"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     

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.     

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.     

Single-digit and two-digit Arabic numerals address the same semantic number line.

Many theories about human number representation stress the importance of a central semantic representation that includes the magnitude information of small integer numbers, and that is conceived as an abstract, compressed number line. However, thus far there has been little or no direct evidence that units and teens are represented on the same number line. In two masked priming experiments, we show that single-digit and two-digit Arabic numerals are equally well primed by an Arabic numeral with the same number of digits as by an equally distant Arabic numeral with a different number of digits (e.g. the priming effect of 7 on the target 9 is the same as the priming effect of 11 on the target 9). The finding was obtained both with a number naming task and with a parity judgement task. This is in line with the hypothesis that units and teens are part of a continuous number line.     

Developmental functions for speeds of cognitive processes

In Experiment 1, 8- to 21-year-olds were tested on a visual search task in which they determined whether a target digit was present in a set of one to five digits and a memory search task in which they determined whether a target digit was a member of a previously presented set of one to five digits. Increases with age in speeds of visual and memory search were both described well by exponential functions, and the rate of developmental change was similar for the two processes. In Experiment 2, 8- to 22-year-olds were tested on a memory search task, a mental rotation task in which they judged whether a stimulus presented in various orientations was a letter or a mirror image of a letter, an analogical reasoning task in which they judged whether sets of pictures were related to one another according to the same rule, and a mental addition task in which they judged the accuracy of problems such as 2 + 3 = 6. Here, too, for three of the four tasks developmental change was described well by exponential functions with a common rate of change. Results are interpreted in terms of a central mechanism that limits speeded performance and that changes with age.     

One, two, three, four, nothing more: an investigation of the conceptual sources of the verbal counting principles

Since the publication of [Gelman, R., & Gallistel, C. R. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.] seminal work on the development of verbal counting as a representation of number, the nature of the ontogenetic sources of the verbal counting principles has been intensely debated. The present experiments explore proposals according to which the verbal counting principles are acquired by mapping numerals in the count list onto systems of numerical representation for which there is evidence in infancy, namely, analog magnitudes, parallel individuation, and set-based quantification. By asking 3- and 4-year-olds to estimate the number of elements in sets without counting, we investigate whether the numerals that are assigned cardinal meaning as part of the acquisition process display the signatures of what we call "enriched parallel individuation" (which combines properties of parallel individuation and of set-based quantification) or analog magnitudes. Two experiments demonstrate that while "one" to "four" are mapped onto core representations of small sets prior to the acquisition of the counting principles, numerals beyond "four" are only mapped onto analog magnitudes about six months after the acquisition of the counting principles. Moreover, we show that children's numerical estimates of sets from 1 to 4 elements fail to show the signature of numeral use based on analog magnitudes - namely, scalar variability. We conclude that, while representations of small sets provided by parallel individuation, enriched by the resources of set-based quantification are recruited in the acquisition process to provide the first numerical meanings for "one" to "four", analog magnitudes play no role in this process.     

Attention blinks for selection, not perception or memory: reading sentences and reporting targets

In whole report, a sentence presented sequentially at the rate of about 10 words/s can be recalled accurately, whereas if the task is to report only two target words (e.g., red words), the second target suffers an attentional blink if it appears shortly after the first target. If these two tasks are carried out simultaneously, is there an attentional blink, and does it affect both tasks? Here, sentence report was combined with report of two target words (Experiments 1 and 2) or two inserted target digits, Arabic numerals or word digits (Experiments 3 and 4). When participants reported only the targets an attentional blink was always observed. When they reported both the sentence and targets, sentence report was quite accurate but there was an attentional blink in picking out the targets when they were part of the sentence. When targets were extra digits inserted in the sentence there was no blink when viewers also reported the sentence. These results challenge some theories of the attentional blink: Blinks result from online selection, not perception or memory.     

Nonverbal Counting in Humans: The Psychophysics of Number Representation

In a nonverbal counting task derived from the animal literature, adult human subjects repeatedly attempted to produce target numbers of key presses at rates that made vocal or subvocal counting difficult or impossible. In a second task, they estimated the number of flashes in a rapid, randomly timed sequence. Congruent with the animal data, mean estimates in both tasks were proportional to target values, as was the variability in the estimates. Converging evidence makes it unlikely that subjects used verbal counting or time durations to perform these tasks. The results support the hypothesis that adult humans share with nonverbal animals a system for representing number by magnitudes that have scalar variability (a constant coefficient of variation). The mapping of numerical symbols to mental magnitudes provides a formal model of the underlying nonverbal meaning of the symbols (a model of numerical semantics).     

Cross-linguistic regularities in the frequency of number words.

We examine the frequency of numerals and ordinals in seven different languages and/or cultures. Many cross-cultural and cross-linguistic patterns are identified. The most striking is a decrease of frequency with numerical magnitude, with local increases for reference numerals such as 10, 12, 15, 20, 50 or 100. Four explanations are considered for this effect: sampling artifacts, notational regularities, environmental biases and psychological limitations on number representations. The psychological explanation, which appeals to a Fechnerian encoding of numerical magnitudes and to the existence of numerical points of reference, accounts for most of the data. Our finding also has practical importance since it reveals the frequent confound of two experimental variables: numerical magnitude and numeral frequency.     

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。     

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.     

Semantic priming in number naming

The issue of semantic and non-semantic conversion routes for numerals is still debated in numerical cognition. We report two number-naming experiments in which the target numerals were preceded by another numeral (prime). The primes and targets could be presented either in arabic (digit) notation or in verbal (alphabetical) notation. The results reveal a semantically related distance effect: Latencies are fastest when the prime has the same value as the target and increase when the distance between prime and target increases. We argue that the present results are congruent with the idea that the numerals make access to an ordered semantic number line common to all notations, as the results are the same for within-notation priming (arabic-arabic or verbal- verbal) and between-notations priming (arabic-verbal or verbal-arabic). The present results also point to a rapid involvement of semantics in the naming of numerals, also when the numerals are words. As such, they are in line with recent claims of rapid semantic mediation in word naming.     

