Tracking the Development of Whole Number Bias and Rational Magnitude Representation in Decimal Comparison

• In Event: From Proportions to Symbolic Rational Numbers

Abstract

Rational numbers are a vital inflection point in the elementary school mathematics curriculum. Current theorizing proposes that mastery of this challenging class of numbers requires both overcoming bias (Ni et al. 2005) from whole number knowledge and building up appropriate magnitude representations of rational quantities (Siegler 2016). Yet, the developmental progression of these two capacities remains largely unknown. Distance effects – faster and more accurate performance for Far relative to Near comparison – have been used in fraction tasks to contrast whole number interference and sensitivity to rational quantities (Obersteiner et al. 2013). Crucially, in proficient adults, neural signals track the distance between rational quantities rather than whole number components (Ischebeck et al. 2009), suggesting that rational magnitude comes to dominate rational number processing. However, fractions comparison tasks inherently confound whole and rational distance (Rosenberg-Lee Forthcoming), as fraction pairs with larger numerator distance will have larger rational distance, while larger denominator distance leads to smaller rational distance. To address these concerns, we developed a decimal comparison task, which orthogonally manipulates whole and rational distance, enabling us to track the progression of these capacities over the course of development and learning.

A robust effect in decimal comparison is slower and less accurate performance when comparing incongruent pairs, (where the larger number quantity has fewer digits i.e. 0.8 vs. 0.27), than congruent pairs (i.e. 0.2 vs. 0.87). Here, we define rational distance as the actual distance between decimal pairs (e.g. 0.8 vs. 0.27 differ by 0.53), and whole number distance as the difference when ignoring the decimal point (8 vs. 27 differ by 19). With this terminology, we can contrast problems with comparable rational distance, but differing whole difference (see Table 1 for example stimuli). For instance, both inconsistent example pairs differ by 0.18, but have whole difference of 27 and 63, respectively. If interference on this task is driven, not just by the number of digits, but also by their whole distance, we would expect worse performance on 0.9 vs 0.72 than 0.5 vs. 0.32. Indeed, that is what we find in two adult, in-person samples: for incongruent problems, participants’ accuracy decreased as whole distance increased. Accuracy was also driven by rational distance, suggesting both factors are being processed in adults.

Next, we compared adults (n=73, age 18-21) and children (n=21, age 11-14) in an online, remote version of the task. Overall performance was quite comparable between groups in terms of accuracy (adults = 92.2%, children = 92.8%) and, in both groups whole distance negatively modulated accuracy on incongruent pairs, with no differences in slope or absolute level. However, adults were more sensitive than children to rational distance for incongruent problems, although the difference did not reach significance. These preliminary results begin to fill in the developmental trajectory of both whole number bias and rational number representation: a prolonged development of rational number magnitude, accompanying persistent processing of whole number magnitude. Together, these results point to the need to flexibly activate appropriate representations as a crucial capacity for rational number mastery.

Authors

• Miriam Rosenberg-Lee, Rutgers University - Newark
  Presenting Author

• Roberto Abreu-Mendoza, Rutgers University - Newark
  Non-Presenting Author

• Shannon Cahalan, Rutgers University - Newark
  Non-Presenting Author