It can be argued, as I have [1], that it should not be an aim of elementary school mathematics to have students attain proficiency in any of the standard paper-and-pencil algorithms for the four arithmetic operations. But however strong or weak you may think the argument about addition, subtraction and multiplication, most people would agree that a stronger argument can be made about division. Still there are those who not only favor the continued teaching of the long division algorithm (hereafter LDA) but argue, in addition, that knowledge of LDA is essential for the further study of mathematics. In this paper, after some general remarks about LDA, I shall try to refute this argument.

**
Common Ground
**

Division plays a unique role in arithmetic calculation. Not only does doing division by any method without a calculator presuppose a knowledge of subtraction and multiplication but, uniquely among the four operations, it also requires the ability to estimate. LDA is, therefore, by far the most difficult of the pencil-and- paper algorithms to learn. Indeed, it is likely that trying – and failing - to learn LDA turns off many people to studying any further mathematics. If this is so, it is a strong argument by itself for trying to avoid teaching it. Indeed, this was one of the motivations for the declaration over 20 years ago in the Cockcroft Report [2] that "it is not profitable for pupils to spend time practicing the traditional method of setting out long division on paper".

This recommendation for schools in England and Wales appears to have been widely ignored in Britain, in the U.S. and, it would seem, in just about all other countries. One reason for this is that, however much or little calculators may be used in elementary school, the proponents of teaching the standard paper-and- pencil algorithms still hold sway in (virtually?) all countries.

However, despite the fact that there are strong opinions on both sides of the argument about whether or not to teach LDA, there are some aspects of this matter which will, I think (hope?), be agreed on by those on both sides:

- All students need to understand what division is, that is, what the quotient of two numbers is, what the remainder is and when division is the appropriate arithmetic operation for solving a problem. This understanding would include knowing that division can always be (tediously) accomplished by repeated subtraction.
- LDA, whether you believe it should be taught or not, is now almost never used by anyone for actual calculational purposes and will not be in any foreseeable future.
- Whatever reduced emphasis there might be on LDA in the elementary school curriculum, this should not be used as an excuse for any reduction in the total time allocated to mathematics in the elementary school curriculum.
- The purpose of reducing emphasis on LDA must not be to make the elementary school mathematics curriculum easier ("dumbed down") for students. Quite the contrary, perhaps.

Starting then from what I hope is a level playing field, we may perhaps discuss the teaching of LDA without the emotive language that often is used in discussions of the use of calculators in elementary school. In the next section I shall present the arguments that have been made for retaining instruction in LDA and try to refute each of these.

**
The Arguments in Favor of LDA and their Refutations
**

**Division, Fractions and Repeating Decimals**LDA is a way to turn fractions into decimals [3]. But is LDA the best way – or even a very good way - to show students how to convert fractions into decimals?

Most students will, anyhow should, learn the short division algorithm (SDA) before they would learn LDA. In SDA all multiplication, subtraction and estimation is done mentally with, normally, one-digit divisors. (But, in fact, if mental arithmetic is stressed in elementary school as I think it should be [1], then it should be possible to do SDA with two-digit divisors since this requires only mental multiplication of a two-digit number by a one-digit number.) Indeed, with one-digit divisors there really is no estimation since, assuming that the multiplication table has been memorized, each quotient digit can be written down without any need to estimate. Starting with 2 and then 3-digit dividends, the student should learn just to write down the quotient and remainder and then should be able to extend this skill to dividends with more digits. Thus, for example, to divide 46847 by 8, the quotient digits 5855 and the remainder 7 should be written down sequentially by the student without any need to record any other data.

For proper fractions a method completely equivalent to SDA or LDA is, first, to write the fraction as

#### A/B = .abcdefg… .

Then, at the first step, multiply by 10, record the integer portion of 10A/B as a, subtract a from both sides to get .bcdefg… on the right and then repeat this idea to get b, c, … . When B has one, or even two, digits, the subtraction can be done mentally. Thus, for example, to compute the decimal equivalent of 3/7 we would set 3/7 = .abcdefg… and proceed as follows:

30/7 = 4.bcdefg… so 2/7 = 30/7 – 4 = .bcdefg…

20/7 = 2.cdefg… 6/7 = 20/7 – 2 = .cdefg…

60/7 = 8.defg… 4/7 = 60/7 – 8 = .defg…

40/7 = 5.efg… 5/7 = 40/7 – 5 = .efg…

50/7 = 7.fg… 1/7 = 50/7 – 7 = .fg…

10/7 = 1.g… 3/7 = 10/7 – 1 = .g…

30/7 = 4. …

and now it is clear that the sequence 428571 just repeats since once one digit of the quotient is repeated, subsequent digits must also repeat. Note that in practice students should be expected to write down just the successive quotient digits with all other calculation done mentally.Of course, applying LDA to such fractions also gives the quotient and the period of repetition but at the expense of considerably more recording of results.

