“Breakthrough Made in Superconduction” declares the headline on the front page of the morning paper. The story reports the development of a new ceramic material that can carry electrical current with minute resistance, at temperatures warm enough to be of practical use. One possible outcome, the story reports, is that desk-sized supercomputers could “come down to the size of a football, and probably operate 10 times faster.”
Statements like that overwhelm me. I struggle to grasp their meaning. Several years ago, when supercomputers were arriving on the scene, I read that one of them could do 4 million “long” multiplications in a second. In an attempt to give some relevance to that statement, I decided to figure out how long it would take me to do that many multiplications, using the old-fashioned paper-and-pencil method I learned in school. I’m sure, for the computer, a “long” multiplication was more than multiplying a five-digit number by a four-digit number, but that was “long” by my standards. I timed myself and found I could do such a multiplication, with reasonable care, in about 75 seconds. At that rate, it would take me 300 million seconds to do 4 million multiplications. That’s 83,333 hours. If I limited myself to an eight-hour workday, five days a week, 50 weeks a year, it would take me about 42 years. That’s a lifetime’s work, and a supercomputer does it in one second! Now I’m told, the next generation of computers will do 10 lifetimes’ work in a second.
So what’s my reaction? It ranges from amazement and celebration to peevishness. The advances in computational tools I have used in my lifetime astound me--from the paper and pencil of school arithmetic, the slide rule of high-school physics, and the hand-cranked calculating machine of college statistics, to those marvelous tools, the solar-powered calculator and personal computer--and there’s more to come. In my wildest schoolboy imagination, I would never have dreamed of possessing such powerful and tractable computing devices. But I do--and when I think of all the time and effort they save me and the possibilities they afford me, I revel in their use.
So what peeves me? Schoolchildren aren’t allowed to join in the celebration. Despite all these magnificent advancements that have brought untold computational power to one’s fingertips, school is “more like it’s been than it’s ever been before.” As in the days of my youth, and my grandparents’ youth, schoolchildren are drilled for hours to perform paper-and-pencil computations in machine-like fashion. There are no breakthroughs here. There are no superconductors that have managed to overcome the resistance that will allow any electricity to permeate the mathematics classroom.
Instead, there are staunch defenders of the status quo. Those that insist on turning students into paper-and-pencil computing machines. For what purpose? It’s ridiculous to expect there is a future for a paper-and-pencil machine that takes a lifetime to do what another machine can do in a moment. It’s no longer a matter of economics when a hand-held calculator costs little more than a good mechanical pencil. And solar power has stilled those ominous voices that warned us about batteries going dead. So why insist on centering a mathematics curriculum on teaching students how to use outmoded computational tools?
The ostensible reason proffered by those resisting change is that students must learn the “basic mathematical skills.” And, in order to do that, the use of electronic tools must be curtailed or even banned. As the author of one current textbook series explained, in a national advertisement, " ... calculators should not be permitted in elementary schools, for this is the time and the place for learning fundamental concepts and mastering paper-and-pencil skills. If students are permitted to use calculators too early, many of them will block and refuse to do the drudgery necessary to perfect the necessary paper-and-pencil skills.”
Such cries for skill drudgery come close to skullduggery. In the first place, paper-and-pencil skills are not “necessary.” Secondly, anything that can remove drudgery from school mathematics programs ought to be extolled instead of condemned.
Designating a mathematical skill as “necessary” implies that it is needed to function mathematically. That is not the case with paper-and-pencil arithmetical skills; one can function mathematically quite well without them. You may object, pointing out that one can’t get through school mathematics without them. That is very likely true, but that doesn’t mean they are basic mathematical skills. That only means they are school-survival skills.
To determine whether or not a mathematical skill is necessary, one ought to examine its essentialness in the nonschool parts of the world. Over the half-century I have been doing mathematics--as a schoolboy, as a college and graduate student, in any number of odd jobs that paid my way though school, as an industrial mathematician, as a university teacher and researcher, in everyday life, and just for fun--there is nothing I have done, apart from schoolwork, that today requires the use of paper-and-pencil arithmetical procedures. Calculators provide an economical and efficient way of doing computations I can’t do in my head. And knowledge of these paper-and-pencil procedures does not provide me with mathematical insight of any significance.
