Algorithms in Everyday Mathematics

  An algorithm is a step-by-step procedure designed to achieve a certain objective in a finite time,
often with several steps that repeat or “loop” as many times as necessary. The most familiar
algorithms are the elementary school procedures for adding, subtracting, multiplying, and
dividing, but there are many other algorithms in mathematics.

Algorithms in School Mathematics

  The place of algorithms in school mathematics is changing. One reason is the widespread
availability of calculators and computers outside of school. Before such machines were invented,
the preparation of workers who could carry out complicated computations by hand was an
important goal of school mathematics. Today, being able to mimic a $5 calculator is not enough:
Employers want workers who can think mathematically. How the school mathematics
curriculum should adapt to this new reality is an open question, but it is clear that proficiency at
complicated paper-and-pencil computations is far less important outside of school today than in
the past. It is also clear that the time saved by reducing attention to such computations in school
can be put to better use on such topics as problem solving, estimation, mental arithmetic,
geometry, and data analysis (NCTM, 1989).
 

  Another reason the role of algorithms is changing is that researchers have identified a number of
serious problems with the traditional approach to teaching computation. One problem is that the
traditional approach fails with a large number of students. Despite heavy emphasis on paper-andpencil
computation, many students never become proficient in carrying out algorithms for the
basic operations. In one study, only 60 percent of U.S. ten-year-olds achieved mastery of
subtraction using the standard “borrowing” algorithm. A Japanese study found that only 56
percent of third graders and 74 percent of fifth graders achieved mastery of this algorithm. A
principal cause for such failures is an overemphasis on procedural proficiency with insufficient
attention to the conceptual basis for the procedures. This unbalanced approach produces students
who are plagued by “bugs,” such as always taking the smaller digit from the larger in
subtraction, because they are trying to carry out imperfectly understood procedures.
An even more serious problem with the traditional approach to teaching computation is that it
engenders beliefs about mathematics that impede further learning. Research indicates that these
beliefs begin to be formed during the elementary school years when the focus is on mastery of
standard algorithms (Hiebert, 1984; Cobb, 1985; Baroody & Ginsburg, 1986). The traditional,
rote approach to teaching algorithms fosters beliefs such as the following:
 • mathematics consists mostly of symbols on paper;
 • following the rules for manipulating those symbols is of prime importance;
 • mathematics is mostly memorization;
 • mathematics problems can be solved in no more than 10 minutes — or else they cannot
be solved at all;
 • speed and accuracy are more important in mathematics than understanding;
 • there is one right way to solve any problem;
 • different (correct) methods of solution sometimes yield contradictory results; and
 • mathematics symbols and rules have little to do with common sense, intuition, or the
real world.

  These inaccurate beliefs lead to negative attitudes. The prevalence of math phobia, the social
acceptability of mathematical incompetence, and the avoidance of mathematics in high school
and beyond indicate that many people feel that mathematics is difficult and unpleasant.
Researchers suggest that these attitudes begin to be formed when students are taught the standard
algorithms in the primary grades. Hiebert (1984) writes, “Most children enter school with
reasonably good problem-solving strategies. A significant feature of these strategies is that they
reflect a careful analysis of the problems to which they are applied. However, after several years
many children abandon their analytic approach and solve problems by selecting a memorized
algorithm based on a relatively superficial reading of the problem.” By third or fourth grade,
according to Hiebert, “many students see little connection between the procedures they use and
the understandings that support them. This is true even for students who demonstrate in concrete
contexts that they do possess important understandings.” Baroody and Ginsburg (1986) make a
similar claim: “For most children, school mathematics involves the mechanical learning and the
mechanical use of facts — adaptations to a system that are unencumbered by the demands of
consistency or even common sense.”

  A third major reason for changes in the treatment of algorithms in school mathematics is that a
better approach exists. Instead of suppressing children’s natural problem-solving strategies, this
new approach builds on them (Hiebert, 1984; Cobb, 1985; Baroody & Ginsburg, 1986; Resnick,
Lesgold, & Bill, 1990). For example, young children often use counting strategies to solve
problems. By encouraging the use of such strategies and by teaching even more sophisticated
counting techniques, the new approach helps children become proficient at computation while
also preserving their belief that mathematics makes sense. This new approach to computation is
described in more detail below.

  Reducing the emphasis on complicated paper-and-pencil computations does not mean that paperand-
pencil arithmetic should be eliminated from the school curriculum. Paper-and-pencil skills
are practical in certain situations, are not necessarily hard to acquire, and are widely expected as
an outcome of elementary education. If taught properly, with understanding but without demands
for “mastery” by all students by some fixed time, paper-and-pencil algorithms can reinforce
students’ understanding of our number system and of the operations themselves. Exploring
algorithms can also build estimation and mental arithmetic skills and help students see
mathematics as a meaningful and creative subject.