How to use this book

Mathematical Writing - Vivaldi Franco 2014


How to use this book

Students

This book is appropriate for self-study at undergraduate or master level. It will be of interest not only to dissertation-writing students looking for specific advice, but also to students at earlier stages of a mathematics programme, and for various purposes.

For example, a student may wish to raise the quality of their written output in coursework and exams, or may need to expand their vocabulary to reflect a growing mathematical maturity. Others may be interested in writing about a particular construct, or may seek to improve the form and style of their proofs.

One year of university mathematics should provide sufficient background for reading this book. Parts of it may be read earlier, even at the start of university if one has a good sense of conceptual accuracy. There is a large supply of exercises; solutions and hints are provided to facilitate self-assessment in absence of a teacher.

We explain succinctly the purpose of each chapter:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Chapter 1 should be read by everyone. Chapters 2 and 3 are fundamental but introductory; an advanced student should be familiar with much of their content, and will consult them only if necessary. Chapter 4 develops the basic logic needed, say, for a first analysis course. This material is a pre-requisite for Chaps. 5 , 7 , 9 , while Chaps. 6 , 10 , and most of Chap. 8 may be read without it. Chapter 10 requires the mathematical maturity of a final year student. With this in mind, we suggest some possible paths through this book:

·  To improve the use of basic terms: Chaps. 2 — 5 .

·  To improve conceptual accuracy: Chaps. 4 , 6 , 7 , 9 .

·  To write better proofs: Chaps. 7 , 8 .

·  To write a dissertation: Chaps. 6 , 10 .

Teachers

This book has been used for several years as the textbook for a second-year course in mathematical writing at the University of London, in classes of 50—70 students. The syllabus covers Chaps. 1 — 9 , excluding the last section of Chaps. 2 — 4 , which contain specialised material not needed elsewhere. Chapter 1 is assigned as a self-study exercise at the beginning of the course; Chap. 5 is given as a reading assignment during a pause in the teaching mid-way through the course.

The students are given weekly exercises taken from the book. In the early part of the course, the writing is limited to short phrases and sentences. Once the students have consolidated their mathematical vocabulary, more complex assignments are introduced. In a typical one, the students are given a two-page excerpt from a standard first-year textbook introducing a mainstream topic (the logarithm, Euclid’s algorithm, complex numbers, etc.). They are asked to write a short document comprising a title, a few concise key points, and a short summary (150—200 words) without using any mathematical symbol . This form of writing is demanding but short, hence manageable in large classes. The requirement that no symbols be used has several educational virtues, and the added bonus of making plagiarism more difficult. An example of this assignment is given at the beginning of Sect. 6.6 , in the form of a summary of a section of the present book. Related exercises are found at the end of Chap. 6 .

The students submit their weekly work electronically, as a single pdf file. They are given complete freedom in the choice of the electronic medium used to generate their files (anything from scanning hand-written pages to L A T E X). This policy works well, and it also encourages independence and sense of responsibility. Each assignment requires only a limited use of symbols, to minimise the lure and distraction of electronic typesetting. Invariably, the best students are keen to learn L A T E X, and they do so without any supervision.

The coursework is marked by postgraduate students, who receive specialised training and are provided with detailed marking schemes. Lecturer and markers meet weekly to fine-tune the marking and resolve unusual cases. Postgraduate students tend to find this experience more instructive than marking conventional exercises.

The coursework constitutes 20—30 % of the assessment for the course, the rest being the final exam. In a small class, it may be desirable to adjust the assessment balance, increasing the weight of coursework and adding a mini-project or a short presentation at the end of the course.

Finally, this book could be used as supplementary material for various courses and programmes: a first-year introduction to mathematical structures, a course in analysis or in logic, a workshop on writing dissertations.