Conceptual Knowledge of Arithmetic for Chinese- and Canadian-Educated Adults

Conceptual knowledge of arithmetic is a critical component for the acquisition of complex mathematical knowledge, including algebra and calculus skills (National Mathematics Advisory Panel, 2008; Nunes et al., 2008). Most research on conceptual knowledge has focused on children, and only a few studies have examined adolescents' and adults' understanding of arithmetic concepts (e.g., Dube, 2014), so only a little is known about how well arithmetic concepts are learned by adulthood. Conceptual knowledge is defined as the understanding of the underlying structure of arithmetic and that the understanding of this structure can be used as the basis for problem-solving procedures (Bisanz & LeFevre, 1990; Robinson & LeFevre, 2012) and, along with factual and procedural knowledge, is considered to be one of the three basic types of mathematical knowledge (Bisanz & LeFevre, 1990). Recent research has shown that conceptual knowledge may be weaker in children than was previously thought.Researchers have found that some preschoolers understand the inversion concept (Klein & Bisanz, 2000; Rasmussen, Ho, & Bisanz, 2003), but other studies have found that even by late adolescence and early adulthood, the inversion concept is not always understood (Dube, 2014; Dube & Robinson, 2010; Robinson & Ninowski, 2003). The inversion concept is the understanding that addition and subtraction are inversely related operations, as are multiplication and division (Bisanz & LeFevre, 1990). These large age differences across studies are at least partially due to how the inversion concept is assessed.The inversion concept has traditionally been assessed using addition and subtraction three-term problems, such as 4 + 8 - 8. In these instances, even young children can successfully apply their knowledge of the inverse relationship between addition and subtraction to successfully solve the problem without performing any calculations (Rasmussen et al., 2003). However, when the inversion concept is assessed using multiplication and division three-term problems, such as 4 X 8 -b 8, performance typically drops. Participants from middle school to high school to even university often resort to using a left-to-right calculation-based approach, suggesting that their understanding of the inverse relation between multiplication and division is weak (Robinson & Dube, 2009; Robinson & Ninowski, 2003).The same performance drop occurs when the associativity concept is investigated. The associativity concept is the understanding that addition and subtraction can be solved in any order, as can multiplication and division (Robinson & Ninowski, 2003). On addition and subtraction three-term problems such as 4 + 8 - 2, children often make use of their understanding of the associative relation between addition and subtraction to solve the subtraction part of the problem first (i.e., 8 - 2), then adding on the first number (i.e., 4 + 6), which is often faster and yields fewer calculation errors in comparison to solving the problem from left to right. However, performance on multiplication and division three-term problems such as 4 X 8 -t- 2 is much weaker, suggesting less understanding of the associativity concept (Robinson, Ninowski, & Gray, 2006).Research with adults on both the inversion and associativity concepts using both additive and multiplicative problems has shown a pattern of findings similar to that for children (Robinson & Ninowski, 2003; Robinson et al., 2006). First, adults perform more poorly on multiplicative than additive inversion and associativity problems. Second, just like children, adults have a stronger understanding of the inversion than the associativity concept. In other studies using only multiplication and division problems (e.g., Dube & Robinson, 2010), adults used the inversion and associativity concepts infrequently (on two thirds of inversion problems and on less than half of associativity problems), suggesting that by adulthood the inversion and associativity concepts, particularly the multiplicative versions, are not fully understood. …