Less precise representation of numerical magnitude in high math-anxious individuals: An ERP study of the size and distance effects☆
Introduction
The task of comparing a pair of numbers is subject to two highly robust phenomena, the distance effect and the size effect, which were first described in Moyer and Landauer (1967) seminal paper. The distance effect refers to the fact that it is easier to compare two numbers (i.e., telling which is the largest or the smallest) if they are far apart than if they are close (e.g., a comparison of 1 and 9 will be faster and less error prone than a comparison of 8 and 9). The size effect refers to the fact that a comparison of two numbers which are equated for numerical distance is more difficult for large numbers than for small numbers (e.g., a comparison of 8 and 9 will be slower and more error prone than a comparison of 1 and 2). Both effects occur with numbers presented in Arabic format (Dehaene, Dupoux, & Mehler, 1990), verbal format (Koechlin, Naccache, Block, & Dehaene, 1999), and nonsymbolic format (Buckley & Gillman, 1974), and they have been observed in humans, even early in childhood (Duncan & McFarland, 1980), as well as in animals (Dehaene & Changeux, 1993).
The distance and size effects are attributed to the access to the mental number line along which the numerical magnitudes are represented. To account for the distance effect it has been suggested that close magnitudes are represented in the mental number line with overlapping distributions of activation and, therefore, they are more difficult to discriminate than are distant magnitudes (Libertus and Brannon, 2010, Restle, 1970). The activation pattern of each numerical magnitude is proposed to follow a Gaussian distribution that peaks at the target magnitude and decreases with increasing distance from the target. Thus, magnitudes that are numerically closer to each other will have more representational overlap than will magnitudes that are numerically farther apart, and, as a consequence, the former will produce slower and less accurate responses than will the latter. To account for the size effect it has been suggested that large numbers are represented in the mental number line more vaguely than are small numbers and, hence, it is more difficult to discriminate between larger numbers than between smaller numbers when the numerical distance between them is equal (Antell and Keating, 1983, Strauss and Curtis, 1981). In other words, the standard deviation of the Gaussian distribution of each magnitude increases as number size increases and, therefore, the activation distributions overlap more for larger numerical magnitudes than for smaller ones. Since the distance and size effects are related to the access to the numerical magnitude representation in the mental number line, they have usually been used as a behavioral measure of the precision of numerical estimation.
Recently, Maloney, Ansari, and Fugelsang (2011) studied the precision of numerical estimation in high and low math-anxious individuals. Math anxiety is defined as a negative reaction to math and to mathematical situations that is negatively related to math achievement or competence (Ashcraft & Ridley, 2005). In their Experiment 1, Maloney et al. (2011) used a number comparison task with a fixed standard (i.e., telling whether a single-digit Arabic number was larger or smaller than 5), while in their Experiment 2 they used a simultaneous number comparison task (i.e., telling which of two single-digit Arabic numbers presented simultaneously had larger numerical magnitude). In both experiments they found that the distance effect was greater for the high math-anxious group than for the low math-anxious group (this effect was found for response time but not for accuracy), leading them to conclude that high math-anxious individuals suffer from a low-level numerical deficit that compromises the development of more complex mathematical skills (see also, Maloney, Risko, Ansari, & Fugelsang, 2010). This conclusion is based on the fact that the size of the distance effect is related to differences in math achievement: the smaller the distance effect, the higher the level of mathematical skill (De Smedt et al., 2009, Holloway and Ansari, 2009). Maloney et al.’s results are important because they cast some doubts on the explanation proposed by Ashcraft et al. (e.g., Ashcraft and Kirk, 2001, Ashcraft et al., 1998, Ashcraft and Krause, 2007) for why math anxiety affects math performance.
According to Ashcraft et al., math anxiety consumes working memory resources and, as a consequence, reduces the resources necessary to solve complex math problems (i.e., those that require working memory resources in order to be performed). Ashcraft et al.’s proposal is based on two premises. First, math anxiety is related to performance in complex arithmetical problems (e.g., multi-digit additions that require working memory resources in order to be solved) but not to performance in simple arithmetical problems (e.g., single-digit additions) (anxiety-complexity effect; Ashcraft & Faust, 1994). And second, in accordance with the theory of Eysenck and Calvo (Eysenck, 1992, Eysenck, 1997, Eysenck and Calvo, 1992), anxiety produces intrusive thinking that consumes working memory resources. Thus, if math-anxious individuals are presented with mathematical problems that require working memory resources in order to be performed, they will have fewer working memory resources available because some of them will be occupied with intrusive thinking generated by their high level of anxiety, the consequence being that their performance on these problems will be poorer. Distinguishing between the proposal of Maloney et al. (i.e., basic numerical deficit in math-anxious individuals) and that of Ashcraft et al. (i.e., anxiety affecting working memory through intrusive thinking) is relevant because understanding the cognitive determinants of math anxiety is necessary in order to design appropriate interventions that can prevent math anxiety effects on math performance.
The present study was designed to investigate the number magnitude representation in high and low math-anxious individuals. It differed from previous investigations in two ways. First, the size effect, in addition to the distance effect, was studied. Second, we used event-related brain potentials (ERPs) to address the question of whether high and low math-anxious individuals differ in their processing of numerical magnitude, such that the former show a deficit in the approximate calculation system, as Maloney et al. (2011) have suggested.
