Some Thoughts about College Algebra — Blog 1

The textbook I am currently using for College Algebra is pretty much the same as every other College Algebra Textbook. The colors used on the cover vary from book to book but that is hardly a contributing factor to the book’s effectiveness. The book is much too large, contains a lot of useless information, trains the student to solve certain classical textbook problems rather than teaching mathematics, and has far too many exercises. These textbooks give students the false impression that you learn mathematics by repeating dozens of insipid textbook problems rather than teaching that one learns mathematics in order to solve problems. Students are not given even a hint that the “detached language” nature of mathematics is what makes it a powerful problem solving tool in every discipline. Students leave College Algebra believing it to be a collection of processes and procedures of no real value. In view of our current teaching practices they might be right.
The purpose of early college level algebra courses is 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. Along the way students should increase their ability to use critical thinking to solve the variety of problems they will surely encounter during a lifetime. No College Algebra book currently on the market (to my knowledge) addresses those goals.
My intention is to provide alternative methods through a series of blogs. As a mathematician I feel strongly that I should present a series of essays that are so arranged as to provide a complete and linearly ordered argument. I have struggled unsuccessfully with that for years and will not attempt it here, rather the order in which topics are presented follow no rhyme or reason.
Traditionally we seem to always begin by discussing equations and inequalities in one variable and those topics are usually compartmentalized as :
Linear Equations Quadratic Equations
Rational Equations Equations with Square Roots
Linear Inequalities
The only thing we do in these isolated compartments is train students to perform a prescribed set of steps which solves the problem. There is very little emphasis on what a solution is. There is no distinction made between solution and solution set. There is no mention of graph. Linear equations and linear inequalities are not related to each other. Quadratic inequalities nor any other kind of inequality is mentioned.
We then move on to a discussion of the these same creatures (Linear Equations, Quadratic Equations, Rational Equations, Equations with Square Roots, Linear Inequalities) in two variables. There is no attempt to relate these compartments to the earlier single variable creatures. The concept of solving and solution set seems to disappear from the discussion and graph is introduced. A graph is never presented as a “picture” of the solution set of the equation. A graph is simply something a student “does”. Again each of these topics is presented in isolation of any of the other topics. By this time any thinking student will have concluded that algebra consists of a large collection of procedures one uses to solve math problems like the ones in the textbook.
The next step is to reconsider these two variable creatures (Linear Equations, Quadratic Equations, Rational Equations, Equations with Square Roots, Linear Inequalities) but this time we use the language of functions. We also extend our discussion to polynomial functions, rational function, log functions, exponential functions, and sequences (but we hardly ever treat them as functions). The astute student soon realizes that he can pass the course if he/she simply identifies the term function with the term equation. Again there are a bunch of isolated rules, procedures, and processes to memorize. In the end we have taught the student that algebra is a large collection of procedures one uses to pass tests but which have little value beyond that.
Four important properties that we all know and should use extensively in College Algebra to provide a more unified and comprehensive view of the beautiful structure of elementary algebra and to examine examples of exciting logic which will awaken even the most lethargic mind. They are:
1 The Distributive Property
2 The Law of Trichotomy
3 The Transitive Property of Equality
4 The Zero Factor Property
The Distributive Property links factoring (to write as a product) and multiplication. If the Distributive Property is understood to be the underlying principle for multiplying multi-term expressions, then multiplication becomes more uniform throughout mathematics. I will present further illustrations of the importance of The Distributive Property in future blogs.
The Law of Trichotomy should be used to link an equation to its two siblings (the inequalities formed by replacing the = symbol with the symbol). We should unify the study of these three mathematical objects by emphasizing the fact that every element of the domain of the equation is a solution to exactly one of the three and that the equation is the boundary between the two inequalities. The graph of the equation is always a boundary between the graph of the other two siblings. I will further illustrate this concept in future blogs.
The Transitive Property of Equality should be interpreted as: It two expressions represent the same quantity, the two expressions must be equal. This interpretation/rewording of the Transitive Property is correct and is easier for the student to understand and use. The model for every traditional word problem is produced by establishing two different expressions for the same quantity and then applying the Transitive Property. The process of finding the intersections of two graphs is explained by using the Transitive Property. More applications and illustrations will be presented in future blogs.
The Zero Factor Property is a property of the real number system (not of quadratic equations as presumed by many students). The fact that this property is used for many equations other than quadratic should be emphasized. It should also be emphasized that the Zero Factor Property produces two equations joined with the conjunction OR which are equivalent to the previous single equation. This approach helps unify the process of solving linear, quadratic, and some other equations. It should be pointed out that such unification decreases the number of processes for solving equations. The student should be shown that this process of solving quadratic equations is logically the same as was used for solving linear equations.
We should use the Zero Factor Property to teach that the process to solve an equation in one variable is to generate a sequence of equations each equivalent to the previous equation until simplest equation(s) is(are) obtained. On occasion (rational equations, equations with square roots) we must make slight modifications to this process, but we never loose sight of the fact that generating a sequence of ever simpler equivalent equations is our strategy for solving equations. It is my intent to elaborate on this point in future blogs.


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