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Martha Alibali

August 2005

As children develop their problem-solving abilities, what comes first: their conceptual knowledge or their procedural knowledge? Or do these grow in tandem?

The debate has long continued. But recent research shows that conceptual and procedural knowledge appear to develop in a hand-over-hand process. Gains in one type of knowledge support gains in the other, which in turn support gains in the first.

These findings emerge from recent research conducted by UW-Madison education professor Martha Alibali and colleagues.* Their study has implications for teaching, because understanding the process of knowledge change is a central goal in the study of development.

Conceptual knowledge is flexible and not tied to specific problem types and is therefore generalizable. It involves understanding the principles that govern a domain and interrelations between units of knowledge in a domain.

Procedural knowledge is the child’s ability to execute action sequences to solve problems. Procedural knowledge is tied to specific problem types and therefore is not widely generalizable.

Linking concepts and procedures

Competence in mathematics requires children to develop and link their knowledge of concepts and procedures. Competing theories have been proposed about the developmental relations between conceptual knowledge and procedural knowledge. According to concepts-first theories, children initially develop (or are born with) conceptual knowledge in a domain and then use this conceptual knowledge to generate and select procedures for solving problems in that domain. This theory and evidence in support of it have been used to justify reforms in mathematics education that focus on inculcating conceptual knowledge before teaching procedural knowledge.

In contrast, procedures-first theories hold that children first learn procedures for solving problems in a domain and later extract domain concepts from repeated experience solving the problems.

Alibali and her colleagues maintain that the debate over which type of knowledge develops first may obscure the gradual development of each type of knowledge and the interactions between them during development. The iterative model indicates how such a process may occur: Increases in one type of knowledge lead to increases in the other type of knowledge, which in turn lead to further increases in the first.

Problem representation may be key

Alibali has found that improved problem representation is one pervasive mechanism of cognitive development. Alibali defines problem representation as the internal depiction or recreation of a problem in working memory during problem solving. Students form a problem representation each time they solve a problem. Problem representation refers to this transitory, internal representation of individual problems.

Improved problem representation may underlie the relations between conceptual and procedural knowledge. First, it may underlie the link from conceptual knowledge to improved procedural knowledge. Children’s conceptual knowledge may guide their attention to relevant features of problems and help them to organize this information in their internal problem representation. This well-chosen problem representation may then support generation and use of effective procedures.

Second, improved problem representation may underlie the link from procedural knowledge to improved conceptual knowledge. Use of correct procedures could help children represent the key aspects of problems, which could led to improved conceptual understanding of the domain.

In their research, Alibali and colleagues evaluated the first pathway: the link from improved conceptual knowledge to improved problem representation to improved procedural knowledge.

Decimal fractions exercise

In a recent study of fifth-grade students, Alibali and colleagues evaluated the iterative model of the development of conceptual and procedural knowledge in two experiments on children’s learning about decimal fractions. Decimal fraction knowledge is a central component of mathematical understanding. But children struggle to understand decimal fractions, and some never master them. Interventions that eliminate misconceptions and improve understanding of decimal fractions are greatly needed, Alibali says. Because of the potential power of the number line for representing decimal fractions, Alibali and colleagues developed an intervention using number line problems.

To evaluate the iterative model, Alibali and colleagues assessed children’s conceptual and procedural knowledge of decimal fractions before and after a brief instructional intervention. In Experiment 1, they examined individual differences in prior knowledge and in amount of learning. The goal was to provide correlational support for each of the links in the iterative model. In Experiment 2, they manipulated support for forming correct problem representation during the intervention. The goal was to evaluate the causal link from formation of correct problem representation to improved procedural knowledge. The first experiment provided correlational evidence for the relations proposed within the iterative model. The second experiment provided causal evidence for one link in the model—that from improved problem representation to improved procedural knowledge.

In both experiments, children’s initial conceptual knowledge predicted gains in procedural knowledge, and the gains in procedural knowledge predicted improvements in conceptual knowledge. Correct problem representation was an important link between conceptual and procedural knowledge.

Neither conceptual nor procedural knowledge developed in an all-or-nothing fashion, with acquisition of one type of knowledge always preceding the other. Nor was either type of knowledge fully developed at the beginning or at the end of the study; rather, they appeared to develop in a gradual, hand-over-hand process. These iterative relations highlight the importance of examining conceptual and procedural knowledge together, Alibali says.

Implications for education

These findings have at least three important implications for education. First, competence in a domain requires knowledge of both concepts and procedures. Developing children’s procedural knowledge in a domain is an important avenue for improving children’s conceptual knowledge in the domain, just as developing conceptual knowledge is essential for generation and selection of appropriate procedures. Current reforms in education focus on teaching children mathematical concepts and often downplay the importance of procedural knowledge. Furthermore, some educators treat the relations between conceptual and procedural knowledge as unidirectional. But this study found that the relations between conceptual and procedural knowledge are bidirectional and that improved procedural knowledge can lead to improved conceptual knowledge.

A second implication of these findings is that identifying the learning processes of good learners and supporting these processes in students who use weaker methods can enhance children’s learning. A third instructional implication is that supporting correct representation of problems is an effective tool for improving problem-solving knowledge. Alibali develops this discussion to include the mechanisms of change in reasoning that involve expressing knowledge in language and gesture in “Mechanisms of change in the development of mathematical reasoning,” Advances in Child Development and Behavior (in press). Alibali can be contacted at mwalibali@wisc.edu.

Funding provided by the National Science Foundation, the National Institute of Child Health and Human Development, and the Spencer Foundation.

Material in this article originally appeared in different form in Journal of Educational Psychology, June 1, 2001, Vol. 93, Issue 2.

*Alibali’s colleagues in this study are Bethany Rittle-Johnson (Peabody College, Vanderbilt University) and Robert S. Siegler (Carnegie Mellon University).