Concept–procedure interactions in children’s addition and subtraction
Introduction
In addition and subtraction, there is a systematic and pervasive relationship between schoolchildren’s understanding of basic concepts and their procedural skills (Canobi, 2004, Canobi, 2005, Canobi et al., 1998, Canobi et al., 2003, Cowan, 2003, Cowan and Renton, 1996, Farrington-Flint et al., 2007, Martins-Mourão and Cowan, 1998, Rasmussen et al., 2003, Robinson et al., 2006), but the causal basis of this relationship is unclear. One promising way to address this issue is to explore how children learn from problem-solving practice. Previous research has established that children make considerable procedural gains as a result of experience in executing procedures to solve basic problems (e.g., Geary et al., 1991, Goldman et al., 1988, Siegler and Shrager, 1984) but has not addressed how such learning is related to the conceptual structure of the domain. For example, it is not known whether concept-based sequencing of practice problems affects children’s procedural gains and whether procedural practice leads to conceptual advances. Exploring these issues is likely to provide insight into mathematical development by indicating whether children’s conceptual understanding influences their procedural learning and whether they make conceptual inferences based on executing procedures.
Section snippets
Measuring conceptual and procedural knowledge
Procedural knowledge can be defined in terms of the skills required to solve individual mathematical problems. Procedures are routes to problem solutions. To arrive at an accurate characterization of children’s procedural skills in solving addition and subtraction problems, it is useful to explore their self-reported procedures along with their problem-solving accuracy on a set of randomly ordered problems (Robinson, 2001, Siegler, 1987, Siegler, 1989.
Conceptual understanding involves knowledge
Exploring concept–procedure interactions
Despite strong relations between schoolchildren’s conceptual understanding and their reported problem-solving procedures and accuracy (Canobi, 2004, Canobi, 2005, Canobi et al., 1998, Canobi et al., 2003), the nature of concept–procedure interactions in basic addition and subtraction is not yet well understood. For example, cross-sectional research indicates that children who correctly apply concepts such as inversion and commutativity to related problems solve randomly ordered problems with
The current study
In the current study, children were randomly allocated to either a conceptually sequenced condition, a randomly ordered condition, or a no-practice condition to explore concept–procedure interactions in basic addition and subtraction. Children’s procedural skill and conceptual understanding were tested before and after a practice phase. During the practice phase, children practiced their problem-solving procedures on basic two-term problems (e.g., 5 + 6) without discussing their solution
Participants
The participants were 41 boys and 31 girls with a mean age of 8 years 2 months (SD = 4 months). They attended primary schools in a middle socioeconomic status suburb of a large Australian city. The participants had previously learned how to solve single-digit addition and subtraction problems in school. Although there was some variation in their teachers’ approaches, all children had initially been taught to solve problems using advanced counting procedures such as counting on (e.g., adding 7 + 2
Pretest scores
Initially, children’s scores at pretest were examined so as to assess their entering procedural and conceptual competences in basic addition and subtraction.
Discussion
This study was designed to test whether there is an iterative relationship between conceptual knowledge and procedural knowledge in children’s addition and subtraction development by assessing the effects of problem-solving experience on key conceptual and procedural indexes. The research makes a significant new contribution by suggesting that in basic addition and subtraction, concept–procedure interactions influence both conceptual and procedural development. The outcomes of the practice
Acknowledgments
I thank the children, teachers, and parents who supported this study. I also thank Narelle Bethune for her dedicated work on data collection and Kate Reid for her assistance with coding. This research was supported under the Australian Research Council’s Discovery funding scheme (Project No. DP0449498).
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