How to Study Mathematics

This essay describes a number of strategies for studying college level mathematics. It has sections entitled

How is college mathematics different from high school math?

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 the ``theory'' 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.

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What should you do with a definition?

Step 1. Make sure you understand what the definition says.

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.

Step 3. Memorize the exact wording of the definition.

Solid knowledge of definitions is more than a third of the battle. Time spent gaining such knowledge is not wasted.

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Theorems, Propositions, Lemmas, and Corollaries

The relative importance and the intended use of statements which are then proved is hinted at by the names they are given. Theorems are usually important results which show how to make concepts solve problems or give major insights into the workings of the subject. They often have involved and deep proofs. Propositions give smaller results, often relating different definitions to each other or giving alternate forms of the definition. Proofs of propositions are usually less complex than the proofs of theorems. Lemmas are technical results used in the proofs of theorems. Often it is found that the same trick is used several times in one proof or in the proof of several theorems. When this happens the trick is isolated in a lemma so that its proof will not have to be repeated every time it is used. This often makes the proofs of theorems shorter and, one hopes, more lucid. Corollaries are immediate consequences of theorems either giving special cases or highlighting the interest and importance of the theorem. If the author or instructor has been careful (not all authors and instructors are) with the use of these labels, they will help you figure out what is important in the subject.

The steps to understanding and mastering a theorem follow the same lines as the steps to understanding a definition.

Step 1. Make sure you understand what the theorem says.

Next you need to understand the logical structure of the theorem: what are the hypotheses and what are the conclusions? If you have several hypotheses, must they all be satisfied (that is, do they have an and between them) or will it suffice to have only some of them (an or between them)? In most cases theorems require that all of their hypotheses be satisfied. A theorem tells you nothing about a situation which does not satisfy the hypothesis. The hypothesis tells you what you must show in order for the theorem to apply to a particular case. The conclusions tell you what the theorem tells you about each case.

Step 2. Determine how the theorem is used.

Step 3. Find out what the hypotheses are doing there.

In some cases a hypothesis is included just because it makes an otherwise complicated proof easy. This means that you may not be able to find examples which illustrate that each hypothesis is essential.

Step 4. Memorize the statement of the theorem.

Fitting the subject together

Step 1. Working backwards

Mean Value Theorem Rolle's Theorem Candidate Lemma Meaning of the sign of the derivative Definition of derivative Definition of max and min Existence of max and min for continuous functions on [a, b] Definition of max and min Definition of closed interval Least upper bound axiom Definition of continuity

Step 2. Make a definition-theorem outline.

How to make sense of a proof

Step 1. Make sure you know what the theorem says.

Step 2. Make a general outline of the proof.

Most theorems have the form of implications: if the hypotheses are true, then the conclusion follows. The easiest structure for a proof to use is to assume the hypotheses and combine them, using previous results, to reach the conclusion through a chain of implications. Some proofs use other strategies: contrapositive argument, reductio ad absurdum, mathematical induction, perhaps even Zorn's lemma (a form of the axiom of choice). The more complicated kinds of proofs will need to be discussed in class.

Step 3. Fill in all of the details.

Why bother with proofs at all?

But many of you will say ``I'm not a math major; I want applications so that I can use tools from mathematics in my field'' or ``I'm just taking this course because it's a requirement in my major and I sort of liked math in high school.'' Why should you learn about proof?

The applications you meet in other fields are not likely to look exactly like the math textbook applications, which are chosen for their appeal to a traditional audience (largely engineers) and for their representative character. Other applications work similarly, though not exactly the same way. This means that you need to learn how to apply the concepts in your math courses to situations not discussed in those courses. (There is no way that a course could discuss every possible known application: about 500 papers appear every two weeks with applications, and those are just the applications published in the ``mathematical'' literature!) To do so you need the best possible understanding of the mathematics you want to apply. Certainly this means that you need to know the hypotheses of theorems so that you don't apply them where they won't work. It is helpful to know the proof so that you can see how to circumvent the failed hypothesis if necessary. One of the major pitfalls of applied mathematics, particularly as practiced by nonmathematicians, is the danger of conveniently overlooking the assumptions of a mathematical model. (Mathematicians trying to do applied mathematics are more likely to fall into the trap of making models which have no relationship to reality.)

Many applications consist of recognizing the definition of a mathematical concept phrased in the terms of another discipline---the more familiar you are with the definition, the more likely you are to be able to recognize the disguised version elsewhere. The nuances of definitions are made most clear in the proofs of propositions relating definitions and pointing out unexpected equivalent variants, some of which may look more like a situation in another discipline than the precise form used in your math class.

Arguments for theory as an aid to application rest on an obvious premise: it is much easier to apply something you understand thoroughly. This is, however, a better argument for care in learning the statements of theorems than it is for spending time understanding proofs. The best justification for the inclusion of proof in math classes is more philosophical:

Proof is the ultimate test of validity in mathematics.

Once one accepts the logical processes involved in a proof no further observation or change in fashion will change the validity of a mathematical result. No other discipline has such an immutable criterion for validity.

The major benefit derived from an education is the ability to think clearly and make considered judgment. Each discipline should teach a body of material, appropriate modes of thought in dealing with that material, and a means for determining the validity of the conclusions reached. A chemistry curriculum with no lab work would be seriously deficient since experiment is the test of validity in science. Similarly mathematics without proof is severely deficient, indeed it is not mathematics.

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Developing technique

Step 1. Read through the theorems and examples.

Step 2. Work enough problems to master the technique.

Step 3. Work a few problems in as many different ways as possible.

Step 4. Make yourself a set of randomly chosen problems.

A few final suggestions

Few students write fast enough to get complete and readable notes in class. For this reason it is useful to go back over your class notes shortly after each class and make a complete, clean copy with all of the definitions and theorems clearly stated. This practice will also help you identify parts you don't understand so you can ask your professor about them in a timely fashion.

Do not let yourself fall behind. Mathematics requires precision, habits of clear thought, and practice. Cramming for an exam will not only fail to produce the desired result on the exam, it will also reinforce a bad habit---that of trying to do mathematics by memorization rather than understanding. A good night's sleep and a clear head will serve you better than last minute memorization.

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lstout@sun.iwu.edu