Long Division Dead as a Dodo Bird?
Long Division Dead as a Dodo Bird?
By Dr. Eugene Maier
Over sixteen years ago, the banner headline on page 4 of the January 19, 1982, edition of Education Week proclaimed, New Technology to Render Long Division 'Dead as a Dodo Bird.' The accompanying article reported on remarks made by Richard Anderson, then president of the Mathematical Association of America, during a symposium on "The Changing Role of the Mathematical and Computer Sciences Precollege Education" at a meeting of the American Association for the Advancement of Science.
According to the article, Anderson predicted: "when computers and calculators truly come of age in the schools, paper-and-pencil long division will probably be 'as dead as a dodo bird'." He maintained that calculators eliminate the need for laborious paper-and-pencil computations and are "changing the nature of what is important in arithmetic." He also predicted that opposition to these changes is likely and educators will stick to the traditional methods they themselves learned.
Anderson was pretty much right on all counts. Many educators are sticking to traditional methods and calculators do eliminate the need for laborious pencil-and-paper procedures. And as far as the extinction of long division, Anderson knew enough about the educational system to add that "coming of age in the classroom" caveat.
Coming of age seems to take a lot longer in the classroom than in other parts of the world. When I was in high school in the early 40's, I was forbidden to turn in a typewritten essay even though typewriters had been around for over 50 years, and there had been one in my family's household for as long as I could remember. Using a typewriter, we were told, would atrophy our handwriting skills (my handwriting skills atrophied anyway and I never did learn to type properly).
The tenacity with which long division holds sway in the classroom and the prejudice against the acceptance of the calculator are epitomized in recent documents issued by the California State Board of Education. Under the heading NUMBER SENSE in the grade 5 section of the California Mathematics Academic Content Standards as adopted by the California State Board of Education and posted on the WEB February 2, 1998, is the statement: "By the end of the fifth grade... students ...are proficient with division, including division with positive decimals and long division with multiple digit divisors." In a chapter on the use of technology in the September 5, 1997 draft of the Mathematics Framework for California Public Schools, the statement is made that "everyone should be able to do arithmetic with facility and without reliance on calculators."
These statements, I think, reflect some common misconceptions. They suggest that skill at long division promotes the development of number sense while the use of a calculator—like, I was told, the use of a typewriter—is debilitating. To promote arithmetical facility, they ban the calculator which, of all universally available computational tools, enables one to do arithmetic with the greatest ease. In the interest of developing arithmetical facility, it makes more sense to me to ban paper and pencil.
Consider the process of division. You might reflect on the last time you had occasion to find the quotient of numbers outside of a school setting. The first instance that occurred to me was when, some days ago, I wanted to know the gas mileage I was getting on my new car. The trip indicator, which I had set at 0 the last time I got gas, said 347 and the gas pump switched off at 13.7 gallons. So, to determine the mileage, I wanted to divide 347 by 13.7.
There are a lot of computational tools I could have used to help me make the computation—an abacus, base10 pieces, a calculator, or paper and pencil, to mention a few. However, driving away from the station, using any of these tools was inconvenient, so I did it in my head. (There are a lot of ways to do this mentally. I don't recall how I did it at the time - as I sit here writing this, I did the following: ten 13.7's are 137, so 20 are 274. Thus I got 20 miles to the gallon with 73—the difference between 347 and 274—miles to spare. Half of 137 is 68.5, so there are five more 13.7's in 347. Thus I got 25 miles per gallon with 4.5 miles unaccounted for which is about one-third of13.7, so I knew I got about 251/3 miles to the gallon—my calculator says 347 ÷ 13.7 = 25.32846.)
If I were at home sitting at my desk and I wanted to perform that computation, I would have used a calculator. The computational mode I use depends on the circumstances and what's available. If I am teaching and want to develop understanding and number sense, I use base ten pieces. If I want to perform a multidigit computation, I use a calculator. If I have a series of computations to perform, I can create a spreadsheet. And, yes, I might use paper and pencil, to record my thinking or to draw a sketch or diagram that aids my thinking, but rarely for the purpose of carrying out some algorithmic method I learned in school. I can't remember any instance when I have used the paper and pencil long division algorithm I learned in school—I suspect I have only used it a handful of times in my adult life. I have no use for it, I have other quicker and more efficient ways to calculate and besides, if I ever have need for a paper and pencil algorithm for long division, I have enough knowledge—as does any moderately mathematically literate person—to create my own.
In most cases, however, when I want to do a computation and the numbers aren't too large, I do it mentally. I find that the most convenient—I don't need paper and pencil, a calculator or any other tool—all I need is my mind, which hopefully is with me most of the time. And least of all, do I need the paper-and-pencil algorithms I learned in school. Rather, I have to thrust them from my mind—if I want to multiply 37 x 25 in my head, starting out by thinking "5 x 7 is 5 and carry 3" gets me nowhere. (On the other hand, a bit of number sense tells me that it takes four 25's to make a 100 and there are 9 groups of four and 1 more in 37, so 37 x 25 is 925.) That's why I say it's a misconception that learning a long-division algorithm, or some other paper-or-pencil procedure, has something to do with developing number sense. It is very likely to have the opposite effect.
Ask the adults in your neighborhood to mentally multiply 37 by 25. I suspect you will find that many of them have been so heavily schooled in paper-and-pencil algorithms they believe that carrying out these algorithms is what arithmetic is all about. Their natural number sense, rather than being nurtured, has been so constricted that, when it comes to arithmetic, they have no recourse but to reproduce the paper-and-pencil processes they have been drilled in. It's just as well we had taught them to do their arithmetic on a calculator, their number sense is likely to have suffered less and we would have saved a lot of time and energy.
So I await the day when calculators come of age in the schools, as they have in other parts of the world. My children marvel at my being forbidden to use a typewriter in school. Hopefully, my grandchildren will marvel at why their parents weren't handed a calculator instead of spending the fifth grade drilling on long division.