Semantic numerical representation in blind subjects: the role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.     

Numerical estimation in blind subjects: evidence of the impact of blindness and its following experience

Vision was for a long time considered to be essential in the elaboration of the semantic numerical representation. However, early visual deprivation does not seem to preclude the development of a spatial continuum oriented from left to right to represent numbers (J. Castronovo & X. Seron, 2007; D. Szücs & V. Csépe, 2005). The authors investigated the impact of blindness and its following experience on a 3rd property of the mental number line: its obedience to Weber's law. A group of blind subjects and a group of sighted subjects were submitted to 2 numerical estimation tasks: (a) a keypress estimation task and (b) an auditory events estimation task. Blind and sighted subjects' performance obeyed Weber's law. However, blind subjects demonstrated better numerical estimation abilities than did sighted subjects, especially in contexts involving proprioception, indicating the existence of better mapping abilities between the symbolic representations of numbers and their corresponding magnitude representations, obeying Weber's law (e.g., J. S. Lipton & E. Spelke, 2005). These findings suggest that blindness and its following experience with numbers might result in better accuracy in numerical processing.     

How do people apprehend large numerosities?

People discriminate remarkably well among large numerosities. These discriminations, however, need not entail numerical representation of the quantities being compared. This research evaluated the role of both non-numerical and numerical information in adult judgments of relative numerosity for large-numerosity spatial arrays. Results of Experiment 1 indicated that judgments of relative numerosity were affected by the amount of open space in the arrays being compared. Further, the accuracy of verbal estimates of the numerosities of the arrays made upon completion of the comparison task bore little relation to performance on that task. Experiment 2, however, showed that numerical estimates for individually presented arrays were affected in much the same way by open space within or around the edges of the array as were the comparative judgments examined in Experiment 1. The findings suggest that adults heuristically utilize non-numerical cues as well as numerical information in apprehending large numerosities.     

Implicit response-irrelevant number information triggers the SNARC effect: Evidence using a neural overlap paradigm

There is evidence from the SNARC (spatial–numerical association of response codes) effect and NDE (numerical distance effect) that number activates spatial representations. Most of this evidence comes from tasks with explicit reference to number, whether through presentation of Arabic digits (SNARC) or through magnitude decisions to nonsymbolic representations (NDE). Here, we report four studies that use the neural overlap paradigm developed by Fias, Lauwereyns, and Lammertyn (2001) to examine whether the presentation of implicit and task-irrelevant numerosity information (nonsymbolic arrays and auditory numbers) is enough to activate a spatial representation of number. Participants were presented with either numerosity arrays (1–9 circles or triangles) to which they made colour (Experiment 1) or orientation (Experiment 2) judgements, or auditory numbers coupled with an on-screen stimulus to which they made a colour (Experiment 3) or orientation (Experiment 4) judgement. SNARC effects were observed only for the orientation tasks. Following the logic of Fias et al., we argue that this SNARC effect occurs as a result of overlap in parietal processing for number and orientation judgements irrespective of modality. Furthermore, we found stronger SNARC effects in the small number range (1–4) than in the larger number range (6–9) for both nonsymbolic displays and auditory numbers. These results suggest that quantity is extracted (and interferes with responses in the orientation task) but this is not exact for the entire number range. We discuss a number of alternative models and mechanisms of numerical processing that may account for such effects.     

The role of vision in the development of finger-number interactions: Finger-counting and finger-montring in blind children

Previous research has suggested that the use of the fingers may play a functional role in the development of a mature counting system. However, the role of developmental vision in the elaboration of a finger numeral representation remains unexplored. In the current study, 14 congenitally blind children and 14 matched sighted controls undertook three different test batteries that examined (a) general cognitive abilities, (b) the spontaneous use of finger-counting and finger-montring strategies (where “finger-montring” is a term used to characterize the way people raise their fingers to show numerosities to other people), and (c) the canonicity level of the finger-counting and finger-montring habits. Compared with sighted controls, blind children used their fingers less spontaneously to count and in a less canonical way to count and show quantities. These results demonstrate that the absence of vision precludes the development of a typical finger numeral representation and suggest that the use of canonical finger-counting and finger-montring strategies relies on the visual recognition of particular hand shapes.     

