Conceptual and Procedural Knowledge of a Mathematics Problem: Their Measurement and Their Causal Interrelations

Conceptual and Procedural Knowledge of a Mathematics Problem: Their Measurement and Their Causal Interrelations Michael Schneider (mschneider@mpib-berlin.mpg.de) Elsbeth Stern (stern@mpib-berlin.mpg.de) Max Planck Institute for Human Development Lentzeallee 94, 14195 Berlin, Germany possibility, i.e. their Iterative Model: there may be bi- directional causal links between conceptual and procedural knowledge. Increase in one kind of knowledge will, then, prompt increase in the other one as well. Abstract Some learning theories see conceptual knowledge as a source of children’s procedural knowledge. Others assume the opposite to be true or posit bi-directional causal relations. Empirical tests of these assumptions are hampered by the lack of knowledge on how to obtain valid measures of these two constructs. We assessed four different measures of both constructs before and after an intervention and modelled the two kinds of knowledge as underlying latent factors in SEM. This enabled us to test the quality of our measures as well as the adequacy of the above-mentioned assumptions. Conceptual knowledge was a source of children’s procedural knowledge, but not vice versa. In contrast to procedural knowledge, conceptual knowledge could be assessed with high internal consistency. The Measurement Problem Keywords: learning theories; conceptual knowledge; procedural knowledge; declarative knowledge; SEM. Introduction Several theories of learning and cognition posit that our behaviour is shaped by at least two different kinds of knowledge: one providing an abstract understanding of the principles and relations between pieces of knowledge in a certain domain, and another one enabling us to quickly and efficiently solve problems. In recent empirical research on mathematics learning the former is frequently named conceptual knowledge, while the latter is labelled procedural knowledge (e.g., Baroody, 2003). Cognitive models of the relations between these different kinds of knowledge can facilitate our general understanding of the human mind, but may also be helpful for designing the contexts in which knowledge is to be conveyed. However, different theories make different predictions as to the interrelations between conceptual and procedural knowledge. One major difference is described by Rittle- Johnson, Siegler, and Alibali (2001) who distinguish between concepts-first and procedures-first theories. According to concepts-first theories, children will initially acquire conceptual knowledge, for example by listening to verbal explanations, and will then, by practice, derive procedural knowledge from it. Procedures-first theories, on the contrary, posit that children initially acquire procedural knowledge in a specific domain, for example by trial-and- error learning, and then gradually abstract conceptual knowledge from it by reflection. Based on the fact that there is empirical evidence for both kinds of theories, Rittle-Johnson et al. propose a third Despite these controversies and the importance of the field, there are comparatively few empirical studies addressing the relationships between the two kinds of knowledge. As a review by Rittle-Johnson and Siegler (1998) shows, these studies yielded partly inconsistent results and had serious methodological limitations. While some of these limitations, such as the use of correlational designs and non-gradual, dichotomous measures, can easily be overcome, one problem was of a more general nature: it is as yet unclear how conceptual and procedural knowledge can be measured independently of each other and with a sufficient degree of validity. Since it would seem that some cognitive procedure is always needed to derive actions from (static) conceptual knowledge representations, how can we find out whether any given action, for example a subject’s response to a test item, is rooted in conceptual or in procedural knowledge or, to different degrees, in both? While this question is largely ignored in the literature, Rittle-Johnson et al. (2001) gave a well-founded answer in the context of a study they conducted to test the Iterative Model. They measured children’s conceptual and procedural knowledge before and after an intervention that was designed to increase both kinds of knowledge, and showed that children’s initial conceptual knowledge predicted gains in procedural knowledge, and gains in procedural knowledge predicted improvements in conceptual knowledge. They distinguished between assessments of conceptual and procedural knowledge on the basis of the novelty of the tasks at posttest, the assumption being that children will apply acquired procedural knowledge when solving the routine tasks they already know from the intervention, but will resort to conceptual understanding when challenged to produce new solutions to hitherto unknown transfer tasks. But even this elaborated answer fails to account for certain parts of the problem because we cannot tell with any certainty that children do not use conceptual knowledge to generate answers to routine tasks. Furthermore, if there are routine tasks only in the posttest, conceptual and procedural knowledge can only be independently assessed for the