It seems appropriate that a series of articles intended to help parents better understand Algebra's basic concepts should begin with a definition of Algebra or some kind of discussion of what Algebra is. What I discovered is that this is easier said than done.
Over the years I have taught out of several different textbook series both in high school and in college and I have evaluated many other Algebra texts, and one characteristic they share is the lack of an understandable explanation of what Algebra is or does. I asked Google the question "What is Algebra?" Result? Not one meaningful answer. I headed to the dictionary and found one response that includes some truth: A generalization of arithmetic in which letters representing numbers are combined according to the rules of arithmetic. So, do you understand what Algebra is now based on that? I didn't think so.
It is actually kind of funny (funny ha ha, funny odd, and funny sad) that there seems to be no agreed upon definition of Algebra or even a generalized understanding of what Algebra is, and yet every high school across this country offers a course in Algebra. So maybe the more appropriate question should be "What do these courses cover?" If all the courses cover the same topics, then at least part of the answer is there. I'll came back to this idea later.
I could just tell you my definition of Algebra, but I know that you will have a much better understanding of Algebra if you create the image in your own brain. You and I both know that "being told" does not lead to long-term retention. I am about to help you create in your mind two different images for understanding Algebra. Hopefully, at least one of these images with be the "light bulb" for you. Ideally, both images will stay with you and give you slightly different perspectives on the meaning of Algebra. Please be patient--I am headed to the definition of Algebra.
I tend to be a visual learner--at least, as I am learning new things, my brain puts this information into pictures rather than a list of details; although occasionally a detail catches my attention and the picture comes back. An example of this is the detail 3/8" (three eighths of an inch). Most likely, as you read that number, an image comes into your mind. Maybe on a ruler, or sewing a hem, or something completely different.
For me, at this moment in time, the number 3/8" is bringing into my head the picture of the new Hoover Dam ByPass Bridge. This bridge, which just opened this past October, 2010, is the culmination of many recommendations dating back to 1968, 28 different studies, the 9/11 attacks showing the Hoover Dam as a major target, years of research into earthquake concerns, years of construction starting Feb., 2005, a wind gust accident in 2006, and a construction fatality in 2008. But, in 2009, after starting concrete footers on opposite sides of the river 1060 feet apart, and working from both sides at the same time, the arch of the bridge slowly came together--almost. Amazingly, the two parts met with an error of only 3/8 of an inch! To fully understand just how amazing this is, understand that many manufacturing companies are thrilled with a 1% error. On this bridge, a 1% error would have been 10.6 inches. They considered being off by half an inch to be perfect, but they were even closer than that! Incredible. The bridge is now complete and is as much of an engineering marvel as is the Hoover Dam.
(When do we get to the Algebra part? Right now. Although weren't the statistics great? You just have to appreciate math even if you don't love it.)
As I look at pictures of this bridge, I see layers of mathematics. On top, I see Calculus in the issues of wind stresses, weight stresses, necessary strength, and earthquake stresses. Under that, I see Trigonometry in all of the angles and distances that couldn't actually be measured by hand. Under that, I see Geometry--so much Geometry! Angles, triangles, and curves everywhere you look. Not to mention 30,000 cubic yards of concrete (volume). So much Geometry! And supporting it all is Algebra. Algebra is the set of basic skills making all of the others possible.
The final picture I'd like you to put in your brain is a very long line--similar to a time line. On the extreme left end of the line, put a mark and label it "learning to count." Then, as you move to the right on the line, add marks and labels for "understanding numbers," "addition." subtraction," multiplication," "division," fractions," "decimals," and "percents." Now draw a very long bracket across the top of everything you just listed and add a label to the bracket that says ARITHMETIC. Now, continue to extend your mental line to the right. (The actual order of the topics will depend on the textbook used.) You will continue to mark and label with items like evaluating expressions, operating with expressions, solving equations, numbers systems, properties, linear equations, quadratic equations, graphing equations, and so on. When we have listed every skill needed for the next level of mathematics, we can then add that bracket with its label--ALGEBRA.
Hopefully you see that Algebra is not really a "thing" to be defined. Just as Arithmetic is a set of basic skills for the math courses that follow, so to is Algebra. And while the textbooks mentioned earlier seemed not to provide a written definition or understanding of Algebra, what they do share is the basic skills needed for all of the math to follow.
Whether it is the bridge image with its layers of mathematics or the time/number line with its brackets, I sincerely hope that at least one of these pictures stays with you with meaning that is helpful; because this really is very important. These subjects, from Arithmetic to Calculus make it possible to build roads and bridges, to divert water, to make electricity, to send missiles to Mars, to cure diseases...to do whatever we humans need or want.
