Mathematicians are often considered to be a world apart, more Archimedes than average Joe. However, a new book sets out shortcuts that can give anyone with an interest a head start in mathematical thinking.
How to Think Like a Mathematician, by Dr. Kevin Houston, a maths lecturer at the University of Leeds, aims to be a one-stop guide to the tricks of the trade in mathematics.
"As an undergraduate I noticed that there were quite a few hints and strategies that are used by successful mathematicians," said Dr. Houston.
"But there was no one place where I could look these strategies up. I thought it would be great to have a book that revealed these hidden techniques."
Dr. Houston advises always trying to show a statement is false, instead of proving it is true. For example, a letter to a newspaper stated that time travel is impossible because of logic, because if it were possible one would meet lots of people from the future.
If you think about this though, you might be able to come up with some ideas of your own about why this is wrong.
Maybe time travel will only allows us to travel forward in time. Perhaps time travellers are not allowed to communicate with us. Maybe time travel has a range, you can’t travel back more than a year and time travel is still a number of years away.
Being sceptical and trying to find a counterexample can be very illuminating, not just with mathematics but in everyday life.
The book also suggests that maths should be written in sentences using words, punctuation and symbols.
How to Think Like a Mathematician contains over 300 exercises and a range of techniques to help students unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. The book has broader appeal in that it encourages readers to construct logical arguments and think critically.
Originating from a pamphlet given to students at the University of Leeds, the strategies outlined in the book have gone through a long refinement process.
"Since I began teaching undergraduates back in October 1990 I have tried out several of these mathematical tips with my students. The book brings together the best ideas that have been tried and tested while I’ve been teaching at university," said Dr. Houston.
How to Think Like a Mathematician is ideal for anyone starting a degree in mathematics and it can also help students studying engineering or physics who need to access information on undergraduate mathematics topics, or for students studying logic as part of computer science, philosophy, or linguistics.
Dr. Houston is currently working on short videos for YouTube to accompany the book. This is a project he likes to call How to Think Like a Mathematician – The Movie.
February 9, 2009
