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 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.