Mathematics is axiomatic. It begins with definitions and then builds on these using inductive arguments to see what properties can be deduced. This is not only true for Calculus or Algebra, but virtually all branches of Mathematics. Lectures (or textbooks) in mathematics begin with definitions, derive theories from these definitions, and often have exercises on the material to test one’s understanding. These guided introductions are great opportunities for the student to learn a subject without having to derive everything from scratch. However, listening (or reading) the theory is not enough to develop expertise since this approach is usually very passive. It is necessary to do mathematics actively if one hopes to develop expertise in mathematics.
Suppose one wants to learn how to play a sport (or learn a game). One way to get started is to watch other people play the game and learn from their actions. This can be very helpful in learning the rules and developing some basic strategies, but you can’t possibly expect to become very skilled at the sport (or game) without actually playing it. The more you play the game, the more you will develop skills that allow you to become better at this activity. This principle applies equally well to learning mathematics. The more exercises you do, the more you will develop expertise and a deeper understanding of the subject. This will take time, a lot of time, and it will not always be easy, but it’s in these struggles that you challenge yourself, test your knowledge, and can learn something very well. When you are stuck, you often deduce what you need to know in order to solve the problem. Maybe you will figure it out yourself, or maybe someone will help you, but you will probably understand it much better because you tried and thought about it for a while.
When you are a novice in a subject you might not have any idea where to begin in solving a given problem. After you learn the different techniques that are possible you might want to use all of them. It is through the experiential (active) process of doing mathematics that one develops expertise and intuition to know what techniques you should use in what situation. To paraphrase G.H. Hardy, a good chess player will think of a dozen or so moves, a great chess player will only think of a few. The more mathematical problems you do, and the more complex, the more of an expert you should become and the deeper your understanding becomes. Spending time trying and failing in these problems should enable you to realize what technique to use at what time.
In my many years of doing and teaching mathematics, I have not learned any way to avoid this process, I have only learned to do it faster. The more practice you get, the faster the process becomes when you learn something either in the subject, or in a different subject. You acquire adept problem solving skills that are useful for mathematics, but also extend to other areas as well.
One question that I am often asked is, “Why do I need to show my work? I have the right answer, that should be enough.” That all depends on what your goals are in doing the question. I do hope the goals are not just simply getting marks. Having the right answer is the bare minimum that you should ask of yourself. In addition, you should work to express your solution as clearly and concisely as possible, using words and/or equations where necessary. This requires thinking clearly through the logical argument and determining what are the essential bits in the problem and then expressing this on paper or orally, so that someone else can understand it. If you have the right answer, but you can’t convince other people that it is in fact correct, than what is the value of that answer?
Mathematics is based on a community of people each working to solve problems and sharing their results with others in the community. If someone proves something but can’t convince anyone else that this is correct, this does not add to our communal knowledge. In all of your mathematics courses you will have opportunities to answer problems and challenge yourself not only to get the right answer, but also give the best explanation that you can. If you ever end up marking someone else’s assignments or essays, you will soon learn to appreciate the value of a well-presented problem or article. As with understanding mathematics, learning to express your solutions clearly is something that takes much time and practice, but we do get better. After we have achieved some level of mastery, we appreciate the time spent to get there, even though it might not have always been fun along the way.
Francis Poulin is Associate Dean, Undergraduate for the Faculty of Mathematics and Computer Science at the University of Waterloo