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.     

Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.     

Working memory and two-digit number processing

The processing of two-digit numbers in comparison tasks involves the activation and manipulation of magnitude information to decide which number is larger. The present study explored the role of different working memory (WM) components and skills in the processing of two-digit numbers by examining the unit–decade compatibility effect with Arabic digits and number words. In the study, the unit–decade compatibility effect and different WM components were evaluated. The results indicated that the unit–decade compatibility effect was associated to specific WM skills depending on the number format (Arabic digits and number words). We discussed the implications of these results for the decomposed view of two-digit numbers.     

Across-notation automatic numerical processing

In this article, the authors explored the existence of across-notation automatic numerical processing using size comparison and same-different paradigms. Participants were Arabic speakers, who used 2 sets of numerical symbols -- Arabic and Indian. They were presented with number pairs in the same notation (Arabic or Indian) or in different ones (Arabic and Indian). In the size comparison paradigm, 2 digits differing both numerically and physically were compared on the physical dimension. Nevertheless, there was evidence that participants automatically processed the irrelevant numerical dimension in different notation pairs. In the same-different paradigm, 2 digits were presented either in the same or in different notations. Participants had to indicate whether the 2 digits were physically the same. The results again showed evidence for the automatic processing of numerical magnitude for pairs in different notations. Findings of both experiments suggest that numbers in different notations are automatically translated into a common representation of magnitude, in line with M. McCloskey's (1992) abstract representation model.     

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。     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

Sequential or parallel decomposed processing of two-digit numbers? Evidence from eye-tracking

While reaction time data have shown that decomposed processing of two-digit numbers occurs, there is little evidence about how decomposed processing functions. Poltrock and Schwartz (1984) argued that multi-digit numbers are compared in a sequential digit-by-digit fashion starting at the leftmost digit pair. In contrast, Nuerk and Willmes (2005) favoured parallel processing of the digits constituting a number. These models (i.e., sequential decomposition, parallel decomposition) make different predictions regarding the fixation pattern in a two-digit number magnitude comparison task and can therefore be differentiated by eye fixation data. We tested these models by evaluating participants' eye fixation behaviour while selecting the larger of two numbers. The stimulus set consisted of within-decade comparisons (e.g., 53_57) and between-decade comparisons (e.g., 42_57). The between-decade comparisons were further divided into compatible and incompatible trials (cf. Nuerk, Weger, & Willmes, 2001) and trials with different decade and unit distances. The observed fixation pattern implies that the comparison of two-digit numbers is not executed by sequentially comparing decade and unit digits as proposed by Poltrock and Schwartz (1984) but rather in a decomposed but parallel fashion. Moreover, the present fixation data provide first evidence that digit processing in multi-digit numbers is not a pure bottom-up effect, but is also influenced by top-down factors. Finally, implications for multi-digit number processing beyond the range of two-digit numbers are discussed.     

Exploring the developmental changes in automatic two-digit number processing

Even when two-digit numbers are irrelevant to the task at hand, adults process them. Do children process numbers automatically, and if so, what kind of information is activated? In a novel dot-number Stroop task, children (Grades 1-5) and adults were shown two different two-digit numbers made up of dots. Participants were asked to select the number that contained the larger dots. If numbers are processed automatically, reaction time for dot size judgment should be affected by numerical characteristics. The results suggest that, like adults, children process two-digit numbers automatically. Based on the current findings, we propose a developmental trend for automatic two-digit number processing that goes from decomposed sequential (activation of decade digit followed by that of unit digit) to decomposed parallel processing (simultaneous activation of decade and unit digits).     

The differential influence of decades and units on multidigit number comparison

What role do the magnitudes of the constituent digits play in three-digit number comparison (e.g., choosing the larger one of two numbers)? The present study addressed this question by examining compatibility effects between hundred and decade digits and between hundred and unit digits. For example, the number pair 372–845 is hundred–decade incompatible because the larger number contains the smaller decade digit, but hundred–unit compatible because the larger number contains the larger unit digit. We obtained significant effects of hundred–decade and hundred–unit compatibility on number comparison times. However, the effect of hundred–unit compatibility was largely restricted to the hundred–decade-compatible condition. These results suggest that place-value information, through decomposition, is automatically taken into account when multidigit numbers have to be compared. Implications of our findings for models of number processing are discussed.     

The link between mental rotation ability and basic numerical representations

Mental rotation and number representation have both been studied widely, but although mental rotation has been linked to higher-level mathematical skills, to date it has not been shown whether mental rotation ability is linked to the most basic mental representation and processing of numbers. To investigate the possible connection between mental rotation abilities and numerical representation, 43 participants completed four tasks: 1) a standard pen-and-paper mental rotation task; 2) a multi-digit number magnitude comparison task assessing the compatibility effect, which indicates separate processing of decade and unit digits; 3) a number-line mapping task, which measures precision of number magnitude representation; and 4) a random number generation task, which yields measures both of executive control and of spatial number representations. Results show that mental rotation ability correlated significantly with both size of the compatibility effect and with number mapping accuracy, but not with any measures from the random number generation task. Together, these results suggest that higher mental rotation abilities are linked to more developed number representation, and also provide further evidence for the connection between spatial and numerical abilities.     

