数学探索：解题与证明的艺术

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"Exploring Mathematics -- Problem-Solving and Proof, Springer 2018" 是一本面向本科生和中学生以及教师的数学书籍，由Daniel Grieser撰写，旨在引导读者发展解决问题的策略和逻辑推理能力，同时介绍证明的重要性和各种证明技术。书中涵盖了递归、数学归纳法、图论、计数、初等数论以及鸽巢原理、极值原理和不变性原理等主题。这本书适合无微积分背景但具备基本中学数学知识的读者，包括对数字和基础几何的理解。书中包含大量练习并提供提示，适合自学或与讲座课程配合使用。关键词包括数学问题解决、数学证明、数学问题解决策略、数学探索、问题解决策略、数学、极值原理、不变性原理、鸽巢原理、计数原则、数学归纳法、逻辑证明、初等数论。该书属于Springer的Undergraduate Mathematics Series，由一系列知名学者担任顾问。作者Daniel Grieser在德国奥尔登堡卡尔·冯·奥西茨基大学的数学研究所工作。本书的目的是帮助学生从中学过渡到大学阶段，提供对数学的全新视角，并深入理解超越本书范围的普遍原则。在本书中，读者可以期待以下内容： 1. **问题解决策略**：学习如何面对数学难题，如何分解问题，以及如何寻找适当的解题方法。 2. **逻辑推理**：培养逻辑思维能力，理解数学论证的重要性，学习如何构建和评估有效的论证。 3. **证明技巧**：介绍不同的证明方法，如直接证明、反证法、构造性证明等，以增强证明能力。 4. **递归**：探讨如何利用递归关系解决问题，特别是在数学和计算机科学中的应用。 5. **数学归纳法**：学习如何通过归纳推理来证明数学命题对所有自然数都成立。 6. **图论**：介绍图的基本概念，如顶点、边、路径和循环，以及在解决实际问题中的应用。 7. **计数原理**：讲解组合计数，包括乘法原理、加法原理和鸽巢原理，理解计数问题的解决策略。 8. **初等数论**：涉及整数性质、同余、质数和素因数分解等基础理论。 9. **极值原理和不变性原理**：学习如何利用这些原理解决最优化问题和保持某些性质不变的问题。通过阅读和实践本书中的例子和练习，读者将能够提升数学素养，掌握关键的数学思维工具，这对于进一步的学术研究和日常生活中遇到的数学挑战都大有裨益。

Introduction 17

by step: What are we looking for? What could be a route of approach?

Not every route that we try will lead to a solution; sometimes we need

to backtrack and start again. Here you experience how mathematics

is born, invented, developed. While discovering mathematics you

learn how mathematicians think. Here you learn problem-solving

strategies. These are collected in the

Toolbox

sections. In Appendix A

you will ﬁnd an overview of all the strategies introduced in the book.

What kind of problems do you ﬁnd in this book? Mathematical

problems for which no straightforward method of solution offers

itself. Problems where you need an idea. Problems that lead to an

interesting mathematical concept. Some chapters introduce an area

of mathematics, for example graph theory in Chapter 4 or number

theory in Chapter 8. These topics are also introduced via a sequence

of problems, and at places where it is natural to state a theorem

and its proof, the proof is developed from the point of view of a

problem-solver.

Here is a more detailed outline. In Chapters 1 and 2 you will ﬁnd

problems where you count things. Counting is the most basic mathe-

matical activity. By learning to advance step by step you will soon

have the wonderful experience of discovering mathematics yourself.

An important technique for counting is to use recurrence relations,

which are discussed in Chapter 2. The basic idea, called recursion,

is to reduce a problem to a smaller problem of the same kind. This

is also behind the principle of mathematical induction introduced in

Chapter 3. Using induction you will prove Euler’s famous formula

in Chapter 4 and then use it to show a remarkable impossibility

result: you cannot connect ﬁve points in the plane pairwise so that

the connecting curves don’t intersect. That we can prove such an

assertion rigorously is actually quite amazing: how can we control

the inﬁnitely many ways to draw the points and curves? After this

excursion into graph theory we return to counting in Chapter 5, this

time formulating general systematic counting principles which are

often useful.

