Do Two Halves Really make a Whole?

By Thomas Clark

Such a simple question! Is the answer that obvious? Not when it comes to high school algebra! I am not talking about some new way to add algebraic fractions. I am referring to the age-old practice of teaching two years of algebra in high school, which, presumably, makes up a complete course in algebra. These courses may have been called Algebra 1 and 2, or they may have been called Beginning Algebra and Advanced Algebra. In either case, the implication was that each class comprised one-half of a complete algebra course. However, if you look at the table of contents in any “second-year algebra” book, you will find that at least fifty percent of the book is a repeat of “first-year algebra.”

So really, there are no such things as Algebra 1 and Algebra 2. These are courses (or names for courses) that came about as a result of school scheduling. Many years ago, when it was the norm to require only two high school math credits to graduate from high school, a study of algebra was a natural beginning credit. Of course, since it was generally taught “mechanically,” utilizing many formulas and rules, a lot of practice and repetition was involved, and the study was not completed in one year. So, for another math credit, geometry was taught for a year. It was considered another discipline, involved a significant amount of logical reasoning and proof, and gave students “another math experience.” That took care of the required credits.

Then, the next year, students interested in going farther in their study of mathematics were offered the opportunity to continue, and finish, their study of algebra. Of course, because of the “procedural” way it was taught initially, students did not remember much of that first year. So, they started over, re-studying many of the same things. This time, however, it was called Advanced Algebra. Something of a contradiction, don’t you think? In fact, the word “advanced” is a relative term anyway. Chapter Two of an algebra book is “advanced,” compared to Chapter One, is it not?

This practice has been perpetuated through the years primarily because of that traditional implementation. When you try to memorize rules, formulas, tricks, and shortcuts without really knowing why they work, it will take a lot of drill and review just to remember the material for a test. Yet, even today that approach is considered to be the “normal way” to teach algebra.

Therefore, I would suggest to you that one of the most fragmenting things we have done in mathematics education is to forcibly insert a geometry course into the middle of an algebra course. Algebra is a single course, a complete course, divided only by concept areas. It is the study of relations (equations and inequalities), and it develops by degrees (as defined by the exponents). It begins, very logically, with a study of relations (all of the exponents are “1”) and continues to develop by exploring other types of exponents. Included in the study of algebra are higher-order relations (with integer exponents), rational-degree relations (with fractions as exponents), and literal-degree relations (when the exponents are variables, or “letters”). As such, algebra is the basic language of all upper-level mathematics courses, including geometry. Not only is geometry not a prerequisite for Advanced Algebra (whatever that is supposed to be), but you really need a good understanding of algebra, as a complete course, before you can fully understand a complete geometry course. That means there is a disadvantage, from an instructional point of view and from the viewpoint of subject integrity, when you study geometry in the middle of an algebra course. The analogy may be somewhat oversimplified, but it is a little like someone beginning to learn English, and before they reach a reasonable level of mastery in the structure and syntax of the language, we introduce them to a study of classical literature. They are just not ready for that yet.

Of course, all of this discussion would be irrelevant if algebra were taught analytically, without dependence on rules and shortcuts. If students were taught the why of algebraic principles, less repetition and practice would be necessary, and algebra could be studied in one school year. Then the two “halves” would truly “make a whole.”