Recent research has reported a link between the distance effect and neurophysiological signatures revealed through ERPs. First, a positive peak, with latency around 200 ms post-stimulus and maximum amplitude over posterior electrode sites, has been reported in comparison tasks with a fixed standard (e.g., compare the presented number with 5: Dehaene, 1996, Libertus et al., 2007, Temple and Posner, 1998; compare the number with 15: Turconi, Jemel, Rossion, & Seron, 2004; or compare the number with 65: Pinel, Dehaene, Rivière, & LeBihan, 2001). This component, known as P2p, is sensitive to the distance between the two numbers to be compared: the smaller the distance, the greater the positivity. It has been reported in both symbolic (Arabic numerals and written number words: Dehaene, 1996, Libertus et al., 2007, Temple and Posner, 1998) and nonsymbolic comparison tasks (dot patterns: Dehaene, 1996, Libertus et al., 2007, Smets et al., 2013, Temple and Posner, 1998), as well as in passive viewing tasks in an adaptation context (Hyde and Spelke, 2008, Hyde and Spelke, 2012, Hyde and Wood, 2011). Second, in simultaneous number comparison tasks (i.e., telling which of two simultaneously presented numbers has the larger numerical magnitude), a positive peak with latency around 200 ms post-stimulus has also been found This ERP distance effect also shows greater positivity when the distance between the two numbers to be compared decreases. However, its scalp distribution is less clear than that of the P2p. There are reports of an ERP distance effect with diffuse topography (Szűcs & Soltész, 2008) or with a fronto-central distribution (Szűcs & Soltész, 2007). The link between the distance effect and these early ERP components makes them good markers of approximate numerical magnitude processing (Hyde & Wood, 2011), and they could be useful instruments for shedding more light on whether math anxiety is related to a deficit in elementary numerical cognition.
In the present study, high and low math-anxious individuals were asked to perform a simultaneous comparison task. They were presented with pairs of single-digit Arabic numerals and were required to decide which number had the larger magnitude. The size of the numbers and the numerical distance between them were manipulated in order to study the size and distance effects. The simultaneous comparison task was used instead of the ‘comparison with five’ task because the latter would only have allowed us to study the distance effect. Moreover, there are two more reasons to prefer the simultaneous comparison task. First, according to Maloney et al. (2011) it is possible that the ‘comparison with five’ task requires more working memory resources than the simultaneous comparison task, which would make it difficult to rule out the possibility that working memory plays a role in the results. Second, Maloney, Risko, Preston, Ansari, and Fugelsang (2010) demonstrated that whereas the symbolic simultaneous comparison task is reliable in response time, the ‘comparison with five’ task is not (as for errors, reliability indexes were marginally significant in both tasks).
In the present study, groups were formed in such a way that they differed in math anxiety but not in trait anxiety, thereby ruling out the possibility that the latter factor might account for any differences between groups. Behavioral and electrophysiological measures were recorded. Based on previous studies our predictions were as follows. In terms of response time, if math-anxious individuals have a less precise representation of magnitude they should present not only a larger distance effect than do their low math-anxious peers (as previously reported by Maloney et al., 2011) but also a larger size effect. We did not expect differences between groups in hit rate. As for ERPs, if math anxious individuals have a less precise representation of magnitude they should show a larger ERP distance effect than do their low math-anxious peers. We also sought to examine whether a similar ERP effect would be found for the size effect.
Section snippets
Participants
Fifty-three healthy volunteers were tested in this study, 26 with a high level of math anxiety and 27 with a low level. They were selected from among a sample of 629 students from the University of Barcelona who were assessed for math anxiety, trait anxiety, and arithmetic ability (see Section 2.2).
The low math-anxious group (henceforth, LMA) comprised 27 participants (age range = 19–31, mean = 21.59, SEM = .63) who scored below the first quartile on the Shortened Mathematics Anxiety Rating Scale
Behavioral data
Regarding response times, trials with small numbers were solved faster (344 ms) than were those with large numbers (380 ms), F(1,51) = 51.498, p < .001, , thereby showing the well-known size effect. Moreover, HMA individuals were slower (376 ms) than their LMA counterparts (348 ms), F(1,51) = 4.419, p = .041, . The Size × Group interaction was marginally significant, F(1,51) = 3.531, p = .066, , showing that the size effect (large-small difference) was larger in the HMA group (46 ms) than
Discussion
Our central aim in this study was to examine whether high math-anxious individuals present a poorer mental representation of numerical magnitude than do their low math-anxious counterparts. To this end, ERPs were recorded while HMA and LMA individuals performed a number comparison task, with the numerical distance and size effects being analyzed. These effects were used to address the research question because they provide measures for indexing the representation of numerical magnitude, both
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Support: This research was supported by the Spanish Ministry of Economy and Competitiveness under grant PSI2012-35703, the Spanish Ministry of Science and Technology under grant BES-2010-036859, and the Generalitat de Catalunya under grant SGR2014-177.