Children's understanding of the inverse relation between multiplication and division

Reconsidering conceptual knowledge: Heterogeneity of its components

Cognitive arithmetic classically distinguishes procedural and conceptual knowledge as two determinants of the acquisition of flexible expertise. Whereas procedural knowledge relates to algorithmic routines, conceptual knowledge is defined as the knowledge of core principles, referred to as fundamental structures of arithmetic. To date, there is no consensus regarding their number, list, or even their definition, partly because they are difficult to measure. Recent findings suggest that among the most complex of these principles, some might not be "fundamental structures" but rather may articulate several components of conceptual knowledge, each specific to the arithmetic operation involved. Here, we argue that most of the arithmetic principles similarly may rather articulate several core concepts specific to the operation involved. Data were collected during a national mathematics contest based on an arithmetic game involving a large sample of 9- to 11-year-old students (N = 11,243; 53.1% boys) over several weeks. The purpose of the game was to solve complex arithmetic problems using five numbers and the four operations. A principal component analysis (PCA) was performed. The results show that both conceptual and procedural knowledge were used by children. Moreover, the PCA sorted conceptual and procedural knowledge together, with dimensions being defined by the operation rather than by the concept. This implies that "fundamental structures" rather regroup different concepts that are learned separately. This opens the way to reconsider the very nature of conceptual knowledge and has direct pedagogical implications.

Conceptual knowledge of the associativity principle: A review of the literature and an agenda for future research

Individuals use diverse strategies to solve mathematical problems, which can reflect their knowledge of arithmetic principles and predict mathematical expertise. For example, '6 + 38 - 35' can be solved via '38 - 35 = 3' and then '3 + 6 = 9', which is a shortcut-strategy derived from the associativity principle. The shortcut may be critical for understanding algebra, however approximately 50% of adults fail to use it. We review the research to consider why the associativity principle is challenging and highlight an important distinction between shortcut identification and execution. We also discuss how domain-specific skills and domain-general skills might play an important role in shortcut identification and execution, and provide an agenda for future research.

Multifaceted assessment of children's inversion understanding

The current study was aimed at examining various theoretical issues concerning children's inversion understanding (i.e., its factor structure, development, and relation with mathematics achievement) using a multifaceted assessment. A sample of 110 fourth to sixth graders was evaluated in three different measures of inversion understanding: evaluation of examples, explicit recognition, and application of procedures. The participants were also evaluated on their mathematics achievement. A one-factor structure best explains inversion understanding involving different arithmetic operations. Grade-related improvements were observed in some facets of inversion understanding. Latent profile analysis using the three inversion measures revealed seven classes of children with different inversion profiles. Furthermore, classes with better inversion understanding also had higher mathematics achievers. The current findings provide evidence to support the multifaceted nature of inversion understanding, grade-related improvements in children's inversion understanding as well as the relation between inversion understanding and mathematics achievement.

Children's understanding of multiplication and division: Insights from a pooled analysis of seven studies conducted across 7 years

Research suggests that children's conceptual understanding of multiplication and division is weak and that it remains poor well into the later elementary school years. Further, children's understanding of fundamental concepts such as inversion and associativity does not improve as they progress from grades 6 to 8. Instead, some children simply possess strong understanding while others do not. Other studies have identified an increase across these grades. The present investigation analyses data from seven studies of Grade 6 (n = 226), Grade 7 (n = 221), and Grade 8 (n = 216) children's three-term problem-solving (e.g., 3 × 24 ÷ 24 and 3 × 24 ÷ 6) and provides a unified account of multiplication and division understanding, one in which grade differences and individual variability coexist and are moderated by sex. Statement of contribution What is already known on this subject? Children's conceptual understanding of multiplication and division is weak and it is unclear whether it increases across the key grades of 6-8. Understanding of the inversion and associativity concepts is characterized by high individual variability, but grade and sex have never been found to be a contributing factor. What does this study add? A meta-analysis of seven data sets (n = 643) indicates that grade differences and individual variability coexist and are moderated by sex. Understanding increases across grade only for boys, but an equal number of boys and girls are in the top 10% of conceptual problem-solvers.

Understanding arithmetic concepts: Does operation matter?

Most research on children's arithmetic concepts is based on (a) additive concepts and (b) a single concept leading to possible limitations in current understanding about how children's knowledge of arithmetic concepts develops. In this study, both additive and multiplicative versions of six arithmetic concepts (identity, negation, commutativity, equivalence, inversion, and associativity) were investigated in Grades 5, 6, and 7. The multiplicative versions of the concepts were more weakly understood. No grade-related differences were found in conceptual knowledge, but older children were more accurate problem solvers. Individual differences were examined through cluster analyses. All children had a solid understanding of identity and negation. Some children had a strong understanding of all the concepts, both additive and multiplicative; some had a good understanding of equivalence or commutativity; and others had a weak understanding of commutativity, equivalence, inversion, and associativity. Associativity was the most difficult concept for all clusters. Grade did not predict cluster membership. Overall, these results demonstrate the breadth of individual variability in conceptual knowledge of arithmetic as well as the complexity in how children's understanding of arithmetic concepts develops.