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.     

Roman digit naming: evidence for a semantic route

Earlier research with monolinguals and bilinguals showed that numbers may be named through both a semantic and a phonological route, depending on the number's language and format (Arabic or verbal), task demands, and naming language. The present study investigated the importance of the semantic route for the processing of a third representation of magnitude, namely Roman digits. Using an interference paradigm, we showed that the processing of Roman target digits is influenced by Arabic digit distractors, both in a naming task and a parity judgment task. Roman digits were processed faster if the target and distractor were of the same magnitude. If this was not the case, processing speed slowed down as the numerical distance between target and distractor increased. This strongly suggests that semantic access is mandatory when naming Roman digits. Implications are discussed for the number processing domain and for models of translation in bilinguals.     

Repetition priming in simple addition depends on surface form and typicality

Repetition priming and recognition memory for numbers were measured in four experiments using single-digit addition. Results of the first two experiments indicate that when numbers were presented as number words and dot configurations, preexposure of the same problem in the same notation produced greater reaction-time benefit than did preexposure of the same problem in Arabic-digit notation. In contrast, when numbers were presented as Arabic digits, preexposure of the same problem in Arabic digit, number word, and dot notation produced the same amount of priming. In the third experiment, priming was shown to be greatest, for all three notations, when the task performed on preexposure trials (addition or multiplication) matched the task performed on repetition trials (addition). Results of the fourth experiment, measuring recognition memory, were comparable to the priming results in the sense that memory was superior when notation matched across repetitions if the test involved number words and dot configurations but not Arabic digits. These data are interpreted in terms of models of numerical cognition, and they support the hypothesis that the influence of surface form on repetition priming depends on the typicality of the input for the task.     

Semantic processing of Arabic,Kanji, and Kana numbers: Evidence from interference in physical and numerical size judgments

Two experiments were conducted with the following objectives (1) to replicate the finding of similar semantic representation of Arabic and written-word (Kanji and Kana) numbers with a direct numerical task, (2) to investigate the automatic semantic processing of Arabic and written-word numbers, and (3) to verify whether the assumption of a common semantic representation is valid in an indirect numerical task. Subjects were asked to judge which of two numbers (e.g, 6-8) was larger either in its numerical size (Experiment 1) or in its physical size (Experiment 2) using the three notations. Effects of two factors were analyzed: the congruity between numerical and physical size and the numerical distance. The effects of these factors were very similar across the three notations in Experiment 1, but were drastically different in Experiment 2. The results of Experiment 2 demonstrated the nonsemantic processing of Kana numbers, and suggest that there may be separate semantic representations for Arabic and Kanji numbers.     

Naming digits in a semantic blocking paradigm

It is widely agreed that word numerals are processed similar to other words, and, thus, they can be named without semantic mediation. However, there is no consensus about Arabic digits. Although digits seem to have a preferential link to magnitude representation, there is some evidence indicating a possible asemantic route to access phonological information. In the present study, we used a semantic blocking paradigm to explore this question. In Experiment 1, participants were asked to name digits and pictures or numeral words and name of objects in a semantic blocked context and in a mixed context. For both types of numerical notation we found facilitation in the blocked condition relative to the mixed condition. In Experiment 2, participants named two-digit numbers in a blocked condition (short numerical distance) or in a mixed condition (large numerical distance). Again, facilitation was found for the blocked condition relative to the mixed condition. This pattern of results seems to indicate that Arabic digits, like number words, might be named through an asemantic route.     

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.     

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.     

Naming digits in a semantic blocking paradigm

It is widely agreed that word numerals are processed similar to other words, and, thus, they can be named without semantic mediation. However, there is no consensus about Arabic digits. Although digits seem to have a preferential link to magnitude representation, there is some evidence indicating a possible asemantic route to access phonological information. In the present study, we used a semantic blocking paradigm to explore this question. In Experiment 1, participants were asked to name digits and pictures or numeral words and name of objects in a semantic blocked context and in a mixed context. For both types of numerical notation we found facilitation in the blocked condition relative to the mixed condition. In Experiment 2, participants named two-digit numbers in a blocked condition (short numerical distance) or in a mixed condition (large numerical distance). Again, facilitation was found for the blocked condition relative to the mixed condition. This pattern of results seems to indicate that Arabic digits, like number words, might be named through an asemantic route.     

Automaticity of two-digit numbers

Automatic processing of 2-digit numbers was demonstrated using the size congruency effect (SiCE). The SiCE indicates the processing of the irrelevant (numerical) dimension when 2 digits differing both numerically and physically are compared on the relevant (physical) dimension. The SiCE was affected by the compatibility between unit and decade digits but was unaffected by the global magnitude of the numbers. Together these results suggest automatic processing of the magnitudes of the components of the 2-digit numbers but not of whole numbers. Finally, the SiCE was affected more by the magnitude of the decade digits compared with the unit digits, indicating that the syntactic roles of the digits were represented. The implications of these results for understanding the numerical representations are discussed.     

