Semantic alignment and number comparison

Are the quantity representations activated by Arabic digits influenced by semantic context? We developed a novel paradigm to examine semantic alignment effects (e.g., Bassok et al. in J Exp Psychol Learn Mem Cogn 34:343–352, 2008) in number comparison. A horizontal word pair (either less more or few many) appeared for 480 ms to prime either relative magnitude (less more) or quantity (few many). Then a horizontal pair of single digits that were either successors (near) or differed by at least four (far) appeared above the word pair. Participants indicated verbally whether or not the word and digit pairs were congruent with respect to left-to-right ascending or descending relative magnitude. The RT advantage for far number pairs compared to near pairs (the distance effect) was greater with magnitude primes (81 ms) than quantity primes (17 ms), demonstrating a semantic alignment effect. This effect disappeared in Experiment 2 in which participants received identical stimuli but named the larger of the two digits and were free to ignore the primes. Nonetheless, mean RT in Experiment 2 was faster with prime and target pairs both ascending or both descending, but only with quantity primes. This prime-dependent order-congruity effect suggests that semantic alignment with respect to numerical order affected number comparison in Experiment 2. The results thereby demonstrate that number comparison exhibits task-dependent semantic alignment effects and recruits distinct numerical representations as a function of semantic context (e.g., Cohen Kadosh and Walsh in Behav Brain Sci 32:313–373, 2009).     

两位数的整体与局部加工

关于两位数的加工方式有整体加工说和局部加工说,实验证据主要来自数字数量控制/主动加工任务。本研究主要考察在数字数量自动加工任务中两位数的加工方式。实验一要求被试完成数量大小比较和物理大小比较两个任务,实验二只要求被试完成物理大小比较任务。结果是在数量比较任务和物理比较任务中都存在显著的个位十位一致性效应和数量物理一致性效应,这表明在两位数的数量主动和自动加工任务中均存在整体加工和局部加工两种方式。     

Coding strategies in number space: memory requirements influence spatial-numerical associations

The tendency to respond faster with the left hand to relatively small numbers and faster with the right hand to relatively large numbers (spatial numerical association of response codes, SNARC effect) has been interpreted as an automatic association of spatial and numerical information. We investigated in two experiments the impact of task-irrelevant memory representations on this effect. Participants memorized three Arabic digits describing a left-to-right ascending number sequence (e.g., 3-4-5), a descending sequence (e.g., 5-4-3), or a disordered sequence (e.g., 5-3-4) and indicated afterwards the parity status of a centrally presented digit (i.e., 1, 2, 8, or 9) with a left/right keypress response. As indicated by the reaction times, the SNARC effect in the parity task was mediated by the coding requirements of the memory tasks. That is, a SNARC effect was only present after memorizing ascending or disordered number sequences but disappeared after processing descending sequences. Interestingly, the effects of the second task were only present if all sequences within one experimental block had the same type of order. Taken together, our findings are inconsistent with the idea that spatial-numerical associations are the result of an automatic and obligatory cognitive process but do suggest that coding strategies might be responsible for the cognitive link between numbers and space.     

The role of part—whole information in reasoning about relative size

Models of comparative judgment have assumed that relative magnitude is computed from knowledge about absolute magnitude rather than retrieved directly. In Experiment 1, participants verified the relative size of part-whole pairs (e.g., tree-leaf) and unrelated controls (e.g., tree-penny). The symbolic distance effect was much smaller for part-whole pairs than for unrelated controls. In two subsequent experiments, participants determined either which of two objects was closer in size to a third object or which of two pairs had a greater difference in the size of its constituents. In contrast to the paired comparison task in Experiment 1, judgments of part-whole items were more sensitive to the influence of symbolic distance than were unrelated controls. The fact that the part-whole relation attenuates the effects of symbolic distance in a paired comparison task but not in tasks that require an explicit comparison of size differences suggests that the part-whole relation provides a source of information about relative magnitude that does not depend on knowledge about absolute magnitude.     

Dual-task interference effects on cross-modal numerical order and sound intensity judgments: the more the louder?

