We could go on for days about what we really think of calculators in K-12 math classes (we think they're great for science classes, especially stochiometry in Chemistry), but suffice to say it just might be one of the signs of the Apocalypse.A recent entry by Joanne Jacobs quoted a study demonstrating that students who use calculators do poorly in arithmetic when forced to do without. While School of Education wonks insist that students be "exposed to technology" at every level of K-12 education, we know that putting calculators in the hands of students learning arithmetic is incredibly shortsighted and harmful.
Our analogy relates to a child learning to walk.
Imagine taking a child who's pretty good at crawling, and starting to pull himself up on things, and placing him in a modified tricycle, one with a supportive cradle for a seat so he can't fall out. Soon he'd learn to push the pedals, randomly at first, then with purpose, as with steering the handle bars.
In no time he'd be zooming around on his little three-wheeler, after which he can be moves up to a real tricycle with a bicycle-style seat. From this he could evolve to a bicycle with training wheels, which can be raised incrementally so that more and more of the time he's on two wheels, after which they can be removed completely.
Now, if the child were placed in these three- and two-wheelers, each and every time he was attempting to stand or walk, there's little doubt that he'd quickly be far more skillful in pedaling himself around than two-footing it. Sure, the kid might learn to walk (probably looking like Frankenstein's monster), but he'd probably never learn to run.
Why bother when it's faster (and easier) to bike?
And so it is with calculators.
Personally, we think calculators (and just about any other technology made possible by the transistor) are great, and they sure do save a lot of time.
But that's not the point behind teaching children arithmetic in school.
The point of doing paper-and-pencil arithmetic (for example, finding the product of 47 and 9) isn't to actually find the product (like we didn't know what the answer was, and that's why we have kids do these problems). The whole purpose of this exercise is to practice a skill accurately to the point of automaticity.
Memorizing the times tables (for example, up to 12x12) is a gateway to so much more. Once a child has memorized every single fact contained in this grid of 144 products, then simple division (where both divisor and quotient are numbers between 1 and 12) is only a matter of backwards thinking.
For example, in picturing the number 24 one might think of 4x6, 3x8, or 2x12 (and of course the commutative versions of those three). Thus 24 divided by 8 is a fairly simple matter.
But doing longer division, especially when it involves remainders, is harder, because the dividend often is not a number found in the times tables. And very quickly, if the lesser skills of simple addition, subtraction, multiplication, and division are not mastered to automaticity, it quickly becomes a chore.
One reason why paper-and-pencil long division is going the way of the dodo bird is because it involves too much computation, 100% of which has to be done accurately, or the whole answer is wrong.
Take a simple problem of dividing a three-digit number by a one-digit number, for example dividing 301 by 7.
7 into 3? No.7 into 30? Yes, 4 times.
7x4 = 28
Subtract 8 from 0? No.
Borrow a 1 from 3 (make it 2), write it next to 0 to make 10.
10-8 = 2
2-2 = 0
Bring down the 1.
7 into 21? Yes, 3 times.
7x3 = 21
21-21 = 0
Depending on how you count, it takes about ten steps, most of which are calculations, but even so this problem should be done in under 15 seconds.
Truth be told, it should take fewer steps the more fluent one is with numbers, for example the step of subtracting 28 from 30 can be done as a quick mental math step without all the borrowing. And of course there's no real need to actually subract 21 from 21 at the end.
But the point is that this fluency cannot come if students aren't asked to solve a great many paper-and-pencil problems (yes, this means worksheets and flashcards) over a number of years of studying mathematics.
Unfortunately in a great many places, the handheld calculator is seen as a way of relieving this drudgery. Students may still be given worksheets, but are permitted to use calculators; a pointless exercise.
We mentioned the shortsightedness of this strategy earlier. Many fresh math teachers (especially in our inner cities where calculator use seems to be at pandemic levels) see nothing wrong with the use of calculators to do simple basic arithmetic, for the simple reason that the calculator is fast and accurate. So much time is freed up for higher-order thinking skills!
Unfortunately, middle and high school math desperately requires a certain fluency with numbers which dependence on a calculator simply cannot permit.
In late elementary school and middle school, students learn things like manipulating fractions (where finding the lowest common denominator is a key skill, one that requires being able to instantly look at several numbers and mentally seeing the greatest common factor or least common multiple) and using square roots (also requiring the ability to instantly factor numbers).
And in late middle school or early high school there's algebra, one place where the calculator isn't much help.
While we're sure a calculator exists which can factor the binomial 49x² - 25 such a beast is not in wide circulation, and if it were, most kids wouldn't know how to tell it to solve such a problem. But if a student has some number sense the answer of (7x + 5)(7x - 5) should be completely obvious just by inspection.
Unfortunately most of the folks advocating rampant calculator usage in the early grades probably can't do algebra, let alone trig or calculus.
In having a child master the calculator, that child just might become its hobbled slave.