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。     

Differences between digit naming and number word reading in a flanker task

Numbers can be represented as number words or as digits, but are the two notations processed differently? Two experiments in which a flanker paradigm with a naming task was used were conducted, with digits and number words as targets and flankers. Reaction times were shortest when the flanker denoted the same numerical value as the target. The numerical distance between the target and a numerically different flanker modulated reaction times in all conditions, except for number word targets with digit flankers. The direction of this effect—targets were named faster when the flanker was numerically close than when it was far—indicates that the numerical magnitude representations of numbers are associatively connected. When the target and the flanker were presented in the same format, no difference was observed in the distance effects for the two formats. This indicates that number words activate the abstract representation of their numerical value in a way that is very similar to that for digits.     

Developing number–space associations: SNARC effects using a color discrimination task in 5-year-olds

Human adults’ numerical representation is spatially oriented; consequently, participants are faster to respond to small/large numerals with their left/right hand, respectively, when doing a binary classification judgment on numbers, known as the SNARC (spatial–numerical association of response codes) effect. Studies on the emergence and development of the SNARC effect remain scarce. The current study introduces an innovative new paradigm based on a simple color judgment of Arabic digits. Using this task, we found a SNARC effect in children as young as 5.5 years. In contrast, when preschool children needed to perform a magnitude judgment task necessitating exact number knowledge, the SNARC effect started to emerge only at 5.8 years. Moreover, the emergence of a magnitude SNARC but not a color SNARC was linked to proficiency with Arabic digits. Our results suggest that access to a spatially oriented approximate magnitude representation from symbolic digits emerges early in ontogenetic development. Exact magnitude judgments, on the other hand, rely on experience with Arabic digits and, thus, necessitate formal or informal schooling to give access to a spatially oriented numerical representation.     

Physical similarity (and not quantity representation) drives perceptual comparison of numbers: evidence from two Indian notations

Numerical quantity seems to affect the response in any task that involves numbers, even in tasks that do not demand access to quantity (e.g., perceptual tasks). That is, readers seem to activate quantity representations upon the mere presentation of integers. One important piece of evidence in favor of this view comes from the finding of a distance effect in perceptual tasks: When one compares two numbers, response times (RTs) are a function of the numerical distance between them. However, recent studies have suggested that the physical similarity between Arabic numbers is strongly correlated with their numerical distance, and that the former could be a better predictor of RT data in perceptual tasks in which magnitude processing is not required (Cohen, 2009a). The present study explored the Persian and Arabic versions of Indian numbers (Exps. 1 and 2, respectively). Na?ve participants (speakers of Spanish) and users of these notations (Pakistanis and Jordanians) participated in a physical same–different matching task. The RTs of users of the Indian notations were regressed on perceptual similarity (estimated from the Spanish participants’ RTs) and numerical distance. The results showed that, regardless of the degree of correlation between the perceptual similarity function and the numerical distance function, the critical predictor for RTs was perceptual similarity. Thus, participants do not automatically activate Indian integers’ quantity representations, at least not when these numbers are presented in simple perceptual tasks.     

Single-digit Arabic numbers do not automatically activate magnitude representations in adults or in children: Evidence from the symbolic same–different task

We investigated whether the mere presentation of single-digit Arabic numbers activates their magnitude representations using a visually-presented symbolic same–different task for 20 adults and 15 children. Participants saw two single-digit Arabic numbers on a screen and judged whether the numbers were the same or different. We examined whether reaction time in this task was primarily driven by (objective or subjective) perceptual similarity, or by the numerical difference between the two digits. We reasoned that, if Arabic numbers automatically activate magnitude representations, a numerical function would best predict reaction time; but if Arabic numbers do not automatically activate magnitude representations, a perceptual function would best predict reaction time. Linear regressions revealed that a perceptual function, specifically, subjective visual similarity, was the best and only significant predictor of reaction time in adults and in children. These data strongly suggest that, in this task, single-digit Arabic numbers do not necessarily automatically activate magnitude representations in adults or in children. As the first study to date to explicitly study the developmental importance of perceptual factors in the symbolic same–different task, we found no significant differences between adults and children in their reliance on perceptual information in this task. Based on our findings, we propose that visual properties may play a key role in symbolic number judgements.     

Basic numerical skills in children with mathematics learning disabilities: a comparison of symbolic vs non-symbolic number magnitude processing

Forty-five children with mathematics learning disabilities, with and without comorbid reading disabilities, were compared to 45 normally achieving peers in tasks assessing basic numerical skills. Children with mathematics disabilities were only impaired when comparing Arabic digits (i.e., symbolic number magnitude) but not when comparing collections (i.e., non-symbolic number magnitude). Moreover, they automatically processed number magnitude when comparing the physical size of Arabic digits in an Stroop paradigm adapted for processing speed differences. Finally, no evidence was found for differential patterns of performance between MD and MD/RD children in these tasks. These findings suggest that children with mathematics learning disabilities have difficulty in accessing number magnitude from symbols rather than in processing numerosity per se.     

The representation of negative numbers: Exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

The representation of negative numbers: exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.     

Semantic processing in the production of numerals across notations

In the present work, we conducted a series of experiments to explore the processing stages required to name numerals presented in different notations. To this end, we used the semantic blocking paradigm previously used in psycholinguist studies. We found a facilitative effect of the semantic blocked context relative to the mixed context for Arabic digits and number words. However, the blocked context produced an interference effect for physical numerosity and Roman numbers. Our results provided further evidence to models suggesting that Arabic digits may be named through an asemantic route similar to that of number words, whereas a semantic route is mandatory to name physical numerosity and Roman numerals.     

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.