How to Learn Mathematics
• You must study every day.
• You should expect to spend about two hours outside of class for each hour spent in class. It is your responsibility to learn the material.
• Read the sections before I discuss them.
• Don’t be afraid to ask questions. I reward especially interesting questions with bonus points!
• Do homework problems, including some with the book closed.
• Accumulate a set of flash cards where one side has the problem and the other side has the first step of the solution and the page of the book where the problem is stated.
• Study with a friend - either on campus or by phone.
• Take practice tests (pick ten flash cards at random) with the book closed.
• Get a good night’s sleep before each exam. Cramming for math tests does not work.
How to Study Higher Math and Do Proofs
Higher Math has been traditionally defined as any subject more advanced than partial differential equations. For example Real Analysis, Abstract Algebra, Topology, Set Theory, etc.. The main difference between higher math and basic math is that higher math relies solely on formal proofs to develop the material.
I used to have trouble writing even the simplest proofs, I understood the basics of how to write proofs, I even understood some of the proofs in my textbooks, but I could not write correct proofs. I went to office hours, I worked with other students, I looked at other textbooks, I even paid tutors to help me with analysis homework, but nothing helped.
I might have understood how one proof was written, or how to write one type of proof for one specific homework problem, but when it came to new homework assignments or a slight twist in the conditions I could not write correct proofs. I thought I could never write correct proofs or learn higher math. I blamed my professors and the textbooks for not making proofs easy enough, until I realized that the problem was not with them or the textbooks (though some textbooks are bad). The problem was approaching proofs like one would approach homework exercises in basic mathematics.
I thought that if I simply understood the concepts and proofs in the section that I could write proofs in the homework, how wrong I was. Writing proofs requires that you not only understand the concepts and proofs in your book but, that you realize how you could have derived those proofs, from your understanding of the concepts. Once I realized this, I began to search for and develop practical techniques to implement this idea. When I consistently put the techniques into practice, I noticed that little by little I could write more and more complex proofs, until I could write difficult proofs. With more practice it became faster and easier to write difficult proofs.
Some might think the techniques take too long to be effective, but just think how much time you will spend on your homework without practicing the techniques. Just think how much time you’ve spent on past homework writing incorrect proofs, or how much time you’ve spent relying on others to help you write correct proofs. So give it a shot, you have everything to gain: