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The Essence of Mathematics Through Elementary Problems
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Understanding mathematics cannot be transmitted by painless entertainment … actual contact with the content of living mathematics is necessary. The present book … is not a concession to the dangerous tendency toward dodging all exertion.

Richard Courant (1888–1972) and Herbert Robbins (1915–2001) Preface to the first edition of What is mathematics?

Interested students of mathematics, who seek insight into the “essence of the discipline”, and who read more widely with a view to discovering what the subject is really about, may emerge with the justifiable impression of serious mathematics as an austere, but distant mountain range – accessible only to those who devote their lives to its exploration. And they may conclude that the beginner can only appreciate its rough outline through a haze of unbridgeable distance. The best popularisers sometimes manage to convey more than this – including hints of the human story behind recent developments, and the way different branches and results interact in unexpected ways; but the essence of mathematics still tends to remain elusive, and the picture they paint is inevitably a broad brush substitute for the detail of living mathematics.

This collection takes a different approach. We start out by observing that mathematics is not a fixed entity – as one might unconsciously infer from the metaphor of an “austere mountain range”. Mathematics is a mental universe, a work-in-progress in our collective imagination, which grows dramatically over time, and whose eventual extent would seem to be unconstrained – without any obvious limits. This boundlessness also works in reverse, when applied to small details: features which we thought we had understood are repeatedly filled in, or reinterpreted, in new ways to reveal finer and finer micro-structures.

Hence whatever the essence of the discipline may be, it is clearly not something which can only be accessed through the complete exploration of some fixed corpus of knowledge. Rather the essential character of mathematics seems to be related to

  • the kind of material that counts as mathematical,
  • the way this material is addressed,
  • the changes in perspective that occur as our understanding grows and deepens, and
  • the unexpected connections that regularly emerge between separate strands and layers.

There are a number of books giving excellent general advice to prospective students about how university mathematics differs from school mathematics. In contrast, this collection – which we hope will be enjoyed by interested high school students and their teachers, by undergraduates and postgraduates, and by many others is more like a messy workshop than a polished exposition. Here the reader is asked to tackle a sequence of problems, to reflect on what they discover, and mostly to draw their own conclusions (though some key messages are explicitly discussed in the text, or in the solutions at the end of each chapter). This attempt to engage the reader as an active participant along the way is inevitably untidy – and may sometimes prove frustrating. In particular, whereas a polished exposition would break up the text with eye-catching diagrams, an untidy workshop will usually leave the reader to draw their own figures as an essential part of the struggle. This temporary untidiness and frustration is an integral part of “the essence” that we seek to capture – provided it leads to occasional glimpses of the power, and the elegance of mathematics.

Young children and students of all ages regularly experience the power, the economy, the beauty, and the elegance of mathematics and of mathematical thinking on a small scale, through struggling with certain elementary results and problems (or groups of problems). For example, one of the problems we have included in Chapter 3 was mentioned explicitly in an interview1 with the leading Russian mathematician Vladimir Arnold (1937–2010):

Interviewer: Please tell us a little bit about your early education. Were you already interested in mathematics as a child?

Arnold: [...] The first real mathematical experience I had was when our schoolteacher I.V. Morotzkin gave us the following problem [VA then formulated Problem 89in Chapter 3].

I spent a whole day thinking on this oldie, and, the solution (based on what are now called scaling arguments, dimensional analysis, or toric variety theory, depending on your taste) came as a revelation.

Notices of the AMS, vol 44, no. 4.

The feeling of discovery that I had then (1949) was exactly the same as in all the subsequent much more serious problems – be it the discovery of the relation between algebraic geometry of real plane curves and four-dimensional topology (1970), or between singularities of caustics and of wave fronts and, simple Lie algebras and Coxeter groups (1972). It is the greed to experience such a wonderful feeling more and more times that was, and still is, my main motivation in mathematics.

This suggests that school mathematics need not be seen solely as an extended apprenticeship, which is somehow different from the craft of mathematics itself. Maybe some aspects of elementary mathematics can be experienced as if they were a part of mathematics proper, in which case suitably chosen elementary material, addressed in the appropriate spirit, might serve as a microcosm, or mini-universe, in which many features of the larger mathematical cosmos can be directly, and faithfully experienced by a relative novice (at least to some extent).

This collection of problems (and solutions) is an attempt to embody this idea in a form that might offer students, teachers, and interested readers a glimpse of “the essence of mathematics” – where this insight is experienced, not vicariously through the authors’ elegant prose, or broad-brush descriptions, but through the reader’s own engagement with carefully chosen, accessible problems from elementary mathematics.

