Musings on Algebra

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.



Overcoming Math Anxiety

Non-math folks think that those in the math biz never suffer from math anxiety. After all, since we’ve voluntarily decided to spend out lives studying the subject, we must love it unconditionally and have no fear, right? Wrong. I have a bachelor’s degree in math, am currently working a master’s, and hope to soon be working on a PhD, but one branch of mathematics has struck terror in my heart for nearly five years: differential equations.

The horror began in my first semester of college. I had taken Calc 1 during my senior year of high school and Calc 2 the summer before college. Calc 3 was the natural next course for me to take, but it happened to be offered at exactly the same day and time as a freshman seminar that I was required to take. Since this was a small school, there was only one section of Calc 3, and all sections of the seminar met at the same time, so I was in a bind. I needed another math class, so my adviser suggested Intro to Differential Equations, which he was teaching. The prerequisite was only Calc 2, but when I entered the class, I realized a lot more knowledge was expected. Without Calc 3 and–especially–Linear Algebra, I was lost. The material quickly surpassed my understanding, and I struggled to keep up. I barely passed with a C, and I was relieved just to have survived. Yet the experience was so traumatic that I changed advisers and never took my school’s second, more advanced course in differential equations. It was the only math class my college offered that I did not take.

Now fast forward nearly five years. This coming fall, I am taking a course in Fourier analysis, a subject that has significant overlap with “diffy” and requires some diffy knowledge. So reluctantly I pulled out my old diffy textbook, knowing I would need to finally get a handle on that subject if I hoped to have any success with Fourier. And what did I find as I began to do some problems? Not only did I finally understand the material, I actually enjoyed doing it. I am happy to say that I am finally beginning to banish the old diffy ghosts.

Having this experience gives me an understanding of and sympathy for math anxiety when I see it in my students. I hope I can now share my new success story as another tool to help my students overcome that anxiety, just as I finally have.

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My name is Erin Milne, and I am a mathematics student and teacher. I earned my master's in math from the University of Vermont, and I received my undergraduate education from Lyndon State College. My goal for this blog is to make mathematics interesting, useful, and non-frightening, as well as to inspire other low-income and first-generation students to continue their education. I hope this blog will be helpful, inspiring, and thought-provoking for high school and college students facing the same challenges that I have faced.

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