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.     

Redundancy gain for categorical targets depends on display configuration and duration

Redundancy gain is an improvement in speeded target detection when the number of targets associated with a single response is increased within a single display. The effect has been clearly demonstrated with specific targets, but it is not clear if it occurs in categorization tasks with non-identical targets. The current study tested the effect of target redundancy on speed and accuracy in a go/no-go categorization task. Targets were digits tilted 45° to the left, and were displayed in unilateral, bilateral, or central displays for either 1500?ms or 100?ms. Redundancy gain only occurred for brief targets displayed bilaterally in the upper visual field. The results indicate that redundancy gain is possible for categorization tasks with some bilateral configurations, supporting a role for interhemispheric processing in redundancy gain. Additionally, the results may indicate that processing strategies mask redundancy gain when participants can view targets for a long period of time.     

Use of numerical symbols by the chimpanzee (Pan troglodytes): Cardinals, ordinals, and the introduction of zero

An adult female chimpanzee with previous training in the use of Arabic numerals 1–9 was introduced to the meaning of "zero" in the context of three different numerical tasks. The first two were cardinal tasks where the subject was required either to select numerals corresponding to the number of items presented on a computer screen (productive use of numerals) or to match sets of the appropriate size to numerals presented as samples (receptive use). The third task addressed the ordinal meaning of the same symbols where the subject was required to respond to numerals sequentially, arranging them into an ascending series. The subject mastered the recognition of the meaning of zero in all three tasks. However, details of her usage of the symbol revealed that transfer of the meaning between different kinds of tasks was incomplete, suggesting that the level of abstraction characteristic of human numerical ability was not attained in the chimpanzee. Over the course of acquisition leading to the high levels of accuracy eventually observed, the newly introduced zero appeared to shift along the length of a continuous numerical scale toward the lower end, while confusions with 1 remained the most frequently encountered mistakes. Such patterns of error thus suggest that Ai's understanding of the meaning of zero in relation to the rest of the number symbols was not consistent with an "absence of items versus presence of items" scheme. Electronic Publication     

Semantic access in number processing investigated with Japanese Kanji and Kana numerals

In number processing, semantic representations are efficiently activated. These representations frequently affect task performance, as demonstrated by semantic effects such as the distance effect (longer reaction times to closer numerical pairs in comparison tasks). The objective of this study was to investigate whether efficient semantic access is a distinctive feature of number processing using the Japanese language. Japanese was chosen to investigate possible effects of ideographic processing and word frequency: Kanji numerals in Japanese are ideograms and are used frequently, while Kana numerals are phonograms and are used infrequently. The results confirmed a distance effect in the numerical and physical matching tasks with both Arabic and Kanji numerals regardless of notation or task (Experiment 1). However, the effect was not observed with the Kana pairs (Experiment 2), thereby suggesting that semantic access is not always efficient in number processing. The results can be explained coherently within a general framework of word recognition by assuming that both the type of character and the word frequency play a critical role in determining the efficiency of semantic access.     

Verbal structure of numerals and digits handwriting: New evidence from kinematics

Two experiments used a digitizing tablet to analyse the temporal, spatial, and kinematic characteristics of handwritten production of arabic numbers. They addressed a specific issue of the numerical domain: Does the lexical and syntactic structure of verbal numerals influence the production of arabic numerals (Experiments 1 and 2), even after enforced semantic processing in a comparison task (Experiment 2)? Subjects had to write multi-digit arabic numerals (e.g., 1200) presented in two different verbal structures: a multiplicative one (e.g., teen-hundred, douze cents (twelve hundred)) or an additive one (e.g., thousand-unit-hundred, mille deux cents (one thousand two hundred)). Results show differences in the inter-digit jumps that reflect the influence of the structure of verbal numerals, even after the semantic task. This finding is discussed with regard to different models of number transcoding (McCloskey, Caramazza, & Basili, 1985; Power & Dal Martello, 1990, 1997).     

Ordinal judgments of numerical symbols by macaques (Macaca mulatta)

Two rhesus monkeys (Macaca mulatta) learned that the arabic numerals 0 through 9 represented corresponding quantities of food pellets. By manipulating a joystick, the monkeys were able to make a selection of paired numerals presented on a computer screen. Although the monkeys received a corresponding number of pellets even if the lesser of the two numerals was selected, they learned generally to choose the numeral of greatest value even when pellet delivery was made arrhythmic. In subsequent tests, they chose the numerals of greater value when presented in novel combinations or in random arrays of up to five numerals. Thus, the monkeys made ordinal judgments of numerical symbols in accordance with their absolute or relative values.     

Adults’ number-line estimation strategies: Evidence from eye movements

Although the development of number-line estimation ability is well documented, little is known of the processes underlying successful estimators’ mappings of numerical information onto spatial representations during these tasks. We tracked adults’ eye movements during a number-line estimation task to investigate the processes underlying number-to-space translation, with three main results. First, eye movements were strongly related to the target number’s location, and early processing measures directly predicted later estimation performance. Second, fixations and estimates were influenced by the size of the first number presented, indicating that adults calibrate their estimates online. Third, adults’ number-line estimates demonstrated patterns of error consistent with the predictions of psychophysical models of proportion estimation, and eye movement data predicted the specific error patterns we observed. These results support proportion-based accounts of number-line estimation and suggest that adults’ translation of numerical information into spatial representations is a rapid, online process.