In this post, I specifically address how I studied for Calculus III, but the methods I discuss can be used for other math or math-heavy classes (e.g., physics) as well. While intended for college math, ambitious high schoolers might also appreciate this advice. I’ve taken 5 college-level math classes so far (Statistics, Calculus I-III, Differential Equations), so I’d like to think I’ve got the hang of this because I put a lot of effort into improving my techniques, allowing me to ace Calculus III this summer!
Includes:
Reviewing Prerequisite Material
My Homework Process
Keeping a Practice Notebook
Resources to Help with Homework
Exam Studying
This post is very detailed, so I hope it helps you out!
– Melissa (@hexaneandheels )
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Reviewing Prerequisite Material
I think that, by far, the best way to review prerequisite material for math classes is through Khan Academy Missions.
The Missions are pretty much like online homework. Which sounds awful to many, I’m sure, but practice problems are a great way to learn and prove that you remember the prerequisite material enough to use it. Khan Academy makes it better than regular homework by showing you relevant videos alongside each question in case you encounter something you need to refresh your memory on (or learn for the first time if you’ve never seen it). You can also get hints if you still have no idea what to do.
You earn energy points for completing questions. And you get badges and medals for different achievements such as completing a topic or hitting a new milestone in energy points. That part isn’t super important, but it’s fun. :)
So how should you go about using it? I would recommend choosing specific topics from prerequisite classes that you know you need work on. For example, in reviewing for Calculus III I know I needed to work on integration, so I found those skills in the Integral Calculus section of the website. Almost all of the maths you can think of on Khan Academy have Missions available, so you should be set!
Alternatively, if you have the time (or feel like doing it for fun), you can work your way up on the “World of Math” mission. That one literally starts you from the beginning. Like 1+1 beginning. But it’s a nice refresher for things you might have forgotten over the years. And you could even use it to practice doing mental math faster.
There are other things you can do to review. But I think this is the most efficient because it’s curated to your specific needs and you don’t even have to decide what to work on next (unless you want to). It decides for you automatically. Easy.
My Homework Process
As I outline in more detail with this post, even if you don’t start your homework the moment it’s assigned, you should at least skim through it ASAP to see what to expect.
Ideally, I’d start the homework a couple days before it’s due. This is especially important if your professor has office hours or offers weekly problem sessions. My Calculus III class was online, so most of my discussion with my professor was through email, meaning I had to give him enough time to respond if I wanted an answer before the online homework was due.
I’ve had some math professors that didn’t have due homework at all. There was “recommended” homework, but none of it had to be turned in for a grade. Although it’s easy to skip it, DON’T. If you feel like you “don’t have time” to complete all of it, at least do some of the problems so you’re getting a little bit of practice. Because you won’t succeed if you do no practice.
Now when it comes to actually completing the homework, these are the sort of steps I would take:
Attempt problem cold without assistance.
Look in lecture notes for equations needed.
Look in lecture notes for similar examples.
Search online for similar, solved problems (to reference, NOT to copy, especially if it’s the same set of numbers).
Ask professor, tutor, friend, or online forum for help.
Practice is THE most important part of applied math classes. So I think it’s very important to keep a detailed and organized notebook to do your homework in (which I explain in the next section).
Keeping a Practice Notebook
You could just as easily do your homework on scrap paper. Only write down what you need to quickly solve the problem. Etc. But I did those things for Differential Equations and it was a HUGE mistake. So I spent my summer with Calculus III perfecting my practice notebook. These are the guidelines I followed.
Number your pages. Optional, but if you skip a question and don’t have enough room to finish it later, marking which page you continued from is helpful. This can also generally make it easier to keep track of everything in your notebook. Not super important, but helpful. I don’t think a table of contents is necessary, though add one if you feel so inclined.
Write out the full question and every step as legible as necessary. It does not need to be perfect note taking quality, but you need to be able to read it in case you want to come back later. (And you will want to come back later.) Do your best to record every step so it’s easier to catch mistakes.
Have a color code and/or symbol system to mark down questions. This is probably the most important part for when you go back later. Was the question so easy you breezed through it with no help? Or did you need to reference other sources to get through it because you were struggling? If you struggled, mark down what you struggled with (and why, if you can figure it out) by underlining or circling it. If you used a specific resource to figure out a step or have a question specifically for your professor, underline it in that color.
When marking down questions, be as specific as possible! If you’re too vague you’ll have no idea what you meant to correct later, which makes all of this completely meaningless. Take the extra few seconds to write it out (e.g., “forgot negative sign”). If you know what kind of problem it is (e.g., “line integral given parameterization”), writing that down can help when you go back to it later, as well!
You can use it for more than just homework! Need to go back and review an old technique? Write it in the notebook. Want to write down a quick note? Into the notebook. Want to check if you memorized conversions to spherical coordinates? Try to write them down cold in the notebook. This notebook is not supposed to be some clean, pristine set of notes. Use it for whatever seems functional.
I figure it would be helpful to show some examples, so here is my color coding scheme:
I stuck this on the inner cover of my notebook in case I needed to reference what each color symbolizes. The ones on the left are for the different resources I used to help me solve problems. The ones on the right are just other notes I made to myself. In the future, I’ll also add colors to mark how easy the problem was for me (which will help when reviewing for an exam). I just didn’t get to it with this class.
