I was recently asked by someone how they should be taking notes in math class. I could immediately relate because I once asked this very same question. In both undergrad and grad, I had to ask myself how to take notes because I often would leave class with a bunch of sentence fragments based on what the professor said, but without anything I could use as a study guide. At best, it would be a garbled thing that I could combine with somebody else’s notes and try to make some legible out of it. Generally though, I would just ignore my notes and go to the text book if it was well written, or the library if the text book was not well written.
So how did I get past this? Well, after taking the Set Theory course, I started seeing mathematics as more of a construction job, like building a house. Mathematics is based on proofs which is nothing more than logical reasoning, and logical reasoning is just a series of statements that are either assumptions, definitions, or conclusions drawn from those assumptions based on known facts. The main purpose of classes is to present these “known facts” to the students and help them become more informed mathematicians. So what is a mathematical fact?
There are two important types of facts we generally learn in mathematics. The first gives us a language we can work from – these are our definitions. These are the most important things in a mathematics class because if it is a class on “group theory”, then one of the first definitions will be of a “group,” and not knowing this definition will cause problems when you are trying to prove that something is or is not a group. Similarly, if your class is on solving quadratic equations, then it is very important to be able to define what a “quadratic equation” is. Definitions in mathematics are important because, unlike in an English class, you cannot always derive the meaning of a mathematical term from its usage. Mathematical definitions are very precise and your notes should include this precision.
As a side note, I will state (if not stress) that examples are not definitions. Examples are meant to bring the definition to life, and to help connect the definition to the student, but if you are asked what is a group and respond that “the set of integers mod 7 under the operation of addition is a group”, you’d be incorrect because you gave an example of a group without telling why this is a group. Similarly, if I were to ask you to define a quadratic equation and you said “x^2 + 2x + 1 = 0 is a quadratic equation”, you’d be giving me an example but not a definition. It is important to understand the distinction between examples and definitions because when we understand the definition we can clearly explain why (ala prove) that the given instance is in fact an example.
Some professors will briefly mention a definition and then focus most of their time on examples in an attempt to make the concepts easier to understand. As a student though, it is important to ask the question “what is the definition of _____” because the exams, or the research that follows, or the applications of this concept in real life are not likely to be that same example. If the professor does not write out the definition of whatever concept you are studying, or you are unclear of this definition you should ask questions, go to an online resource, or a text book for supplementary reference.
The second type of fact presented in mathematical classes is the theorem (aka lemma or corollary). A theorem is a statement that is provable by mathematical reasoning. In a classroom setting, theorems will generally be presented in an orderly fashion such that if Theorem 1 is presented before Theorem 2, then Theorem 1 does not require Theorem 2 in order to be proven.
Because theorems are provable statements, it may be tempting to jump directly into the proof, and particularly to only take notes on the proof. I cannot stress against this enough. Before writing the proof, you should make sure to begin the proof with a clear statement of the theorem. This should be a declarative statement that is thus provable. Do not confuse the questions a professor may ask before proving a theorem with the theorem itself. An example of a theorem from group theory is “Any group that has a prime number of elements of elements in that group is cyclic (or can be generated by a single element). Also, do not confuse the nicknames of theorems with the theorem itself. For instance Lagrange’s Theorem says that “the number of elements of any subgroup of a finite group divides the number of elements in the original group.” Remembering this as Lagrange’s Theorem is fine, but it is much more important to remember the declarative statement proved.
I’m sure there are other things people use to take notes in math classes. Feel free to leave a comment sharing some of this advice.