Introduction. As your instructor, my job is to encourage you to succeed in all your endeavors related to the class you are taking. Everyone who receives a passing grade (admittedly a disputable measure of success) in my classes makes me proud, and makes me look good. Many students could probably use some advice for how to succeed, and that is why I wrote these notes. You may consider this as a set of my expectations of you if you wish to succeed in my class. Here's my first piece of advice: Read on.

Math vs. Fine Restaurants. Knowledge is power, and you are in college to acquire knowledge and therefore power. Knowledge is truly a premier commodity. Most people in the world are genuinely and often desperately starved for knowledge. Taking a college course in the United States of America is therefore one of the most luxurious things you will ever do. This is why college costs so much money. Your professor and his/her superiors are your servants in this regard. No matter how much you feel at odds with your professor when s/he dumps the homework on you or gives you a grade you didn't like, s/he is there to serve you with rich, high-quality knowledge. However, as much as your instructor may try to encourage you, it is your duty to engage in acquiring it. If you go to a fine gourmet restaurant with astronomical prices, you are the one who must eat all you can and enjoy every morsel. The same can be said for college. "You can lead a horse to water, ...."

Math vs. Phys-Ed. One may rightfully compare mathematics to a physical activity. There are many parallels. If you want to stay in shape, you must train every single day you can. You must always try to do a little better than you did before, but, on the other hand, you shouldn't try so hard as to burn yourself out. If you're training for a particular sport, there are always a number of different skills your body must acquire. The same can be said for learning mathematics. If you are aware of a weakness in your abilities, that is exactly what you should practice. Another cliche from phys-ed is "no pain, no gain". Learning mathematics can often be frustrating, unpleasant, painful, traumatic, and so on. I know because I've seen countless students cry after taking an exam. You can always learn more, however, and you will most likely recover from all your stumbling. Just like a physical activity, it often takes time to heal from such injuries, but eventually, if you have the right attitude, you will be able to get back on the field. Finally, if you accept this analogy, you should consider your instructor as a sort of coach, another type of servant.

Be Prepared. There are many things you can do to prepare yourself for each class meeting. A class is usually taught with a syllabus in mind, and usually the syllabus is available to the students. (In fact, your instructor is usually required to provide one.) The syllabus for a course is merely an outline of the topics to be covered throughout the term. Thus, before coming to class, you should have some idea of what the instructor will be discussing. You should review this material before coming to class, and do so every day you plan to attend. A class meeting is a type of meeting, and, as such, you should expect that your instructor or your peers might wish to speak with you about the subject at hand. You should bring your textbook, maybe your calculator, and any other materials you might need to every class meeting. Each day is different, so you should plan accordingly. If you know that the only thing you will do is take an exam, then you probably don't need your textbook. However, you should always bring something to write with. Another way to prepare for class is to attend every class meeting. Your instructor may offer suggestions for how to do some of the homework, or s/he may even tell you what will be on an upcoming quiz or exam. S/he may even tell you what not to study before the next meeting!

Homework, Homework, Homework! You will never learn everything in any class you take, so you really can't get enough homework. You should work through every homework problem which is assigned, and then some. You should read every problem carefully and repeatedly as you work through them. There is a rule of thumb which says that the harder problems in any assignment appear later in the assignment. The first several problems are usually routine computations which can be performed with a few lines of scratchwork. The harder problems naturally require more time. They occasionally may require some creativity and resourcefulness. Sometimes you may even have to look outside the textbook for help. In addition, you should try to be curious and inquisitive about the problems. Many problems are designed to be routine, but many others were written by professional mathematicians with specialties closely related to the material in your text. For these latter types of problems, there are often whole volumes devoted to different aspects of them, so you should have an open mind that there may be much more below the surface.

If you have any questions or doubts, ask. Your instructor is a type of servant. Although s/he may take great care in determining exactly what you need in order to learn the material, there will inevitably be gaps in his/her best efforts. Your instructor certainly can't do everything, but if you are aware of some piece of understanding which you need, then you must have the self-confidence to ask about it. Although you single yourself out in class by asking questions, most likely about a third of the class has the exact same problem as you. In fact, by asking questions, you may even gain the favor of your peers by pointing out what a lousy job your instructor has been doing. You have nothing to fear by asking questions, although you may feel stupid doing so. Your instructor already knows that you don't know the material very well. After all, you are in the class to learn it, and s/he is there to teach it.