In the current study, cross-task interactions between number order and sound intensity judgments were assessed using a dual-task paradigm. Participants first categorized numerical sequences composed of Arabic digits as either ordered (ascending, descending) or non-ordered. Following each number sequence, participants then had to judge the intensity level of a target sound. Experiment 1 emphasized processing the two tasks independently (serial processing), while Experiments 2 and 3 emphasized processing the two tasks simultaneously (parallel processing). Cross-task interference occurred only when the task required parallel processing and was specific to ascending numerical sequences, which led to a higher proportion of louder sound intensity judgments. In Experiment 4 we examined whether this unidirectional interaction was the result of participants misattributing enhanced processing fluency experienced on ascending sequences as indicating a louder target sound. The unidirectional finding could not be entirely attributed to misattributed processing fluency, and may also be connected to experientially derived conceptual associations between ascending number sequences and greater magnitude, consistent with conceptual mapping theory.     

Processing of order information for numbers and months

Despite a great deal of research on the processing of numerical magnitude (e.g., the quantity denoted by the number 5), few studies have investigated how this magnitude information relates to the ordinal properties of numbers (e.g., the fact that 5 is the fifth integer). In the present study, we investigated order-related processing of numbers, as well as months of the year, with a novel ordering task to see whether the processing of order information differs from the processing of magnitude information. In Experiments 1 and 2, participants were shown three numbers (Experiment 1) or three months (Experiment 2) and were required to indicate whether the stimuli were in the correct order. In Experiment 3, participants were again shown three numbers; however, now they were instructed to indicate whether the three numbers were ordered in a forward, backward, or mixed direction. Whereas number comparison tasks typically reveal distance effects (comparisons become easier with increased distance between two numbers), these three experiments reveal a different pattern of results. There were reverse distance effects when the stimuli crossed a boundary (i.e., when numbers crossed a decade or months crossed the year boundary) and no effect of distance when the stimuli did not cross a boundary (i.e., when numbers were within a decade and months were within the January—December calendar year). These data suggest that additional mechanisms are involved in the processing of order information: a scanning mechanism and a long-term memory checking mechanism.     

Representation of the numerosities 1-9 by rhesus macaques (Macaca mulatta)

Three rhesus monkeys (Macaca mulatta) were trained to respond to exemplars of 1, 2, 3, and 4 in an ascending, descending, or a nonmonotonic numerical order (1-->2-->3-->4, 4-->3-->2--1, 3-->1-->4-->2). The monkeys were then tested on their ability to order pairs of the novel numerosities 5-9. In Experiment 1, all 3 monkeys ordered novel exemplars of the numerosities 1-4 in ascending or descending order. The attempt to train a nonmonotonic order (3-->1-->4-->2) failed. In Experiment 2A, the 2 monkeys who learned the ascending numerical rule ordered pairs of the novel numerosities 5-9 on unreinforced trials. The monkey who learned the descending numerical rule failed to extrapolate the descending rule to new numerosities. In Experiment 2B all 3 monkeys ordered novel exemplars of pairs of the numerosities 5-9. Accuracy and latency of responding revealed distance and magnitude effects analogous to previous findings with human participants (R. S. Moyer & T. K. Landaeur, 1967). Collectively these studies show that monkeys represent the numerosities 1-9 on at least an ordinal scale.     

Is three greater than five: The relation between physical and semantic size in comparison tasks

In this study, subjects were asked to judge which of two digits (e.g., 3 5) was larger either in physical or in numerical size. Reaction times were facilitated when the irrelevant dimension was congruent with the relevant dimension and were inhibited when the two were incongruent (size congruity effect). Although judgments based on physical size were faster, their speed was affected by the numerical distance between the members of the digit pair, indicating that numerical distance is automatically computed even when it is irrelevant to the comparative judgment being required by the task. This finding argues for parallel processing of physical and semantic information in this task.     

Chinese kindergartners’ automatic processing of numerical magnitude in Stroop-like tasks

Using Stroop-like tasks, this study examined whether Chinese kindergartners showed automatic processing of numerical magnitude. A total of 36 children (mean age 5 5 years 10 months) were asked to perform physical size comparison (i.e., “Which of two numbers is bigger in physical size?”) and numerical magnitude tasks (i.e., “Which of two numbers is bigger in numerical magnitude?”) on 216 number pairs. These number pairs varied in levels of congruence between numerical magnitude and physical size (for Stroop effect) and numerical distance (for distance effect). On the basis of analyses of response time and error rates, we found that Chinese kindergartners showed automatic processing of numerical magnitude. These results are significantly different from previous studies’ findings about the onset age (ranging from around the end of first grade to third grade) for automatic processing of numerical magnitude.     

