Guide for Students of Mathematics
I will first describe briefly how each year looks for a student in the Math department.
Freshmen Year: In order to be able to declare mathematics as a major students need to complete the two-course sequence MAT 215/217 or MAT 216/218. The latter sequence is designed for students with a significant exposure to rigorous proof-writing and covers more material at a faster pace. The first course is on Analysis and the second course is on Linear Algebra. Most of the students intending to study mathematics take these two classes their freshmen year, and a few other students will start with MAT 202/204 in their freshmen year and take the 215/217 sequence in their sophomore year.
Many students face challenges during the introductory sequence, especially if they have less exposure to rigorous mathematics, have never been to math camps, or have not taken college-level mathematics classes before. For students that have little exposure to theoretical math classes and proof-writing, freshmen year is an important year for them to develop and experiment with the strategies of learning and problem-solving in mathematics.
The introductory sequence is designed with the following goals in mind. Firstly students learn basic concepts that will show up in all other mathematics upper-level classes. Then students are introduced to rigorous mathematical proofs and learn both how to read them and how to write their own.
Sophomore Year: In their sophomore year students usually take classes that introduce them to the various branches of mathematics, the most famous ones being Complex Analysis, Algebra and Topology. Taking as many of these classes as possible in their sophomore year allows students to understand better which branch of mathematics they want to work on for their junior independent work. These classes are also introductory in nature, though they assume prior experience with proof-writing and knowledge of some basic concepts of analysis/linear algebra that have been developed in the introductory sequence.
Junior Year: In their junior year students can continue to take the introductory courses described in the previous paragraph, as well as more specialized courses within a branch of math after they have taken the introductory class in that branch. For their junior independent work students can choose between a junior seminar and writing a junior paper. The junior paper is the same as for all other departments: the student works together with a professor to write a short paper. However, students in the math department are not requested to produce original research for their junior paper, so the process usually entails reading a somewhat complex math paper/textbook and writing an expository paper. The junior seminar runs in the form of student presentations and is guided by a professor. Every students presents on a topic twice during the course of the semester and writes a 10 page paper at the end of the seminar. Math majors have to take at least one semester of junior seminars, as these seminars teach students how to expose and communicate clearly to their peers complicated mathematical concepts.
Aspects of a mathematics class and strategies for each
Despite the various levels, every math class at Princeton has these four components that students interact with: lecture, office hours, problem sets, and exams. I will describe some strategies that students can use to approach each of these components.
Lecture / lecture notes:
Most of the lectures have a formal style: the professor talks and writes on the board. Ideally, the student concentrates on the mathematical argument and tries to fully understand it. In order to juggle writing down notes as well as understanding the argument, one approach is to write down only the main points of the argument during lecture and fill in the gap afterwards, provided one does this before they forget everything. However, sometimes one may find it hard to keep up with the lecture’s reasoning. In this case rather than coming out of the class with little understood and little written down, it is better to copy whatever is on the board, plus anything else which seems important. Copying down notes will at least help concentrate the mind, ensure that something is absorbed, and provide with something personally written which one can work on afterwards.
Thus if certain lectures are harder to understand or less prepared than others, it is nonetheless important to keep trying to take notes and make out of it as much as possible. It will be much easier to understand the material while reading the book having been to lecture before than having skipped it.
After the lecture, as soon as there is time, and before the next lecture, it pays a lot to read the notes carefully to make sure everything is understood. For future reading, it is helpful to add notes on the margins that make the lecture material more clear and complete.
Reading the book ahead, (or at least skimming the main results), makes it easier to follow the lecture and therefore benefit more from it, than if the lecture material is encountered for the first time.
TAS and office hours:
Office hours can be used both for questions about the theory or for help with problem sets. TAS and professors expect that the student has tried solving a problem before they ask for help, so these resources should be used after some amount of work. When asking for help with a problem, the professor or TA will usually try to guide through the solution giving hints, so engagement from the student is expected. Therefore it is best to make sure that one understands the theory exposed in lecture, or asks questions about that, before asking for help with a problem. The solutions arrived at during office hours will be largely sketched, so as soon as possible, after office hour, one should try to make sense of the notes and write out the full solution.
