I’ve been teaching mathematics for some forty years. That entitles me to make a few observations about mathematics students—at least mathematics students in the United States. Actually, this essay is based on anecdotal observations of my own northwest Ohio classrooms over the years. But! I have had a lot of students, and I have the perspective of time.
Many students dislike mathematics. No, let me change that—many students dislike arithmetic. They know little of actual mathematics beyond that which uses basic computational skills. Unfortunately they do not even know basic computation very well. Calculators came into use early in my career. I was in about my second or third year teaching junior high mathematics when I purchased my first calculator, a basic, four-function TI calculator, for $65. The price had come down considerably since the first models had come on the market, so I jumped in.
Electronic calculators were liberating for students. The difficulty of working through hand computations with multi-step algorithms disappeared. Students could work problems with more realistic numbers. They could tackle complicated problems confident they were getting good answers unmarred by minor arithmetic errors.
During this time I taught several groups of low-achieving students whose arithmetic skills were so poor they struggled with the slightly more abstract idea of area and volume formulas. These formulas have implied, rather than explicitly stated, computations. I found a box of calculators, and we tackled areas and volumes problems relatively free of manual arithmetic. How much did they learn? I have no research data, but for the first time they worked out how many gallons of paint they would need to buy to paint a room.
Calculators have become as much a part of a student’s school supplies as a pencil and paper. Has this all been for the good? I don’t think so. Yes, students can now get answers to complicated arithmetic problems, but is getting an answer the entire point of the process?
I have taught mathematics at the community college level for almost thirty years. It used to embarrass me to say to my College Algebra classes, “You really must know your multiplication tables.” It used to embarrass me, but not any more.
If asked, most (but not all!) students can multiply three times eight. That is not really the point. The problem is they do not know the multiplication tables backwards as well as forwards. They cannot look at a number like twenty-four and immediately see the multiplication facts that multiply to make twenty-four. Without this mental computational fluidity, they will never be able to factor polynomials easily. Factoring polynomials? Is this a major life skill? Well, no, but it is an important gateway to learning algebra well enough to move on to take trigonometry and Calculus. These are the math courses required for the high-paying, high-tech jobs we are trying to promote to our students.
Why has this happened? Students are using their calculators as a crutch rather than a tool. When asked to do a simple computational problem, my current students reflexively reach for their “electric brain” rather than evaluating whether or not they really need it.
They avoid fractions like the plague by either immediately converting everything to decimal fractions or by using a calculator that is programed to do fraction computations. Fraction arithmetic is no fun, but the simple common fraction computational algorithms are exactly the same patterns used with algebraic fractions and equations. Again, students are not practicing the arithmetic skills necessary to understand the more abstract manipulations of algebra.
So am I just being an old curmudgeon about this? Perhaps, but abstract, higher-level thought requires a solid foundation of knowledge. Think what it would be like taking a course in French literature without having a good working vocabulary of the French language. How much analysis of character, conflict, and theme could you do if you had to stop to look up every other word in a French-to-English dictionary? How much higher level mathematics can you learn if you have to do every little arithmetic computation on a calculator? Algebra is a basic lexicon of higher mathematics, and fluent algebra requires arithmetic—mental arithmetic.