Coding strategies in number space: Memory requirements influence spatial–numerical associations

The tendency to respond faster with the left hand to relatively small numbers and faster with the right hand to relatively large numbers (spatial numerical association of response codes, SNARC effect) has been interpreted as an automatic association of spatial and numerical information. We investigated in two experiments the impact of task-irrelevant memory representations on this effect. Participants memorized three Arabic digits describing a left-to-right ascending number sequence (e.g., 3–4–5), a descending sequence (e.g., 5–4–3), or a disordered sequence (e.g., 5–3–4) and indicated afterwards the parity status of a centrally presented digit (i.e., 1, 2, 8, or 9) with a left/right keypress response. As indicated by the reaction times, the SNARC effect in the parity task was mediated by the coding requirements of the memory tasks. That is, a SNARC effect was only present after memorizing ascending or disordered number sequences but disappeared after processing descending sequences. Interestingly, the effects of the second task were only present if all sequences within one experimental block had the same type of order. Taken together, our findings are inconsistent with the idea that spatial–numerical associations are the result of an automatic and obligatory cognitive process but do suggest that coding strategies might be responsible for the cognitive link between numbers and space.     

"1-2-3": is there a temporal number line? Evidence from a serial comparison task

Evidence suggests that numbers are intimately related to space (Dehaene, Bossini, & Giraux, 1993; Hubbard, Piazza, Pinel, & Dehaene, 2005). Recently, Walsh (2003) suggested that numbers might also be closely related to time. To investigate this hypothesis we asked participants to compare two digits that were presented in a serial manner, i.e., one after another. Temporally ascending digit pairs (such as 2-3) were responded to faster than temporally descending pairs (3-2). This effect was, in turn, qualified by a local SNARC (spatial numerical association of response codes) effect and a local semantic congruity effect (SCE). Moreover, we observed a global numerical SCE only for temporally descending digit pairs. However, we did not observe a global SNARC effect, i.e., an interaction of numerical magnitude and the right/left response hand. We discuss our results in terms of overlearned forward-associations ("1-2-3") as formed by our ubiquitous cognitive routines to count off objects or events.     

Response-retrieval and negative priming

The existence of across-notation automatic numerical processing of two-digit (2D) numbers was explored using size comparisons tasks. Participants were Arabic speakers, who use two sets of numerical symbols—Arabic and Indian. They were presented with pairs of 2D numbers in the same or in mixed notations. Responses for a numerical comparison task were affected by decade difference and unit-decade compatibility and global distance in both conditions, extending previous findings with Arabic digits (Nuerk, Weger, & Willmes, 2001). Responses for a physical comparison task were affected by congruency with the numerical size, as indicated by the size congruency effect (SiCE). The SiCE was affected by unit-decade compatibility but not by global distance, thus suggesting that the units and decades digits of the 2D numbers, but not the whole number value were automatically translated into a common representation of magnitude. The presence of similar results for same- and mixed-notation pairs supports the idea of an abstract representation of magnitude.     

Common magnitude representation of fractions and decimals is task dependent

Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction–decimal mixed pairs and decimal–fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.     

Asymmetries in the processing of Arabic digits and number words

Numbers can be represented as Arabic digits ("6") or as number words ("six"). The present study investigated potential processing differences between the two notational formats. In view of the previous finding (e.g., Potter & Faulconer, 1975) that objects are named slower, but semantically categorized faster, than corresponding words, it was investigated whether a similar interaction between stimulus format and task could be obtained with numbers. Experiment 1 established that number words were named faster than corresponding digits, but only if the two notation formats were presented in separate experimental blocks. Experiment 2 contrasted naming with a numerical magnitude judgment task and demonstrated an interaction between notation and task, with slower naming but faster magnitude judgment latencies for digits than for number words. These findings suggest that processing of the two notation formats is asymmetric, with digits gaining rapid access to numerical magnitude representations, but slower access to lexical codes, and the reverse for number words.     