Almost every math class has a weekly or biweekly problem set assigned. This is where the students get to practice the very theoretical concepts they have learnt in lecture. In solving a mathematical problem there are usually three stages:
- i) understanding the problem
- ii) experimentation – to find a method which looks like it will work
iii) verifying and writing out the solution
For many of the problems, students do not come to a solution immediately. It is common to feel stuck or frustrated during the process of problem solving. These are some suggested steps to deal with the situation:
- i) While searching for a method to solve the problem, do not worry about the details; fill them in later.
- ii) Try working backwards from the answer; this may show the connection between the assumptions and the conclusions; turn the argument around later on
iii) If the problem is a general one, try special cases, or simpler versions of the problem
- iv) Read (and understand) as much as possible of the relevant theory. Look again at the textbook and the lecture notes and put on a sheet of paper all the results that seem relevant.
- v) Look in the notes and the book for similar solved examples.
- vi) Ask a friend or classmate: students in the math department are generally open to being asked for help.
It might also happen that after finding a solution that seems to work, while trying to fill in the details and write the proof, one finds a mistake in the method, a contradiction or something else that was neglected. This is why it is important to fully verify the solution, even write it down in full, before thinking that the problem is solved and moving on to the next one.
Cooperation on problem sets is highly encouraged for math classes, but cooperation can be highly beneficial only if done selectively. One should always try the problems on their own first and ponder about the problem for as long as possible. If after many attempts the student still feels stuck then asking a classmate is probably the best first resort. The process of collaboration should always start by explaining to the classmate what one has tried and where that has gotten them – even doing this might help one solve the problem on their own! Then ask for a hint from the fellow student and try to solve the problem with the hint that has been given. If still stuck, then ask them to outline the solution rather than asking to look at their solution, and write down the steps on one’s own. This process is beneficial for both students: the other student will understand and appreciate the solution even better, whereas the one asking for help will be able to understand the thinking behind the solution.
Textbook and Theory:
In the first two years doing mathematics consists mainly in studying the theory and solving problems. These activities need to work together for a proper formation: when doing problems we apply the theorems and formulas we have learnt or prove some results of smaller scope, but doing problems will also be necessary to understand the significance or the vast applications of the theory we learn.
In order to develop stronger mathematical skills, studying theory should be more than just reading and understanding – it should also be a process of experimentation. To achieve the previous, one can try the following steps when reading theory (from notes or textbook):
- i) making sure one understands the formal statement
- ii) compare different versions of the theorem; does each imply the other?, is one version more general?
iii) try to prove the theorem yourself, without reading any given proof
- iv) make sure that you understand the proof line by line and that you understand why each statement follows from the previous one
- v) identify where in the proof each assumption is used
- vi) identify the crucial ideas in the proof
vii) try omitting one of the assumptions; to what extent does the conclusion of the theorem still hold?
These steps help understand why each assumption of the theorem is important, convince that the theorem is true, develop some problem-solving skills and gain an insight of the proof that will be necessary to reproduce the proof later. In some of our exams we might be asked to reproduce proofs (professors will usually warn you about this) so it is better to start earlier on looking for the main points of the proofs in the book/lecture and trying to reproduce the proofs from memory.
Exams in math classes are usually either 3 hour written exams, or take-homes of longer duration. Unfortunately doing well in a math exam will be really hard unless a good amount of work has been put in the class during the semester. Therefore the minimal way to prepare for an exam during the semester is to make sure that one understands most of the theory and that one has made honest and patient attempts at solving the problems in the problem sets independently. Preparation before the exam usually consists of going through the theory once more and practicing with problems. When going through the theory for an exam, one can try to ask questions different from the first time when they encountered the theory, such as for example:
-What are the main theorems we covered in this class and what makes them so relevant? The answer usually lies in the many implications that important theorems have.
-What are the key points/observations that enable the proofs of these main theorems?
-How does changing the assumptions of each theorem influence the results?
Answering these questions helps tie together the many results that have been proved during the course of a class and understand how they relate to each other or fit in a picture. Moreover, understanding and being able to reproduce proofs of theorems, helps with coming up with strategies to solve the new problems one encounters in an exam.