There are times when I find these paper-and-pencil computational skills useful--although I can’t remember the last time I used them for long division other than at school. Also, there are those who prefer these methods of computation. But even if these procedures are occasionally useful, or preferred by some, it does not follow that they are necessary skills. In my mathematical life, I can get along without them. Most adults do.
This doesn’t only apply to the paper-and-pencil procedures of elementary-school arithmetic. It also applies to the paper-and-pencil procedures of high-school algebra, college calculus, and all other math courses. These days, any step-by-step procedure involving the manipulation of mathematical symbols, according to a fixed set of rules, can be done by a calculator or computer. Some procedures are simple enough that they are best done mentally or by hand, but any that require more than a modicum of time and energy to do manually are most economically done by machine. And, outside school, they are.
Thus, school math programs that center on the mastery of mechanical paper-and-pencil procedures are not necessary skills. They are vestiges of another age, when human beings, in conjunction with paper and pencil, were the computing machines of the day. To gear a math program to producing such machines does indeed reduce students to drudges.
I suspect the resistance to calculators in classrooms is not a tenacity for teaching basic skills, but rather an anxiety about what to do if existing programs are abandoned. I suspect many educators share the feelings of that 5th-grade teacher whose immediate response to the suggestion that he allow calculators to be freely used in his classroom was, ''But that would destroy my whole program!” It would. However, once one sees the truth of that statement, lets the initial shock wear off, and asks what ought to happen next, one can envision a mathematics program that recognizes current technology, is economically feasible, and provides pertinent mathematics for purposeful students, without drudgery.
Such a program does not require that classrooms be equipped with the latest in electronic computing devices. Rather, it requires that the existence of these devices be recognized, and time and energy not be wasted teaching students pencil-and-paper procedures that, except in school, are done electronically. For most school computational purposes, inexpensive calculators will do. And, since calculators are easy to use, math programs needs not devote much attention to computational skills.
Thus, computation plays a minor role in a pertinent math program. Such a program will emphasize meaning rather than symbolic manipulation. It will reduce the mathematical creativity innate in every student; it will develop mathematical insight and intuition; it will stress cooperative problem solving; and it will allow students to compute by whatever means they can--mentally, counting on their fingers, with an abacus, using pencil and paper, or punching the keys of a calculator. As students grow in mathematical maturity, they will find the computational methods that work best for them.
I recognize that there is a vast difference between listing characteristics of a math program that is appropriate for the electronic age and implementing such a program in the schools. Doing the latter is as exciting and challenging as searching for superconductors.
For the past several years, I have been involved with a coalition of school and college math educators who are working to instill the above characteristics into portions of the school mathematics program. The vehicle we have chosen is visual thinking--the use of sensory perception, models, sketches, and imagery to provide insight into mathematical concepts and bring meaning to mathematical symbolism. It’s gratifying to watch general-mathematics students--who have become accustomed to perfunctory paper-and-pencil drill carried out with little meaning, mediocre success, and no interest--make contact with their mathematical instincts and come alive mathematically. It’s rewarding to see math-anxious elementary teachers overcome their doubts about ever understanding mathematics, or tackling an open-ended mathematical question that they cannot solve mechanically. And it’s encouraging to know they no longer will pass on an apprehensive and distorted view of mathematics to their students.
There are other people, scattered throughout the country, engaged in similar activities. These are the people who celebrate calculators and computers for the computational power they bring to all students. They are in contact with their own mathematical spirit, ignite the mathematical spark in others, and know the essentials needed to nourish it. These are the people who, despite my peevishness, give me hope. It is their energy that can overcome the resistance of those who impede mathematical power with the drudgery of mechanical paper-and-pencil drill. It is their energy, conducted into classrooms, that can electrify the mathematical potential inherent in every student.
