So how should you study calculus? It doesn鈥檛 work the same way for everyone, but here鈥檚 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鈥檚 probably just enough to read and understand them. But for others you鈥檒l 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鈥檝e seen before coming to college was presented to you as fact with little or no justification. College mathematics is different鈥攁 logical justification is required before any mathematics can be accepted. You don鈥檛 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 鈥渇ormal鈥�, 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 鈥渨hy鈥� the theorem is true.
When you come to a theorem, you鈥檒l see first the statement of the theorem. You鈥檒l nearly always be able to understand the statement of the theorem without understanding the proof. In other words, you鈥檒l know what it means even if you don鈥檛 know why it鈥檚 true.
That鈥檚 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鈥檒l 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鈥檒l find 鈥渁nswers鈥� 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鈥檙e answer should be complete. (More about that below.) The course software for on-line exercises checks your answer and does more. After you鈥檝e worked out your solution to the question on paper (or in your head if it鈥檚 particularly easy), enter your final answer. For practice exercises, if you don鈥檛 have the right final answer, then you may get some help in finding the right answer.
For some of the homework exercises you鈥檒l 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鈥檚 true that some of the problems are simple computation, and for those it鈥檚 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鈥攕ince, therefore, but, thus we have, substituting (some expression for a variable) we find鈥攊n 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鈥檒l need whole sentences to explain what you鈥檙e doing. It鈥檚 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鈥檛 consult other students until you鈥檝e 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鈥檚 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鈥檛 do the assignments, or if you get someone else to do your homework for you, you won鈥檛 learn.
- Try to understand the principles. The particular problem you鈥檙e 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鈥檛 help just by giving an answer, but you will if you can lead someone to the answer. It takes longer, but it鈥檚 worth while. Rather than giving the next step to solve the problem, explain what鈥檚 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 鈥渨hy鈥� of something, then you鈥檒l understand it better yourself. Formulating an explanation helps set it in your mind. Sometimes, even, you鈥檒l find that even though you know how to solve a problem, when you try to explain it you might find that you don鈥檛 know why your method works, and that鈥檚 an important discovery. You鈥檒l 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鈥檛 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鈥檝e 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鈥檚 about three hours per class.
Clark University