Scratchwork vs. Homework Solutions. This is a delicate matter. Sometimes your instructor wants to see your scratchwork, but not usually. Your instructor wants you to write solutions which explain your scratchwork. So what's the difference between "scratchwork" and "homework solutions"? A perfectly good example is the technique of substitution for finding antiderivatives. When one performs a substitution, one usually declares u to be some function of x, so one writes

Exams. Whether you like it or not, hour-long and final exams are with us, and will likely be for generations to come. Often exams are worth a significant part of your grade. You should therefore determine your best strategies for preparing for exams. This really varies a lot for different people, although generally it is a bad idea to try to learn everything the night before an exam. If you attend every class meeting and work the homework, this is often still not enough to perform well on an exam. Perhaps you are the type to be nervous during an exam. Perhaps this is natural, but if you can find ways to reduce your anxiety, you should try them. Sometimes this is merely a problem of regulating the correct amount of caffeine in your diet. Another strategy for preparing for an exam is to have a good idea of what sorts of problems may appear on it. If you have a question during an exam, then you should ask. Perhaps a question is not worded clearly enough. If you ask, the instructor, while not giving the solution away, may be able to point you in the right direction. You should also have a reasonable idea about your instructor's expectations of what a "solved" problem looks like. Generally, it is a good idea to show all your work, and explain as much as possible. Describe an outline for how to solve a problem even if you can't remember how to work all the details. If you agree with the analogy that mathematics is like a physical activity, it follows that one of the things you should do before an exam is to spend about 30-60 minutes working through and solving problems immediately before taking the exam. That way you will be "warmed up" to what you will be expected to do. Alternatively, you can look through the exam and determine which ones are easier, and trust that these will warm you up to the others.

Graphite vs. Ink. If you are taking a college mathematics course, then you should know by now that mathematics can be very unforgiving for mistakes. This may be a matter of personal style, but I believe it is a good idea to work on mathematics with a pencil and an eraser, as opposed to an ink pen. We live in a marvelous age with designer mechanical pencils which never dull. Moreover, I don't mean the eraser on the other end of your pencil. I mean one of those big clunky erasers that form a noticeable bulge in your pocket. Sometimes, as you work through a problem, you may realize that you are completely off base, with totally bad assumptions, and you've written something tremendously ugly and wrong. With an eraser in hand, you can inspect your work, learn your mistakes, and erase from that moment onwards. Using ink, you must either cross out what you wrote or start again from scratch. Your decision.

Math vs. English Composition. Many people say that learning mathematics is like learning a whole new language. My perspective is a little different. I see mathematics merely as a supplement to whatever language you are using. The difference is that I don't consider mathematics to be a separate language which is communicated only with a random assembly of confusing notation. Mathematicians over the years have developed tremendously varied confusing notation for all sorts of things, but if you pick up any professional mathematics journal, you will notice that everything is still written using all the usual customs of writing in that language. Every sentence must have a subject and verb. You organize sentences to explain ideas into paragraphs. You assemble paragraphs into cogent arguments called essays or reports. This is true no matter what subject you study, and this includes mathematics. The only reason mathematicians have all those strange symbols is to make their writing more succinct. (For instance, no one in their right minds wants to write the long stringy "the first derivative of the function with respect to the independent variable" when one could simply write "df/dx".) The next time you open your textbook, take special care to notice punctuation. Generally, if the book is well-written, you will notice punctuation scattered amidst all the formulas. The author does this because s/he is forming complete sentences, and s/he wants to make sure the reader can feel the appropriate cadence as s/he reads. Perhaps you can tell by now, but I could probably write a whole essay on this subject alone. Here's a summary: You can always try to improve your writing skills, and you have ample opportunity in every math class you take.

Conclusion. Well, there they are, my ramblings about taking mathematics courses in college. I hope it helps. Maintain a positive attitude, never take anything for granted, never assume anything, and try to remember that when you succeed, everyone succeeds.