Our understanding of the human body and how it works owes much to those (such as the ancient Greeks from 500 BC to Galen in the 2nd century AD, and much later Vesalius in the 16th century AD), who went beyond merely writing about such things in high-sounding prose, and who got their hands dirty by procuring cadavers, and cutting them up in order to see things from the inside – while asking themselves all the time how the different parts of the body were connected, and what function they served. In a similar way, the European discovery of the New World in the 15th century, and the confirmation that the Earth can be circumnavigated, depended on those who dared to set sail into uncharted waters and to keep a careful record of what they found.

The process of trying to understand things from the inside is not a deterministic procedure: it depends on a mixture of experience and inspiration, intelligence and inference, error and self-criticism. At any given time, the prevailing view may be incomplete, or misguided. But the underlying approach (of checking current ideas against the reality they purport to describe) is the only way we human beings know that allows us to gradually overcome errors and to gain fresh insight.

Our goal in this book is universal (namely to illustrate the idea that a suitably selected elementary microcosm can capture something of the essence of mathematics): hence the problems have all been chosen because we believe they convey something universal in a relatively elementary setting. But the particular set of problems chosen to illustrate the central goal is personal. So we encourage the reader to engage with these problems and results in the same way that old anatomists engaged with cadavers, or old explorers set out on voyages of discovery – getting their hands dirty while asking questions, such as:

How do the things we see relate to what we know?

What does this tell us about the subject of mathematics that we want to understand better?

In recent years schools and teachers in many countries have been under increasing political pressure to concentrate on measurable, short term “improvements”. Such pressures have often been linked to central testing, with negative consequences for low scores. This has encouraged teachers to play safe, and to focus on backward-looking} methods that allow students to produce answers to predictable one-step problems. The effect has been to downgrade the more important challenges which every student should face: namely

  • of developing a robust mastery of new, forward-looking techniques (such as fractions, proportion, and algebra), and
  • of integrating the single steps students have at their disposal into larger, systematic schemes, so that they can begin to tackle and solve simple multi-step problems.

Focusing on short-term goals is incompatible with good mathematics teaching. Learning mathematics is a long game; and teachers and students need the freedom to digress, to look ahead, and to build slowly over time. Teachers at each stage must be free to recognise that their primary responsibility is not just to improve their students’ performance on the next test, but to establish a firm platform on which subsequent stages can build.

The pressures referred to above will be recognised in many countries, where well-intentioned, but ill-considered, centrally imposed accountability mechanisms have given rise to short-sighted “reforms”. A didactical and pedagogical framework that is consistent with the essence, and the educational value of elementary mathematics cannot be rooted in false alternatives to mathematics (such as numeracy, or mathematical literacy). Nor can it be based on tests measuring cheap success on questions that require only one-step routines. We need a framework that encourages a rich combination of childlike curiosity, persistence, fruitful frustration, and the solid satisfaction of structural sense-making.

A problem sequence such as ours should ideally be distilled and refined over decades. However, the best is sometimes the enemy of the good:

Striving to better,

Oft we mar what’s well.

(William Shakespeare, King Lear)

Hence, as a mild contribution to this process of rediscovering the essence of elementary mathematics,\ix{mathematics!elementary} we risk this collection in its present form. And we encourage interested readers to take up pencil and paper, and to join us on this voyage of discovery through elementary mathematics.\ix{mathematics!elementary}

Those who enjoy watching professional football (i.e. soccer) must sometimes marvel at the way experienced players seem to be instinctively aware of the movements of other players, and manage to feed the ball into gaps and spaces that we mere spectators never even noticed were there. What we overlook is that the best players practise the art of constantly looking around them, and updating their mental record – “viewing the field of play, with their heads up” – so that when the ball arrives and their eyes have to focus on the ball, their ever-changing mental record keeps updating itself to tell them (sometimes apparently miraculously) where the best tactical options lie. Implementing those tactical options depends in part on endless practice of skills; but practice is only one part of the story. What we encourage readers to develop here is the mathematical equivalent of this habit of “viewing the field of play, with one’s head up”, so that what is noticed can continue to guide the choice of tactical options when one is subsequently immersed in the thick of calculation.

Ours is a unique discipline, which is so much richer than the predictable routines that dominate many contemporary classrooms and assessments. We hope that all readers will find that the experience of struggling with, and savouring, this little collection reveals the occasional fresh and memorable insight into “the essence of mathematics”.

We should not worry if students don’t know everything,

but only if they know everything badly.

Peter Kapitsa, (1894–1984)

Nobel Prize for Physics 1978}

To ask larger questions is to risk getting things wrong.

George Steiner (1929– )


Our thanks for suggestions, corrections comments and other contributions go to: Jean Bacon, Ay\c{s}e Berkman, Anna Borovik, Raul Cordovil, Serkan Dogan, Gwyneth Gardiner, Dick Hudson, Martin Hyland, Hovhannes Khuderverdyan, Ali Nesin, Martin Richards, Simon Singh, Gunnar Traustason, Ozge Uklem, Yusuf Ulum, and numerous students from the 2014 UKMT Summer School in Apperley Bridge.