Originally published in Literacy across the curriculum sometime in 1988–1989. At the time this article was written, parents and educators were concerned that allowing students the use of calculators in school would rob them of such fundamental skills as doing arithmetic the then prevalent way.

Remember Pebbles? The tiny tot of Flintstone fame? Well, time passes quickly and here we observe her as a charming young lady—Wilma's spitting image really—at her final examination in college math.  210922-1

Strewn about are stone chips, probably from the quarry where Fred Flintstone still labors. The examiners are Professors Rock and Hardplace. And this is what we hear:  210922-2

      Prof.Rock: Pebbles, tell us, how much is three times four?  210922-3

      Pebbles (barely giving the question a moment's thought): Twelve.  210922-4

      Prof. Hardplace: I'm sorry Pebbles, that won' do. We have no alternative but giving you a failing grade.  210922-5

Pebbles is shocked. She had worked very hard for this exam and she feels her performance was tops.  210922-6

      Prof.Rock: Pebbles, you have come to this examination wih an unauthorized device. You have memorized a tablet of multiplication. We, however, must insist on the application of fundamentals.  210922-7

      Prof. Hardplace: To have passed this test, you might have made three collections of four chips each and then count all the chips in those three collections.  210922-8

      Prof. Rock (kindly): Only thus would we have obtained visible evidence you understand the problem. Only then could we have given you the credit for college math.  210922-9

Knowing, understanding,professors. Troublesome words, bothersome people .... No wonder college is giving Pebbles a rough time. She knows how to multiply numbers and, yet, that knowledge does not demonstrate understanding.  210922-10

Let's say that understanding begins with knowing fundamentals. Maybe that is how those who understand understanding understand understanding. And counting, it seems reasonable to say, is more fundamental than multiplication. Professor Hardplace's counting method for doing a multiplication is closer to understanding than recalling some entry from a memorized table.  210922-11

But is counting fundamental?  210922-12

It has been found that a goldfinch is usually able to tell three seeds from two, four seeds from three, but not four seeds from five. Goldfinches, and some other animals as well, have some number sense. And with evolution as a Leitmotiv and experience as a clincher, I would guess that without counting or the arranging of seeds in neat patterns, people can do about just as well as goldfinches. So, which is then the more fundamental, counting or number sense? Can we pinpoint fundamentals—sometimes? always?. Can we pinpoint understanding—sometimes? always? I rather doubt it. Teachers, I have it on good auhority, can examine students for all sors of knowing, from the factual right up to judgment, but I, for one, cannot test for understanding, unless, perhaps, I know how to identiffy fundamentals. And if I cannot get to the bottom of fundamentals, then what? Should I seek some primitive level at which to assess the quality of a student's knowledge?  210922-13

Pofessors Rock and Hardplace had an answer. They felt that settling at the level of counting would be just fine. Little did they realize that Pebble's distant offspring actually learned to pass exams by merely recalling entries in memorized multiplication tables.  210922-14

Today's handheld electronic calculators can replace memorized tables by the memorized use od push-buttons. In fact, these calculators more than simply replace those tables. They replace mental arithmetic and long-hand calculations as well. Or, perhaps, not altogether? Maybe the reader knows of any use for long-hand multiplication a calculator cannot satisfy? If this is indeed the case, then it may well be desirable to keep on instructing the young in long-hand multiplication.  210922-15

Long-hand multiplication is learned by memorizing a sequence of well-defined steps. Any such sequence is called an algorithm. A mathematical equation, too, is usually solved by employing some memorized sequence of steps, some algorithm. But evn though a knowing of the right algorithms is convenient, it is not at all essential for solving mathematical problems.  210922-16

An algorithm is a trick passed on by the person who first managed to solve a particular kind of problem more expeditiously than did others before him. He may not have known any algoritm worthy of the name and proceeded by trial and error, a process called heuristic. Heuristic methods require thinking, algorithmic methods require memory.  210922-17

Many memorized algorithms have their use. but with certain modern calculators and pocket computers others are losing their value. Calculators are, and pocket computers can be filled to the brim with tricks. Twinning man and microchip, homo sapiens and pulex exiguus, can free a lot of mental capacity for gathering knowledge and skills more sublime than collecting a headfull of algorithms. What is considered fundamental today may well appear prehistoric tomorrow.  210922-18

Typically, students are scared stiff of having to use heuristic methods. These just don't come with a nice guarantee that if a problem is not solved wihin a given time slot, the student will pass the exam anyway. Students consider exam problems needing a heuristic approach unfair.  210922-19

And so they are. Unfair, that is, to students who don't have sufficient experience in solving problems heuristically. And unfair if examiners set inflexible time limits. Of course, no examiner wants to be unfair! Therefore, let's just continue examining students for the rote application of memorized algorithms. We keep on training them for their exams as well we can and thereby, at great expense to the community, provide them with those most excellent skills yesterday's $100 calculators and pocket computers already had on tap-tap-tap.  210922-20

It is possible, however, that in our attempts to be fair we may be denying our kids an advantage in a threatening and threatened world. Or let students elsewhere get ahead by having electronics handle algorithms while they amass heuristic experiences by trying to solve real problems.  210922-2

And now, my dear reader, you may guess three times who then will move ahead in this cruel world and who may not even get to do any profitable work at all!  210922-22