If you wanted to know the result of multiplying 5463 by 78912, it is very likely that you would reach for a calculator. No one in the world outside of schools would do this problem with pencil and paper. Children know this. They know that adults at work do arithmetic by machine, whether in a fast food joint, or a bank, or a lab, or an engineering firm – anywhere at all. It is obvious to students that doing multidigit arithmetic by hand is not a useful skill. No one will hire you to do long division.
Still, being able to predict that the result of that multiplication is not too far from 400,000,000 is a worthwhile skill, if only to confirm that the answer given by a calculator is in the right range. I got this result by multiplying 5000 by 80,000. I could gain a little more accuracy by following this up with an estimate for 55 × 79. Well, 55 × 80 is (5 × 80) + (50 × 80), or 4400. Subtract 55, and get 4345. So the result of the original calculation should be close to 434,500,000.
How did I do? The actual answer is 431,096,256. In millions, my error was a number less than 4, divided by a number greater than 400. In other words, I was within 1%. Not too bad! (My calculator says the error was .7895555%.)
Few students would be able to work through this like I did. That is not surprising: this sort of computation is not taught much. If students spent less time with paper and pencil multi-digit arithmetic, that would free up some time to work on mental calculation and estimation. Working on this depends on, and helps develop number sense. For my first estimate, I had to understand rounding, and multiplication by powers of ten. In the second phase I used the distributive law twice. Paper-pencil computation, on the other hand, can be done with little or even no understanding. And almost no students gain deeper understanding by doing it.
But, you say, by practicing multi-digit multiplication on paper, they can gain accuracy and speed! Yes, that is true of many students. But even the best will not be as accurate or as fast as the free app on their phone. And in any case, what good would it do? Speed and accuracy in computation can no longer be a priority in math education. We should spend what limited time we have with students developing their understanding, not wasting it on useless skills.
Mental calculation and estimation need a place of honor throughout the K-12 curriculum As far as I’m concerned, that is the main consequence of the availability of calculators. This can take place in number talks appropriate to what is being taught at each level. Estimates should precede every calculation. Possible strategies should be discussed before and after the introduction of standard or non-standard algorithms.
And this need not stop after middle school, quite the opposite: number sense and operation sense continue to be a priority all the way to graduation and beyond. What is your estimate for the square root of 18? why? A mental math component can be added to topics like algebra: solve 3x + 1 = 13 without paper, pencil, or technology. Or trigonometry: you should know or be able to quickly retrieve cos(60◦). What would you guess for cos(55◦)? cos(65◦)?
One thing that the traditional paper-pencil approach and the mental math approach have in common is that they both require knowing basic addition and multiplication facts. So yes, students should learn their addition and multiplication tables. Of course, those facts are best learned in the context of understanding.
But what about students for whom remembering 6 × 7 is an unreachable goal? No matter what we try, they can’t seem to hang on to those facts. Should we shut the door in their face, and say that if you don’t know your multiplication table you will not be allowed to go on with secondary school math? That is the way it used to be. I don’t think such a policy makes sense any more. If a student needs to use the calculator ”as a crutch”, I let them. I wouldn’t deny a crutch to someone with a broken leg.
None of this is to deny that there is some interesting math to learn when working, within reason, on three-digit addition, or two-digit multiplication. But that is my point: learning some interesting math should be the goal. Speed and accuracy will ensue for some students, but it should not be prioritized.