The differential influence of decades and units on multidigit number comparison

What role do the magnitudes of the constituent digits play in three-digit number comparison (e.g., choosing the larger one of two numbers)? The present study addressed this question by examining compatibility effects between hundred and decade digits and between hundred and unit digits. For example, the number pair 372-845 is hundred-decade incompatible because the larger number contains the smaller decade digit, but hundred-unit compatible because the larger number contains the larger unit digit. We obtained significant effects ofhundred-decade and hundred-unit compatibility on number comparison times. However, the effect of hundred-unit compatibility was largely restricted to the hundred-decade-compatible condition. These results suggest that place-value information, through decomposition, is automatically taken into account when multidigit numbers have to be compared. Implications of our findings for models of number processing are discussed.     

Grasping numbers: evidence for automatic influence of numerical magnitude on grip aperture

Previous research has shown that the fingers’ aperture during grasp is affected by the numerical values of numbers embedded in the grasped objects: Numerically larger digits lead to larger grip apertures than do numerically smaller digits during the initial stages of the grasp. The relationship between numerical magnitude and visuomotor control has been taken to support the idea of a common underlying neural system mediating the processing of magnitude and the computation of object size for motor control. The purpose of the present study was to test whether the effect of magnitude on motor preparation is automatic. During grasping, we asked participants to attend to the colors of the digit while ignoring numerical magnitude. The results showed that numerical magnitude affected grip aperture during the initial stages of the grasp, even when magnitude information was irrelevant to the task at hand. These findings suggest that magnitude affects grasping preparation in an automatic fashion.     

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.     

On the perceptual generality of the unit-decade compatibility effect

Number magnitude is assumed to be holistically represented along a single mental number line. Recently, we have observed a unit-decade-compatibility effect which is inconsistent with that assumption (Nuerk, Weger, & Willmes, 2001). In two-digit Arabic number comparison, we have demonstrated that compatible comparisons in which separate decade and unit comparisons lead to the same decision (32_47, 3 < 4 and 2 < 7) were faster than incompatible trials (37_52, 3 < 5, but 7 > 2). Because overall distance was matched, a holistic model could not account for the compatibility effect. However, one could argue that the compatibility effect was due to the specific vertical perceptual arrangement of the two-digit numbers in Nuerk et al.'s (2001) experiment where the decade digits and unit digits were presented column-wise above each other. To examine this objection, we studied the perceptual generality of the compatibility effect with diagonal presentation. We replicated the compatibility effect with diagonal presentation. It is concluded that the compatibility effect is not due to encoding characteristics imposed by the perceptual setting of the original experiment. In particular, the assumption of an overall analog magnitude representation for two-digit numbers is not consistent with these data.     

Number processing of Arabic and Hebrew bilinguals: Evidence supporting the distance effect

In the current study, a direct assessment of the effect of language lexical‐syntactic structure and magnitude semantic access on numerical processing was made by contrasting the performance of Arabic/Hebrew bilinguals in a digital (Hindi‐digits/Arabic‐digits) and verbal numerical comparison task (Arabic, an inverted language: Units‐Decades, Hebrew, a non‐inverted language: Decades‐Units). Our data revealed in the digital presentation format (Experiment 1) a regular distance effect in Arabic language‐Hindi digits and Hebrew language‐Arabic digits, characterized by an inverse relation between reaction times and numerical distance with no difference in the mean reaction times of participants in Arabic‐L1 and Hebrew‐L2. This indicates that both lexical digits of two‐digit numbers in L1 and L2 are similarly processed and semantically accessed. However in the verbal presentation format (Experiment 2) a similar pattern of distance effect was found, but the mean reaction times in Arabic were lower than in Hebrew in each numerical distance. This indicates that the processing of two‐digit number words in L1 and L2 is semantically accessed and determined by the syntactic structure of each language.     

Holistic representation of unit fractions

The present study examined the processing of unit fractions and the extent to which it is affected by context. Using a numerical comparison task we found evidence for a holistic representation of unit fractions when the immediate context of the fractions was emphasized, that is when the stimuli set included in addition to the unit fractions also the numbers 0 and 1. The holistic representation was indicated by the semantic congruity effect for comparisons of pairs of fractions and by the distance effect in comparisons of a fraction and 0 and 1. Consistent with previous results (Bonato, Fabbri, Umilta, & Zorzi, 2007) there was no evidence for a holistic representation of unit fractions when the stimulus set included only fractions. These findings suggest that fraction processing is context-dependent. Finally, the present results are discussed in the context of processing other complex numbers beyond the first decade.     

