How and why we do mathematical proofs

Abstract

This is a module framework. It can be viewed online or downloaded as a zip file.

As taught in Autumn Semester 2009/10

The aim of this short unit is to motivate students to understand why we might want to do proofs (why proofs are important and how they can help us) and to help students with some of the relatively routine aspects of doing proofs.

In particular, the student will learn the following:

* proofs can help you to really see why a result is true;

* problems that are easy to state can be hard to solve (e.g. Fermat's Last Theorem);

* sometimes statements which appear to be intuitively obvious may turn out to be false (e.g. Simpson's paradox);

* the answer to a question will often depend crucially on the definitions you are working with;

* how to start proofs;

* how and when to use definitions and known results.

The module is organised into three sections: Why; How (Part I); How (Part II)

With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs. A practice sheet is included after students have completed all three sections. Each section is suitable for a different level of audience, as described below:

Suitable for: Foundation, undergraduate year one and undergraduate year two students

Section 1: Why: Anyone with a knowledge of elementary algebra and prime numbers, as may be obtained by studying A level mathematics. (Foundation)

Section 2: How (Part I) – Suitable for anyone with a knowledge of elementary algebra (including odd numbers, multiples of eight and the binomial theorem for expanding powers of (a+b)), and functions from the set of real numbers to itself (odd functions, even functions, multiplication and composition of functions). (Undergraduate year one)

Section 3: How (Part II) – Requires some background knowledge of convergence and divergence of series of real numbers. A revision sheet is available. (Undergraduate year two)

Dr Joel Feinstein, School of Mathematical Sciences

Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras.

Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area