How to Study Mathematics at the University Level
Prof. Naomi J S de Moraes
UFABC, Santo André, Brazil
Since studying at the university level is different than studying in high school, and you probably did not receive a manual with explanations, I will outline here my suggestions on how to do well in university math classes.
Before a lecture (make sure you know what it will be about)
1. Read the section in the textbook, or at least skim it to know what will be taught. You will understand lectures so much better!
2. Look at the questions in the back of the section, to know what kind of questions you will be asked to solve. That will motivate you and help you put what you learn in context.
During a lecture
1. Take good notes. I use a three-column Cornell note format with the first column for headings, the second columns to note down what the professor wrote on the board, and the third column for my comments on the material. I especially note down anything I did not understand, and anything the professor says in answer to a question I asked. This third column is where I interact with the material.
2. If there are pauses to work out exercises in class, do the exercises. Do not think “I will try this at home”. There is no better time than now! Do not procrastinate. If you get stuck, the professor is right there so you can ask a question.
3. If you do not know what section/material the professor will cover in the following class (because the professor did not provide a schedule), ask at the end of class so you can prepare. I was amazed, when taking math classes, by how many math professors treat math lectures like an adventure, in which you have no idea where you are going until you get there at the end of a lecture or a demonstration. I prefer to have a road map so I can note the beautiful scenery along the way. :o) That is why I provide a full syllabus with a schedule of which section I will cover every day. Ask your professor to at least tell you what the next lecture will cover.
After a lecture
1. Try some of easy problems at the end of the section (or on the material covered), even if the professor has not assigned any problems yet. The textbook has problems. Do not wait! The material may look easy or hard, but you will not know until you get your hands dirty by doing problems. The longer you wait, the harder it will be because you will forget everything you heard in the lecture. Do only problems for which you have solutions – either in the back of the book or in a separate solutions manual. I found four different textbooks online for the course I am teaching, and each of them had solutions manuals online too. WolframAlpha and Symbolab can also solve problems for you. If you do several problems, checking the solution for each as you go, by the time the Professor assigns graded homework or a quiz, you will be prepared.
2. Read the textbook carefully now if you did not do so before. Textbooks tend to skip steps. Do them on a post-it if reading a paper book, or with a stylus if an e-book.
3. I have a “toolbox” of summary notes that I have created over the years, summarizing the most important mathematical topics that I have studied. They are a sort of reminder to me of what I have learned, and I refer to them when doing problems and when preparing for the exam. Do not prepare for an exam by re-reading the textbook if you already read it and understood it. Write up summary notes for each section or lecture, and another summary note for each chapter with a big-picture overview of the chapter’s contents. This will really cement the ideas in your mind. By organizing them and writing them down, as if you were teaching them to someone else, you really learn them! These notes can be linear, in a flow diagram, in a mind map, in a table, whatever works for the topic. I use different styles for different material.
4. Do the homework the Professor assigns, especially if it is for a grade. This is obvious, but if you do only this you will probably not succeed. If you are having trouble, do easier problems from the textbook (see above) to warm up your problem-solving skills. Then do more problems. If you have any questions, talk to the professor after class or during office hours. I visited my professors during every single office hour all through my undergraduate degree. I always did the homework before the office hour so I would have some questions. Think of it as free tutoring! I would love to help, and feel very lonely sitting there with no visitors during my office hour every week. :o(
Preparing for the exam
1. Keep a list of every problem you try, get wrong, then learn how to do from the solutions manual. Before the exam, try these problems again without peeking at the solution. Never simply read the solutions manual. This will give you a false sense of security.
2. Before the exam, do more problems. Read the questions at the end of each section/chapter. Ask yourself “do I know how to do this problem?” If not, try it. At least set the problem up, then check the solutions manual to see if you were on the right track. If you are short on time, do not do all the algebra to get the final answer, just the hard part when starting the problem.
3. Starting with a blank piece of paper, try to reproduce your summary notes for the class. Not all of them, but the ones you have found to be most important when working on problems. Especially any flow diagrams. Try doing this every day for a week before the exam. Recalling information cements it in your neural pathways. Do this during time you would normally waste, like waiting for a class to start.
4. Create flash cards for any information you must memorize, like derivatives and integrals, graphs of functions, trigonometric identities, properties of logs and powers, etc. Practice with these flashcards on the bus, when walking, when waiting for class to start, while eating alone, etc.
5. Sleep 8-10 hours every night, and never, ever stay up late the night before an exam. You will be groggy during the exam and do worse than if you had not studied those extra hours. Last quarter, one student stayed up late, woke up late and missed the exam completely!
Make studying automatic
1. Most students have a schedule that tells them where they should be for classes and other activities, but not when to study. At most, they may write “study” from 2pm to 6pm every day. Decide when you will study for your math class, for how long, and where. Add this to your schedule. For example, every Wednesday afternoon from 2 to 6 in the library, every Thursday evening from 6 to 8 at home, and every Sunday from 10 to 2 . You can do this for all your classes, but if you can only plan for one class because your schedule is too crazy, plan for your hardest or least favorite class. This will hopefully prevent you from wasting time during your free hours while deciding what to do next. If you do fall behind and have not studied the material for the previous week(s), schedule twice as much time for the subject.
2. Create a study log. This can be on your phone, or on a piece of paper in your math notebook. When you sit down to study, write a few words stating what your goal is. Then study for the planned time. When finished, note down what you intend to do during the next study session. Remind yourself where you stopped and what is next. This will save you a lot of time, each session, because it will be easy to start where you left off.
3. Try the Pomodoro method. When you start your study session, set a timer for 25 to 45 minutes. Study for that period with no distractions. Put your phone in airplane mode! Then take a little break during which you do not study, but also do not start some other project. Get up and walk around, get some coffee, go to the bathroom, etc. Then sit back down and study for another 25- to 45-minute period. Your brain needs a break every once in a while to process information.
What do most of these tips have in common? You need to be an ACTIVE learner, thinking of the professor as a helpful guide. You must learn the material, by reading, doing problems, summarizing, and organizing it in your mind. Attending lectures and taking notes is a good start, but it is passive. You must do problems, lots of problems, and make the material your own. You need to know it so well you could teach it to someone else.