Digit identity influences numerical estimation in children and adults

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, is what matters for task performance. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different hundreds digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7–11 and adults completed 0–1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of “Hundreds” boundaries (e.g., 698 and 701) and “Fifties” boundaries (e.g., 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non‐speeded NLE task. Under both speeded and non‐speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children's or adults’ placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.     

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.     

Are Arabic numerals processed as pictures in a Stroop interference task?

In a picture-word interference task, picture naming is interfered by an incongruent word, but word naming is hardly hindered by the presence of an incongruent picture. In this study, we investigated whether Arabic digits are processed more like pictures or like words. We report two experiments in which Arabic digits and verbal numerals were confronted in a Stroop task. Arabic digit naming is interfered by the presence of an incongruent verbal numeral, while naming the verbal numeral is not influenced by the presence of an incongruent Arabic digit. In a second experiment, we excluded the hypothesis that the results are due to ignoring the Arabic digits: interferences from an incongruent distracter were similar for both notations in a semantic classification task. It seems that an asemantic conversion for Arabic digits is too slow to influence naming times, and that Arabic digit naming, like picture naming, is semantically mediated.     

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.     

整体-局部范式下的负启动效应

在通常的负启动范式基础上 ,以小数字构成的大数字 ,即以同时具有整体特征和局部特征的复合数字为实验材料 ,采用数字命名任务 ,将刺激画面中复合数字的数目和注意指向作为变量进行实验。大学生充任被试。结果发现 ,在单个复合数字条件下对同一客体的非注意指向的特征进行忽略重复 ,注意整体时出现负启动 ,而注意局部时不出现负启动 ;在两个复合数字条件下对启动刺激中充当干扰项的复合数字进行忽略重复 ,则注意整体与注意局部都出现了负启动。该研究表明 ,在不同注意指向时 ,整体 -局部范式下的负启动效应具有知觉组织的层次性 ,并显示出客体内选择和客体间选择对负启动的不同影响。     

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.     

Long-term retention of the differential values of Arabic numerals by chimpanzees (<Emphasis Type="Italic">Pan troglodytes</Emphasis>)

As previously reported (Beran and Rumbaugh, 2001), two chimpanzees used a joystick to collect dots, one-at-a-time, on a computer monitor, and then ended a trial when the number of dots collected was equal to the Arabic numeral presented for the trial. Here, the chimpanzees were presented with the task again after an interval of 6 months and then again after an additional interval of 3.25 years. During each interval, the chimpanzees were not presented with the task, and this allowed an assessment of the extent to which both animals retained the values of each Arabic numeral. Despite lower performance at each retention interval compared to the original study, both chimpanzees performed above chance levels in collecting a quantity of dots equal to the target numeral, one chimpanzee for the numerals 1-7, and the second chimpanzee for the numerals 1-6. For the 3.25-year retention, errors were more dispersed around each target numeral than in the original study, but the chimpanzees' performances again appeared to be based on a continuous representation of magnitude rather than a discrete representation of number. These data provide an experimental demonstration of long-term retention of the differential values of Arabic numerals by chimpanzees.     

感觉记忆中含语义对象的视觉表征

感觉记忆能够以特征形式构建对视觉对象表面信息的表征, 然而对于具有重要意义的潜在抽象信息, 尚未有研究考察其是否影响感觉记忆的视觉表征。结合掩蔽技术与部分报告法范式考察了感觉记忆的视觉表征中是否包含语义信息。以阿拉伯数字作为目标, 实验1与实验2分别采用正立与倒置的汉字大写数字作为掩蔽刺激, 比较其与白噪音刺激的掩蔽效应。结果显示, 汉字大写数字的掩蔽效应均强于白噪音刺激, 且正立汉字大写数字的掩蔽效果较倒置汉字大写数字更强。实验3采用汉字大写数字与汉字常用字分别作为掩蔽刺激, 结果显示汉字大写数字的掩蔽效果强于汉字常用字。上述结果显示, 刺激间的语义相似性带来了视觉表征冲突, 表明感觉记忆的视觉表征中包含了刺激的语义信息。     

Cross-notation number priming investigated at different stimulus onset asynchronies in parity and naming tasks

In this paper, a parity judgment task and a number naming task were used to investigate cross-notational number priming. Primes and targets could be verbal (e.g., seven) or Arabic numbers (e.g., 7), and were always presented in a different notation within the same trial (either a verbal prime and an Arabic target or an Arabic prime and a verbal target). Previous experiments showed that response latencies increase when the distance between prime and target increases (for example, in a naming task, seven is pronounced faster after 6 than after 5). This semantic distance priming effect was the same for Arabic and verbal targets and was the same for within-notation trials as for cross-notation trials. In the present experiments, we wanted to investigate whether the cross-notational priming effect also occurs at SOAs shorter than the ones used in previous experiments. Therefore, we used SOAs of 43, 57, 86, and 115 ms. Semantic distance effects were indeed present at these shorter SOAs: Processing times in the semantic parity judgment task and in the non-semantic naming task increased when the distance between prime and target increased. The results are discussed and integrated within an interactive dual-route model of number processing that postulates that the impact of the semantic and the non-semantic route depends on the task and the notation of the stimuli.     

