## What Are We Teaching

In arithmetic we use and manipulate numbers to create concrete answers to specific numeric questions.

Algebra is the theory of arithmetic. The two main tools used to create this theory are abstraction and generalization. First we try to describe abstractly, without reference to specific numbers, the manipulations and processes that are used in arithmetic. The goal of such abstraction is to arrive at a general rule which will be applicable to many questions. Secondly we attempt to generalize the abstractions. The hope here is to arrive at an even higher level mathematical structure which may be useful in describing a wider array of questions.

Algebra is one small part of a much larger discipline called mathematics.

According to The North Central Regional Education Library: “Mathematics is a study of patterns and relationships; a science and a way of thinking; an art, characterized by order and internal consistency; a language, using carefully defined terms and symbols; and a tool.”^{(1)} Clearly this has very little to do with numbers and computation.

Dr. Robert H. Lewis, Professor of Mathematics at Fordham University correctly observes: “The great misconception about mathematics — and it stifles and thwarts more students than any other single thing — is the notion that mathematics is about formulas and cranking out computations. It is the unconsciously held delusion that mathematics is a set of rules and formulas that have been worked out by God knows who for God knows why, and the student’s duty is to memorize all this stuff.”^{(2)}

According to the National Council of Teachers of Mathematics, “One of the primary goals of mathematics education is to enhance students’ ability to reason deductively. The capability to think logically is needed in every discipline, and it is particularly important in mathematics.”^{(3)}

According to Illana Weintraub, “Algebra is a very unique discipline. It is very abstract. The abstract-ness of algebra causes the brain to think in totally new patterns.”^{(4)}

Smith, Eggen, and St. Andre voice a basic tenet of all mathematics when they state: “The characteristic thinking of the mathematician, however, is **deductive reasoning**, in which one uses logic to draw conclusions based on statements accepted as true.”^{(5)}

The purpose of early college level algebra courses is therefore to introduce the student to the use of abstraction, generalization, and deductive reasoning while exploring the patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions.

1. http://www.ncrel.org/sdrs/areas/issues/content/cntareas/math/ma3ques1.htm

2. http://www.fordham.edu/mathematics/whatmath.html

3. http://illuminations.nctm.org/LessonDetail.aspx?ID=L384

4. http://www.mathmedia.com/whystudal.html

5. Smith, Eggen, St. Andre. A Transition to Advanced Mathematics. Monterey: Brooks/Cole (1986). p.1.

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