1) Question everything
For me one of the truly great beauties of mathematics is that it can be checked. You don’t have to take anyone’s word. If someone says something is true, then you can ask them to prove it. Better still, if you want to really think like a mathematician, then you can try to prove it. Don’t let people spoon feed you!
Your reaction to someone’s statement should be to disbelieve them and attempt to find an example that shows it’s false. Even if it’s true, the mental workout that this process gives is beneficial. It also helps develop a feel for a statement. (Note that constantly doing this in real life situations can lose you friends - people tend to get upset if you are constantly finding fault with what they say!)
A letter to a newspaper stated that time travel is impossible because of logic: If time travel were possible, then one would meet lots of people from the future. I had some ideas why this might be wrong. Maybe time travel will only allows us to travel forward in time (by amounts larger than we do already!) Maybe 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 (and time travel machines can’t be transported).
2) Write in sentences
Write in sentences? How is that going to help me think like a mathematician, you may be asking. Well, sentences are the building blocks of arguments. And higher-level mathematics is about arguments in the form of proof (not just about getting the right numerical answer!) .
Too often students don’t see the need for sentences. They often say things like ‘I didn’t come to university to write essays’, ‘But I got the right answer’ or ‘You know what I meant’. In the past they could submit a collection of unconnected symbols as homework and still achieve almost full marks. But if you want to understand mathematics and to think clearly, then the discipline of writing in sentences forces you to think very carefully about your arguments. If you can’t write the sentence properly, then probably you don’t
truly understand what you are writing about. That’s a great opportunity to learn more and develop your skills. And writing well in any subject is a useful skill to possess. [Bonus: One easy way to improve your mathematical writing and thinking is to know how to use the implication symbol =>]
3) Ask ‘What happens if. . . ?’
Good mathematicians like to ask ‘What happens if . . . .’ For example, what happens if I drop that assumption? By thinking about this we can see better why a result is true or why a definition is the way it is. Sometimes we can produce a new theorem by making the assumptions weaker!
When Sir Christopher Zeeman founded the Mathematics Institute at the University of Warwick one of his key ideas for fostering a mathematical atmosphere was that the institute should have plenty of blackboards in the corridors – not just in the lecture rooms – so that people could talk with each other and explain their work. This would foster collaboration but crucially allow people to have their work tested by others.
The Isaac Newton Institute in Cambridge went further with blackboards. They have them in the toilets and even one in the lift – which only serves two floors!
There are many advantages of communicating with others. Explaining your work forces you to think clearly. And you learn from others, they can find mistakes in your thinking or suggest ideas for solving a problem. You can even learn just from explaining. So get yourself someone to talk to. Don’t have one? Do a search.
Dr. Kevin Houston:
Obviously you don’t want my life story but do need to know I’m qualified to write this stuff. Well, I have a PhD in Mathematics from the University of Warwick, England and currently I am a Senior Lecturer at the University of Leeds, England. I’ve been teaching mathematics in one form of another to a variety of students since 1990.