Implicit response-irrelevant number information triggers the SNARC effect: Evidence using a neural overlap paradigm

There is evidence from the SNARC (spatial-numerical association of response codes) effect and NDE (numerical distance effect) that number activates spatial representations. Most of this evidence comes from tasks with explicit reference to number, whether through presentation of Arabic digits (SNARC) or through magnitude decisions to nonsymbolic representations (NDE). Here, we report four studies that use the neural overlap paradigm developed by Fias, Lauwereyns, and Lammertyn (2001) to examine whether the presentation of implicit and task-irrelevant numerosity information (nonsymbolic arrays and auditory numbers) is enough to activate a spatial representation of number. Participants were presented with either numerosity arrays (1-9 circles or triangles) to which they made colour (Experiment 1) or orientation (Experiment 2) judgements, or auditory numbers coupled with an on-screen stimulus to which they made a colour (Experiment 3) or orientation (Experiment 4) judgement. SNARC effects were observed only for the orientation tasks. Following the logic of Fias et al., we argue that this SNARC effect occurs as a result of overlap in parietal processing for number and orientation judgements irrespective of modality. Furthermore, we found stronger SNARC effects in the small number range (1-4) than in the larger number range (6-9) for both nonsymbolic displays and auditory numbers. These results suggest that quantity is extracted (and interferes with responses in the orientation task) but this is not exact for the entire number range. We discuss a number of alternative models and mechanisms of numerical processing that may account for such effects.     

Audiovisual links in exogenous covert spatial orienting

Subjects judged the elevation (up vs. down, regardless of laterality) of peripheral auditory or visual targets, following uninformative cues on either side with an intermediate elevation. Judgments were better for targets in either modality when preceded by an uninformative auditory cue on the side of the target. Experiment 2 ruled out nonattentional accounts for these spatial cuing effects. Experiment 3 found that visual cues affected elevation judgments for visual but not auditory targets. Experiment 4 confirmed that the effect on visual targets was attentional. In Experiment 5, visual cues produced spatial cuing when targets were always auditory, but saccades toward the cue may have been responsible. No such visual-to-auditory cuing effects were found in Experiment 6 when saccades were prevented, though they were present when eye movements were not monitored. These results suggest a one-way cross-modal dependence in exogenous covert orienting whereby audition influences vision, but not vice versa. Possible reasons for this asymmetry are discussed in terms of the representation of space within the brain.     

Is approximate numerical judgment truly modality-independent? Visual, auditory, and cross-modal comparisons

The numerosity of any set of discrete elements can be depicted by a genuinely abstract number representation, irrespective of whether they are presented in the visual or auditory modality. The accumulator model predicts that no cost should apply for comparing numerosities within and across modalities. However, in behavioral studies, some inconsistencies have been apparent in the performance of number comparisons among different modalities. In this study, we tested whether and how numerical comparisons of visual, auditory, and cross-modal presentations would differ under adequate control of stimulus presentation. We measured the Weber fractions and points of subjective equality of numerical discrimination in visual, auditory, and cross-modal conditions. The results demonstrated differences between the performances in visual and auditory conditions, such that numerical discrimination of an auditory sequence was more precise than that of a visual sequence. The performance of cross-modal trials lay between performance levels in the visual and auditory conditions. Moreover, the number of visual stimuli was overestimated as compared to that of auditory stimuli. Our findings imply that the process of approximate numerical representation is complex and involves multiple stages, including accumulation and decision processes.     

Enumeration of small and large numerosities: the effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1-8) or by tens (10-80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

Cross-notational semantic priming between symbolic and nonsymbolic numerosity

Symbolic and nonsymbolic numerosities produce similar behavioural effects and activate the same brain areas. These results have usually been interpreted in terms of a common, notation-independent magnitude representation. However, semantic priming between symbolic and nonsymbolic inputs has been somehow elusive (e.g., Koechlin, Naccache, Block, & Dehaene, 1999). In Experiment 1, we looked at whether cross-notational semantic priming depends on exact numerical meaning. Dice faces and digits were mixed as prime and target. Semantic priming occurred when prime and target were in the same notation as much as when they were in different notation. In Experiment 2, we found cross-notation semantic priming even when the nonsymbolic numerosity was presented as a set of random dots. Priming, however, occurred only from sets of dots to digit, not vice versa. These data support the computational model recently proposed by Verguts and Fias (2004; Verguts, Fias, & Stevens, 2005).     

