How to Study Calculus

During the fall, I have been teaching a section of calculus. As finals are approaching, I find myself repeatedly providing students with tips on how to study for my class. Here will be a central location for my recommendations. This will be a dynamic post that will hopefully increase in length over time.

1. Go to class.

   This should go without saying. I (or your teacher) work hard thinking about how to present the information to in an easily digestible format. It takes time to craft meaningful examples that draw out the finer details of what is being discussed. Not only should you make a point to attend your lectures, and this could potentially be a second point, you should pay close attention and try to follow the entire lecture. If you lose concentration and fall off the bus, try to get back on as soon as possible. Ask questions. I can’t tell you how great it feels when a student asks a terrific question. Don’t worry about whether your question is spectacular. Chances are, if you don’t understand something, others in class don’t either.

2. Do your homework.

   Again, this should be a given, but I’m afraid it is all too common for students to just skip doing their assigned homework problems.

   Even if your teacher isn’t collecting the homework, you should make a point to do it. I do not collect the homework I assign. Rather, I give a quiz about once a week over the material from the week. None the less, I expect that you HAVE done the homework problems.

   Math classes are tricky because the content can appear to come about so easily and naturally. This is a direct result of how logical mathematics is. Just because you can follow the lecture, DOES NOT mean you will automatically know how to do all the homework problems. You need to do them to help nail down what has been discussed. Which leads to another point.

3. Do the homework problems immediately after class.

   If you can’t do them immediately after, do them as soon as possible. I have found this equates to less over all time spent on homework and a better understanding of the material. When you follow this suggestion, the material is fresh in your mind from class and you are able to remember the examples worked in class better. All of this helps you solve the problems faster and then synthesizes the information quicker.

4. Go over all your homework problems again before an exam.

   These are often the best examples of what is going to be on the test. I encourage my students to do all the problems again the weekend before a test.

5. Go to office hours.

   The one lecturing knows what they expect of their students better than anyone else. You should take advantage of this resource whenever possible. I make a point to be abundandtly available to my students and I availability known often at the beginning of the course. Many students are still too intimidated (or something) to come and ask for help. The few who do however, end up doing very well since I can see exactly where they are struggling and help them individually.

   A side benefit to this advice is that, at least in my case, the lecturer is also the one who writes the exams. So when you get help from them, they may hint at what you actually need to know (and to what depth) for those exams.

6. Connect old ideas with new ones.

   For example, when you learn the product rule for differentiation, make sure you understand where it comes from in terms of the definition of the derivative. Try to prove where the result comes from. This will aid your understanding. When you are learning L’Hospital’s Rule, try working the same examples WITHOUT his rule to see if you remember how to do these limits “the old way”. Doing this will hopefully help you see where the new methods are extending the types of problems you are able to solve.

7. Try studying early in the morning.

   This advice applies more to the class I am currently teaching at 8 AM. Most people are not used to using the type of thinking required to do well in calculus when they first wake up. Try to make a habit of getting up and solving one problem before you get started with the rest of your day. It will get your brain moving in the right direction and will make solving calculus problems second nature.

8. Read the book.

   Most students like to complain about the textbook. Just read it. Most of the calculus books today are NOT that terrible. The examples can often provide another point of view, or at least another look at the the techniques discussed in class.

9. Read the “For Dummies” version.

   I found this to be a very good strategy. Often the “For Dummies” version or a book from the “Demystified” series will be even easier to understand than your textbook and will provide a plethora of worked examples. They often also provide exercises with answers so that you can check your work. (I’ve never used either of the specific books I linked to here so I can’t speak about these books in particular. I have had one student use the Calculus for Dummies and they seemed happy with it.)

10. Use online resources.

    Khan Academy provides numerous worked examples and explains the material in a more colloquial fashion that some mathematicians don’t appreciate. I and, I think, my students have found this to be a useful resource.

    Another good resource that I feel isn’t utilized enough is MIT Open Courseware. These are lectures from the leading experts in their fields. Their lectures are good. They explain things in a way only subject matter experts are able. You may find the material is at a level slightly deeper than what is expected for your class however, but don’t let this scare you away. There is A TON of great information in these videos.

Well, I think that is a good list to get started. I almost certainly have forgotten something. Hopefully nothing too obvious. I will update this as I think of more.

Feel free to leave a comment with your ideas: either how you study or, if you’re a teacher, how you encourage your students to study for your class.