Eye-movement patterns during nonsymbolic and symbolic numerical magnitude comparison and their relation to math calculation skills
Introduction
Recent years have witnessed an increase in attention paid to the relations between basic numerical capacities and the development of math skills. Humans possess the ability to process basic numerical magnitude information, allowing them to compare, order, add, and subtract quantities of objects (Feigenson, Dehaene, & Spelke, 2004). This so-called ‘approximate number system’ (ANS) is observable in infancy (Xu & Spelke, 2000) and is shared with non-human species (Cantlon and Brannon, 2006, Cantlon and Brannon, 2007). The ANS shows individual differences in precision, which are, typically, indexed by the effect of numerical ratio on nonsymbolic number comparison tasks (i.e. deciding which of two sets of dots is the more numerous). The numerical ratio effect (NRE) refers to the robustly observed effect that as the ratio of the smaller over the larger number increases, error rates and response times for comparing those two numbers increase (Moyer & Landauer, 1967). In other words, the closer two numbers are to one another, the more difficult it is to compare their relative magnitude. This effect is thought to reflect a greater degree of representational overlap for quantities that are closer together on a logarithmically compressed mental number line (Dehaene, 2003).
The NRE is also observed (and in fact was originally observed) when individuals compare the relative numerical magnitude of Arabic digits (Moyer & Landauer, 1967). This overlap has led some researchers to believe that Arabic digits acquire their semantic referents by being mapped onto the ANS (Mundy & Gilmore, 2009) which in turn may lead to refining of the ANS itself (Mussolin et al., 2015, Piazza et al., 2013). An alternative perspective suggests that Arabic digit knowledge is acquired independently of the ANS and is instead based on a developmental interplay between linguistic and object-attention systems (Carey, 2001). Whether or not nonsymbolic and symbolic number processing are based on the same underlying neurocognitive mechanisms remains an issue of some debate (Cohen Kadosh, Lammertyn, & Izard, 2008). However, it seems reasonable to presume that behavioral performance on number comparison tasks is not solely driven by the precision of underlying cognitive representations, but instead, comprises a combination of visuo-perceptual processes, response selection mechanisms, and other cognitive processing. Although some recent findings suggest that numerosity processing of multiple modalities and presentation conditions (sequential vs. simultaneous) may depend on a general representation of number (Arrighi, Togoli, & Burr, 2014), and that numerosity information is extracted from visual displays spontaneously (Cicchini, Anobile, & Burr, 2016), at present, the extent to which visual-perceptual processing contributes to number comparison performance for nonsymbolic versus symbolic number formats is unknown.
At the same time, a growing body of research suggests that individual differences in the processing of nonsymbolic (Halberda et al., 2008, Mazzocco et al., 2011) and symbolic (Bugden and Ansari, 2011, Holloway and Ansari, 2009) numerical magnitude are related to individual differences in the acquisition of math skills. There is some conflict across the extant literature as to whether nonsymbolic or symbolic skills are the stronger predictor of math performance (for a review see De Smedt, Noël, Gilmore, & Ansari, 2013), with recent evidence suggesting that symbolic skills may mediate the relation between nonsymbolic skills and math (Fazio et al., 2014, Lyons and Beilock, 2011, Price and Fuchs, 2016). Given the current lack of knowledge regarding the component mechanisms underlying numerical comparison performance, it is unclear what is driving the relation between number comparison and math outcomes at the mechanistic level. Despite this lack of knowledge, educational interventions have already been developed that seek to improve ANS precision as a method to improve math outcomes (Park and Brannon, 2013, Wilson et al., 2006). It is critical that the component mechanisms underlying numerical magnitude processing and their relation to math performance be elucidated if such interventions are to achieve optimal efficacy.
