Number symbols are processed more automatically than nonsymbolic numerical magnitudes: Findings from a Symbolic-Nonsymbolic Stroop task

Are number symbols (e.g., 3) and numerically equivalent quantities (e.g., •••) processed similarly or distinctly? If symbols and quantities are processed similarly then processing one format should activate the processing of the other. To experimentally probe this prediction, we assessed the processing of symbols and quantities using a Stroop-like paradigm. Participants (NStudy1 = 80, NStudy2 = 63) compared adjacent arrays of symbols (e.g., 4444 vs 333) and were instructed to indicate the side containing either the greater quantity of symbols (nonsymbolic task) or the numerically larger symbol (symbolic task). The tasks included congruent trials, where the greater symbol and quantity appeared on the same side (e.g. 333 vs. 4444), incongruent trials, where the greater symbol and quantity appeared on opposite sides (e.g. 3333 vs. 444), and neutral trials, where the irrelevant dimension was the same across both sides (e.g. 3333 vs. 333 for nonsymbolic; 333 vs. 444 for symbolic). The numerical distance between stimuli was systematically varied, and quantities in the subitizing and counting range were analyzed together and independently. Participants were more efficient comparing symbols and ignoring quantities, than comparing quantities and ignoring symbols. Similarly, while both symbols and quantities influenced each other as the irrelevant dimension, symbols influenced the processing of quantities more than quantities influenced the processing of symbols, especially for quantities in the counting rage. Additionally, symbols were less influenced by numerical distance than quantities, when acting as the relevant and irrelevant dimension. These findings suggest that symbols are processed differently and more automatically than quantities.

Further insights into the operation of the Chinese number system: Competing effects of Arabic and Mandarin number formats

Here we report the results of a speeded relative quantity task with Chinese participants. On each trial a single numeral (the probe) was presented and the instructions were to respond as to whether it signified a quantity less than or greater than five (the standard). In separate blocks of trials, the numerals were presented either in Mandarin or in Arabic number formats. In addition to the standard influence of numerical distance, a significant predictor of performance was the degree of physical similarity between the probe and the standard as depicted in Mandarin. Additionally, competing effects of physical similarity, defined in terms of the Arabic number format, were also found. Critically the size of these different effects of physical similarity varied systematically across individuals such that larger effects of one compensated for smaller effects of the other. It is argued that the data favor accounts of processing that assume that different number formats access different format-specific representations of quantities. Moreover, for Chinese participants the default is to translate numerals into a Mandarin format prior to accessing quantity information. The efficacy of this translation process is itself influenced by a competing tendency to carry out a translation into Arabic format.

Non-numerical Distance and Size Effects in an Ant

The distance effect (the fact that the individuals’ discrimination between two similar elements increases with the magnitude of the distance between them) as well as the size effect (the fact that the individuals’ discrimination between two similar elements decreases with the size of these elements) have been largely reported in vertebrates but not in invertebrates. Here, we demonstrate their existence in an ant, using operant conditioning to visual cues (black circles) of different dimensions. The two effects were obvious and differed from one another. Both effects could be accounted for Weber’s law, but it was here not tempted to verify if they are in line with this law by defining the just noticeable difference the ants can perceive between the cues.

Weber’s Law Applies to the Ants’ Visual Perception

Non-numerical distance and size effects have been previously observed in the ant Myrmica sabuleti. As such effects can be theoretically in line with Weber’s law, we presumed that this law, until now examined in vertebrates, could also apply to ants. Using operant conditioning we trained then tested M. sabuleti workers faced with black circles having fixed diameters of 2, 3 and 4 mm against circles with diameters increasing by 0.5 mm until the ants perceived a difference between the smaller and the larger circles. This just noticeable difference occurred when the larger diameter reached 3.5, 5.5 and 7 mm respectively, what corresponded to a ratio larger/smaller surface of 3.06, 3.36 and 3.06. Owing to the degree of accuracy of the experimental methodology, this ratio is sufficiently constant for being consistent with Weber’s law.

Professional mathematicians do not differ from others in the symbolic numerical distance and size effects

The numerical distance effect (it is easier to compare numbers that are further apart) and size effect (for a constant distance, it is easier to compare smaller numbers) characterize symbolic number processing. However, evidence for a relationship between these two basic phenomena and more complex mathematical skills is mixed. Previously this relationship has only been studied in participants with normal or poor mathematical skills, not in mathematicians. Furthermore, the prevalence of these effects at the individual level is not known. Here we compared professional mathematicians, engineers, social scientists, and a reference group using the symbolic magnitude classification task with single-digit Arabic numbers. The groups did not differ with respect to symbolic numerical distance and size effects in either frequentist or Bayesian analyses. Moreover, we looked at their prevalence at the individual level using the bootstrapping method: while a reliable numerical distance effect was present in almost all participants, the prevalence of a reliable numerical size effect was much lower. Again, prevalence did not differ between groups. In summary, the phenomena were neither more pronounced nor more prevalent in mathematicians, suggesting that extremely high mathematical skills neither rely on nor have special consequences for analogue processing of symbolic numerical magnitudes.

Verbal working memory load dissociates common indices of the numerical distance effect: Implications for the study of numerical cognition

In four experiments, we explore the role that verbal WM plays in numerical comparison. Experiment 1 demonstrates that verbal WM load differentially impacts the two most common variants of numerical comparison tasks, evidenced by distinct modulation of the size of the numerical distance effect (NDE). Specifically, when comparing one Arabic digit to a standard, the size of the NDE increases as a function of increased verbal WM load; however, when comparing two simultaneously presented Arabic digits, the size of the NDE decreases (and here is eliminated) as a function of an increased verbal WM load. Experiment 2, using the same task structure but different stimuli (physical size judgments), provides support for the notion that this pattern of results is unique to tasks employing numerical stimuli. Experiment 3 demonstrates that the patterns observed in Experiment 1 are not an artifact of the stimulus pairs used. Experiment 4 provides evidence that the differing pattern of results observed between Experiment 1 and Experiment 2 are due to differences in stimuli (numerical vs. non-numerical) rather than to other differences between tasks. These results are discussed in terms of current theories of numerical comparison.