Numerical and physical magnitudes are mapped into time

In two experiments we investigated mapping of numerical and physical magnitudes with temporal order. Pairs of digits were presented sequentially for a size comparison task. An advantage for numbers presented in ascending order was found when participants were comparing the numbers' physical and numerical magnitudes. The effect was more robust for comparisons of physical size, as it was found using both select larger and select smaller instructions, while for numerical comparisons it was found only for select larger instructions. Varying both the digits' numerical and physical sizes resulted in a size congruity effect, indicating automatic processing of the irrelevant magnitude dimension. Temporal order and the congruency between numerical and physical magnitudes affected comparisons in an additive manner, thus suggesting that they affect different stages of the comparison process.     

Interference effects in a numerical Stroop paradigm in 9- to 12-year-old children with ADHD-C.

Attention deficit hyperactivity disorder (ADHD) is characterized by deficient self-regulation, poor attentional control, and poor response inhibition. To date, however, the extent to which these deficits affect basic interference control remains a matter of controversy. Secondly, ADHD has been reported to be associated with arithmetic deficits. It remains unclear whether such deficits are a secondary consequence of the above-mentioned characteristics of ADHD or whether basic numerical magnitude representations are also affected. In the present study we attempted to investigate these issues using a basic numerical interference paradigm. Nine- to twelve-year-old children with ADHD-C (attention-deficit hyperactivity-disorder combined type) and control children without ADHD (each n = 16) were presented with two digits of possibly different physical sizes (e.g., 3 7). This numerical Stroop task requires subjects to make a magnitude classification concerning either the physical or the numerical stimulus dimension. The irrelevant dimension can be congruent (same response), incongruent (different response), or neutral (no response association). Children with ADHD-C performed worse than control children in most analyses. The most important finding was a significant interaction of congruity effects with group in the numerical comparison task. Children with ADHD-C tended to show larger congruity and interference effects than controls, and these were not attributable to a speed-accuracy trade-off. The results might reflect differential processing speeds, or a different degree of automatic activation of physical and numerical magnitudes in children with and without ADHD-C. Alternative explanations, such as insufficient inhibition of selective (domain-specific) attention are also discussed.     

Interference Effects in a Numerical Stroop Paradigm in 9- to 12-year-old Children with ADHD-C

Attention deficit hyperactivity disorder (ADHD) is characterized by deficient self-regulation, poor attentional control, and poor response inhibition. To date, however, the extent to which these deficits affect basic interference control remains a matter of controversy. Secondly, ADHD has been reported to be associated with arithmetic deficits. It remains unclear whether such deficits are a secondary consequence of the above-mentioned characteristics of ADHD or whether basic numerical magnitude representations are also affected. In the present study we attempted to investigate these issues using a basic numerical interference paradigm.

Nine- to twelve-year-old children with ADHD-C (attention-deficit hyperactivity-disorder combined type) and control children without ADHD (each n = 16) were presented with two digits of possibly different physical sizes (e.g., 3 7). This numerical Stroop task requires subjects to make a magnitude classification concerning either the physical or the numerical stimulus dimension. The irrelevant dimension can be congruent (same response), incongruent (different response), or neutral (no response association).

Children with ADHD-C performed worse than control children in most analyses. The most important finding was a significant interaction of congruity effects with group in the numerical comparison task. Children with ADHD-C tended to show larger congruity and interference effects than controls, and these were not attributable to a speed-accuracy trade-off.

The results might reflect differential processing speeds, or a different degree of automatic activation of physical and numerical magnitudes in children with and without ADHD-C. Alternative explanations, such as insufficient inhibition of selective (domain-specific) attention are also discussed.     

Are Arabic and verbal numbers processed in different ways?

Four experiments were conducted in order to examine effects of notation--Arabic and verbal numbers--on relevant and irrelevant numerical processing. In Experiment 1, notation interacted with the numerical distance effect, and irrelevant physical size affected numerical processing (i.e., size congruity effect) for both notations but to a lesser degree for verbal numbers. In contrast, size congruity had no effect when verbal numbers were the irrelevant dimension. In Experiments 2 and 3, different parameters that could possibly affect the results, such as discriminability and variability (Experiment 2) and the block design (Experiment 3), were controlled. The results replicated the effects obtained in Experiment 1. In Experiment 4, in which physical size was made more difficult to process, size congruity for irrelevant verbal numbers was observed. The present results imply that notation affects numerical processing and that Arabic and verbal numbers are represented separately, and thus it is suggested that current models of numerical processing should have separate comparison mechanisms for verbal and Arabic numbers.