Algorithms in Everyday Mathematics

algorithms as substitutes for thinking and common sense.

Alternative Algorithms

Over the centuries, people have invented many algorithms for the basic arithmetic operations.

Each of these historical algorithms was developed in some context. For example, one does not

need to know the multiplication tables to do "Russian Peasant Multiplication" - all that is

required is doubling, halving, and adding. Many historical algorithms were "standard" at some

time and place, and some are used to this day. The current "European" method of subtraction, for

example, is not the same as the method most Americans learned in school.

The U.S. standard algorithms-those that have been most widely taught in this country in the

past 100 years-are highly efficient for paper-and-pencil computation, but that does not

necessarily make them the best choice for school mathematics today. The best algorithm for one

purpose may not be the best algorithm for another purpose. The most efficient algorithm for

paper-and-pencil computation is not likely to be the best algorithm for helping students

understand the operation, nor is it likely to be the best algorithm for mental arithmetic and

estimation. Moreover, if efficiency is the goal, in most situations it is unlikely that any paperand-

pencil algorithm will be superior to mental arithmetic or a calculator.

If paper-and-pencil computation is to continue to be part of the elementary school mathematics

curriculum, as the authors of Everyday Mathematics believe it should, then alternatives to the

U.S. standard algorithms should be considered. Such alternatives may have better cost-benefit

ratios than the standard algorithms. Historical algorithms are one source of alternatives. Studentinvented

procedures are another rich source. A third source is mathematicians and mathematics

educators who are devising new methods that are well adapted to our needs today. The Everyday

Mathematics approach to computation uses alternative algorithms from all these sources.

In Everyday Mathematics, as students explain, compare, and contrast their own invented

procedures, several common alternative methods are identified. Often these are formalizations of

approaches that students have devised. The column-addition method, for example, was shown

and explained to the Everyday Mathematics authors by a first grader. Other alternative

algorithms, including both historical and new algorithms, are introduced by the teacher or the

materials. The partial-quotients method, for example, first appeared in print in Isaac

Greenwood's Arithmeticks in 1729.

Many alternative algorithms, whether based on student methods or introduced by the teacher, are

highly efficient and easier to understand and learn than the U.S. traditional algorithms. For

example, lattice multiplication requires only a knowledge of basic multiplication facts and the