What is algebra? To most of us, it’s that x and y stuff that we suffered through in high school. To the person who studies algebra on the college or grad school level–people like me–algebra is a much more abstract topic, a study of sets endowed with certain operations, entities that come with names like “group,” “ring,” and “field.” And to the most brilliant mathematicians, according to one of my professors, algebra means a set of overarching principles that govern the universe. So algebra obviously has different meanings to different people. And yet, the college and universal algebras would be nothing, says my professor above, if not for their foundation in that familiar old x and y high school algebra. I must say I agree with him whole-heartedly. I can’t count the number of times I have gotten a homework or test problem wrong not because I misunderstood the concept, but because I made some “silly” algebraic error. I’m sure other mathematicians can share this frustration; sometimes one becomes so focused on the “hard” concept that they carelessly mess up the “easy” algebra part. I often see this with my calculus students as well. Sometimes the problem is the carelessness mentioned above, but other times the problem is that these students never got comfortable with basic algebra to begin with. They may intuitively understand the concepts of calculus, but they don’t have the algebraic “toolkit” to solve problems. This hinders their advancement in mathematics.
Perhaps the best description of algebra I ever heard was from an Air Force Veteran I had the pleasure of tutoring in the subject two summers ago: “Algebra is just finding an unknown quantity.” This idea, which underlies all of mathematics, is what we as math teachers must impart to our students if we want them to develop the skills and the confidence needed for success. I like to think I am doing my part. Hopefully others will join me, and together we can improve math education.