Thinking Like a Mathematician
1. Question everything
When looking at something that is said to be true, see if you can prove it. If someone tells you something is true, ask them to prove it to you. Trying to prove something to yourself will help you exercise your math brain and start thinking about all math problems in this way.
2. Study logic.
Logic is the foundation of a mathematical argument and necessary to developing a proper proof. There are self-study guides online or textbooks you can purchase to study logic. You can also sign up to take a logic course at the local college.
Spend time working through logic problems and cultivating that type of thought process.
Logic will help you write proofs in an ordered way that people won’t be able to find error with.
3. Write in sentences
Although it may seem strange to use sentences to explain equations and numbers, a proper mathematical proof is written out in a grammatically correct sentence. If you can’t write your argument in a sentence, you may not fully understand what you are trying to say and need to think about it more.
When writing a proof, always write in full sentences.
Study old proofs to get an idea of the proper format to use.
4. Rewrite statements with the contrapositive
The contrapositive of a statement “If A, then B” is “If not B, then not A”. As an example, the contrapositive of the statement “If I’m from New York, I am American” is “If I am not American, I am not from New York”. The contrapositive is always true which can sometimes be difficult to wrap your head around.
Always rewrite statements with the contrapositive to exercise that thought process.
5. Ask yourself about the converse of a statement
The converse of a statement such as “If A, then B” is “If B, then A”. The converse is not necessarily a true statement. When studying a problem, always ask yourself if the converse is true and then try to prove it. Even if it’s not true, it’s a good exercise to keep in mind when thinking about problems.
