Section 0.3 How to Study Math
Math is a subject best learned in small increments over a long period of time. Cramming for hours at a time is not an effective way to learn mathematics. The best thing you can do to help you study is to schedule relatively short (30 minutes - 1 hour) study sessions for yourself spaced out over the course of several days or weeks, as often as needed based on the amount of improvement you wish to achieve and how much material you hope to cover.
Leading up to an exam, you may wish to lengthen your study sessions, but do not push yourself past the point of mental fatigue. You will simply get frustrated and will not retain the material you review. Avoid working on math when tired. Think about the times of day when you are best able to focus and try to study during them.
Most importantly, find other students with whom to study. Do not work with someone far above or below your own level of understanding, but rather someone with whom you can exchange ideas equitably. Take turns suggesting problems or topics to review and explaining solutions and ideas to each other. Ideally, each of you understands what the other does not. However, you will also benefit from combining your partial understandings. The social interaction involved in working with another person will help you retain the information you learn. At the very least, making plans with another person to study will hold you accountable for using that time as intended.
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Resources you can use to study.Plan when you begin studying to give yourself enough time to attend office hours or tutoring to ask questions. The resources below are available to you in order to identify what you need to know:
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Your notes from reading and from class
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Problems from in-class worksheets and the posted solutions
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Old Check-Your-Understanding problems. Some have solutions that you can view after the due date.
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Homework problems and posted solutions
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Online resources, such as YouTube or online textbooks. There are specific suggestions in Section 0.1. Seeing an alternate explanation and additional examples may help you understand a topic. Just be aware that different sources do not all cover topics in the same way and material that you find online may not be relevant to the course; if something seems entirely unfamiliar and you can’t find something similar in course materials, chances are it’s not something you are responsible for knowing.
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Explanations and examples in the textbook. Warning: Many people try to study by simply reading through a text from start to finish. This is not an effective way to learn. If you plan to read the textbook to study, you need to read actively. See below for details on active reading as it pertains to mathematical texts.
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How to study effectively.A great way to begin your studying is to create a concept map of the big ideas being tested on the exam. Include all of the relevant definitions, theorems, and formulas. Reread comments left on homework and see which Check-Your-Understanding problems took you the most attempts to get correct. Do not study everything, but rather focus on those topics that give you the most trouble.
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Practice Problems Math is best learned by actively solving problems. When practicing problems, complete them on your own with as little help from your notes or another person as possible. Only resort to consulting other resources when you are stuck and have already tried to get un-stuck.
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Use Homework as Assessment Similar to the above point, use the work that you already have to do as a way to assess your understanding as you do it. Avoid copying methods from examples and their solutions but rather review your notes and then try the problem without looking.
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Check your Solutions Use solutions to check your own work. If you get the same final answer, make sure that the intervening steps are the same or at least equivalent. If you use a different method to complete a problem than the solutions, compare them: Where do they differ? Which is more efficient? Which is more applicable to other, similar problems? If you do not get the same final answer, determine how much of your work is correct and precisely what kind of mistake you made.
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If your mistake was based on faulty conceptual understanding, review or make a note to review material related to that concept.
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If your mistake was computational, what can you do in the future to avoid the same mistake? If you find yourself consistently making the same computational mistakes, make a mental note to double-check your work whenever doing similar calculations.
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Write Your own Practice Exams Combine problems from homework and worksheets and either do them yourself, or trade with a study partner. Having to solve a problem out of the context of a section of the text is a test-like challenge that is good to practice. Moreover, anticipating which problems are "test-worthy" and thinking from the perspective of the instructor can help you to anticipate the types of problems that will be on the exam.
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Learn by Teaching and Learn as if to Teach Explaining an idea will make it stick better in your mind, so take turns in your study group explaining problems to each other. When working on your own, don’t study with the intention of solving problems: study with the intention of teaching other people to solve problems. How much more thoroughly would you prepare to give a 50-minute lecture than you would to take a 50-minute test? Study as if you will be standing at the front of the class giving explanations, rather than sitting in a seat working on paper.
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Review the Theory Review the definitions and theorems covered in class. Do not simply read through them, however. Find examples (from homework, the book, online) that relate back to them and explicitly connect the example with the ideas in the theory. Give a few examples that fit the definition or theorem, and try to find examples that do not fit. Compare them and determine what differences make the definition or theorem apply/not apply.
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Review the Computation Review all of the formulas and methods covered in class. Ideally, homework has given you enough practice with these so that you retain them. If not, find additional problems to practice. If you find yourself having a very hard time applying these quickly, make yourself a study sheet (maybe a notecard or a separate piece of paper) where you copy over the formula/method and write out an example or two using that method. Briefly review this study sheet several times a day.
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If using a notecard, you may wish to put the formula/method on one side and examples on the other side.
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If using a piece of paper, fold it down the middle and put the formula/method and examples on opposite sides of the crease.
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Read Actively Many people like to review their notes or the textbook to study. However, simply reading through explanations and examples is not an effective way to learn material. Do not confuse familiarity with understanding. You must read actively to benefit from reading, which for mathematics means:
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Rework any examples or computations (with a pen or pencil on a separate piece of paper). Make a note of any steps you do not follow but don’t let them get in your way of continuing to read. It’s possible something included later in the text will clear up your confusion.
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Identify the main idea of the section as well as the main idea of each subsection or paragraph. Connect these ideas: how does the material in the section support the main idea? How does the main idea of the section tie together the ideas of each subsection? There’s no need to formally summarize your reading, but it’s very useful to think about the big picture ideas as a way of understanding the details.
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Connect the theory and ideas discussed to explicit examples that relate to them. Why do these examples fit? What change in the example would make it no longer fit? How are the specific details of the theory represented in the problem (for example, if there are constants used in the theorem, like a and b, that are given specific values in the example).
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Connect the theory and ideas discussed to other sections covered in the text. What ideas does this section use? Where are these ideas used in subsequent sections?
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If there are any words that you do not understand, look them up! If they were previously defined in the book, they will be listed in the index. If you can’t find where they are defined, you can always google them. Just make sure that you are looking at a relevant mathematical definition. Oftentimes, words are defined to have very specific mathematical meanings that are not the same as their colloquial use. Sometimes, different areas of mathematics will use the same words to mean different things, so make sure the definition you find is familiar or seems relevant.
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Seek Help from your InstructorThe best place to get help is office hours. These are times your instructor has set aside to meet with students. You do not need an appointment to go. Questions can be about specific problems but can also be more general. Do not go with a laundry list of questions to get answered right before an exam, but rather attend office hours regularly and ask a few questions each time. Formulate your question (and write it down) ahead of time so that you can use the face-to-face time with your instructor productively. Bring any work you have attempted on a problem and show your instructor so that they can pick up precisely where you need help.
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