COMPUTATION VERSUS UNDERSTANDING? by John Allen Paulos is a slight expansion of an OpEd that appeared in FORBES magazine (copyright January, 2000).
"Ambition, Distraction, Uglification, and Derision" is how Lewis Carroll referred to the basic arithmetic operations of addition, subtraction, multiplication, and division. Although most people resonate with this whimsically expressed repugnance toward computation, most would also grant that a facility with computational procedures is frequently necessary.

This tension underlies the latest skirmish in the simmering Math War that rages periodically. Like the Science War between hard scientists and post-modernists, the Math War tends to create more heat than light. The passion aroused is notable given that the issue is the proper place of computation and algorithms (step-by-step procedures) in the school curriculum. What, in particular, is their relation to such often neglected skills as understanding graphs, interpreting probability, modeling situations, applying mathematical concepts in other domains, estimating and comparing magnitudes, etcetera?

Various groups of educators have devised textbooks and curricula that attempt to foster these latter skills, and their efforts have been criticized by many as insufficiently rigorous. When the Department of Education recently endorsed some of these new curricula as "exemplary," a group of prominent mathematicians published an open letter to Education Secretary Riley claiming that many of the recommended books and programs do not deal at all adequately with the basic algorithms of arithmetic.

This might seem a parochial controversy were it not for the social cost of our arithmetical failings - clerks who are perplexed by discounts and sales taxes, medical personnel who have difficulty reckoning correct dosages, quality control managers who don't understand simple statistics, voters who can't recognize trade-offs between contrary desiderata, and a media that seems oblivious to serious risks but frequently becomes apopleptic over trivial ones.

A problem with this common formulation of the disagreement between "reformers" and "fundamentalists" is that there is no real opposition between conceptual understanding and the mastering of basic algorithms. The extreme positions are easy to parody. Assigning 500 long division problems to elementary school students is a sure way to stultify them. So, in later grades, is requiring students to factor 500 polynomials in algebra class or to differentiate 500 functions in calculus class. A martinet's mindless discipline will deaden anyone's enthusiasm.

On the other hand, the reformist endeavor (with which I've been associated) to tell stories, describe applications, play games, and naturally embed mathematical insights and ideas into everyday life can also be mocked. Thoughtlessly implemented, it can lead to a feel-good wishy- washy ineffectiveness. A new "aha" experience and engaging vignette can't be required for every problem, and a mere glimmer of the idea generally isn't sufficient to secure numerical answers.

The proper balance depends, of course, on the topic, the student's age, the area of mathematics, and the specific algorithm. There is no royal road to mathematical pedagogy, certainly not one capable of being reduced to a column. Despite common belief, arithmetic is not easy (see problem below), nor are "higher-level" subjects necessarily difficult. Some "elementary" algorithms, such as those for dealing with fractions, may be drudgery if they are not presented well, but they are mathematically significant and essential to real understanding. No stories about dividing or combining parts of pies or salaries, for example, can replace the formal rules for finding 2/7 3/11 or 2/7+3/11.

Acknowledging that there are glaring weaknesses in some of the new recommended programs, I'm nevertheless pleased that, unlike many conventional curricula, the new programs do stress conceptual understanding and engaging applications of mathematical ideas. I'm also pleased that they do not place an undue emphasis on repetition of rote procedures. The universal availability of calculators and computer software makes this emphasis on drill all the more dispiriting and inappropriate.

In arithmetic, as in most areas of mathematics, the stories and applications should set the stage and create an engaging and lively matrix that provide motivation for understanding the algorithms and performing the calculations. The many good people on opposite sides of the Math War should re-compute their strategies.

Problem: Imagine buying 100 pounds of potatoes and being told that they're 99% water. After the potatoes have been left outdoors for a day, you're told that they're now 98% water. What is the weight of these slightly dehydrated potatoes?

Back to Home Page ---------------------------------- Professor of Mathematics at Temple University, John Allen Paulos is the author of Innumeracy and, most recently, of Once Upon a Number. He writes the Who's Counting column for