Chapters 6 and 7 are central: Chapter 6 introduces general problem-

solving strategies, that is, strategies which are useful beyond mathe-

matics. Chapter 7 is about logical foundations and the most important

general types of proof. These are illustrated by many examples. Chap-

ter 8 is about number theory, that is, prime numbers, divisibility etc.

18 Introduction

Here you can use the various types of proof to answer interesting

questions about numbers. Number theory also provides useful tools

and pretty examples for the following chapters. In Chapter 9 you

learn about the pigeonhole principle, a simple idea which spawns

amazing consequences when used aptly. Chapters 10 and 11 intro-

duce the extremal principle and the invariance principle. These are

versatile tools for solving mathematical problems, and also turn up

again and again in all exact sciences. Here you also learn about the

fundamental notions of permutations and their signature.

Sections titled

Going further

at the end of Chapters 4, 5, 6, 10 and

11 encourage you to delve deeper into the matters touched upon

in these chapters and give you a glimpse of the higher summits of

mathematics. Occasionally you may ﬁnd here an unfamiliar term

without detailed explanation, but you should read on, just to get an

idea of what else mathematics has to offer. The references at the end

of the book guide you to literature that will be useful for further

exploration.

The mathematical topics treated in the book are elementary, that

is, they can be understood with very few prerequisites. However, we

take a higher perspective than you may be used to: often problems

are formulated in greater generality than is common at school, we

pay attention to logically complete arguments, and although the

tone is generally informal, we use modern mathematical terminology,

in particular the language of sets and maps. This is explained in

Appendix B.

The chapters are mostly independent of each other, so you may

read them in a different order. However, later chapters tend to be

more demanding than earlier chapters.

Hints for using the book

You learn problem-solving by solving problems, and by understand-

ing other people’s solutions. You learn proof by proving, and by

carefully reading and understanding proofs. The problems in the

text and the exercises at the end of each chapter give you ample

opportunity for this. Be an active reader! After reading a problem,

ﬁrst try to solve it yourself. After reading part of the solution, put the

book aside and think about how to go on. The symbol

20 Introduction

4. Review

These are similar to the four steps of problem-solving introduced by

olya in his classic book

How to solve it

olya, 2014). Steps 1. and

2. are closely interwoven and carried out in the sections marked by

the symbol

G



. The solution is marked by the symbol

, the review

by the symbol



. The end of each of these sections is marked by the

same symbol, and the end of a proof is marked q. e. d. (quod erat

demonstrandum = which was to be proven).

Notes for the teacher

This book arose from a course whose main aims were to let students

see mathematics from a new perspective (not rules to be followed

but a world to be discovered), to give them conﬁdence that they can

discover mathematics for themselves and to lead them to appreciate

proofs.

Along the way students acquire tools for problem-solving,

get to know the main types of proofs, learn to write solutions and

proofs properly, learn about fundamental ideas that occur everywhere

in mathematics, and also pick up important concepts like graphs,

congruences, permutations and their signatures. These are introduced

in informal, often playful contexts, so they can be shared with friends

and will be remembered more vividly than if learned in more abstract

ways.

If you want to teach such a course my main piece of advice is:

Less

is more.

Don’t try to ‘cover’ a certain curriculum. Take time to solve

problems together with the students in class, to develop ideas for

solutions, to try out different approaches, to suffer the frustration

of getting stuck and enjoy the satisfaction of success together. Share

with the students your experience of how to approach a mathematical

problem, rather than presenting prefabricated solutions.

Always start

with an easy problem. There will be enough challenges for everyone.

I am happy to acknowledge the help and valuable feedback that

This course, titled Mathematisches Probleml¨osen und Beweisen (Mathematical

problem-solving and proving), was introduced in the mathematics curriculum at

Carl von Ossietzky University of Oldenburg, Germany, in 2011, with the goal

of smoothing the difﬁcult transition from high school mathematics to university

mathematics.

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