Automaticity for numerical magnitude of two-digit Arabic numbers in children

This paper examines the automatic processing of the numerical magnitude of two-digit Arabic numbers using a Stroop-like task in school-aged children. Second, third, and fourth graders performed physical size judgments on pairs of two-digit numbers varying on both physical and numerical dimensions. To investigate the importance of synchrony between the speed of processing of the numerical magnitude and the physical dimensions on the size congruity effect (SCE), we used masked priming: numerical magnitude was subliminally primed in half of the trials, while neutral priming was used in the other half. The results indicate a SCE in physical judgments, providing the evidence of automatic access to the magnitude of two-digit numbers in children. This effect was modulated by the priming type, as a SCE only appeared when the numerical magnitude was primed. This suggests that young children needed a relative synchronization of numerical and physical dimensions to access the magnitude of two-digit numbers automatically.     

The role of general and number-specific order processing in adults’ arithmetic performance

Digit order processing is highly related to individual differences in arithmetic performance. To examine whether serial scanning or associative mechanisms underlie order processing, order tasks (i.e. deciding whether three digits were presented in an order or not) were administered in two experiments. In the first experiment, digits were presented in different directions namely ascending, descending and non-ordered. For each direction, close and far distance sequences were presented. Results revealed reversed distance effects for ordered sequences, but ascending sequences elicited faster performance and stronger reversed distance effects than descending sequences, suggesting that associative mechanisms underlie order processing. In the second experiment, it was examined to which extent the relation between order processing and arithmetic is number-specific by presenting order tasks with digits, letters and months. In all order tasks similar distance effects were observed and similar relations with arithmetic were found, suggesting that both general associative mechanisms and number-specific mechanisms contribute to arithmetic.     

Numerical processing in the two hemispheres: studies of a split-brain patient

Neuroimaging and lesion studies have provided insights into the neural mechanisms underlying numerical processing, yet the roles of the right and left hemispheres have not been systematically investigated within a single study. To address this issue, we investigated subitizing and magnitude comparison abilities in a split-brain patient. The first experiment examined the two hemispheres' abilities to enumerate briefly presented sets of one to four stimuli. Both hemispheres were equally able to perform this task. The second and third experiments examined the hemispheres' abilities to make magnitude judgments about two simultaneously presented stimuli that were either identically coded (i.e., two Arabic numerals, two number words, or two arrays of dots) or differently coded (e.g., an Arabic numeral and a number word). Although the left hemisphere was more accurate than the right when the task involved number words, both hemispheres were able to make comparisons between numerical representations regardless of stimuli coding. In addition, both hemispheres exhibited a distance effect. The results are discussed in the context of Dehaene's triple-code model.     

The associations between space and order in numerical and non-numerical sequences

Spatial-numerical associations have been found across different studies, yet the basis for these associations remains debated. The current study employed an order judgment task to adjudicate between two competing accounts of such associations, namely the Mental Number Line (MNL) and Working Memory (WM) models. On this task, participants judged whether number pairs were in ascending or descending order. Whereas the MNL model predicts that ascending and descending orders should map onto opposite sides of space, the WM model predicts no such mapping. Moreover, we compared the spatial-order mapping for numerical and non-numerical sequences because the WM model predicts no difference in mapping. Across two experiments, we found consistent spatial mappings for numerical order along both horizontal and vertical axes, consistent with a MNL model. In contrast, we found no consistent mappings for letter sequences. These findings are discussed in the context of conflicting extant data related to these two models.     

What can the same–different task tell us about the development of magnitude representations?

We examined the development of magnitude representations in children (Exp 1: kindergartners, first-, second- and sixth graders, Exp 2: kindergartners, first-, second- and third graders) using a numerical same-different task with symbolic (i.e. digits) and non-symbolic (i.e. arrays of dots) stimuli. We investigated whether judgments in a same-different task with digits are based upon the numerical value or upon the physical similarity of the digits. In addition, we investigated whether the numerical distance effect decreases with increasing age. Finally, we examined whether the performance in this task is related to general mathematics achievement. Our results reveal that a same-different task with digits is not an appropriate task to study magnitude representations, because already late kindergarteners base their responses on the physical similarity instead of the numerical value of the digits. When decisions cannot be made on the basis of physical similarity, a similar numerical distance effect is present over all age groups. This suggests that the magnitude representation is stable from late kindergarten onwards. The size of the numerical distance effect was not related to mathematical achievement. However, children with a poorer mathematics achievement score seemed to have more difficulties to link a symbol with its corresponding magnitude.