Before Class

Prerequisite understanding

The cumulative nature of math means that you need to revisit previous concepts often enough that you don’t forget them. Taking College Algebra, passing, and then immediately trying to push aside all the math equations you had memorized will do you no good in Precalculus when you have to build upon that previously learned material. The same thing applies throughout the duration of a course. Completing the first assessment does not mean you can promptly forget what you were just tested on. Learning everything from Module 1 and then moving on and forgetting that content will be a huge disservice to yourself when you get to Module 5 and you have to use those concepts again.

This is why prerequisite understanding is so important. Before every lecture or new section, review any previous content that will be key to mastering the new material. Your knowledge should be like a rubber band ball and keep layering on top of itself. Spend some time prior to each class reviewing material from the class before that will serve as the foundation for the upcoming material and older concepts that you may be rusty in using, but will need to be used to master the upcoming material (TIP: ask your professor for a list of these prerequisite skills if you are unsure).

Preview the upcoming topics

There are two friends who have never been snowboarding, heard of snowboarding, or seen anyone snowboard before. Their roommate Ashlee is taking them snowboarding for the first time tomorrow and she hopes it will go well. One of the friends, Janaan, decides to watch some YouTube videos of people snowboarding, and goes to look at the gear that is packed in the truck. The other friend, Kyle, decides that he’s just going to wing it the next day on the slopes. Who do you think is better equipped to learn this new skill of snowboarding?

In math, the same concept applies. By looking ahead at the content that you will be taught or are about to learn, you give yourself a tiny head start and a framework for where to start storing newly learned information. This preview doesn’t mean that you need to complete all the homework before it’s even assigned or study the section in great detail (though it can’t hurt!). It simply means that you familiarize yourself with the vocabulary, new equations/theorems, and the type of questions that will be asked within the next lesson.

The first step in previewing the material is to simply learn the key terms. It’s hard to learn a new concept in any subject if it’s being taught to you in a foreign language. If you don’t have the basic vocabulary memorized for the course, you will struggle to follow your instructor’s lesson or the textbook prompts. At the bare minimum, learn the terms and basic theorems and rules that will be covered in class. One step further to be really prepared to get the most out of class is to do the following: skim the chapter text once through, note anything that seems especially important or difficult, and develop a rough understanding of what you should learn in more detail the next day.

During Class

Actively participate

Learning is not sitting and listening to someone explain something to you. In fact, sitting and listening is one of the most ineffective ways to retain information. You need to actively engage with the speaker by taking notes, responding to questions even if you’re not sure of the answer, and staying focused on the task at hand. If you are sitting through a lecture, make sure you are taking the time to reflect on what’s being said. If you are tasked with group work, make sure you are completing the activity in a way that you are both helping and learning from others. When you actively participate in class, it automatically leads to behaviors that help you become a better student.

Ask questions

This step at first seems to be an obvious and simplistic piece of advice. But we are not referring to things you can look up on your own like definitions of words, or when the midterm is, we are referring to conceptual questions. The art of asking a good conceptual question is a very important math study skill. A good question is not, “I don’t get it, why is that the answer?” A good question is, “I see how we combined like-terms to get to the third step, but why did the negative sign in front of the [latex]3[/latex] change to a positive after we distributed and multiplied?” The act of identifying your uncertainty, thinking of what specifically you are struggling to grasp, formulating how to ask that question, and receiving the answer as feedback to process is a powerful tool.

Another aspect of cumulative course information in math is that waiting to ask for help and falling behind can make or break you in that class. It’s possible in some other subjects to miss a chunk of content and then come back strong and redeem yourself in that course. In math, that is almost impossible because the next concept depends on your understanding of the first ones.

Becoming active in your learning journey means developing the confidence to ask for help. Speak up when you don’t understand something; you are the only one who knows you are confused until you say something. Even if you’re not comfortable asking a question right then, at least think of questions and write them down so that you can get those questions answered soon after.

Annotate your notes

While you are taking notes during class, it is important to take detailed notes. These notes should include more than just what your professor is writing down; they should include bits of your inner dialogue as you were sitting through class. If something seemed really crucial, put a star next to it. If one of the worked examples was unclear, put a question mark next to it and a brief note of why you were confused. If the professor reworded their explanation of a concept and you had an ah-ha moment, write those new phrases down. Without these additional entries, looking at your notes will be just like looking at a textbook. Your notes should be in your own words rather than an identical copy of what your professor chose to write on the board. If you write your notes in a way that makes sense to you, you won’t  have to spend additional time deciphering them and your annotations will act as a guide once you leave class.

After Class

Confidence in the new material

Now that you have attended class and actively participated, asked questions, and taken annotated notes, what do you do with all this new information? For your brain to hold onto this new information tightly enough to apply later on, now is a good time to review all the notes you took, follow through with getting your questions answered, and get your murky spots solidified. You can try to answer your own questions, but you should reach out for help if you can’t figure it out yourself. Professors often hold office hours for just these types of questions, or you could organize a study session with some of your peers. Either option will keep you from falling behind. Your goal is to be super confident in everything you’ve just learned. In fact, helping other students with their questions is a great way to develop confidence in what you’ve just learned. If you can answer other students’ questions, you have a strong grasp of the material.

Chase a deeper understanding

To truly understand a concept and be able to apply it as knowledge in the future, you must understand the why. Why does it behave that way? Why do we use it for that? Why is it important? This way, when you see a new problem that looks scary, you can figure out how to tackle it step by step based on your understanding of why you can and why you do certain actions in math. Deeply understanding a concept allows you to fully put it into practice and develop new ideas on top of it having strong roots.

Carry out practice, practice, practice

Although parents will often say they don’t have a favorite child, most math instructors will say that they have a favorite study skill. And I am willing to bet that ninety-nine percent of math instructors would agree that THE most important study skill is practice.

As we’ve discussed, math is an active learning subject. You cannot sit passively in a lecture, read the textbook, and then go in and ace an exam. If you have not practiced the application of the content regularly through practice problems (hint hint, that’s why teachers give you homework!), the assessment will be a struggle. You might be able to recall some of what you saw or read, but it will be very hard to apply it.

In math, you are lucky to have a subject that typically provides many worked examples with correct answers for you to look over. However, simply looking at these examples will do you no good. Remember, active learning. Write down a few example problems and try to do them yourself without the guidance of the book unless you get stuck or end up with the wrong answer. Then use that detailed solution to guide your learning and understanding of how to solve future problems of that type. In fact, go find more of that problem type and just keep trying until you are getting to correct solutions without looking at the worked steps in the book.

Another critical part of practicing math problems is showing all your work. Write down all details of your work instead of just the answer. If you are trying to solve a multistep problem (which is pretty much all college-level math) in your head, you are much more likely to make mistakes. Solving in your head will also make it impossible to go back and figure out which step, or part of the problem, you are introducing an error to. Learning from your own mistakes is one of the most powerful ways to learn, but you won’t be able to without first being able to see and search for the mistake.

If you practice enough, you will master a concept backwards and forwards and be able to tackle all different versions and types of questions.

Connect ideas

The best way to keep these study skills rolling into your next math class is to have a big picture of how all the math concepts you’ve been learning fit together. Can you make a web of math ideas and how they are all connected? Remember the example at the beginning of the section about the relationship between addition and multiplication. If you had created a concept map as you learned addition and then multiplication, you would better understand how they fit together, and how knowing one skill will help you conquer the other skill. This map allows you to visualize and identify connections between concepts, so that every new concept you learn won’t feel like you’re starting to build your understanding from scratch.