Calculators: A gift from the angels or agents of evil?
Here is a true story: A high school student was working an algebra problem that required use of geometry to determine the value of x. He had reached an impasse and so he asked his teacher for help. His teacher assisted him to the point where x could be evaluated. The teacher then prodded the student with the question “Now what is 10 + 6?” The student got a befuddled look on his face and proceeded to punch the arithmetic in to his calculator. After a few seconds the student declared “60” and proceeded to write the erroneous answer on his paper. Besides the disgusted amazement of his teacher, what’s wrong here?
The error analysis is simple, the student accidentally multiplied 6 and 10 rather than adding them. But that is an understandable mistake. Is the problem that the student feels he needs a calculator to do simple arithmetic? In part. But far more alarming is the willingness of the student to accept as fact a woefully incorrect answer. The student had no “number sense”, no concept that the sum of 10 and 6 is nowhere near 60. Many brilliant mathematicians have made, and continue to make similar mistakes. But their sense of in what vicinity of the real number system the correct answer lies, tells them immediately that they have made an error, and they correct it. This may be an extreme example. But it should be an alert to the fact that we as a society, cannot let this tech-savvy generation pass in to adulthood without a sense of what’s greater than what.
Still, it’s not time for a calculator burning party. If the problem our geometry/algebra student was working required him to determine a distance that was miles, he would have to use a calculator to approximate the square root of 10. Telling a police dispatcher that you had witnessed a crime miles back, is likely to get you arrested yourself. But what our student and our crime witness, and indeed all citizens need is a sense that the square root of 10 is somewhere between the square root of 9, or 3, and the square root of 16, or 4. Further, they need to be able to instantly reason that is closer to than it is to because 10 is closer to 9 than it is to 16. Then, if our student, or witness, or good citizen accidentally punches in to a calculator and gets 10, they would immediately realize they had made a mistake and correct it.
So what’s to be done? Calculators do, in fact, make life more productive by freeing our minds for higher order thinking. But basic arithmetic is not higher order thinking. While memorization is anathema to modern educational philosophers, it does breed confidence. Our student should have been certain that 10 + 6 is 16, not because he reasoned it out, or because he counted on his fingers and toes, but because the fact had been burned in to his brain while still in Elementary School. This “number sense” is not acquired by doing Differential Calculus, or Linear Algebra, or Euclidean Geometry. Number sense should be established before tackling these subjects. Memorization is tedious. But its very tedium makes students appreciate the power of calculators once they are permitted use them. Until that number sense is established, calculator use should not be permitted.
We, as tutors, should be very reluctant to permit students to use calculators. However, when it comes to checking an answer, we should be enthusiastic about using calculators. If, for instance, a student from Booth Middle School, uses linear interpolation to approximate and gets 3.1, He or she should be shown that a better approximation is 3.1623, found by using a calculator. Now the student can see that they have done well (after all, they are just getting an approximation), they get a confirmation of their number sense, and they are compelled to think on a higher level in reasoning why their method works but is less precise than the calculator.