How is college mathematics different from high school math?
In high school mathematics much of your time was spent learning algorithms and manipulative techniques which you were expected to be able to apply in certain well-defined situations. This limitation of material and expectations for your performance has probably led you to develop study habits which were appropriate for high school mathematics but may be insufficient for college mathematics. This can be a source of much frustration for you and for your instructors. My object in writing this essay is to help ease this frustration by describing some study strategies which may help you channel your abilities and energies in a productive direction.
The first major difference between high school mathematics and college mathematics is the amount of emphasis on what the student would call theory—the precise statement of definitions and theorems and the logical processes by which those theorems are established. To the mathematician this material, together with examples showing why the definitions chosen are the correct ones and how the theorems can be put to practical use, is the essence of mathematics. A course description using the term rigorous'' indicates that considerable care will be taken in the statement of definitions and theorems and that proofs will be given for the theorems rather than just plausibility arguments. If your approach is to go straight to the problems with only cursory reading of thetheory’’ this aspect of college math will cause difficulties for you.
The second difference between college mathematics and high school mathematics comes in the approach to technique and application problems. In high school you studied one technique at a time—a problem set or unit might deal, for instance, with solution of quadratic equations by factoring or by use of the quadratic formula, but it wouldn’t teach both and ask you to decide which was the better approach for particular problems. To be sure, you learn individual techniques well in this approach, but you are unlikely to learn how to attack a problem for which you are not told what technique to use or which is not exactly like other applications you have seen. College mathematics will offer many techniques which can be applied for a particular type of problem—individual problems may have many possible approaches, some of which work better than others. Part of the task of working such a problem lies in choosing the appropriate technique. This requires study habits which develop judgment as well as technical competence.
We will take up the problem of how to study mathematics by considering specific aspects individually. First we will consider definitions—first because they form the foundation for any part of mathematics and are essential for understanding theorems. Then we’ll take up theorems, lemmas, propositions, and corollaries and how to study the way the subject fits together. The subject of proofs, how to decipher them and why we need them, comes next. And finally, we will discuss development of judgment in problem solving. To contents
What should you do with a definition?
A definition in mathematics is a precise statement delineating and naming a concept by relating it to previously defined concepts or such undefined concepts as number'' orset.’’ Careful definitions are necessary so that we know exactly what we are talking about. Unfortunately, for many of the concepts in undergraduate mathematics the definition is rather difficult to understand, so often at low levels an intuitive feeling for the meaning of a term is all that is given or required. This intuitive feeling, while necessary, is not sufficient at the college level. This means that you need to grapple with and master the formal statement of definitions and their meanings. How do you do it?
Step 1. Make sure you understand what the definition says.
This sounds obvious, but it can cause some difficulties, particularly for definitions with complicated logical structure (like the definition of the limit of a function at a point in its domain). Definitions are not a good place to practice your speed reading. In general there are no wasted words or extraneous symbols in established definitions and the easily overlooked small words like and, or, if … then, for all, and there is are your clues to the logical structure of the definition. First determine what general class of things is being talked about: the definition of a polynomial describes a particular kind of algebraic expression; the definition of a continuous function specifies a kind of function; the definition of a basis for a vector space specifies a kind of set of vectors.
Next decipher the logical structure of the definition. What do you have to do to show that a member of your general class of things satisfies the definition: what do you have to do to show that an expression is a polynomial, or a function is continuous, or a set of vectors is a basis.
Step 2. Determine the scope of the definition with examples.
Most definitions have standard examples that go with them. While these are useful, they may lead you to expect that all examples look like the standard example. To understand a definition you should make up your own examples: find three examples that do satisfy the definition but which are as different as possible from each other; find two examples of items in the general class described by the definition which do not satisfy it. Prove that your five examples do what you think they do—such proofs are usually short, follow the structure of the definition quite closely, and help immensely in understanding the definition. These examples should be neatly written up so that you can refer to them later. Your own examples will have more meaning for you than mine or the book’s when it comes time to review. Step 3. Memorize the exact wording of the definition.
This step may sound petty, but the use of definitions demands knowledge of exactly what they say. For this reason you can count on being asked for the statement of any definition on an exam. The importance of precise wording should have been made clear by your examples in step 2 and it certainly is essential in the proof of theorems. Solid knowledge of definitions is more than a third of the battle. Time spent gaining such knowledge is not wasted. To contents