Box 4–2 A common error in multidigit subtraction

this error.15 Further, when students learn a procedure without understanding, they need extensive practice so as not to forget the steps. If students do understand, they are less likely to forget critical steps and are more likely to be able to reconstruct them when they do. Shifting the emphasis to learning with understanding, therefore, can in the long run lead to higher levels of skill than can be attained by practice alone.

If students have been using incorrect procedures for several years, then instruction emphasizing understanding may be less effective.16 When children learn a new, correct procedure, they do not always drop the old one. Rather, they use either the old procedure or the new one depending on the situation. Only with time and practice do they stop using incorrect or inefficient methods.17 Hence initial learning with understanding can make learning more efficient.

When skills are learned without understanding, they are learned as isolated bits of knowledge.18 Learning new topics then becomes harder since there is no network of previously learned concepts and skills to link a new topic to. This practice leads to a compartmentalization of procedures that can become quite extreme, so that students believe that even slightly different problems require different procedures. That belief can arise among children in the early grades when, for example, they learn one procedure for subtraction problems without regrouping and another for subtraction problems with regrouping. Another consequence when children learn without understanding is that they separate what happens in school from what happens outside.19 They have one set of procedures for solving problems outside of school and another they learned and use in school—without seeing the relation between the two. This separation limits children’s ability to apply what they learn in school to solve real problems.

Also, students who learn procedures without understanding can typically do no more than apply the learned procedures, whereas students who learn

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