In the context of school mathematics, an algorithm is a finite sequence of explicitly defined, step-by-step computational procedures which ends in a clearly defined outcome. The purpose of this chapter is to give an overview of the so-called standard algorithms for the four arithmetic operations among whole numbers. The succeeding chapters will provide the mathe- matical explanations.
At the outset, we should make clear that there is no such thing as the unique standard algorithm for any of the four operations +, −, ×, ÷, because minor variations have been incorporated into the algorithms by various coun- tries and ethnic groups. It may also be mentioned in passing that computer programs often make use of diverse algorithms that are derived from vari- ous cultures’ pencil-and-paper algorithms.1 Such variations notwithstand- ing, the underlying mathematical ideas always remain the same and, to the extent that our focus is on the underlying mathematical ideas and not the explicit procedures, the nomenclature of “standard algorithms” is eminently justified. Now, to say that we will focus on the mathematical ideas is not to say that the algorithms themselves—the computational procedures—are of no interest. On the contrary, they are, because computational techniques are an integral part of mathematics. Furthermore, the conciseness of these algorithms, especially the multiplication algorithm and the long division al- gorithm, is a marvel of human invention, and one of the goals of this chapter is to make sure that you come away with a renewed respect for them. An appreciation of the standard algorithms from a slightly different perspective is given in section 11.3, Arithmetic in Base 7, (page 161).
1I am indebted to Ken Ross for this observation.
57 http://dx.doi.org/10.1090/mbk/079/03
A fundamental question about these arithmetic algorithms is, Why should you bother to learn them? Take a simple example: What is 17× 12? By definition, this is 12 added to itself 17 times and, in the 1990s, there was a curriculum that would have you count 17 piles of birdseed with 12 in each pile to get an answer. Indeed, if we are limited to such simple computations, one may be able to get away with not knowing any algorithm at all. But we are concerned with the computations for all whole numbers, no matter how large. For example, what about 34,609 × 549,728? Are you going to tell your students to count 34,609 piles of birdseed with 549,728 in each pile? So you see that in this case, a shortcut is clearly called for. This is where the algorithms come in: they provide a shortcut in lieu of direct counting. It takes mathematical insight into the Hindu-Arabic numeral system to arrive at these shortcuts. To learn the algorithms is to learn such insight.
The preceding discussion also explains why we would be interested in the efficiency of an algorithm, i.e., how to get the answer as simply and as quickly as possible. At this point you may ask, Why worry about efficiency if pushing buttons on a calculator is a very efficient way to perform a com- putation such as 34,609× 549,728? There are at least two reasons why, from the point of view of mathematics education, we cannot afford to let students be completely dependent on the calculator for whole-number computations. First, without a firm grasp of the place value of our numeral system and the logical underpinning of the algorithms, it would be impossible to detect mistakes caused by pushing the wrong buttons on a calculator.2 A more important reason is that in mathematics learning a fact is synonymous with learning why it is true. In the case at hand, learning the reasoning behind these efficient algorithms is a very compelling way to acquire many of the fundamental skills in mathematics, including abstract reasoning with the basic laws of operations in Chapter 2 and the ability to make deductions from precise definitions. These are skills that are absolutely essential for the understanding of fractions and decimals in the subsequent chapters and, looking further ahead, for the understanding of algebra in middle school. One can flatly state that, if students do not feel comfortable with the kind of mathematical reasoning used to justify the standard algorithms for whole numbers, then their chance of success in algebra is minimal.
More can be said along this line. If we want to expose students to mathematical reasoning early, then exposing them to the inner workings of these algorithms would be a splendid starting point. A dominant theme that runs through these algorithms is one that is also part of the basic tools of research mathematicians, namely, that whenever possible, break down a
2I trust that it would be unnecessary to recount the many horror stories related to a finger- on-the-wrong-button computations.
3. The Standard Algorithms 59
complicated task into simple subtasks. To be specific, we formulate a leitmotif of the standard algorithms:
Leitmotif. To perform a computation with multidigit numbers, break it down into several steps so that each step (when suitably interpreted) is a computation involving only a single digit.
Therefore, a virtue of the standard algorithms is that, when they are prop- erly executed, they allow students to ignore the individual numbers being computed, no matter how large, and concentrate instead on one digit at a time. This is an excellent example of the kind of abstract thinking that is critical to success in mathematics learning. If students can learn from this leitmotif how to break down the complex into the simple, they will gain a foothold in mastering algebra and advanced mathematics.
Ironically, it is precisely this virtue of being able to perform whole num- ber computations by ignoring place value that has stirred up controversy in mathematics education. One objection to the teaching of the standard algorithms is that by making children focus on one digit at a time, they lose all sense of place value and consequently become prone to computational errors. The other is that the routine nature of the single-digit computations promotes the suspension of thinking, and if anything can be done without thinking, then it does not belong in a mathematics classroom. As a class- room teacher, you must confront both kinds of erroneous thinking.
The fear that teaching the standard algorithms would cause a loss of the sense of place value can only be founded on the common educational practice of teaching the standard algorithms without also teaching the rea- soning underlying these algorithms. The fact that such harmful practices are common is well known, but less known is the fact that universities have not given prospective teachers the mathematical support they need in order to get them out of these harmful practices (see [Wu99b] for a more ex- tended discussion). A main reason for the writing of the present book is, in fact, to address this very issue of mathematical support for teachers. This is why we have stressed throughout this book the importance of mathematical reasoning. Furthermore, while educators may look askance at the routine and nonthinking nature of these algorithms, the fact remains that it is the routine nature that accounts for their usefulness. To teach these algorithms without emphasizing their routine character is to miss the point of these algorithms completely, to say nothing of falsifying the mathematics.
The other objection is based on the perception that if anything can be done without thinking, then it does not belong in a mathematics classroom. This is wrong. If mathematicians are forced to do mathematics by having to think every step of the way, then little mathematics of value would ever get done and all research mathematics departments would have to close shop. What is closer to the truth is that deep understanding of a topic tends to
reduce many of its sophisticated processes to simple mechanical procedures. The ease of executing these mechanical procedures then frees up mental en- ergy to make possible the conquest of new topics through imagination and mathematical reasoning. In turn, many of these new topics will (eventually) be themselves reduced to routine or nearly routine procedures, and the pro- cess repeats itself. There is nothing to fear about the ability to execute a correct mathematical procedure without thinking unless the fluency of exe- cution is not supported by a thorough understanding of why the procedure is valid. A teacher’s charge in the classroom is therefore to promote both the facility with procedures and the ability to reason. The teaching of these algorithms must emphasize both their routine, nonthinking nature as well as the logical reasoning that lies behind the procedures.
The preceding discussion is about the kind of mathematical understand- ing teachers of mathematics must have in approaching the basic arithmetic algorithms. The pedagogical issue of how to introduce these algorithms to children in the early grades is something that lies outside the scope of this book and needs to be treated separately. See, however, [Wu99a] for a discussion of this issue from a mathematical point of view concerning the addition and multiplication algorithms.