# 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.