Really, there is no such thing as Algebra 1 and Algebra 2.
The word “legacy” comes from the Latin legare, meaning “to bequeath.” Of course, that generally brings to our minds an inheritance in the form of money or property. As parents, we are sure we will be bequeathing something to our children. Will it be of any more value than the material goods we have acquired? While it may be somewhat narrow in perspective, here is something to consider.
As you educate your students, can you say they are involved in concept development, or are they learning passively? Are they figuring things out for themselves, or are they learning tricks and shortcuts? Do they see the logic in what they are learning, or are they just memorizing information for a test? Are they analyzing their mistakes to find the reasons why they answered incorrectly, or are they just accepting their fate and recording a grade?
A legacy can mean many things, but helping our children learn to think may be one of the longest-lasting tools we can bequeath to our children. Of course, we need to carefully consider the educational materials we use to teach our children, and those materials need to be developed logically. Unfortunately, traditional mathematics instruction is often driven by programs that are developed topically instead of logically.
As we cover the traditional scope and sequence of algebra, I trust you will receive food for thought as you strive to leave an educational legacy to your child.
“Do two halves really make a whole?” seems like such a simple question, but is the answer that obvious? Not when it comes to high school algebra! And I’m not talking about some new way to add algebraic fractions. I’m referring to the age-old practice of teaching two years of algebra in high school, which presumably makes up a complete algebra course. They may have been called Algebra 1 and 2, or Beginning Algebra and Advanced Algebra. In either case, the implication was that each 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 50 percent of the book is a repeat of “first-year algebra.”
So really, there is no such thing 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 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, in fact, the study was not even completed in one year. So, for another math credit, geometry was taught for a year. It was considered “another discipline,” involving a significant amount of logical reasoning and proof, and it gave students “another math experience.”
That took care of the required credits.
Then, the next year, students interested in going further 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 simply didn’t 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 2 of an algebra book is “advanced” compared to Chapter 1, isn’t it?
This 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 often considered to be the “normal way” to teach algebra.
Therefore, I suggest 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, “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 first-degree relations (all of the exponents are “1”), and continues to develop by exploring other types of exponents. Included 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 the viewpoints of instruction and subject integrity, when you study geometry in the middle of an algebra course. The analogy may be somewhat over-simplified, but it is a little like beginning to learn English, and before reaching a reasonable level of mastery in the structure and syntax of the language, being introduced to a study of classic literature. They are just not ready for that yet.
Of course, all this 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.”
Thomas Clark is the president of VideoText Interactive and the author of Algebra: A Complete Course and Geometry: A Complete Course. His Convention workshops will go into greater detail concerning the effective teaching of mathematics. For more information, visit www.videotext.com.