Many students enter their college math courses with the idea that getting "the right answer" is what mathematics is about. This is not surprising, since that's how many high-school courses approach mathematics. Most students have been successful in math classes because they can memorize formulas and procedures for computing answers to specific types of problems. You have probably done this yourself. For example, you probably know the quadratic equation can be used to solve an equality like
x2 + 3x − 2 = 0, though most students don't know why this is true.
Certainly this is an important fact to know, but knowing it only allows you to solve one kind of problem. The quadratic formula would be of no use to you if you wanted to solve the equation
x3 + 3x − 2 = 0or the equation x4 + 3x2 − 2 = 0. (Actually, it can be used to solve the second of these; can you see how?)
Mathematics is not really about how to compute answers to problems that other people have already solved.
Mathematics is about finding ways to use information you already know to solve problems that no one has told you how to solve before. It involves combining things you know how to do in order to do things you don't know how to do.
Mathematics is about finding ways to use information you already know to solve problems that no one has told you how to solve before.
If all you can do is solve the problems that other people have told you how to solve, you are not going to be valuable to an employer. For example, if what you get out of the section of this course on differentiation is only how to compute derivatives, then you have missed the main point of the section. An employer who wants to compute derivatives will buy a computer, not hire you. The computer will always beat you at that game: it is faster, more accurate, and more reliable than you or I will ever be. Anything for which we could write down a procedure in class, someone can program a computer to do, and you will not be successful if that's the skill you use to promote yourself to an employer.
What you have to do is realize that your value comes from knowing what it is that you are computing, and what it is that needs to be computed. The computer is very bad at this; it is not adaptable or insightful. That is where you can beat it, and that's why you need to understand the meanings (the definitions and the theorems and the proofs) of the things we study. You need to practice them, not because you will be called on to compute them, but because you need to know how they work, how they interact with one another, what they mean, when to use them, and when not to.
The thing that separates you from the computer is that you can take a situation that you have never seen before, and can adapt the knowledge you do have to that situation, perhaps breaking it into parts that you can solve, perhaps rewriting it into another form, or perhaps developing a new idea that is analogous to one you already know. It is that flexibility that makes you valuable to an employer. It is what makes you a leader rather than a follower.
In order to become adept at this, you need to practice those problem-solving techniques. You need to learn how to determine for yourself what the important features of a problem are, and how to isolate the crucial information and relationships.
This involves learning how to ask the right questions about a problem. In academics, you are judged not by the answers you can produce, but by the questions you can ask.
You are judged not by the answers you can produce, but by the questions you can ask.
Learning to ask the right questions is not easy, and it does not come quickly, but it is part of the mission of this course to provide you with some of that experience. To that end, some of the problems assigned will not be things we have already solved in class. You will not be able to just look at a problem and recognize it as an example of what we did in class (or what is in the book). You will need to find your own method of solution, using the tools we have developed in class. These are the most important problems in the course, and while they can be frustrating, they can also be the most rewarding. Don't give up, you can do it!