Several students have come to me asking for advice on studying the “conceptual” side of calculus. Many of you have had previous calculus courses that emphasized the computational, and my philosophy is to emphasize the conceptual a little bit more. The textbook and its homework problems are very computational, and computational questions are a dime-a-dozen: you can just invent loads of them and do them one after another after another (and that can be fun, but that’s not all of mathematics). Studying for conceptual understanding is a much more difficult process. There aren’t an easily infinite supply of questions, although I try to give you as many as I can. Here are some tips:
1) Read Nykamp’s conceptual readings as well as your textbook. But don’t just read them, work through them. Be active in your reading. Read a sentence, then ask yourself, why is that true? If it describes something, draw a picture. If it defines something, come up with an example. If it gives a logical step of reasoning, tell yourself why that step is valid. If it references a theorem, state the theorem to yourself. Can’t remember it? Try to figure out from context what it might say, then look it up. If it says “if A then B,” ask yourself, “is it also true that `if B then A’?” If it says “whenever A is true, then B is true,” find an example where A is false and B is false. Whenever it talks about anything, come up with your own examples, examples, examples. The art of doing this — coming up with your own questions and examples — is just that: an art. You’ll slowly get better at it as you practice. But it provides an infinite source of conceptual practice with math, right there in your readings. Finally, when you are done reading a section, close the book and explain what you just learned to your mother. I’m serious: call her up and tell her your newfound knowledge. Or your roommate. Or your hamster. Or just your diary. But explain it (out loud: moving your mouth or your pencil, not just in your inner voice) and you will discover, through explaining, where your hidden misunderstandings lie. If your listener is animate, they can ask you questions and that’s excellent too. I know it sounds outlandish but this is a tremendous way to learn! (Many mathematicians, myself included, will write a book about whatever they’re learning, just to exercise the explaining neurons: the book doesn’t necessarily come out that great, but it’s a log of all the thoughts and ideas that have gone into the head. The more times stuff passes in and out and swirls around in there, the better.)
2) You can also make more out of your computational homework. When you sit down to do your homework, don’t review first. Just dive into the problems. Give yourself only as much help as you need. If you start a problem and think, “hmm, I know we did an example like this in class, but I don’t remember how it goes” then instead of looking it up, try to figure out how it should go. Pretend you’re the first person ever to face this question and you have to invent the mathematics to do it. Forgotten the formula for something? Don’t look it up! Forgetting it is a golden opportunity: now you can re-derive it like a newborn, and that’s a very valuable learning experience. If you find you can’t invent the method for solving the problem or come up with the formula, after giving it a real honest try, then go get a hint, as tiny a hint as you can. Maybe read the first few sentences of a similar problem in your notes or text. Then try again with the books and notes closed. Proceed in this way, giving yourself little hints, little bits of help, just a tiny bit at a time. Your goal is to solve the problem having used the minimum of help from outside your head (its kind of like titrating hints). The learning happens in the process of struggling, not in succeeding. Also, a good indicator is if you find yourself saying “ooooohh! that’s how it has to work!” when you finally figure it out. If you just look up the method right off the bat, there’s no eureka moment, and less learning. (If you’ve been struggling with it, your brain starts to want the answer like a drug, and it gets very excited about it when it gets it; then it stores it better.) And one more thing: many of the textbook questions seem random, but they aren’t. After you do one of those “what the heck is the point of that?” kind of questions in Stewart, ponder a bit what the point is. Sometimes there really is one (not always). For example, this week 17.5 #32 has some real interest to it that you can learn about in the reading I suggested next to it on the homework list.
3) To sum up: you can take the amount of struggle as an indicator of how successful your studying has been. Good studying should feel like good exercise on the treadmill: if you set the level too easy, you can do the 30 minutes with less benefit. Similarly, if you use your textbook or your solutions manual as a crutch, you can do the 30 questions with less benefit. And if you just read the readings, instead of reading them actively, it’s like walking the mile instead of running it. Same distance, less muscle growth.
4) But finally, have fun with it! Math really is supposed to be fun. (So is exercise!) Puzzles are fun — crosswords, sudoku, brainteasers. Math is supposed to be like that. Approach it that way and when you get frustrated, take a break. Study long in advance of the midterm so you can approach it in a relaxed way, doing only as much at one time as you have the interest for. I know this can be hard in a fast-paced academic schedule, but do the best you can.
Good luck!
kg/s
, we are adding up little pieces “
“. The unit normal is unitless; it’s just a way of taking a component of 
, i.e. 
.
, so the zero obscures the dependence on 
. But the correct answer is actually 0 kg/s. (I’m glad you asked. Stupidly, #41 also has answer 0. Obviously the one who designed the questions didn’t think about this type of possible misunderstanding.)
or something.
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