Calculators are helpful, but they can become a crutch
By Jeff Cruzan, math teacher
We math and science teachers are prone to thinking up eulogies for civilization when our students reach for a calculator to do simple arithmetic. It’s partly because every touch of one seems like a missed opportunity to refine the neural connections of a promising brain, and partly because we know that premature use of a calculator can mask the simple beauty of a mathematical problem—and mathematics is far more than just arithmetic.
We want our students to be able to brainstorm as adults. (I thought I had a brainstorm once, but it was just a drizzle). We want them to be the leaders of conversations about new ideas. We want them not only to ask “what if …?”, but to be able to follow that reasoning from possibility toward probability and implementation. We want them to be relentlessly present in their work because they’ll be in charge of the world we’ll leave to them. That often means possessing sharp arithmetic and estimation skills, and having a strong sense of the relative sizes of numbers.
When the calculator becomes a digital crutch, we wonder what happened to all of that good foundation learning: the multiplication tables, adding fractions, mental arithmetic, estimation and number sense. In lower school, our students learn most of the basic arithmetic skills: adding, subtracting, multiplying and dividing, and the reasoning behind our methods of performing them. They’ll need those skills forever, whether for writing the family budget or calculating the best trajectory to Mars. Early on, students become aware of the usefulness of calculators for doing tougher problems, just as any adult would be.
It’s one thing to know that 120 x 7 = 840, but another altogether to work out 983.42 x 121.9. The latter is an example of what calculators are for, but here’s the rub: We’d like our students to understand that anyone at all can punch those numbers into a keyboard and get the right answer (119,878.9), but a mathematical thinker would quickly notice that 983.42 x 121.9 is just about the same as 1000 x 120, or 120,000, which is only 0.1% different than the actual product, and in most cases just as useful.
Middle school teachers are the first to expose our kids to the symbolic mathematical thinking that will form the foundation for higher math reasoning. To solve 2 + 3 = ___ is to solve just one problem. But in solving x + y = 5, you’ve solved an infinite number of problems with the same pattern. Pick any y and I can tell you what x has to be for x and y to sum to 5. Powerful stuff, I know. Because of the symbolic nature of math in middle school, teachers insist that new material be learned first with paper and pencil. It’s only later, say when practical applications of algebra involve multiplication of complicated numbers or when square roots arise, that calculators are justifiably used as timesavers, in the same way that any math teacher would use one.
In upper school, most math is symbolic and arithmetic is just a necessary byproduct of learning it. It’s important that flagging arithmetic skills do not bog us down as we explore new concepts. When multiplying 4/5 by 1/3 is a calculator task, a student can become distracted from what’s really important. Understanding new concepts in mathematical thinking can grind to a halt when arithmetic sharpness declines.
Not all is paper-and-pencil math in upper school, though. We teach our students how to use calculators and computers for things at which they’re really good. I can sketch the graph of f(x) = –e3x – 1 (and so will my Algebra 2 students, by February) but a calculator can do it with much more precision in a fraction of the time, provided I’ve entered it correctly. Calculators can do the kinds of repeated summing that are valuable when paper and pencil methods fail, such as in integral calculus. Calculators can’t think, but they can fill in for us in the speed and precision departments, and that’s their real value. Increasingly, we’re incorporating computer programming into our math curriculum, too.
It’s tempting to blame calculator over-reliance on learning to use them too young, but in researching this post, I’ve learned that teachers across all grade levels have the same basic lament: Kids use calculators because it’s easy and because they lack the confidence (that comes with practice) to know that they can do arithmetic correctly without one. I think that the real problem is that calculators, … well, … exist. So this will be a constant battle for the soul of mental arithmetic and to teach students when it is appropriate to reach for technology.