How to Study Mathematics
How to Study Mathematics
Lawrence Neff Stout,
Department of Mathematics,
Illinois Wesleyan University,
Bloomington, Il 61702-2900
This essay describes a number of strategies for studying college level<br>mathematics. It has sections entitled
How is college mathematics different?
What should you do with a definition
Theorems, Propositions, Lemmas, and Corollaries
Fitting the subject together
How to make sense of a proof
Developing technique
A few final suggestions
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How is college mathematics different from high<br>school math?
In high school mathematics much of your time was spent learning algorithms<br>and manipulative techniques which you were expected to be able to apply<br>in certain well-defined situations. This limitation of material and expectations<br>for your performance has probably led you to develop study habits which<br>were appropriate for high school mathematics but may be insufficient for<br>college mathematics. This can be a source of much frustration for you and<br>for your instructors. My object in writing this essay is to help ease this<br>frustration by describing some study strategies which may help you channel<br>your abilities and energies in a productive direction.<br>The first major difference between high school mathematics and college<br>mathematics is the amount of emphasis on what the student would call theory---the<br>precise statement of definitions and theorems and the logical processes<br>by which those theorems are established. To the mathematician this material,<br>together with examples showing why the definitions chosen are the correct<br>ones and how the theorems can be put to practical use, is the essence of<br>mathematics. A course description using the term ``rigorous'' indicates<br>that considerable care will be taken in the statement of definitions and<br>theorems and that proofs will be given for the theorems rather than just<br>plausibility arguments. If your approach is to go straight to the problems<br>with only cursory reading of the ``theory'' this aspect of college math<br>will cause difficulties for you.
The second difference between college mathematics and high school mathematics<br>comes in the approach to technique and application problems. In high school<br>you studied one technique at a time---a problem set or unit might deal,<br>for instance, with solution of quadratic equations by factoring or by use<br>of the quadratic formula, but it wouldn't teach both and ask you to decide<br>which was the better approach for particular problems. To be sure, you<br>learn individual techniques well in this approach, but you are unlikely<br>to learn how to attack a problem for which you are not told what technique<br>to use or which is not exactly like other applications you have seen. College<br>mathematics will offer many techniques which can be applied for a particular<br>type of problem---individual problems may have many possible approaches,<br>some of which work better than others. Part of the task of working such<br>a problem lies in choosing the appropriate technique. This requires study<br>habits which develop judgment as well as technical competence.
We will take up the problem of how to study mathematics by considering<br>specific aspects individually. First we will consider definitions---first<br>because they form the foundation for any part of mathematics and are essential<br>for understanding theorems. Then we'll take up theorems, lemmas, propositions,<br>and corollaries and how to study the way the subject fits together. The<br>subject of proofs, how to decipher them and why we need them, comes next.<br>And finally, we will discuss development of judgment in problem solving.
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What should you do with a definition?
A definition in mathematics is a precise statement delineating and naming<br>a concept by relating it to previously defined concepts or such undefined<br>concepts as ``number'' or ``set.'' Careful definitions are necessary so<br>that we know exactly what we are talking about. Unfortunately, for many<br>of the concepts in undergraduate mathematics the definition is rather difficult<br>to understand, so often at low levels an intuitive feeling for the meaning<br>of a term is all that is given or required. This intuitive feeling, while<br>necessary, is not sufficient at the college level. This means that you<br>need to grapple with and master the formal statement of definitions and<br>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<br>definitions with complicated logical structure (like the definition of<br>the limit of a function at a point in its domain). Definitions are not<br>a good place to practice your speed reading. In general there are no wasted<br>words or extraneous symbols in established definitions and the easily overlooked<br>small words like and, or, if ... then, for all, and there is<br>are your clues to the logical structure of the definition.<br>First determine what general class of...