Some Words of Advice

Preface: Let me mention something before you read this. This is not a list of laws that you must abide by. The title means what it says, this page contains ADVICE -- meaning: IF you want to do well in a class, IF you want to get the most out of it, IF you want to really learn the material and, ultimately, IF you want to get a good grade (unfortunately what ends up being important to many), THEN the following items may be useful to listen to. I am not trying to tell you how to live your life. I am not trying to lecture or reprimand you. I am not telling you how to set your priorities. I would support you in whatever you feel is most important to you, even if it ends up not being your math class. But IF AT SOME POINT, you feel that it is in your interest to do well in math, then I offer to you the following advice which may make your life easier. My best wishes to you in your math endeavors, and in all others. -GK

1. (NEW!) THE IMPORTANCE OF CLASS TIME. You should think of class time as extremely valuable time. Every minute you have in class with a teacher is a valuable resource that you can't get anywhere else. I understand that not everything that is said in class is new to you, and maybe not all of it is helpful. But what may happen is that at some point during a lecture, something may be said that makes something click inside you, and explains a part of the material in a way that suddenly makes it very clear. You just don't know exactly when that moment may be, so it is VERY important that you are attentive the whole time, waiting for these moments. If you don't treat the class time this way, and are not paying full attention, you may miss them, and there is really no way to get them back. To me, those moments are the whole point of going to lectures. You can always read the material in a book. The purpose of having someone explain it to you in their own way is to try to get a different perspective and to try to make the material "click" inside you. I think you need to be relaxed enough in lecture to allow the most subtle points to penetrate your subconscious (to make you go, "oh!") but attentive enough that you really "hear" everything. (Also, at the risk of sounding pretentious, I will say: ESPECIALLY in math. There is a lot of material and pretty much all of it is tricky.)

2. Possibly the biggest source of frustration for the young, or untrained, math student, is the incorrect preconception of what math is. There seems to be, for example, the idea that all of math can be concentrated into a set of algorithms, meaning that you are told the steps to follow, and that given the right set of steps (memorized to the necessary degree), those steps will lead you to the solution. This is not completely true! There are SOME cases in which it is true. But the truer broad statement would be that math is an application of your intellect. No one can tell you a set of steps for how to apply or sharpen your intellect, and so no one can give you a set of steps that will cover every math "problem".

   In this sense, Math is not so different from any other subject. Suppose, for example, that you were asked to write a term paper for History, or Anthropology. If you asked your teacher, "How am I supposed to make the connections between this aspect of the American Revolution and current events in Southeast Asia?", or, "How can I compare this western-African tribe's customs to those of indigenous tribes of Balinesia?", they would tell you, "Go home and think about it! Try to come up with something."

   I am sorry to break the news, but the same is true in mathematics. Sometimes you just have to look to your intellect, relax your mind, try to make connections and remember everything that you know, and come up with something.

3. Think of math as exercise. Math is for the brain what push-ups, lifting weights, running or other forms of strenuous exercise is for the body. It hurts. It's difficult. It doesn't necessarily feel like the natural thing to do, at least in the beginning. But the more you do it, your brain gets stronger, and the easier it gets.

   The thing that's weird about it is that you have to completely focus your mind and cleanse it of all the stuff that ordinarily wanders in. You need pure, direct, concise thoughts that you are absolutely certain are 100% logically sound and TRUE. This is what math is all about.

4. Find a QUIET PLACE where you can THINK! The best way to get to this level of clarity is first get rid of all the _concrete_ stuff that is around, e.g. noise, people, any other distractions. Then you can concentrate on getting rid of those extraneous thoughts that cloud your mind, the "intangible stuff". That's the hard part. (One good place is a LIBRARY... An example of a bad place may be the hallway of your dorm building...)

5. Ask questions. Once you've had some time to think clearly on your own, when you get to a point where you just can't figure it out (which you will), _then_, at that point, you need to come to ask your _well_thought-out_and_organized_ questions to someone who knows the stuff -- for example, me. (You CAN, and it might be a good idea to, ask your classmates your questions as well, but (a) this should absolutely NOT replace the time you spend thinking about it quietly on your own, and (b) you should realize that your classmates may also have a certain amount of confusion as well -- don't confuse each other! Take what each other say as a _suggestion_ and then return to your quiet place and try to criticize it further and poke holes in it. The same applies to what I say, but there is MUCH less chance that you will find anything wrong with it! But when you start to question everything, you will find out that you gain a much better understanding which brings us to number )

6. BE SKEPTICAL! A healthy level of skepticism is CRUCIAL in math! You must never quietly accept anything, or you will only ever half understand it. However, you don't necessarily have to take up class time to ask every question that hits you -- you should think about your questions before you ask them, that way we don't waste valuable class-time. (Of course, if it is about a calculation that doesn't make sense, ask it -- I may have made a mistake!) But whenever something doesn't make sense to you, don't leave it until someone, either you (the best way), someone else or I make it clear to you.

   WHEN YOU ANSWER YOUR OWN QUESTION, YOU REALLY LEARN SOMETHING

   -- try to do it! Whenever you have a question about something, spend time on your own first before you ask anyone. Struggling is the way to learn. Which brings us to

7. Struggling is the way to learn. Don't EVER think that you are wasting time when you are struggling to understand something. That is the BEST way to learn and remember it. The more you struggle on your own, the better. Giving up early and looking to someone or someplace else for "the answer" without first giving it any thought yourself is the best way _not_ to learn it. However, struggling with one thing _forever_ IS a waste of time -- after a healthy period of meditating, give up your pride and ask someone!

8. Use all of your resources. In particular, read your textbook. Also: talk to other people who are willing to give you their time and who may be able to help you. Your classmates, your teachers... these are also resources that are available to you.

9. Be humble. When you realize that math is something to be reckoned with, when you seriously acknowledge your limitations and you respect the opinions and ideas of those trying to teach the material to you, and try to take what they say and what the books are saying as gospel, you will do much better in the end. If you constantly fight it, or if you are proud, you will not learn.

10. "Close only works in horseshoes and hand-grenades." In math, the truth is all that matters. Not something that's "close to the truth". (Special thanks to my high-school geometry and trig teacher for that one.) -- WELL, in the end it's all that matters. Along the way, a vague understanding, more precisely an _intuition_ about the ideas is actually _very_ important and valuable. So maybe I'll change this one at some point...

11. Don't give up!

12. Try not to fall behind in class. If you do fall behind, "don't give up" will become extremely important. I recommend that as SOON as you feel you have fallen behind, you schedule an appointment with your teacher to help you get back up to speed. This may be very difficult to accomplish on your own, but I'm SURE that any self-respecting teacher would be more than willing to help you do this.

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