Concept–procedure interactions in children’s addition and subtraction

https://doi.org/10.1016/j.jecp.2008.07.008Get rights and content

Abstract

A 3-week problem-solving practice phase was used to investigate concept–procedure interactions in children’s addition and subtraction. A total of 72 7- and 8-year-olds completed a pretest and posttest in which their accuracy and procedures on randomly ordered problems were recorded along with their reports of using concept-based relations in problem solving and their conceptual explanations. The results revealed that conceptual sequencing of practice problems enhances children’s ability to extend their procedural learning to new unpracticed problems. They also showed that well-structured procedural practice leads to improvement in children’s ability to verbalize key concepts. Moreover, children’s conceptual advances were predicted by their initial procedural skills. These results support an iterative account of the development of basic concepts and key skills 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).

References (62)

  • N.C. Jordan et al.

    Arithmetic fact mastery in young children: A longitudinal investigation

    Journal of Experimental Child Psychology

    (2003)
  • J. LeFevre et al.

    What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2

    Journal of Experimental Child Psychology

    (2006)
  • C. Rasmussen et al.

    Use of the mathematical principle of inversion in young children

    Journal of Experimental Child Psychology

    (2003)
  • K.M. Robinson et al.

    Children’s understanding of the arithmetic concepts of inversion and associativity

    Journal of Experimental Child Psychology

    (2006)
  • A.J. Baroody

    Children’s relational knowledge of addition and subtraction

    Cognition and Instruction

    (1999)
  • A.J. Baroody et al.

    The development of the commutativity principle and economical addition strategies

    Cognition and Instruction

    (1984)
  • A.J. Baroody et al.

    The relationship between initial meaningful and mechanical knowledge of arithmetic

  • A.J. Baroody et al.

    Children’s use of mathematical structure

    Journal for Research in Mathematics Education

    (1983)
  • G.M. Boulton-Lewis

    Young children’s representations and strategies for subtraction

    British Journal of Educational Psychology

    (1993)
  • G.M. Boulton-Lewis et al.

    Young children’s representations and strategies for addition

    British Journal of Educational Psychology

    (1994)
  • B. Butterworth et al.

    Basic multiplication combinations: Passive storage or dynamic reorganization?

  • J.P. Byrnes et al.

    Role of conceptual knowledge in mathematical procedural learning

    Developmental Psychology

    (1991)
  • K.H. Canobi et al.

    Number words in young children’s conceptual and procedural knowledge of addition, subtraction, and inversion

    Cognition

    (2008)
  • K.H. Canobi et al.

    The role of conceptual understanding in children’s addition problem-solving

    Developmental Psychology

    (1998)
  • K.H. Canobi et al.

    Young children’s understanding of addition concepts

    Educational Psychology

    (2002)
  • K.H. Canobi et al.

    Patterns of knowledge in children’s addition

    Developmental Psychology

    (2003)
  • T.P. Carpenter et al.

    The acquisition of addition and subtraction concepts in Grades One through Three

    Journal for Research in Mathematics Education

    (1984)
  • C.A. Christensen et al.

    The effectiveness of instruction in cognitive strategies in developing proficiency in single-digit addition

    Cognition and Instruction

    (1991)
  • R. Cowan

    Does it all add up? Changes in children’s knowledge of addition combinations, strategies, and principles

  • R. Cowan et al.

    Do they know what they are doing? Children’s use of economic addition strategies and knowledge of commutativity

    Educational Psychology

    (1996)
  • A. Dowker

    Individual difference in arithmetic: Implications for psychology, neuroscience, and education

    (2005)
  • Cited by (89)

    • Cognitive predictors of children's arithmetic principle understanding

      2023, Journal of Experimental Child Psychology
      Citation Excerpt :

      The roles of working memory and processing speed in the acquisition of APs could be deduced from some previous experimental findings. Various researchers have demonstrated that children can learn APs after solving conceptually sequenced arithmetic problems/equations, for example, solving “6 + 3 = __” immediately after “3 + 6 = __” (Canobi, 2009; Prather, 2012; Siegler & Stern, 1998). To discover the regularities underlying different arithmetic equations, one needs to store various arithmetic equations in one’s mind for comparison.

    • The influence of sex on the relations among spatial ability, math anxiety and math performance

      2022, Trends in Neuroscience and Education
      Citation Excerpt :

      Well-learned small operations are retrieved, automatically, from the language-based format [69]. 2) Procedural knowledge requires knowledge of procedures- solutions of mathematical operations and rules such as multiplications in whole hundreds and, therefore, strongly activated the language-based format [70,71]. 3) Number line knowledge-based on number space mapping and related to spatial abilities and the preverbal spatial representation of quantities [72].

    View all citing articles on Scopus
    View full text