# The Relationship between Adults' Conceptual Understanding of Inversion and Associativity

## Article excerpt

Children's understanding of the mathematical concepts of inversion and associativity are positively related, as measured by the use of conceptually based shortcut strategies on 3-term inversion problems (i.e., a + b - b, d × e ÷ e) and associativity problems (i.e., a + b - c, d × e ÷ f; Robinson & Dubé. 2009; Robinson & Ninowski, 2003). Individuals who use the inversion shortcut (e.g., 3) are more likely to use the associativity strategy (e.g., 3 × 12 ÷ 14. 12 ÷ 4 = 3, 3 ÷ 3 = 9), which is almost never used by an individual who does not also use the inversion shortcut (Robinson & Dubé, 2009). One possible reason for this relationship is that directing attention to the right-most operation during problem solving may be required to prime the conceptually based shortcut strategies for both problem types. This study investigated the relationship between adults' understanding of inversion and associativity. Adults (N = 42) solved inversion and associativity problems in 1 of 2 conditions. The participants were either presented with the left-most operation and then the whole problem or presented with the right-most operation and then the whole problem. A positive relationship between the use of the conceptually based strategies was found, and it was strikingly similar to the relationship found in childhood. There was evidence that the presentation of the right-most operation first primed the inversion shortcut.

Keywords: arithmetic, inversion, associativity, multiplication, division

(ProQuest: ... denotes "strike-through" in the original text omitted.)

Having an understanding of the concepts of inversion and associativity contributes to an individual's understanding of other mathematical concepts, such as the additive nature of numbers and the relationship between operations (Bryant, Christie, & Rendu, 1999; Farrington-Flint, Canobi, Wood, & Faulkner, 2007). Individuals who understand the inversion concept know that adding and subtracting or multiplying and dividing a number by the same number results in no change to the original number, a + b - b = a, d × e ÷ e = d (Robinson & Ninowski, 2003; Starkey & Gelman, 1982). Individuals who understand the associativity concept know that numbers can be decomposed and recombined in various ways and still result in same answer, (a + b) - c - a + (b - c), (d × e) ÷ f= d × (e ÷ f) (Canobi, Reeve, & Pattison, 1998; Robinson, Ninowski, & Grey, 2006). One method by which individuals' understanding of inversion and associativity can be studied is to determine whether an individual can apply the concepts to problem solving, producing fast and accurate strategies called the inversion shortcut (e.g., 3 ...) and the associativity strategy (e.g., 3 × 12 ÷ 4. 12 ÷ 4 = 3, 3 × 3 = 9; Rasmussen, Ho, & Bisanz, 2003; Robinson et al., 2006). Considering how critical both these concepts are to individuals' ability to reason about numbers (Butterworth, 2005; Gilmore & Bryant, 2006), it is not surprising that recent studies have found a relationship between children's use of these conceptually based strategies.

The results of two studies have suggested that children's use of the inversion shortcut is positively related to their use of the associativity strategy. A study of Grade 6 and 8 children by Robinson et al. (2006) found a positive relationship between the use of the two conceptually based strategies for both addition and subtraction and multiplication and division problems. Robinson and Dubé's (2009) study of Grade 2, 3, and 4 children found a similar positive relationship between inversion shortcut and associativity strategy use on addition and subtraction problems. In both studies, inversion shortcut use was far more frequent than associativity strategy use, suggesting that the concept of associativity is either more difficult for children to understand or more difficult to apply to problem solving than the concept of inversion. …

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