Association between basic numerical abilities and mathematics achievement

Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum‐based standardized mathematics achievement test in kindergarteners, first, second, and sixth graders. Children completed a number comparison task and a number line estimation task with a balanced set of symbolic (Arabic digits) and non‐symbolic (dot patterns) stimuli. Associations with mathematics achievement were observed for the symbolic measures. Although the association with number line estimation was consistent over grades, the association with number comparison was much stronger in kindergarten compared to the other grades. The current data indicate that a good knowledge of the numerical meaning of Arabic digits is important for children's mathematical development and that particularly the access to the numerical meaning of symbolic digits rather than the representation of number per se is important.     

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.     

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.     

Place and summation coding for canonical and non-canonical finger numeral representations

Fingers can be used to express numerical magnitudes, and cultural habits about the fixed order in which fingers are raised determine which configurations become canonical and which non-canonical. Although both types of configuration carry magnitude information, it has been shown that the canonical ones are recognized faster and directly linked to number semantics. Here we tested whether this difference is a consequence of differences in the qualitative way of processing the two types of configurations. When participants named Arabic digits (Experiment 1) or verbal numerals (Experiment 2) primed by canonical and non-canonical finger configurations, qualitatively different priming patterns were observed for the two types of configurations. Canonical configurations activated a place coding representation, with priming spreading to close smaller and larger magnitudes as a function of the prime–target distance. Conversely, non-canonical configurations activated a summation coding representation priming smaller and equal magnitudes independently of the prime–target distance, and larger targets depending on this distance.     

"Constructive" enumeration by chimpanzees (Pan troglodytes) on a computerized task

Two chimpanzees used a joystick to collect dots, one at a time, on a computer monitor (see video-clip in the electronic supplementary material), and then ended a trial when the number of dots collected was equal to the Arabic numeral presented for the trial. Both chimpanzees performed substantially and reliably above chance in collecting a quantity of dots equal to the target numeral, one chimpanzee for the numerals 1–7, and the second chimpanzee for the numerals 1–6. Errors that were made were seldom discrepant from the target by more than one dot quantity, and the perceptual process subitization was ruled out as an explanation for the performance. Additionally, analyses of trial duration data indicated that the chimpanzees were responding based on the numerosity of the constructed set rather than on the basis of temporal cues. The chimpanzees' decreasing performance with successively larger target numerals, however, appeared to be based on a continuous representation of magnitude rather than a discrete representation of number. Therefore, chimpanzee counting in this type of experimental task may be a process that represents magnitudes with scalar variability in that the memory for magnitudes associated with each numeral is imperfect and the variability of responses increases as a function of the numeral's value. Accepted after revision: 11 June 2001 Electronic Publication     

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.     

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.     

基于位置和基于语义交互作用的无意识启动

采用线索-靶子范式,同时操纵刺激的位置和语义属性,考察了基于位置和基于语义交互作用的无意识启动。实验1和实验2分别要求被试完成定位任务和辨别任务。结果发现,当刺激位置属性与实验任务需求相关时,无论刺激的语义属性是否一致都产生了基于位置的启动效应;当刺激的语义属性与实验任务相关时,出现了基于位置和基于语义交互的无意识启动,即启动项和目标项的位置不同时,语义辨别任务存在启动效应;而启动项与目标项位置相同时,语义辨别任务存在着抑制趋势。研究结果表明,基于位置和基于语义交互作用的无意识启动效应仅存在于辨别任务中,而不存在于定位任务中。     

不同手指表征方式对数量表征能力的影响

采用距离启动范式,考察中国文化背景下不同手指表征方式对数量表征能力的影响。实验首先验证单手表征中不同手指数量表征方式对小数字（1～5）认知表征的影响;实验2则进一步采用中国人特有的单手手指表征,考察其对大数字（5～9）认知表征的影响。结果表明,小数字中出现了标准手指表征方式语义层面的位置编码、非标准手指表征方式知觉层面总和编码的激活;但大数字中两种手指表征方式均出现了语义层面位置编码的激活。此结果与计算模型理论一致,说明当手指数量从少到多变化时,标准手指表征方式为语义性的符号数量表征;而非标准手指表征方式由知觉性的非符号向语义性符号数量表征过渡。     

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。     

Naming two-digit arabic numerals: evidence from masked priming studies

The authors investigated how 2-digit Arabic numerals are named by looking at the effects of masked primes on the naming latencies. Target numerals were named faster when prime and target shared a digit at the same position (e.g., the target 28 primed by 18 and 21). In contrast, naming latencies were slower when prime and target shared 1 or 2 digits at non-corresponding places (e.g., the target 28 primed by 82, 86, or 72). Subsequent experiments showed that these priming effects were situated at the level of the verbal production of the Arabic numerals. The data point to a non-semantically mediated route from visual input to verbal output in the naming of 2-digit Arabic numerals.