1. Children’s Understanding of Multiplication and Division: Novel effects identified through a meta-analysis of 7 studies Katherine M. Robinson and Adam K. Dubé
INTRODUCTION
Children’s conceptual understanding of multiplication and division is an important precursor for the development of later mathematical skills (Kilpatrick et al. 2001). However, children’s conceptual understanding of these operations lags significantly behind addition and subtraction (National Mathematics Advisory Panel, 2008) as does research on multiplication and division concepts (Robinson & LeFevre, 2012) but a growing body of research on the inversion and associativity concepts exists.
For 3x28÷28, if children understand that multiplication and division are inversely related then no calculations are required to quickly and accurately solve the problem (Robinson & Ninowski, 2003). Children with partial understanding of the inversion concept may use a negation strategy involving multiplying and then realizing that dividing will negate the multiplication.
For 3x28÷7, if children understand that multiplication and division are associatively related then the problem can be quickly and accurately solved by performing the division part first (28÷7=4, 4x3=12) instead of by going left-to-right (Robinson et al., 2006).
DISCUSSION
By combining the data from 7 studies, developmental differences in the understanding of inversion and associativity have been clarified suggesting that increased experience and education are of benefit to most students’ understanding of multiplicative concepts.
There is new evidence of an increasing gender gap in children’s understanding of arithmetic concepts which could explain later gender differences in complex mathematical skills (Stoet & Geary, 2013),
Finally, for the first time there is evidence that some children understand the concept of associativity better than the inversion concept. This finding contradicts previous conclusions that inversion is the easier concept for children and adults.
Overall, the results of this meta-analysis suggest that the pathways to arithmetic concept development are more varied than previously thought.
PRESENT STUDY
For the first time, data from 7 studies with children conducted between are analyzed together for a more robust look at children’s understanding of the inversion and associativity concepts on multiplication and division problems.
RESULTS
Grade Differences: CLARIFIED
Unlike results from individual studies, pattern of grade differences much clearer (see Figure 1). Grade 8 students have the strongest understanding of both Inversion and Associativity and Grade 7 students the most frequent use of Negation suggesting that they are in a period of transition from little to complete understanding of the inversion concept.
Gender Differences: NEW
For the first time, gender differences emerged (see Figure 1) with increasingly better understanding of inversion for males while females fell increasingly behind .
Clusters: NEW CLUSTER IDENTIFIED
In addition to the original Inversion, Negation, and No Concept clusters, (see Figure 2) we also found a High Dual cluster resembling the Dual Concept group previously identified and a new Low Dual cluster who understand both concepts but (a) to a lesser extent than the High Dual cluster and (b) used associativity more than inversion which has not been found previously.
METHODS
Participants: sample size median age (yrs: mos) age range
Grade (82 males) 11: :1-12:4
Grade (89 males) 12:6 11:
Grade (87 males) 13: :1-14:9
Arithmetic Problems:
10-16 Inversion problems (3 x 5 ÷ 5, 3 x 25 ÷ 25)
10-16 Associativity problems (2 x 6 ÷ 3, 2 x 24 ÷ 3)
Measure (reported here):
Immediately Retrospective Strategy Reports
7 STUDIES
Dubé, A. K. (2014). Adolescents’ understanding of inversion and associativity. Learning and Individual Differences, 36,
Dubé, A. K., & Robinson, K. M. (2010). Accounting for individual differences in inversion shortcut use. Learning and Individual Differences, 20,
Robinson, K. M. & Dubé, A. K. (2009a). The multiplication and division inversion concept: Relationships between procedural, conceptual, and factual knowledge. Cognitive Development, 24,
Robinson, K. M. & Dubé, A. K. (2009, July). Children’s inversion and associativity concepts: Can brief instruction improve conceptual understanding? Paper presented at the joint meeting of the Experimental Psychology Society and the Canadian Society for Brain, Behaviour, and Cognitive Science, York, UK.
Robinson, K. M., & Dubé, A. K. (2009, August). Children’s use of arithmetic shortcuts on three-term multiplication and division problems. Paper presented at the biennial meeting of the European Association for Research on Learning and Instruction, Amsterdam, ND.
Robinson, K. M., & Dubé, A. K. (2011, June). Children’s understanding of multiplicative concepts, inhibition, and fluency. Paper presented at the biennial meeting of the Jean Piaget Society, Berkeley, CA.
Robinson, K. M., Ninowski, J. E., & Gray, M. L. (2006). Children’s understanding of the arithmetic concepts of inversion and associativity. Journal of Experimental Child Psychology, 94,
PREVIOUS STUDIES
Adult and child studies have shown that inversion is easier than associativity.
Adults have better conceptual knowledge but grade differences inconsistent across child studies.
3. Four clusters of individuals (children & adults) have been identified based on their conceptually-based strategy use (Robinson & LeFevre, 2012).
Dual Concept: understand both concepts
Inversion Concept: understand inversion
Negation: use the negation strategy demonstrating partial understanding of inversion
No Concept: solve most problems from left-to-right demonstrating little understanding of inversion or associativity.
4. Most children are in the No Concept cluster.
Partially funded by an NSERC discovery grant.
Correspondence to
Figure 1. Inversion, Negation, and Associativity mean percentages and standard errors for Grade 6, 7, & 8 males and females.
Figure 2. Inversion, Negation, and Associativity mean percentages and standard errors for the No Concept (n=243), Negation (n=146), Inversion (n=76), Low Dual (n=23), and High Dual (n=52) Concept clusters.