Math.221.005 Calculus 1 Fall 2012
Course Objectives

Loosely speaking, the goals of the course are as follows:
Admittedly, parts of this list are vague.  Paradoxically, in order to understand a more specific set of objectives, you would already need to know calculus.  The objectives would naturally mention new concepts and use terminology that you will learn in the course, but which would not make any sense to you now (unless you have already studied calculus).

To make an analogy, imagine that you have completed a study of the arithmetic of whole numbers -- that is, addition, subtraction, multiplication, and division of integers greater than or equal to zero.  Next you will take a course on the arithmetic of rational numbers.  The goals of instruction would mention concepts like fractions, negative numbers, reciprocals.  But until you have studied rational numbers, these terms have no meaning, or what is more confusing, a meaning in everyday usage that is quite different from the intended meaning in the course. 

In a sense, a math course is similar to a course in a foreign language.  A fundamental goal is to master enough of the vocabulary and grammar and idiom to carry on a conversation.   You might say that every mathematical subject corresponds to a different language, say Algebraish for algebra, Trigish in trigonometry, and Calculish for calculus.  Of course, these aren't really separate languages.  Each mathematical language is actually a specialized part of your normal language.  That can give you a misleading impression.  You can listen to me conversing in Calculish and think you are understanding it, because you understand the regular English parts of it.  However, true understanding only occurs when we can communicate our ideas to each other, and that is impossible if my ideas concern concepts you have not yet learned.  From this perspective, the point of studying calculus is to learn a new dialect of English, namely Calculish, and to learn it well enough to understand spoken and written conversation.  In a word, your goal is fluency.

Many students are required to take calculus as they pursue majors in other disciplines, like biology, or physics, or economics.  If you are one of these students, fluency in Calculish is probably the most important goal of the course.   That is because there are important applications of calculus in your major.  Your completion of calculus 1 is not necessarily so that you can actively apply it in your future work, or even so you can make direct use of results others have obtained using calculus.  Rather, it is so you can participate meaningfully in the conversation.

This is a key observation about the objectives of the course.  Learning the methods and memorizing the definitions and being able to solve homework problems accomplishes little or nothing if you do not gain fluency in the process.   That is why the first objective listed above concerns conceptual understanding, and why writing coherently and correctly are also included.

The last objective on the list is probably the vaguest of them all.  It reflects the fact that mathematical knowledge is developed in a highly stylized form.  Think of it as a combination of grammatical structure and idiomatic usage that underlies all mathematical language.  True fluency demands an understanding of the grammar and idiom of the language.  And this is especially true for mathematics, because its structure is so different from normal conversational language.  Over the course of the semester, you will see how this structure appears in the specific context of calculus.  The better you understand the structure, the greater your ability to converse fluently in Calculish.  I refer to these as general characteristics of the mathematical method because they pervade all of mathematics.  Developing an understanding of this structure of mathematical knowledge is an important general education goal, because it gives you access to a broad literature of mathematical analyses and methods.