Abstract
Multiplication and division are conceptually inversely related: Each division problem can be transformed into as a multiplication problem and vice versa. Recent research has indicated strong developmental parallels between multiplication and division in primary school children. In this study, we were interested in (i) whether these developmental parallels persist into secondary school, (ii) whether similar developmental parallels can be observed for simple and complex problems, (iii) whether skill level modulates this relationship, and (iv) whether the correlations are specific and not driven by general cognitive or arithmetic abilities. Therefore, we assessed performance of 5th and 6th graders attending two secondary school types of the German educational system in simple and complex multiplication as well as division while controlling for non-verbal intelligence, short-term memory, and other arithmetic abilities. Accordingly, we collected data from students differing in skills levels due to either age (5th < 6th grade) or school type (general < intermediate secondary school). We observed moderate to strong bivariate and partial correlations between multiplication and division with correlations being higher for simple tasks but nevertheless reliable for complex tasks. Moreover, the association between simple multiplication and division depended on students' skill levels as reflected by school types, but not by age. Partial correlations were higher for intermediate than for general secondary school children. In sum, these findings emphasize the importance of the inverse relationship between multiplication and division which persists into later developmental stages. However, evidence for skill-related differences in the relationship between multiplication and division was restricted to the differences for school types.
Highlights
IntroductionChildren are introduced to addition, subtraction, multiplication, and division in elementary school and consolidate their arithmetic skills throughout secondary school
Basic arithmetic operations constitute a milestone of numerical development
We were interested in (i) whether these developmental parallels persist into secondary school, (ii) whether similar developmental parallels can be observed for simple and complex problems, (iii) whether skill level modulates this relationship, and (iv) whether the correlations are specific and not driven by general cognitive or arithmetic abilities
Summary
Children are introduced to addition, subtraction, multiplication, and division in elementary school and consolidate their arithmetic skills throughout secondary school. With schooling, they become more proficient and use more efficient strategies to solve both simple and complex arithmetic operations. The focus of the current study is on multiplication and division, two of the four basic operations that are mathematically inversely related. By inversion of the operands, each division problem (e.g., 28 ÷ 4 = 7) can be recast as a multiplication problem (e.g., 4 × 7 = 28) and vice versa. We looked at how this interaction is influenced by skill level as the correlation between multiplication and division may not be the same for all children
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