Here are some examples of actual markup:
I was very bad about marking up problems earlier in the class, so I don’t have as many examples as I’d like. But these should give you a decent idea.
I’ll be honest. A few times I got desperate and looked up the answers to certain online homework problems on Chegg. For those situations, I left enough space to go back and do the problem but I marked the question number with an orange asterisk and bookmarked the web page I used with the chapter and problem number (e.g., “5.4 #2).
I strongly urge you NOT to just look up answers without going through the work. Because although you might say you’re going to go back later, there’s a good chance you won’t. Especially if it’s a busy time of the semester. So do it right the first time. That little bit of advice in the preceding paragraph is meant for emergencies only.
Resources to Help with Homework
This is not meant to be an exhaustive list. These are just the references I have used that I personally find to be useful. No fluff.
Symbolab: A website that gives you FREE step-by-step solutions to almost any type of math you can think of. I found it especially helpful for lengthy integrations with lots of steps so I could mark down my mistakes more easily. The topics range from Algebra to Calculus and Statistics.
Wolfram Alpha: There’s so much you can do with WA. It can plot (even 3D and contours!), factor, simplify, differentiate, integrate, etc. And there are even some nifty calculators on there (the easiest way to find them is through Google; e.g., “differential equation calculator”).
Chegg: This website requires a subscription but it’s a HUGE help. I detail more about Chegg in this post, but I mostly use it to reference similar problems when I’m having trouble solving mine, textbook solutions, or if I’m that stuck I might as a tutor for help. Do NOT use it to cheat. It won’t help you in the end.
Online Graphing: For basic 2D graphs, I would recommend Desmos. It’s simple and easy to use. I used this one for graphing vector fields. Those are two examples, but you can find plenty on Google. If you have access to something like Maple or MATLAB, learning how to use those effectively can be a big help for when other resources fail you or you want to plot more than one function on the same graph.
Other Calculators: Want to calculate a unit vector? A determinant? Solve a matrix? Google “____ calculator”. You will most likely find some. I have a long list of them I like, but they were the first few results on Google. So no need to throw my list at you. But again, Do NOT use them to cheat. You will fail your exams. Use them to check.
YouTube: If your professor didn’t explain something well or you just need extra help, there are SO MANY videos on YouTube. I recommend Khan Academy (though I recommend using their full website instead), PatrickJMT, Krista King, and Professor Leonard (for full lectures).
If you find any online resources that helped you solve a problem or showed a complete solution, bookmark them for later (e.g., “5.4 #6″ or “2.3 #17 (similar)” if it was for a specific homework problem, or “unit vector calculator”). Even if it was easy to Google when you found it, you make it easier on yourself by bookmarking it so it’s faster to access. So if you’re studying for your exam and want solutions to reference, it’s all right there.
Exam Studying
This post was supposed to be about studying, so why is this the only section about actually studying? Because studying for applied math classes pretty much compromises of doing the same stuff you did for your homework.
Do. Practice. Problems. I suspect some of you are groaning at this point, but seriously. You can never do too much practice. Since applied math exams are all problems, that’s how you have to study.
But what problems should I practice? If you didn’t complete all the homework, finish it. If you struggled with certain problems, try them again (this is where your detailed practice notebook comes in handy), but don’t redo the same one several times. Do similar ones. If you have old exams, try those. Otherwise, you could look for problems under the sections you went over in your textbook (and check your answers with Chegg’s textbook solutions) or use something like Khan Academy Missions which will provide you with questions and grade them for you.
Some things just have to be memorized. A lot of people like making flashcards to flip through. What I personally do is make a reference flashcard (stick all the equations for a topic on one flashcard) then use the Blank Sheet Method to check if I memorized it. You could do this either in your notebook or on a whiteboard. I find this method more efficient than flip-through flashcards because oftentimes I memorize something fast enough that the process of making flip-through flashcards takes longer than actually memorizing the information. You could also search for premade flashcard sets on Quizlet.
Don’t practice everything unless you have the time. You should absolutely prioritize practicing the problems you have the most trouble with. Some people like doing easy problems to boost their confidence, but I personally find that a waste of precious time. If you’ve already gone through the trickier problems thoroughly, then maybe go through 1-2 each of the easier problems you already have down pat to solidify them even more. But if you’re running out of time, don’t waste it!
Ask your professor about the exam. Know what topics are going to be on the exam, of course. If there seems to be a lot to memorize, you could ask your professor if certain topics are priorities or not. Some professors may not tell you, but it’s worth asking.
Don’t rewrite your notes. Unless your notes are so bad that they’re not usable and you can’t do without them, avoid it. There’s really no educational benefit to rewriting notes for a practice-heavy class. Do it right the first time. And remember that notes are not really meant to be perfect anyway, despite the impression studyspo might give you.
Know when to stop. Cramming within the hour or two before your exam is not going to help, and you might start overthinking things. Give your brain some time to rest and consolidate all that you studied so you can more clearly think through your exam. If you put a lot of work into studying, you will probably do fine. There’s literally no functional purpose to worrying about it at that point, so try to relax and do your best.
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This might sound like a lot, but trust me on this, I am only including all of this because I truly think it is helpful and an efficient way to study. While it might seem like some of these methods would take a lot of time (the notebook, in particular), spending that extra time up front will actually save you a lot of frustration later on. If you manage your time well and are diligent and conscientious in your work, you can still succeed in math classes you thought were impossible to get through.
Good luck with your studies!