What is mathematics?

Mathematics is a way of organising our experience of the world. It enriches our understanding and enables us to communicate and make sense of our experiences. It also gives us enjoyment. By doing mathematics we can solve a range of practical tasks and real-life problems. We use it in many areas of our lives.

In mathematics we use ordinary language and the special language of mathematics. We need to teach students to use both these languages. We can work on problems within mathematics and we can work on problems that use mathematics as a tool, like problems in science and geography. Mathematics can describe and explain but it can also predict what might happen. That is why mathematics is important.

For more than two thousand years some familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger.

These opening sentences to the preface of the classical book “What Is Mathematics?”were written by Richard Courant in 1941. It is somewhat soothing to learn that the problems that we tend to associate with the current situation were equally acute 65 years ago (and, most probably, way earlier as well). This is not to say that there are no clouds on the horizon, and by this book we hope to make a modest contribution to the continuation of the mathematical culture.

The first mathematical book that one of our mathematical heroes, Vladimir Arnold, read at the age of twelve, was “Von Zahlen und Figuren”1 by Hans Rademacher and Otto Toeplitz. In his interview to the “Kvant” magazine, published in 1990, Arnold recalls that he worked on the book slowly, a few pages a day. We cannot help hoping that our book will play a similar role in the mathematical development of some prominent mathematician of the future.

We hope that this book will be of interest to anyone who likes mathematics, from high school students to accomplished researchers.

 We do not promise an easy ride: the majority of results are proved, and it will take a considerable effort from the reader to follow the details of the arguments. We hope that, in reward, the reader, at least sometimes, will be filled with awe by the harmony of the subject (this feeling is what drives most of mathematicians in their work!) To quote from “A Mathematician’s Apology” by G. H. Hardy, The mathematician’s patterns, like the painters or the poets, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.

For us too, beauty is the first test in the choice of topics for our own research,

as well as the subject for popular articles and lectures, and consequently, in the choice of material for this book. We did not restrict ourselves to any particular area (say, number theory or geometry), our emphasis is on the diversity and the unity of mathematics. If, after reading our book, the reader becomes interested in a more systematic exposition of any particular subject, (s)he can easily find good sources in the literature.

About the subtitle: the dictionary definition of the word classic, used in the

title, is “judged over a period of time to be of the highest quality and outstandingof its kind”. We tried to select mathematics satisfying this rigorous criterion. The reader will find here theorems of Isaac Newton and Leonhard Euler, Augustin Louis Cauchy and Carl Gustav Jacob Jacobi, Michel Chasles and Pafnuty Chebyshev, Max Dehn and James Alexander, and many other great mathematicians of the past.

Quite often we reach recent results of prominent contemporary mathematicians, such as Robert Connelly, John Conway and Vladimir Arnold.

We started this preface with a quotation; let us finish with another one. Max Dehn, whose theorems are mentioned here more than once, thus characterized mathematicians in his 1928 address [22]; we believe, his words apply to the subject of this essay:

At times the mathematician has the passion of a poet or a conqueror, the rigor of his arguments is that of a responsible statesman or, more simply, of a concerned father, and his tolerance and resignation are those of an old sage; he is revolutionary and

conservative, skeptical and yet faithfully optimistic.

There are four main issues in the teaching and learning of mathematics:

Teaching methods

Students learn best when the teacher uses a wide range of teaching methods. This book gives examples and ideas for using many different methods in the classroom,

Resources and teaching aids

Students learn best by doing things: constructing, touching, moving, investigating. There are many ways of using cheap and available resources in the classroom so that students can learn by doing. This book shows how to teach a lot using very few resources such as bottle tops, string, matchboxes.

The language of the learner

Language is as important as mathematics in the mathematics classroom. In addition, learning in a second language causes special difficulties. This book suggests activities to help students use language to improve their understanding of maths.

The culture of the learner

Students do all sorts of maths at home and in their communities. This is often very different from the maths they do in school. This book provides activities which link these two types of rnaths together. Examples are taken from all over the world. Helping students make this link will improve their mathematics.

Why do we learn mathematics?

Why should anyone study mathematics? Should those in high school or college be forced to take math courses, even if their intended future profession does not require higher level mathematics?

A common argument in favor of forcing math classes to be taken is that it is a necessary part of educational process to make the individual well-rounded in all aspects including science, humanities, writing classes, language, and the other basic fields of study. While this assumption is true, this is not the main reason why mathematics should be a required part of a school’s curriculum. First of all, simply removing the requirement for math classes would lower the standards for everyone either educating or being educated. Simply giving up on a generation of kids who are not good at math is definitely not the right way to go. If anything, people should be pushed harder in mathematics to do the best as possible even if that means ”failing.” Hopefully failure would lead persistence to succeed and motivation to do better.

In high school, math should be required for at least three years. This is because during the high school years, a student rarely knows what future profession he or she may have. Entering high school, one may have dreaded mathematics in previous years, but upon being exposed to higher levels of math, the person might gain curiosity or even enjoyment for mathematics. Dropping the requirement may steer students away from something that they may develop to be very good at or have a future profession in. Another reason that mathematics should be a required course is that fact that it helps to develop the problem solving process. Although one may not directly use mathematics in their everyday lives, math plays an indirect role in how they make many day to day decisions such as finances or time management. The ability to problem solve is not something that can be directly taught, but is rather a discrete skill learned over many years through trial and error, past experiences, and exercises that might help to develop such a skill.

Mathematics is certainly one these exercises. Math is one of the few fields of study where many facts are not necessarily memorized. Such a field where this is necessary is history, or a high school English class.

In math, one only has to know the basic operations, and symbolisms. With this limited knowledge, one can solve an infinite number of problems.Theorems may require a certain amount of memorizing, but even these can be derived from previously gained knowledge. In other words, gaining math skills does not just mean that one knows how to do math,

but rather opens up the brain to all kinds skills and development. Yet another reason why mathematics should be studied is the fact that basically anything that happens in the world can be related to mathematics. Gaining an understanding of this concept can greatly affect how one perceives the world around them. New enlightenment may come about with this understanding. Math is also the only true universal part of human culture.

Searching for some more insight on this project, I came across a quote said by James Caballero that caught my attention: ”I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors.

Everybody a mathematician?” CAIP Quarterly 2 (fall, 1989). I believe that what was said is very apparent. Every time we stop pursuing a subject, we lose another opportunity to develop our thinking to a greater level. Mathematics isn’t just a tool for engineers, chemists, physicists, or economists. It can be used by everyone to gain problem

solving skills, take care of every day tasks, understand the world around them, and to feel a sense of accomplishment. Shutting these possibilities off by not pursuing studies in mathematics severely limits a person’s growth and ability, thus affecting everyone around them.

Reference:

1.Carson, Elizabeth and Denise Haffenden. ”NYC HOLD: Honest Open Logical Debate on math reform”. http://www.mathematicallycorrect.com/nychold.htm.
2.Mathematically Correct. ”What has happened to Mathematics Education?”. 

http://mathematicallycorrect.com/intro.htm#doyou.

3.Ross, David. ”TheMathWars”. http://www.ios.org/articles/dross mathwars.asp.

Stohr-Hunt, Patricia. ”What Good is Math?”. http://oncampus.richmond.edu/academics/19 Oct 2003.