Individual differences in algebraic cognition: Relation to the approximate number and semantic memory systems

Abstract

The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 (92 girls) 9th graders, controlling parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation, but not schema memory. Frequency of fact-retrieval errors was related to schema memory but not coordinate plane or expression evaluation accuracy. The results suggest the ANS may contribute to or is influenced by spatial-numerical and numerical only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest different brain and cognitive systems are engaged during the learning of different components of algebraic competence, controlling demographic and domain general abilities.

Does the Approximate Number System Serve as a Foundation for Symbolic Mathematics?

“…In the first study to show this relationship, Halberda and colleagues demonstrated that Weber fraction measured at 14 years of age retroactively predicted standardized math scores at age 5, even after controlling for verbal IQ (Halberda, Mazzocco, & Feigenson, 2008). This positive correlation has since been found in adults (Agrillo, Piffer, & Adriano, 2013; DeWind & Brannon, 2012; Halberda, Ly, Wilmer, Naiman, & Germine, 2012; Libertus, Odic, & Halberda, 2012; Lourenco, Bonny, Fernandez, & Rao, 2012); in school-aged children (Geary, Hoard, Nugent, & Rouder, 2015; Pinheiro-Chagas et al, 2014); in children just beginning formal math education (Gilmore et al, 2010; Keller & Libertus, 2015; Mundy & Gilmore, 2009); and in preschool aged children before they begin formal math education (Chu, vanMarle, & Geary, 2015; Libertus, Feigenson, & Halberda, 2011, 2013; Mazzocco, Feigenson, & Halberda, 2011b; Soto-Calvo, Simmons, Willis, & Adams, 2015; Starr, Libertus, & Brannon, 2013; van Marle, Chu, Li, & Geary, 2014). Three meta-analyses have concluded there is a significant correlation between ANS and math ability (Chen & Li, 2014; Fazio, Bailey, Thompson, & Siegler, 2014; Schneider et al, 2016).…”

“…This result holds even when controlling for performance on the verbal portion of the SAT (Halberda et al, 2012; Libertus et al, 2012). Geary and colleagues (2015) found that ANS acuity is positively correlated with specific algebra skills in 9 th grade students. ANS acuity was a significant predictor for accuracy of placing a point in the coordinate plane and for accuracy in evaluating simple algebraic expressions, but not for remembering algebra equations.…”

Developmental foundations of children’s fraction magnitude knowledge

“…Most children were assessed individually in a quiet location at the school site for all tasks except the fractions procedural knowledge (7 th grade) and algebra (9 th grade) tests were part of group-administered assessments (Geary et al, 2013; Geary, Hoard, Nugent, & Rouder, 2015). The WMTB-C was administered in a mobile testing van that parked outside the child’s house after school hours or on weekends.…”

Comparing Numerical Comparison Tasks: A Meta-Analysis of the Variability of the Weber Fraction Relative to the Generation Algorithm

“…In other words, w was nor more precise nor more informative than general accuracy. The advantages of using this parameter are thus disputable, yet it is commonly used and referred to in the literature as an appropriate tool to compare data sets from different published studies (e.g., in Castronovo and Göbel, 2012 ; Halberda et al, 2012 ; Geary et al, 2015 ; Libertus et al, 2016 ).…”