A nice property of the approach above is that it can be extended by a combination of calculator and pencil-and- paper (or mental) calculation to dividends and divisors larger than one- or two-digits in a way that also embodies the need for estimation as in LDA. For example, to find the decimal expansion .abcd… of 1264/4329, a can be estimated to be 2 and then 12640/4329 – 2 can be found using a calculator to be 3982/4329 etc. By requiring students to record each step, the calculator cannot just be used to find the answer. So, in effect, LDA has been used without all the usual pencil-and-paper recording.

Another advantage of the method described above is that it leads to an easy association of converting a fraction to a decimal with the dual problem of converting a repeating decimal to a fraction. If the decimal is, say, .abcdefabcdefabcdef… and you wish to find the equivalent fraction A/B, then multiplying A/B by 106 and then subtracting A/B from the result gives

(10^{6}– 1)A/B = abcdef.abcdefabcdef… - .abcdefabcdef… = abcdef

so that A/B = abcdef/(10^{6}– 1). Of course, for other repeating decimals the exponent 6 must be replaced by the period of the repetition. A variation on this theme when the repeating decimal is preceded by a non-repeating part is straightforward.The lessons here are that (1) LDA has no advantages over SDA for one-digit divisors, (2) LDA has no advantage over a combination of calculators and mental arithmetic in any case and (3) SDA can be nicely related to the problem of the conversion of repeating decimals to fractions. All of the above holds also for improper fractions with no more than changes to some details. Moreover, the discussion above is easily adapted to non-integer dividends and divisors because "place value is irrelevant" [4, p.88] in LDA as well as the other methods discussed above.

**The Need for LDA in High School Mathematics**The main purposes of the mathematics taught at any educational level are, first, to impart a useful skill and, second, to provide an entrée to the further study of mathematics. If you accept the Common Ground in the previous section, then, with me, you believe that the skill to perform LDA is of rapidly diminishing importance. But how about it's use as an entrée to portions of high school mathematics?

The obvious place where LDA might help in high school algebra is when studying the division of polynomials [3]. But, perhaps surprisingly, the algorithm for polynomial division is considerably easier to learn than LDA. That sounds like a paradox because LDA is clearly a special case of polynomial division when both numerator and denominator polynomials have degree 0. The paradox disappears, however, when you realize that, almost without exception (does any reader know an exception?), the coefficients in any actual polynomial division are simple (one digit or, very occasionally, two-digit) integers or very simple fractions. Thus, the coefficient division required to divide two polynomials can be done mentally.

Polynomial division, therefore, is a quite simple algorithm compared to LDA since, assuming simple coefficients, not only can all multiplications and subtractions be easily done mentally, but also no estimation is required. This algorithm follows from a generalization of the equation

#### a = bq + r, 0 ≤ r < b

for finding the quotient q and remainder r when dividing a/b to

#### (1)

#### a(x) = b(x) q(x) + r(x)

for the division a(x)/b(x) of two polynomials with degree(a(x)) = degree(b(x)) to get the quotient polynomial q(x) and the remainder polynomial r(x) with degree(r(x)) < degree(b(x)). The algorithm follows by equating coefficients of like powers of x in (1).Using the same example as in [3],

#### (x

^{4}+ 2x^{3}+ 4x + 1)/(x^{2}+ 1),

the quotient must be a quadratic polynomial ax2 + bx + c and the remainder a linear polynomial dx + e. Thus, we must have

#### x

^{4}+ 2x^{3}+ 4x + 1 = (ax^{2}+ bx + c)(x^{2}+ 1) + dx + e.

Equating like powers of x we have

Power 4 a = 1 3 b = 2 2 c+a = 0 1 b+d = 4 0 c+e = 1

from which it easily follows that c = -1, d = 2, e = 2. Other examples would not be quite so simple but the procedure itself is always straightforward. The analog of LDA, which is often taught for polynomial division, is cumbersome and unnecessary.Moreover, learning polynomial division naturally follows learning polynomial addition, subtraction and multiplication, the first two of which are trivially simple algorithms and the third of which is not very difficult. (A good exercise to test whether high school or college students understand polynomial multiplication in other than rote fashion is to ask them to write a program (or algorithm) to compute the product of two polynomials given the coefficients of both. Even good students find this (surprisingly?) hard.) It could even be argued that it would be better to teach polynomial division before LDA since learning the former means that the essential pattern has been learned, leaving for the latter only the adaptation of this pattern to non-trivial numbers.

**LDA and College Mathematics**Suppose a student arrives at college knowing those aspects of division discussed above but not having ever learned LDA. Will s/he have any problems in mathematics or engineering courses without the ability to do LDA? Of course not. One doesn't know whether to laugh or cry when a college or university mathematician says that s/he knows what mathematics students arriving at university need to know and that this includes LDA. Not only can this not possibly be correct but, more generally, as I wrote some years ago, it would be better if students entering college "knew even less than they do now in terms of factual knowledge if only they knew something about the mathematical enterprise – what it means to prove something, why certain approaches tend to be effective in solving certain kinds of problems, what it means to do mathematics instead of just pushing symbols around on a piece of paper" [5].

