Increasing the use of conceptually-derived strategies in arithmetic: using inversion problems to promote the use of associativity shortcuts

Abstract

Declarations of interest: None ACKNOWLEDGEMENTS: This work was supported by a PhD Studentship from Loughborough University Doctoral college. We would like to thank Jayne Pickering who helped to dual-score the data. C.G. is supported by a Royal Society Dorothy Hodgkin Fellowship. Highlights • Associativity is an arithmetic concept that children and adults struggle to apply • Inversion is a simpler concept that older children and adults easily apply • Solving inversion problems ('a + b-b') increased subsequent associativity use • 'a + b-b' inversion, but not 'a + b-a' inversion, increased associativity use • 'a + b-b' problems may direct attention or validate a 'right-to-left' strategy

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

“…Edwards (2013) found that self-reported shortcut use was significantly higher on 'a × b ÷ c' problems that were conducive to a shortcut than those that were not conducive. Eaves et al, (2019) found that self-reported users of the shortcut solved more conducive 'a + b -c' problems in a restricted timeframe than individuals who did not self-report using the shortcut, but that self-reported users and non-users did not differ on non-conducive 'a + b -c' problems. Comparing performance on conducive and non-conducive problems is therefore one way that researchers could investigate implicit associativity shortcut use independently from other concepts.…”

“…Three studies have compared performance on conducive and non-conducive associativity problems (Eaves et al, 2019(Eaves et al, , 2020Edwards, 2013). Edwards (2013) found that self-reported shortcut use was significantly higher on 'a × b ÷ c' problems that were conducive to a shortcut than those that were not conducive.…”

Investigating the role of attention in the identification of associativity shortcuts using a microgenetic measure of implicit shortcut use

“…Second, in a classroom intervention study with adults, Eaves et al (2019) found that individuals who solved “a + b − b” inversion problems were more likely than individuals who solved “a + b − a” inversion problems to subsequently use the associativity shortcut on “a + b − c” problems. They proposed two mechanisms through which “a + b − b” inversion problems helped individuals to identify the associativity shortcut.…”

“…Empirically, three strands of research provide some preliminary evidence that attention could be important for identifying associativity shortcuts (Dubé & Robinson, 2010; Eaves et al, 2019; Landy & Goldstone, 2007a). In the experimental studies by Landy and Goldstone (2007a, 2007b, 2010), adults validated the equivalence of multi-term problems and solved multi-term problems such as “2 + 3 × 4” in conditions where the spacing within and between the operations was manipulated.…”

Characterizing Mathematics Learning in Colombian Higher Distance Education

“…Multiple publications characterize the types of knowledge common to mathematics. Although there are many studies for students in regular education at primary and secondary education levels [20][21][22], very few can be found for higher education and adults [23,24]. Particularly limited is the number of publications in the Colombian context.…”