About studying mathematics in general, and Calculus in particular
So how should you study calculus? It doesn’t work the same way for everyone, but here’s a suggested pattern.
Step one. On your own, read through one section of a chapter. Each section introduces concepts, often through formal definitions, has theorems with proofs, and has worked out examples illustrating the definitions and theorems. Have a notepad with you so you can follow through the examples and proofs. When you get to an example, read and understand the statement at the beginning of the example. An example often has a question or two at the beginning to be answered. Then follow through the exposition of the example. For easier examples it’s probably just enough to read and understand them. But for others you’ll want to use your notepad to write down algebraic equations and do missing intermediate steps in order to understand the example better. A typical section has a dozen or half a dozen examples, starting with easier examples and working up to complicated examples.
We have theorems throughout our course. A theorem is a mathematical statement that can be justified with a logical proof. Probably most of the mathematics you’ve seen before coming to college was presented to you as fact with little or no justification. College mathematics is different—a logical justification is required before any mathematics can be accepted. You don’t just accept a statement on faith, or on the authority of a book or instructor, but because you can prove it yourself. Some of the proofs in our course are “formal”, that is, fairly complete, self-contained, logical justifications of the statements, but many of our proofs are only outlines. Sometimes an abbreviated proof is easier to comprehend, then the details fall into place. Remember, the proofs answer the question “why” the theorem is true.
When you come to a theorem, you’ll see first the statement of the theorem. You’ll nearly always be able to understand the statement of the theorem without understanding the proof. In other words, you’ll know what it means even if you don’t know why it’s true.
That’s the end of step one: read the section, work out the examples and understand the meaning of the theorems. Save any questions you have for step 2.
Step two. Attend the class meeting on the section. You’ll see the concepts explained again, but probably in different words. Only a couple of examples will be presented, and probably different ones, but the proofs will be presented in detail and discussed in class. Ask questions in class.
Step three. Do the homework assignment. Most of the problems on the homework assignment for the section are similar to the examples in the section. Use them as guides.
You’ll find “answers” to the odd problems at the end of the book. These are not complete answers, but just the final line of the answer so you can check to see if you got it right. You’re answer should be complete. (More about that below.) The course software for on-line exercises checks your answer and does more. After you’ve worked out your solution to the question on paper (or in your head if it’s particularly easy), enter your final answer. For practice exercises, if you don’t have the right final answer, then you may get some help in finding the right answer.
For some of the homework exercises you’ll write written answers rather than using the software to check your final answer. Except for the easiest problems, you should work out the problem on scratch paper before writing it on your answer sheet. When you do write your answer sheet, copy the statement of the problem and any given diagram. Then, without cramming in the answer, write it clearly.
There should be as much detail in your written answers as you see in the exposition of the problems in the section. It’s true that some of the problems are simple computation, and for those it’s enough to present the computation. But many of the problems require more than simple computation. Look at the exercises and you see that almost every equation is preceded by a few words explaining what the equation is doing there. There are loads of logical connectives—since, therefore, but, thus we have, substituting (some expression for a variable) we find—in the examples, and you should include them in your answers, too. Pepper your answers with words so that the reader knows why what you claim is the answer actually is the answer. Frequently, you’ll need whole sentences to explain what you’re doing. It’s better to include too much than too little.
Incidentally, staple the pages of your homework together before handing them in.
Getting help and working together
Always do as much of the homework assignment as you can first by yourself. There will be tutors/teaching assistants available to help you as you need it. You may also work together with others in study groups, but please don’t consult other students until you’ve tried the problems yourself first. If you get help from others, or give help to others, follow the following principles:
- Your goal is to learn, not to get answers. That’s the entire goal of the homework assignment. The best way, and perhaps the only way, to learn mathematics is by doing it. If you don’t do the assignments, or if you get someone else to do your homework for you, you won’t learn.
- Try to understand the principles. The particular problem you’re working on is of no importance in and of itself; it only helps you to get the concepts. If you get, or give, help on a problem, try to understand the concepts behind the problem so that you can apply them on other problems and in other situations.
- Help others find the way by themselves. You won’t help just by giving an answer, but you will if you can lead someone to the answer. It takes longer, but it’s worth while. Rather than giving the next step to solve the problem, explain what’s to be done and why. Even better is to ask what the goal is and how to get there.
- You can actually learn by teaching others. The best way to learn something is to teach it. When you explain the concepts, the “why” of something, then you’ll understand it better yourself. Formulating an explanation helps set it in your mind. Sometimes, even, you’ll find that even though you know how to solve a problem, when you try to explain it you might find that you don’t know why your method works, and that’s an important discovery. You’ll have a deeper understanding, and a more long-term understanding, when you know why, and you can explain why, something works.
How much time should this all take?
Don’t skip step 1 where you read the text before coming to class and doing the homework. It will actually save you time. Concentrate only on the parts that are new to you or you’ve had difficulty with before. It should come to less than an hour for each class, even less at the beginning of the course. Step 3, the homework assignment, should take about two hours for each class. Altogether, that’s about three hours per class.
Clark University
