This book is intended to lead students to develop their mathematical ability, to learn the art of mathematics and to create mathematical ideas. This is not a compendium of mathematical facts and inventions to be read over as a connoisseur of art looks over the paintings in a gallery. It is, instead, a sketchbook in which the reader may try his hand at mathematical discovery.
The American painter Winslow Homer is said to have declared that painters should not look at the works of others for fear of damaging their own directness of expression. I believe the same is true of the mathematician. The fresher the approach the better - there is less to unlearn and there are fewer bad thinking habits to overcome. In my teaching experience, some of my best students have been among those who entered my classes with the least previous mathematical course work. On the other hand, I have usually found it very difficult, if not impossible, to get any kind of creative effort from a student who has had many poor courses in mathematics. This has been true in some cases even though, as it developed later on, the student had very unusual mathematical ability.
The development of mathematical ability does not occur quickly. There are no short cuts. This book is written for the person who seeks an intellectual challenge and who can find genuine pleasure in spending hours and even weeks in constructing proofs for the theorems of one chapter or even a portion of one chapter. It is a book that may be useful for the formal student, but is intended also for the person who is not in school but wants to study mathematics independently. A person who has worked through this book can be regarded as a good mathematician.
In this book, I have tried to say exactly what I mean according to my best understanding of the English language. There are fine shades of meaning in the language used. The little words are especially important. For example, if a man says, "I have a son," it is not to be assumed that he does not have two sons. Thus, in this book, a set that contains ten objects contains one object and may contain twenty objects. I start in the first chapter with certain axioms - statements that are taken for granted - and try to lead students to derive other statements as necessary consequences of the axioms - still other and deeper statements may be derived. In this way a structure of ideas is built up. Suppose that a student is unable to supply an argument to establish the truth of some statement upon which further developments depend. Rather than seek help from someone or from some other book, the student should take this unsettled thing for granted temporarily and go on to further developments. With additional experience it may later on be possible to go back and fill in the gap. As long as the question remains unsettled, there is a nice problem to work on which, if the creative spirit of this book has been assimilated, will be regarded by the student not as a frustration but as a challenge!
This book reflects a method of teaching that some of us use at the University of Texas. This method of teaching derives from the notion that mathematics is a creative art and that students should be given the opportunity to develop their ability in this art. Starting with this premise, I have, through constant experimentation, developed a certain way of teaching mathematics. In an effort to interest others in this method, I shall try to describe some of its main features.
I have abandoned the lecture method. That is, I do not state a theorem and then proceed to prove it myself. Instead, I try to get students started creating mathematics for themselves as piano teachers start their students creating music - not by lecturing to them but not by lecturing to them but starting at once to develop coordination of mind and muscle. In mathematics this means training the mind to coordinate the right ideas from a set of axioms and definitions and arriving by logical reasoning at a proof or theorem.
A proof of a theorem consists of a suitable succession of statements each of which is completely justified. It has been my experience that there will be about as many different proofs of certain theorems as there are students who have proved them in my classes. I would not say that one of these proofs is better than another. Different people think in different ways and all should be encouraged. It is thus that new ideas are born!
I have arranged the subject matter of the calculus in such a way that the fundamental ideas may be gradually introduced to, and sometimes even discovered by, students, and all theorems may be proved on the basis of a system of axioms for the number system. This subject matter includes the development of the elementary functions in such a way that trigonometry, for example, is not a prerequisite. Instead of a succession of "obvious" statements as in the lecture method, there is a smaller number of less obvious statements to be proved by students. The questions to be settled become gradually more involved as students develop their powers.
For many of the theorems, I put no definite time limit within which they are to be settled. Frequently, propositions are taken for granted as axioms (with the expectation that they will be settled later on) and freely used, as occasion may arise, in proving other things.
The notion of "covering ground" according to some schedule is completely discarded. The work may seem to progress extremely slowly, especially at the start. Much attention is given to matters of language and logic. Often entire class periods are taken up with these things. To develop clear thinking, it is necessary to develop the ability to make statements that say exactly what is intended. Also, it is necessary to learn to deny statements. Attention to these matters pays large dividends later on. As the work progresses, I am amazed at the accomplishments of the students. In fact, much more ground is covered than under the lecture method.
Very little emphasis is placed upon examinations. I quote from an address given by Professor W B Carver, (Thinking versus manipulation, American Mathematical Monthly 44 (6) (1937, 359-363). "Examination systems, in spite of all efforts to the contrary, seem to influence our teaching in the direction of formalism rather than insight; because it is easy to test a student's manipulative skill and extremely difficult to test his ability to think ... may it not be possible that the only really important objective in our teaching of mathematics is something that we will never be able to measure satisfactorily by any kind of test or examination?" I go so far as to give no examinations whatsoever. Instead of the customary examination at the end of each semester, I present a list of problems or topics from which students are asked to select something for a term paper. This is done about two or three weeks before the end of the semester in order to allow students enough time to accomplish something. I believe it is important to remove the fear of examinations so that students may relax and give their brains a chance!
Mathematics is regarded, not as a body of facts, but as a way of thinking and creating ideas. Even if at a given time, all the useful mathematical facts could be assembled and students taught to use them, a short time later on, new facts would be needed to solve new problems. The basic principle is to teach students to think for themselves and to create their own mathematics to solve problems.
Students are encouraged from the outset to develop their own ideas in their own way. If a person's mind works in a certain way effectively, why should a teacher try to change it and perhaps destroy originality? For instance, if a student presents a proof that seems to me to be strange or unnecessarily involved, it rarely occurs to me to point out a different proof. Furthermore, students are proud of their accomplishments in proving a theorem. Should a teacher hurt and discourage them by pointing out some easier (but perhaps not better) proof?
I try to avoid unnecessary names for things and unnecessary symbols. Attaching the names of persons to a theorem might prevent someone from attempting to prove it. For example, imagine a beginner who would not be in awe of "The Bolzano-Weierstrass Theorem." Also, a word or symbol that is a substitute for an idea may very well bury the idea.
Geometrical formulations of definitions and theorems are preferred throughout. Thus in calculus the simple graph rather than the concept of variable is taken as fundamental. By employing geometrical ideas, the sense of sight is utilized and the chance of stirring the imagination is thereby increased. Also the statements in geometric terms can often be simpler than when expressed in other terms.
I point out to students that it is to their advantage not to read proofs in books or even to look into mathematical texts indiscriminately. Some good problem eventually stated in class may thereby be forever spoiled, and originality may be impaired.
I find that this method of teaching can be an inspiration both to students who discover that they have unusual mathematical ability and decide to specialize in mathematics or a mathematical science, and to those of less ability. Students who are unable to prove any but the simplest propositions still get training in language and logical thinking and also obtain the benefit of seeing the presentations by the students who prove their results in class. Moreover, they find out something about what mathematics really is.
This book is the outgrowth of the ideas and inspirations of my many students over the years and I wish to express to them my thanks and my feelings of admiration.
H S W