Algorithms in Everyday Mathematics

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

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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