It is my theory that the most important component to learning mathematics is homework. It would be nice if we could just attend lecture and learn everything we need to know about calculus, geometry or the meaning of life. But unfortunately for the overwhelming majority of us this is not the case. In order to learn and perform well we need practice and lots of it.
When we are younger we have an incentive to practice our skills. But a lot of the time college professors don't collect and grade our math homework. This encourages apathy in many students. They say things like "It doesn't count in my grade so why should I waste my time doing it?" If this is your opinion, keep reading. If not, read on anyway.
Homework gives us the opportunity to practice what we are learning in class. When we do homework we learn which skills we don't fully understand. We can also discover new ways of approaching a problem. Homework does not just aid our learning it also helps us study for those inevitable tests and quizzes which do determine our grades.
But to reap these benefits we have to do our homework correctly. By "correctly" I don't just mean recording the right answers I mean recording our procedures and thoughts as we doing the problems. When the final comes around it is unlikely that you remember why the answer to question 32 on page 67 is 2.78214645.
So you may be wondering what goes into a correct homework assignment. The short answer is work.
First you need to evaluate what your long-term goals for the assignment are. Some of mine are: to create a good study reference, to get practice with my new skills, to build a strong foundation for future math classes I will take. In order to meet those goals I know that I need to create a map of my thoughts as a work the problem. I know that what I write on the paper is going to have to explain why I am proceeding in the way that I am. So I know I have use clear language. Math isn't just numbers and symbols on a paper. Math should be mostly words. After all, numbers and symbols are only shorthand for words.
Solve 9x^3 + 15x^2 +3x = 0 for x.
9x^3 + 15x^2 +3x = 3x (3x^2 + 5x + 1). By factoring out the 3x it looks like I will be able to use the quadratic formula.
3x (3x^2 + 5x + 1) = 0 implies that 3x = 0 or 3x^2 + 5x +1 = 0.
If 3x = 0, then x = 0.
What if 3x^2 + 5x +1 = 0? Then I can use the quadratic formula to solve for x. The quadratic formula is x = (- b +- sqrt((b^2 - 4ac)))/2a, where a = 3, b = 5 and c = 1. So, x = (-5 - sqrt(5))/6, (-5 - sqrt(5))/6 or 0.
It is an excellent idea to narrate your homework, but not all your math work should be done in this manner. Several times a quarter you will be taking timed tests and quizzes. It's not always feasible to write down all your thoughts as you work through the test. It also makes your work harder for your professor and TA to grade (remember: grading is their homework).
So on a test or quiz it is fine (and many times better) to leave out much of your narration, but make sure you show your work. Sometimes it is not always clear why you go from one step to another, so in these cases it is important to include a brief sentence about why you are proceeding in that manner. You should use your best judgment to determine what needs to be written on a test or quiz. This may sound vague, but once you start writing homeworks the "correct" way you will learn to determine what is necessary and what isn't.
The best way to figure out what you should include on a test or quiz is to ask your professor. This simple task may seem daunting or scary but don't be afraid to do it. We live in a capitalist society. If you give your grader what he or she wants, you might get what you want … a better grade!
Department of Mathematics
University of California
Riverside, CA 92521
tiff@math dot ucr dot edu