Spatial-Numerical Associations of Fractions: Evidence From Internal and External Representations

• In Event: 34.032 - Probing Students' Understanding With Differing External Representations

Abstract

Background & Motivation. Understanding fractions is key to establishing a solid foundation in mathematics, yet little is known about how people’s internal representations of fraction impacts this understanding. One approach to understanding representations of fractions on the mental number line has been to use explicit number line estimation (NLE) tasks, where participants select a point on a physical line that best corresponds to a given number. These tasks, which utilize external representations to probe internal representations of numerical magnitudes, have been shown to be correlated and causally related to improvements in arithmetic (Booth & Siegler, 2008).

We have recently begun exploring another approach: testing implicit associations between numbers and space. The Spatial-Numerical Association of Response Codes (SNARC) effect refers to the association of numerical magnitude with sides of space (Dehaene, Bossini & Giraux, 1993; Fias et al., 1996). In cultures with left-to-right writing systems, this mapping results in smaller numbers being associated with the left side of space, and larger numbers with the right, reflecting the canonical orientation of number lines (Zebian, 2005; Shaki, Fischer & Petrusic, 2009). Adults exhibit a consistent SNARC for fractions, demonstrating an implicit association between holistic fraction magnitudes and space (Toomarian & Hubbard, 2018). However, no studies have explored the relationship between explicit number line estimation and implicit SNARC tasks for fractions.

Methods & Results. To understand this relationship, we tested 94 undergraduate students on SNARC and NLE measures of fraction magnitude, as well as additional numerical and cognitive tasks. To measure the SNARC, students completed a simple comparison task in which they compared the magnitude of single digit, irreducible fractions to ½ (which has previously produced a reliable, group-level SNARC effect). In the NLE task, students placed fractions between 0-1 and 0-5 on bounded number lines. Performance was measured through each participant’ mean Percentage Absolute Error (PAE), where PAE = (|estimated value – given value|/numerical range). NLE performance (PAE) was significantly correlated with individual fraction SNARC slopes (r = 0.23, p = 0.02). See Figure 3. A regression using individual SNARC slopes as the outcome measure and PAE, IQ, average RT, and mean accuracy as predictors, revealed only PAE as a significant predictor of individual SNARC slopes (β = 81.12, SE = 31.10, p = 0.007). See Table 1.

Implications & Future Directions. This study demonstrates a link between internal and external representations of fraction magnitude. We are currently investigating the developmental trajectory of these implicit and explicit representations, as it remains unclear how and when they emerge. Given the predictive and causal power of NLE in improving arithmetic, similar training on implicit spatial-numerical associations may yield comparable results. For example, playing a linear board game theoretically grounded in research on the mental number line improves understanding of numerical magnitudes in preschool children (Siegler & Ramani, 2009; Whyte and Bull, 2008). Training on a circularly arranged board game failed to produce comparable improvements. Similar techniques leveraging explicit spatial associations, but specific to fractions, might aid in developing holistic fraction magnitude representations early on in childhood.

Authors

• Elizabeth Yael Toomarian, Stanford University/Synapse School

• Yunji Park, University of Wisconsin - Madison

• Percival Grant Matthews, University of Wisconsin - Madison

• Edward Michael Hubbard, University of Wisconsin - Madison