The development of ordinal numerical competence in young children

Two experiments assessed ordinal numerical knowledge in 2- and 3-year-old children and investigated the relationship between ordinal and verbal numerical knowledge. Children were trained on a 1 vs 2 comparison and then tested with novel numerosities. Stimuli consisted of two trays, each containing a different number of boxes. In Experiment 1, box size was held constant. In Experiment 2, box size was varied such that cumulative surface area was unrelated to number. Results show children as young as 2 years of age make purely numerical discriminations and represent ordinal relations between numerosities as large as 6. Children who lacked any verbal numerical knowledge could not make ordinal judgments. However, once children possessed minimal verbal numerical competence, further knowledge was entirely unrelated to ordinal competence. Number may become a salient dimension as children begin to learn to count. An analog magnitude representation of number may underlie success on the ordinal task.     

Memory processing of spatial order as transmitted by auditory information in the absence of visual cues

Two experiments were conducted concerning spatial order recall when spatial information is transmitted by auditory stimuli. Temporal order either was congruent with spatial order or was independent of spatial order. In Experiment 1, the comparisons were among normally or partially sighted subjects allowed to look, normally sighted subjects who were blindfolded, and blind children. The main findings were a superiority of the sighted subjects allowed to look (that is, to support auditory information with visual cues) and a smaller advantage for the sighted-but-blindfolded subjects, relative to the blind group. In Experiment 2, normally sighted adults (either seeing or blindfolded) and blind adults were tested. Surprisingly, the blind were not worse than the sighted in this study. Subsequent interviews and detailed analysis of errors suggested that the blind coded spatial information kinesthetically. These indirect analyses also suggested that whereas spatial order was coded temporally in the sighted, it was controlled by both temporal and spatial factors in the blind and blindfolded subjects.     

盲人的时间水平方向隐喻的通道特异性

通过2个实验, 考察了感觉通道对盲人的时间水平方向隐喻的影响。实验1发现, 盲人只在动作水平上存在同阅读方向一致的时间水平方向隐喻, 对表征过去的词或句子按左键反应和对表征将来的词或句子按右键反应时更快, 表明盲人用于隐喻时间的空间信息是知觉性的。实验2比较了明眼人可视组被试和明眼人遮眼组被试的时间空间隐喻, 发现明眼人可视组被试只在动作水平上存在时间水平方向隐喻, 明眼人遮眼组被试只在刺激在右侧时呈现才出现动作水平的时间水平方向隐喻。综合分析表明, 盲人和明眼人可视组被试的反应更为接近, 表明盲人的时间空间隐喻未受听觉信息的影响。盲人通过运动通道获得的空间认知代偿了视觉通道信息的缺失。     

Enumeration of small and large numerosities: The effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1–8) or by tens (10–80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

Components representation of negative numbers: Evidence from auditory stimuli detection and number classification tasks

Past research suggested that negative numbers could be represented in terms of their components in the visual modality. The present study examined the processing of negative numbers in the auditory modality and whether it is affected by context. Experiment 1 employed a stimuli detection task where only negative numbers were presented binaurally. Experiment 2 employed the same task, but both positive and negative numbers were mixed as cues. A reverse attentional spatial–numerical association of response codes (SNARC) effect for negative numbers was obtained in these two experiments. Experiment 3 employed a number classification task where only negative numbers were presented binaurally. Experiment 4 employed the same task, but both positive and negative numbers were mixed. A reverse SNARC effect for negative numbers was obtained in these two experiments. These findings suggest that negative numbers in the auditory modality are generated from the set of positive numbers, thus supporting a components representation.     