One approach to investigating visuo-perceptual processes during cognitive processing is to record and analyze eye-movement patterns during task performance. Eye-tracking data can provide subjective and sensitive information about attentional allocation (Duchowski, 2007), and has been used to gain insights into the mechanistic processes underlying a number of cognitive domains including reading, visual search, memory, language, and problem solving (Henderson, 2013). Eye-tracking has also been used to reveal a number of characteristics of numerical and arithmetic processing (for a review see Mock, Mock, & Huber, 2016) and examination of multiple eye-tracking measures, such as number of fixations and location of first fixation, can provide information regarding sensitivity to top-down cognitive processes or bottom-up stimulus salience respectively (Mock et al., 2016). The majority of numerical cognition eye-tracking studies have investigated multi-digit number processing, with a focus on understanding the ways in which multi-digit numbers are decomposed (Moeller et al., 2009, Huber et al., 2014). While a handful of studies have investigated nonsymbolic numerical processing, they have largely focused on estimation and enumeration (Gandini et al., 2008, Godau et al., 2014, Sophian and Crosby, 2008). Further, despite a widespread focus on the numerical ratio effect in behavioral and neuroimaging studies, only one study to date, to the best of our knowledge, has investigated the effect of ratio on eye-movement patterns during nonsymbolic comparison. Odic and Halberda (2015) reported a decreasing number of fixations on the correct stimulus, number of saccades, and probability of first fixation being on the correct stimulus, as task difficulty increased. In the case of symbolic number processing, to our knowledge, only two studies to date have investigated the effect of ratio on eye-movement patterns. Merkley and Ansari (2010) showed ratio effects for fixation dwell time (FD), fixation count (FC) and number of saccades (SC), with more difficult comparison trials eliciting more fixations, more saccades, and longer fixations. And, while several of their eye-movement measures correlated with task performance (i.e. reaction time (RT) and accuracy rate) when calculated as an overall mean, their ratio effects did not significantly correlate with performance ratio effects, leading the authors to suggest that eye-movement measures may index distinct processes from performance metrics. More recently, in an eye-tracking investigation of common and cross-notational comparison of whole numbers, fractions, and decimals, Hurst and Cordes (2016) found that ratio effects were only apparent for fixation dwell time on the smaller number, although participants generally looked longer at the larger number than the smaller number.
In summary, performance measures of both nonsymbolic and symbolic number comparison have been related to the development of arithmetic skills. The presence of a numerical ratio effect for both formats suggests a potentially shared underlying semantic representation, however, little is known regarding how visuo-perceptual processes are related to the numerical ratio effect, whether they are shared across numerical formats, and whether they relate to math competence independently of performance outcomes. Thus, the present study addresses the following two questions. First, do eye-movement patterns during symbolic and nonsymbolic numerical comparison indicate a shared underlying semantic representation, shared visuo-perceptual processing mechanisms, or both? Second, do eye-movement measures provide unique information about the processing of numerical magnitudes that relates to individual differences in math competence, beyond that accounted for by task performance (i.e., error rate and reaction time)?
In regards to our first question, if cognitive mechanisms for processing symbolic and nonsymbolic number share semantic representation, we would expect to see significant, positive correlations between performance measures across formats and perhaps eye-movement measures as well. If they share visuo-perceptual processing mechanisms, eye-movement measures should be positively correlated, but if only visuo-perceptual processing mechanisms and not semantic representations are shared, then eye-movement measures would be expected to correlate across formats in the absence of cross-format performance correlations. Based on previous research calling into question the reliability of ratio effects (Lyons, Nuerk, & Ansari, 2015), we explore both total means and ratio effects for each measure.
Regarding the second question, if visuo-perceptual processing of dot arrays or Arabic digits is related to math competence, beyond cognitive aspects captured by task performance measures, we would expect to see a relationship between eye-movement patterns and math competence even after controlling for error rate and reaction time, and general cognitive ability. Beyond this, spatial distribution of eye-movement patterns may provide additional details about individual differences in perception. For example, number of fixations on singular stimuli during the nonsymbolic task may serve as a proxy for enumeration strategy. Or, fixation dwell time on single Arabic digits, while controlling for mean reaction time, may be one way to capture visual fluency in processing number symbols.
If eye-movement patterns do prove to be a unique predictor of math outcomes, they could serve as an additional tool for future research investigating individual differences in the cognitive mechanisms underlying representation of number and their relation to math, such as visual fluency with digits apart from the acuity of the semantic representation. Similar research has already begun to make progress for understanding the role of binocular coordination in dyslexia (Hutzler et al., 2006, Kirkby et al., 2011).
Section snippets
Participants
Seventy-three undergraduate students completed participation in the study. Three students were excluded due to incomplete or inaccurate eye-tracking data. Of those three, one student misunderstood instructions, one student did not respond in the appropriate response window, and there was distracting ambient noise during one experimental session. Three other participants were excluded due to low accuracy on the nonsymbolic comparison task. Their performance was not significantly above chance
Error rate
Mean error rate was lower for symbolic compared to nonsymbolic trials [F(1, 55) = 202.41, p < 0.001, ηp2 = 0.79] and lower for large ratio compared to small ratio trials [F(1, 55) = 278.90, p < 0.001, ηp2 = 0.84] (Fig. 1a). The interaction between format and ratio size was also significant, [F(1, 55) = 75.85, p < 0.001, ηp2 = 0.58], revealing that a larger ratio effect was observed for nonsymbolic compared to symbolic trials. Post-hoc analyses revealed a lower error rate for small ratio than for large ratio
Discussion
The current study investigated the relation between eye-movement and performance indices of nonsymbolic and symbolic numerical magnitude comparison and their relation to math calculation skills in order to address two principle questions. First, do eye-movement patterns indicate a shared underlying semantic representation across number formats, shared visuo-perceptual processing mechanisms, or both? Second, do eye-movement measures provide unique information about the processing of numerical
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