditional curriculum. The student should thus be happy to know that many of the traditional formulas are no longers such a must, and that a proper understanding of relatively few basic mathematical facts can help a lot in coping with modern life and science.