Differences in Processing Symbolic Versus Nonsymbolic Representations of Ratios: Behavioral and Neural Evidence

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

Abstract

Rational number knowledge is critical for the development of algebra and other mathematical abilities (Siegler, Duncan et al., 2012). As such, much work has focused on which external representations of rational numbers best support learners’ developing competence (e.g., Cramer & Wyberg, 2009). To the extent that some external representations provide more intuitive access to concepts than others, they may serve as better anchors for introducing those concepts (Rau & Matthews, 2017). Recent evidence suggests that perceptual sensitivity to nonsymbolic ratio magnitudes (e.g., ratios instantiated by juxtaposing two line segments, Figure 4) is widespread, extending throughout the developmental continuum and even to nonhuman animals (Jacob, Vallentin & Nieder, 2012; McCrink &Wynn, 2007). Lewis, Matthews, and Hubbard (2015) have dubbed this sensitivity as the ratio processing system (RPS) and hypothesized that it can be leveraged to aid in the acquisition of symbolic number knowledge. This presentation details early behavioral and neuroimaging findings from a longitudinal study of 2nd and 5th grade students.

A large cohort of 2nd (n = 173) and 5th (n = 124) graders were given a battery of assessments, including measures of symbolic fractions knowledge and other cognitive and non-cognitive factors (e.g., working memory, math anxiety, arithmetic fluency). One particularly interesting measure is a comparison task (Figure 4), whereby children decided which of two ratios or fractions was numerically larger. Comparison stimuli were presented either as symbolic pairs (Frac-Frac), non-symbolic pairs (Line-Line), or as mixed symbolic and non-symbolic stimuli (Line-Frac). The same task was also conducted in the fMRI scanner for a subset of participants.

Data collection is ongoing, but early results are consistent with our hypotheses. Importantly, students at both grade levels are capable of making rapid and accurate comparisons using non-symbolic formats. Indeed, performance for the line-line format was best, and performance for the Frac-Frac format was worst (Figure 5). Moreover, 5th graders performed significantly higher than 2nd grader in all ratio formats. Furthermore, we found a significant distance effect (Figure 5) in each format, whereby children were more accurate as the difference between sizes of compared stimuli increased.

Consistent with previous imaging studies with whole numbers (Ansari et al., 2006), we observed significant distance effects in bilateral intraparietal sulcus (IPS), pre-motor areas (superior frontal lobe), and right dorsolateral prefrontal cortex (PFC) (Figure 6, left). A network of parietal-frontal areas was recruited when processing ratios in various formats. However, there was little overlap in this network with children when compared to adults, for whom areas recruited for processing different formats overlapped significantly.

Our findings demonstrate that ratios in symbolic and non-symbolic formats were processed both similarly and differently at behavioral and neural levels. Moreover, the contrast between child and adult data suggests that processing across formats may become more similar with experience. Continued collection of our longitudinal data will provide more insight into whether nonsymbolic representations are more easily accessible to young children. These findings will inform theories about whether nonsymbolic ratios might serve as anchor representations for teaching about rational number concepts.

Authors

• Yunji Park, University of Wisconsin - Madison

• John Vito Binzak, University of Wisconsin - Madison

• Elizabeth Yael Toomarian, Stanford University/Synapse School

• Priya Kalra, The University of Wisconsin - Madison

• Percival Grant Matthews, University of Wisconsin - Madison

• Edward Michael Hubbard, University of Wisconsin - Madison