One tamed at a time: A new approach for controlling continuous magnitudes in numerical comparison tasks

Abstract

Non-symbolic stimuli (i.e., dot arrays) are commonly used to study numerical cognition. However, in addition to numerosity, non-symbolic stimuli entail continuous magnitudes (e.g., total surface area, convex-hull, etc.) that correlate with numerosity. Several methods for controlling for continuous magnitudes have been suggested, all with the same underlying rationale: disassociating numerosity from continuous magnitudes. However, the different continuous magnitudes do not fully correlate; therefore, it is impossible to disassociate them completely from numerosity. Moreover, relying on a specific continuous magnitude in order to create this disassociation may end up in increasing or decreasing numerosity saliency, pushing subjects to rely on it more or less, respectively. Here, we put forward a taxonomy depicting the relations between the different continuous magnitudes. We use this taxonomy to introduce a new method with a complimentary Matlab toolbox that allows disassociating numerosity from continuous magnitudes and equating the ratio of the continuous magnitudes to the ratio of the numerosity in order to balance the saliency of numerosity and continuous magnitudes. A dot array comparison experiment in the subitizing range showed the utility of this method. Equating different continuous magnitudes yielded different results. Importantly, equating the convex hull ratio to the numerical ratio resulted in similar interference of numerical and continuous magnitudes.

From “sense of number” to “sense of magnitude”: The role of continuous magnitudes in numerical cognition

“…This point has been recently demonstrated even in the subitizing range. A work by Salti et al (2017) revealed that different manipulations of continuous magnitudes influence performance in a non-symbolic Stroop-like task with numerosities in the subitizing range (e.g., 2-4; for an example of a nonsymbolic Stroop-like stimuli, see Fig. 3B).…”

“…However, dissociating perimeter and area from numerosity ignores the possibility that participants rely on other magnitudes or even switch between them. Salti et al (2017) detailed an elaborate taxonomy showing a more complex relationship between continuous magnitudes and numerosity. According to this taxonomy, the inter-correlations between the different continuous magnitudes make the dissociation between numerosity and all continuous magnitudes far from trivial.…”

“…For example, one can calculate the area of a dot or the total area of three dots. In contrast, extrinsic magnitudes (i.e., density and convex hullthe smallest polygon containing all items) provide information on the size and spatial location of the items (Salti et al 2017). Accordingly, extrinsic magnitudes that contain information about both individual items' size and the spatial relationship between the items have been suggested to have greater influence on numerical judgments.…”

Asymmetrical interference between number and item size perception provide evidence for a domain specific impairment in dyscalculia

“…These studies have to some extent arrived at different conclusions, sometimes 130 finding that numerosity, and sometimes that area judgement is more subject to interference, 131 possibly as a consequence of the above mentioned factor of degree of change / 132 discriminability. Indeed when total surface area was claimed to be dominant over the 133 numerical dimension, larger changes in the unattended area dimension were used (Hurewitz 134 et al, 2006;Leibovich et al, 2016b), however when the range of ratio variation across 135 dimension was physically equated, the opposite conclusion was reached (Nys and Content,136 2012; Salti et al, 2016). Indeed the interference arising from numerosity changes in total 137 surface area comparisons was reported to be either similar or stronger with respect to the 138 total surface area interference during numerosity judgments, both when testing the subitizing 139 range (Salti et al, 2016) and much higher numerosities (Nys and Content, 2012).…”

“…Indeed when total surface area was claimed to be dominant over the 133 numerical dimension, larger changes in the unattended area dimension were used (Hurewitz 134 et al, 2006;Leibovich et al, 2016b), however when the range of ratio variation across 135 dimension was physically equated, the opposite conclusion was reached (Nys and Content,136 2012; Salti et al, 2016). Indeed the interference arising from numerosity changes in total 137 surface area comparisons was reported to be either similar or stronger with respect to the 138 total surface area interference during numerosity judgments, both when testing the subitizing 139 range (Salti et al, 2016) and much higher numerosities (Nys and Content, 2012). However, 140 none of these studies took into account the differences that may exist between the 141 perceptual discriminability of different features, as a result of which using identical physical 142 ratios across dimensions may not necessarily translate into equating perceptual salience.…”