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HOW DO UNDERGRADUATES DO MATHEMATICS?
A guide to studying mathematics at Oxford University

Charles Batty
St. John’s College, Oxford

with the assistance of
Nick Woodhouse
Wadham College, Oxford

April 1994
1

CONTENTS ii Preface
PART I

1

Chapter 1: University study
1.1
1.2
1.3
1.4
1.5
1.6

Pattern of work
Lectures
Tutorials
Cooperation with fellow-students
Books and libraries
Vacation work

11

Chapter 2: University mathematics
2.1
2.2
2.3

Studying the theory
Problem-solving
Writing mathematics

11
15
16

Chapter 3: The perspective of applied mathematics by Nick Woodhouse
3.1
3.2
3.3

1
3
5
8
9
10

Pure and applied mathematics
Solving problems in applied mathematics
Writing out the solution

19
19
20
24

PART II
Chapter 4: The formulation of mathematical statements
4.1
4.2
4.3
4.4
4.5

Hypotheses and conclusions
“If”, “only if”, and “if and only if”
“And” and “or”
“For all” and “there exists”
What depends on what?

30
33
37
41
43
46

Chapter 5: Proofs
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

29

Counterexamples
Constructing proofs
Understanding the problem
Experimentation
Making the proof precise
What can you assume?
Proofs by contradiction
Proofs by induction

46
49
50
53
58
64
65
70
76

Appendix: Some symbols i Preface
In one sense, mathematics at university follows on directly from school mathematics. In another sense, university mathematics is self-contained and requires no prior knowledge. In reality, neither of these descriptions is anything like complete. Although it would be impossible to study mathematics at Oxford without having studied it before, there is a marked change of style at university, involving abstraction and rigour.
As an undergraduate in any subject, your pattern and method of study will differ from your schooldays, and in mathematics you will also have to master new skills such as interpreting mathematical