Advice on Success in University Mathematics Courses

Many students ask me how to study. Particularly in freshman mathematics courses, students may experience a sort of culture shock: university courses have far more depth and a faster pace than high school courses. Here are my recommendations to students.

Solve lots and lots of math problems. If you solve problems, you will learn mathematics. If you don’t solve problems, you won’t learn mathematics. This is far and away the most important determinant of success in math courses, and it is universal. Solving problems is the way all human beings learn mathematics, be they six-year-olds or professional mathematicians. Your professor has probably assigned a number of problems, carefully selected as instructive. Solve all of these problems—plus additional, unassigned problems as needed, possibly from supplemental sources—unaided insofar as possible. Resist the temptation to look up a solution to a problem on the Internet! By obtaining someone else’s solution instead of thinking up your own, you deprive yourself of a crucial element of your mathematical development, as well as development of habits such as perseverance. If, after protracted effort, you’re completely stuck, ask your professor about the problem. He or she may be able to point you in the right direction with an appropriate hint while still allowing you to develop your own ideas.
Master the prerequisite material. Later mathematics builds upon earlier mathematics: the further you go in the subject, the more abstract and subtle it becomes. If you have a solid foundation in the background material, your experience will be of new worlds opening up before you. If, however, your understanding of the prior mathematics is particularly weak, learning subsequent mathematics is likely to be an insuperable task. This problem notably arises in freshman calculus classes, where the unprepared student has to struggle to understand every little algebraic manipulation in the course of a solution to a problem. It’s as if someone didn’t understand a word of Russian but decided to read untranslated Tolstoy by looking up every word in a bilingual dictionary. Not too likely to yield a real understanding of—or appreciation for—a literary masterpiece. If you do take a math class without having assimilated the antecedent mathematics, allocate ample time for review and expect to have to work twice as hard.
Schedule sufficient study time. A one-semester math course is likely to require 150 to 200 hours of studying and solving problems outside of class. This work must be evenly distributed throughout the semester.1 It is important to avoid distractions while studying mathematics (and in the classroom): engaging in a secondary task impedes the acquisition of so-called “flexible knowledge,” the type needed to solve novel problems.2 Beware the temptation to curtail your efforts when you think you’ve fully understood the material. There is a natural human proclivity to overestimate our own proficiency.3 Math students with a superficial understanding of the material often lament, “I understood the concepts, but I got lost when I tried to solve problems on the exam.” This statement betrays a fundamental misconception. Understanding the concepts is not the objective of a mathematics course. I understand the concept of running a four-minute mile. I understand the concept of speaking fluent Welsh. I understand the concept of writing a novel that wins the Man Booker Prize. Sad to say, I can’t actually do any of those things. Understanding the concepts is, as a matter of fact, a pretty trivial accomplishment. The true test of your understanding is the ability to solve problems you’ve never seen before using the mathematical ideas and techniques of the course.
Attend all class meetings without fail. You will benefit from preparing for class in advance, for example, by reading ahead in the textbook. Incidentally, it’s impolite to arrive late.
Take careful notes during lectures; then review, annotate, and assimilate your lecture notes. Some students say it’s difficult to concentrate on a lecture while simultaneously taking notes. The reason it’s difficult is also the reason you should do it. Passively listening is easier than actively engaging, but you learn less. Taking notes reinforces understanding while the lecture is in progress, by forcing you to formulate what you are seeing and hearing in your own words (in addition, obviously, to providing you with a resource for later study). Merely recording the lecture or taking photographs of the blackboard robs you of this benefit; even taking notes on a laptop is less efficacious than taking notes with pen and paper.4
Read the textbook slowly and carefully. Ask questions about parts you don’t understand. Auxiliary reading from books in the library is also beneficial. Remember that reading a mathematics textbook is not like reading the newspaper: you may need to pore over each sentence, wherein each word is apt to be significant. Treat the occasion as a conversation between you and the author. Jot down notes, fill in skipped steps, draw accompanying illustrations. Cover up the proof of a theorem or the conclusion of an example and see if you can produce the argument yourself. You should read and digest the textbook material before attempting to solve the relevant problems.
Form study groups with other students. It will make studying more pleasant if it becomes a social activity. By working with others, you create incentives for yourself to prepare. It will help you understand difficult topics to get an explanation from a classmate who has mastered the point in question. It will solidify your understanding to explain topics to your peers. Communicating mathematics is a skill worth cultivating. If you can’t explain some idea to a fellow student who doesn’t understand it, that might suggest that you don’t truly understand it yourself.
Attend office hours. There is great value to individual attention from your professor, who is apt to have long experience resolving common mathematical difficulties, but may not be able to divine the nature of your misunderstandings unless you ask about them. It helps to come prepared, but you shouldn’t let concerns about that stop you. And don’t feel you are somehow intruding. Office hours are your time.
Avail yourself of free tutoring. Most universities—certainly my own—provide tutoring by graduate students or advanced undergraduates; you should take advantage of this valuable resource. You can get times and locations from the math department office. An important caveat bears emphasis. It is educationally harmful, not beneficial, to get a tutor simply to show you an answer to a math problem you couldn’t solve without teaching you the underlying mathematical principles. If you cannot solve a problem—and certain problems can require a great deal of thought and effort—that is an indication that you have not truly understood and absorbed the relevant mathematical concepts as they arise in practice. To simply “get help with a homework problem” without achieving an understanding of the mathematical principles relevant to it and other, similar problems is to treat the symptom and not the disease.
Solve lots and lots of math problems. My first recommendation is also my last recommendation; this vital point bears repeating. You won’t become proficient at serving a tennis ball merely by watching Serena Williams; you won’t become proficient at playing the piano merely by listening to Glenn Gould; you won’t become proficient at mathematics merely by watching a mathematician solve problems. You have to do it yourself, diligently and repeatedly. The reward is inestimable: not simply a good performance on an exam, but a more powerful brain.5

If the way which I have pointed out as leading hither seems exceedingly hard, it can nevertheless be discovered. Needs must it be hard, since it is so seldom found. How would it be possible, if salvation lay ready to our hand, and could without great labor be found, that it should be by almost everybody neglected? But all excellent things are as difficult as they are rare.

—B. Spinoza, Ethics, V

Notes

1. D. Rohrer and K. Taylor, “The effects of overlearning and distributed practice on the retention of mathematics knowledge,” Applied Cognitive Psychology 20 (2006), no. 9, 1209–1224.

2. K. Foerde, B. J. Knowlton, and R. A. Poldrack, “Modulation of competing memory systems by distraction,” Proceedings of the National Academy of Sciences 103 (2006), no. 31, 11778–11783.

3. B. Finn and S. K. Tauber, “When confidence is not a signal of knowing: How students’ experiences and beliefs about processing fluency can lead to miscalibrated confidence,” Educational Psychology Review 27 (2015), no. 4, 567–586.

4. P. A. Mueller and D. M. Oppenheimer, “The pen is mightier than the keyboard: Advantages of longhand over laptop note taking,” Psychological Science (2014): 0956797614524581.

5. B. Draganski, C. Gaser, G. Kempermann, H. G. Kuhn, J. Winkler, C. Büchel, and A. May, “Temporal and spatial dynamics of brain structure changes during extensive learning,” The Journal of Neuroscience 26 (2006), no. 23, 6314–6317.

Greg Marks