Small and large number processing in infants and toddlers with Williams syndrome

Previous studies have suggested that typically developing 6‐month‐old infants are able to discriminate between small and large numerosities. However, discrimination between small numerosities in young infants is only possible when variables continuous with number (e.g. area or circumference) are confounded. In contrast, large number discrimination is successful even when variables continuous with number are systematically controlled for. These findings suggest the existence of different systems underlying small and large number processing in infancy. How do these develop in atypical syndromes? Williams syndrome (WS) is a rare neurocognitive developmental disorder in which numerical cognition has been found to be impaired in older children and adults. Do impairments of number processing have their origins in infancy? Here this question is investigated by testing the small and large number discrimination abilities of infants and toddlers with WS. While infants with WS were able to discriminate between 2 and 3 elements when total area was confounded with numerosity, the same infants did not discriminate between 8 and 16 elements, when number was not confounded with continuous variables. These findings suggest that a system for tracking the features of small numbers of object (object‐file representation) may be functional in WS, while large number discrimination is impaired from an early age onwards. Finally, we argue that individual differences in large number processing in infancy are more likely than small number processing to be predictive of later development of numerical cognition.     

Spatial updating of locations specified by 3-d sound and spatial language

Blind and blindfolded sighted observers were presented with auditory stimuli specifying target locations. The stimulus was either sound from a loudspeaker or spatial language (e.g., "2 o'clock, 16 ft"). On each trial, an observer attempted to walk to the target location along a direct or indirect path. The ability to mentally keep track of the target location without concurrent perceptual information about it (spatial updating) was assessed in terms of the separation between the stopping points for the 2 paths. Updating performance was very nearly the same for the 2 modalities, indicating that once an internal representation of a location has been determined, subsequent updating performance is nearly independent of the modality used to specify the representation.     

两种数量表征系统

数量表征是人类数学能力的基础,数量表征研究中的一个争论焦点在于是否存在两种不同的数量表征系统：对小数的精确表征系统和对大数的近似表征系统。通过综述不同研究领域对数量表征的研究,总结了支持两种表征系统分离的证据：对1~3范围内小数的表征受数量大小的限制,基于指向物体本身的注意,更依赖于物体的知觉特征,对物体及其数量进行精确表征;而对4以上的数量的近似表征系统则受韦伯定律的限制,基于指向数量的模拟幅度的表征,而不依赖单个物体的知觉特征,是对数量的近似的、心理的表征。fMRI、PET和ERP的脑成像研究结果迄今尚无定论,但认知神经科学研究的深入开展将最终阐明数量表征的机制     

亮度对空间-数字反应编码联合效应的影响

本研究探讨亮度对空间-数字反应编码联合效应（Spatial-Numerical Association of Response Codes,简称SNARC效应）的影响及其机制。通过三个实验设计不同的亮度对比水平,要求被试对阿拉伯数字1~9（5除外）进行奇偶判断。实验一将数字亮度设为最高值255时,结果出现了数字的SNARC效应。实验二将数字的亮度值分别设为255和213时,结果仍存在SNARC效应。实验三将亮度值分别设置为213和42时,数字的SNARC效应却消失了。这些结果表明亮度会激活或抑制数字的空间表征,可能与亮度对比值的高低及所消耗认知资源的多少有关。     

The role of crossmodal competition and dimensional overlap in crossmodal attention switching

Crossmodal selective attention was investigated in a cued task switching paradigm using bimodal visual and auditory stimulation. A cue indicated the imperative modality. Three levels of spatial S–R associations were established following perceptual (location), structural (numerical), and conceptual (verbal) set-level compatibility. In Experiment 1, participants switched attention between the auditory and visual modality either with a spatial-location or spatial-numerical stimulus set. In the spatial-location set, participants performed a localization judgment on left vs. right presented stimuli, whereas the spatial-numerical set required a magnitude judgment about a visually or auditorily presented number word. Single-modality blocks with unimodal stimuli were included as a control condition. In Experiment 2, the spatial-numerical stimulus set was replaced by a spatial-verbal stimulus set using direction words (e.g., “left”). RT data showed modality switch costs, which were asymmetric across modalities in the spatial-numerical and spatial-verbal stimulus set (i.e., larger for auditory than for visual stimuli), and congruency effects, which were asymmetric primarily in the spatial-location stimulus set (i.e., larger for auditory than for visual stimuli). This pattern of effects suggests task-dependent visual dominance.