Whether students know any particular fact or algorithm is far, far less important than whether they are aware of what mathematics is and how it can be learned. A student who knows these things but not LDA could quickly rectify this gap if someone really believed it necessary to do so. But a student who knows LDA but who considers mathematics to be just a collection of facts, theorems and procedures will be at a large disadvantage.

- Other Issues
- Place value. Students must understand place value
long before they study division. Could a study of LDA
or SDA or anything about division add substantially to
this understanding? I think not and the tendentious
attempt to do so in [3] supports this.

- Real numbers. Of course, as explained in [3], LDA
can serve as an entrée to an explanation of what real
numbers are. But having discussed, as above, how
rational numbers lead naturally to repeating decimals,
it is easy enough to introduce the notion of
irrationals. Indeed, at the middle school level,
probably no more than this should be done.

- Number sense. Although hard to define precisely,
almost everyone agrees that elementary and secondary
school mathematics should aim at developing number
sense, that is general familiarity with numbers and how
they can be manipulated. There is no way to develop
number sense other than to spend lots of time getting
your hands dirty with numbers. But whether it is
better to do so by learning the classical algorithms of
pencil-and-paper arithmetic or by a combination of
mental arithmetic, calculator use and problem solving
is something reasonable people may disagree about but
about which no one has a definitive answer.

- The Beauty of LDA. "The standard algorithms [i.e.,
those of pencil-and-paper arithmetic] are the only
collection of beautiful, serious, mathematical theorems
you can teach a child in K-6" a research mathematician
wrote to me recently [6]. I think most readers of this
piece would agree with this, perhaps with "the only"
replaced by "a". And as someone who has written two
books with "algorithms" in the title, I would certainly
not disagree with anyone who wished to introduce
children as early as possible to the notion of an
algorithm.
But it cannot be the case that, even with teachers capable of the presentation of these algorithms as "beautiful, serious, mathematical theorems", any but the smallest fraction of 10 and eleven year-olds will appreciate pencil-and-paper arithmetic as such. Even in K-6 it may be possible to serious mathematical ideas such as the notion of what a proof is. But, whatever the values of teaching the algorithms of pencil-and- paper arithmetic, instilling an appreciation of mathematical esthetics in students at this level will be possible, if at all, only in that small fraction of 1% who will become research mathematicians.

- Place value. Students must understand place value
long before they study division. Could a study of LDA
or SDA or anything about division add substantially to
this understanding? I think not and the tendentious
attempt to do so in [3] supports this.

**Conclusion**

The essential case against long division is that there is no case for it as we have tried to indicate above by showing that for each supposed advantage of using LDA, there is a simpler method that teaches the same lessons as LDA. Indeed, there may be reasons adduced for supporting the teaching of LDA other than those given in the previous section. But, I suggest, all such are amenable to straightforward refutations. Thus, Cockcroft [2] was correct in 1982 and still is today.

The status of LDA today is much like the status of the pencil-and-paper algorithm used to teach the computation of square roots to elementary schoolchildren until, I suppose, the 1960s or 1970s when this algorithm was finally consigned to the graveyard of once great but no longer significant algorithms. (Indeed, why that square root algorithm worked was opaque to almost all students.) This doesn't mean, of course, that the notion of a square root should no longer be taught. It means only that teaching an algorithm to compute the square root is no longer an appropriate topic for middle school mathematics. (Iteration is an important mathematical idea that should be introduced in secondary school if not in middle school. When it is introduced, students can learn - and understand - the algorithm for computing square roots by Newton's method.) So it is with LDA today. All the middle school student – or anyone else - needs to learn about division can be taught without LDA.

Yet it still is claimed that no one should be expected to "make sense out of arithmetic if they can't do long division" meaning LDA [7]. What basis can there be for such a dogmatic statement? Arithmetic, just like virtually any other topic in mathematics, can be made sense of from a variety of approaches. The truth is that few can be expected to make sense of arithmetic if elementary and middle school teachers are supposed to act as if the calculator and computer had never been invented.

**References**

- Ralston, A. Let's Abolish Pencil-and-Paper
Arithmetic,
*Journal of Computers in Mathematics and Science Teaching,***18**, 1999, pp. 173-194. - Cockcroft, W., et al.
*Mathematics Counts,*Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty's Stationery Office, 1982, p. 114. - Klein, D. and R. J. Milgram. The Role of Long Division in the K-12 Curriculum, http://math.stanford.edu/ftp/milgram/long-division-try- again.doc, 2000.
- Wu, H-H. Chapter 1: Whole Numbers (Draft), http://www.math.berkeley.edu/~wu/EMI1c.pdf.
- Ralston, A. Questions about Precollege Mathematics
Education,
*Mathematics Intelligencer,***9**, 1987, pp. 65-67. - Wilson, W. S. email 3 March 2005.
- Rude, B. The Case for Long Division, http://www.brianrude.com/calndv.htm, 2004.