The sorry state of mental math

I don't know what to think of calculators. On the one hand, they are a great tool, and I sometimes wonder what else Isaac Newton might have had time to discover if I could go back in time and hand him a $5 scientific calculator. On the other hand, I despair of his ever having learned the facility with numbers he later showed if he had been exposed to it too early in his development.

The fact is, hardly any of my students can do basic arithmetic without the aid of the calculator. I mentioned in a previous post the blind-leading-the-blind aspect of class efforts to solve 22/4 by hand in 4th hour, and I am consistently floored by the reliance on calculators for everything from simplifying fractions to simple addition and subtraction. I am left wondering how much of the sad state of my students' math ability in general is due to a consistent reliance on this tool to perform simple tasks for them. How much practice in basic operations like adding and subtracting, multiplying and dividing, have my kids lost because of this crutch? How much more fluent in the language of numbers would they be if they had never been exposed to a calculator?

I remember learning the multiplication tables in 3rd grade, from 1x1 to 9x9. The whole class made rocket ships out of construction paper, and the teacher put up nine planets increasing in size, labeled 1-9, against a black background on the rear bulletin board. Everyone started at planet 1, and was able to advance through the planets only when they could do all the problems for that number for the teacher, Mr. Murphy. The best part was that, for each planet, we got a number of jolly ranchers equal to that planet's number (that's 45 for the whole circuit, if you're keeping count). We also did algebra in elementary school occasionally -- we just didn't know it, because we were told it was 3 + square = 4, or 9 - triangle = 7, and find what goes in the square or the triangle. We did the "Mad Minute" in grade school too, where we were given a page of simple math problems and were challenged to see how many we could solve in 60 seconds. It is staggering that my students are just now reaching this point -- what did they do in grade school?

There is an argument, a compelling one, that the ability to do mental math is overrated, unnecessary even, and that forcing children to do without calculators is a relic of a bygone era, like learning to do long division. I'm not terribly qualified to comment on this, but here's my take anyway: insofar as mental mathematics for basic operations nurtures a sort of mental flexibility, a comfortable ease working with numbers on the small scale, I think it is invaluable. How many times in our lives will we need to add 8 and 6 without a calculator at hand? Or how about getting a rough idea of whether the gallon of milk at 3.20 or the half-gallon at 1.89 is a better deal?

Long division is an algorithm. That's why no one remembers it when they reach adulthood -- it's because it's a precise series of steps that removes the necessity of thinking about the problem. This is similar in spirit to the manner in which the book teaches my students to add numbers: If the numbers have the same sign, then add their absolute values and attach the common sign. For opposite signs, subtract the smaller absolute value from the larger, and attach the sign of the number with the larger absolute value. Seriously? It's like the intuitive sense of relating numbers to zero and that subtracting more than you have leaves a debt has been willfully excised.

So, while it is troubling that I have a whole class of algebra students who can't divide 22 by 4, it is not entirely surprising.