Although there are a number of important computational aspects to this course, it is still a concept and proof-based course.
As with any upper-level math course, you will be expected to explain what you are doing, not just perform computations, in the work that you hand in. You will be asked to prove theorems on your own, and perhaps even to comment on the proofs given in class.
Clear writing reflects clear thinking.
Clear writing reflects clear thinking, so your ability to explain your work carefully in writing is an important indicator of your level of understanding. Most people find that the process of organizing material well enough to express it clearly is a tremendous help in solidifying their own understanding of the topic.
This has the following consequences for this course:
You must know the definitions precisely. Typically, a quick restatement in your own words, while fine at an intuitive level, will lack the precision necessary for a correct use of the concept. You are responsible for all definitions from class, and you will be asked to reproduce these on exams and quizzes. For example, a quiz question might begin: "Give the precise definition of ..."
You are responsible for understanding the theorems and proofs from class; it is not sufficient simply to know how to apply them. Your statement of a theorem should include all the hypotheses that were present in the statement of the theorem in class. An exam question might begin: "State and prove the theorem about ...".
You must explain your work using words. Written explanation is a crucial part of the learning process, and it is not sufficient simply to write down a series of equations and circle a number or formula at the end. It is important that you be able to give clear and well-organized indications of what you are doing and why. I will try to provide examples of this as we go, and will put copies of the best student answers in the notebook outside my office.
Note that the goal is to explain why you are doing what you are doing, not what you are doing. Saying something like "I took the derivative" usually is insufficient; I can see that you took the derivative, the question is why? How do you know that the derivative is the right thing to use? What differentiation rules did you use?
It is important to realize that writing explanations, while it helps me to grade your work, is mostly for your benefit. It is the best way to make sure for yourself that you fully understand the material. If you do understand the material, it should not be a hardship to write a brief explanation of what you are doing. On the other hand, if you are not entirely sure about the process, trying to write an explanation of what you have done is one of the best ways to recognize that you are not completely clear on the subject. Even if you are right, the organization required in writing about what you have done will help you draw the connections necessary for a full understanding of the material. Don't look at writing as just another hoop to jump through; view at it as an integral part of the learning process. It's one of the things you can do far better than a computer.