Cognitive Skills and Arithmetic Performance: A Reply to Steffe and Cobb

Abstract

Investigating relationships between major domains of knowledge is a complex task. A number of fundamental questions often accompany such attempts, and our study on the relationship between cognitive skills and arithmetic performance is no exception. Steffe and Cobb (1983) identified some of these questions. This interchange, we hope, will help to clarify the issues and the alternative views. Two central questions seem to be of primary concern: (a) What is the relationship between theory and data? and (b) How does one measure whether certain cognitive skills are prerequisites for some level of arithmetic performance? With regard to the theory/data issue, Steffe and Cobb are concerned primarily with how one's theoretical perspective affects the collection and interpretation of empirical data. They argue that observation is not independent of theory and that explanations of observed behavior are influenced by basic theoretical assumptions. We share this orientation. In fact, the first quotation in Steffe and Cobb's critique (from Hiebert, Carpenter, & Moser, 1982, p. 96) was intended as a caution to the reader that our conclusions were not a theory-free statement about the relationship between cognitive abilities and arithmetic performance. As we acknowledge, there may very well be significant relationships operating at some theoretical level that were not assessed in our study. Future advances in how to measure other, potentially important, constructs may make it possible to investigate these relationships. The other side of the theory/data question pertains to the way in which data inform theory. Theories about how children think are developed and modified by testing them against observations of children's behavior (made by oneself and by others with different theoretical orientations). Our specific question in the study was whether children's performance on commonly used Piagetian tasks would provide any insight into the way in which these cognitive constructs operate in children's treatment of arithmetic problems. There are many theoretical statements about how these constructs relate to arithmetic performance; we were interested in whether the Piagetian tasks could help us understand the nature of these relationships. To summarize, we agree that there is an intricate relationship between theory and data, that data do not have a life of their own. Consistent with this