Rapid proportion comparison with spatial arrays of frequently used meaningful visual symbols

It has been shown that when two arrays of Arabic numerals were briefly presented, observers could accurately indicate which array contained the larger number of a target numeral. This study investigated whether this rapid proportion comparison can be extended to other meaningful symbols that share some of notable properties of Arabic numerals. We tested mainly several Japanese Kanji letters, each of which represents a meaning and can work as a word. Using physically identical stimulus sets that could be interpreted as different types of letters, Experiment 1 first confirmed the rapid proportion comparison with Arabic numerals for Japanese participants. Experiment 2 showed that the rapid proportion comparison can be extended to Kanji numerals. Experiment 3 successfully demonstrated that rapid proportion judgments can be found with non-quantitative Kanji letters that are used frequently. Experiment 4 further demonstrated the rapid proportion comparison with frequently used meaningful non-letter symbols (gender icons). The rapid processing cannot be attributed to fluent processing of familiar items, because it was not found with familiar phonograms (Japanese Kana letters). These findings suggest that the rapid proportion comparison can be commonly found with frequently used meaningful symbols, even though their meaning is not relevant to the task.     

Sequential responding to arabic numerals with wild cards by the chimpanzee (Pan troglodytes)

One adult female chimpanzee (Pan troglodytes) was trained to respond serially to three arabic numerals between 1 and 9, presented on a cathode-ray-tube (CRT) screen. To examine the factors affecting her sequential responding behavior, wild-card items were added to the three-item sequences. When this wild-card item remained until the subject responded to the last numeral (i.e., the terminator condition), her response to the terminator at each point of the sequence was controlled by the ordinal distance between numerals. Thus, the number of responses to the terminator increased as the ordinal distance between numerals increased. When the wild-card item was eliminated by the subject’s response (wild-card conditions), the probability of responses to the wild card before the first numeral increased as a function of the serial position of the first numeral. These results were consistent with previous studies of response time and suggest both serial position and symbolic distance effects. It is suggested that the subject might form the integrated 9-item linear representations by training of possible subsets of three-item sequences. Knowledge concerning the ordinal position of each numeral was established through this training. Received: 27 October 1999 / Accepted: 22 November 1999     

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.     

Flexible mental processes in numerical size judgments: The case of hebrew letters that are used to convey numbers

In addition to its primary linguistic function, the Hebrew alphabet is sometimes used as a means of number notation (i.e., the system of gematria). Hebrew letters, Arabic numerals, Hebrew number names, and Hebrew letter names were used in a numerical size comparison task, in which two visually presented symbols were compared for numerical value while irrelevant variations in their physical size had to be ignored. A size congruity effect, indicated by faster responses when differences in physical and numerical size were consistent, was larger for Arabic numerals than for number names. The effect for Hebrew letters was similar to that for Arabic numerals and was stronger than that observed for letter names. These results suggest flexible processing of Hebrew letters, so that they function as ideographic symbols in an arithmetic context. A distance effect, indicated by an inverse relationship between reaction time and numerical distance, was found for all notations but was particularly strong for Hebrew letters.     

Mapping Among Number Words,Numerals, and Nonsymbolic Quantities in Preschoolers

In mathematically literate societies, numerical information is represented in 3 distinct codes: a verbal code (i.e., number words); a digital, symbolic code (e.g., Arabic numerals); and an analogical code (i.e., quantities; Dehaene, 1992). To communicate effectively using these numerical codes, our understanding of number must involve an understanding of each representation as well as how they map to other representations. In the current study, we looked at 3- and 4-year-old children’s understanding of Arabic numerals in relation to both quantities and number words. The results suggest that the mapping between quantities and numerals is more difficult than the mapping between numerals and number words and between number words and quantities. Thus, we compared 2 competing models designed to investigate how children represent the meanings of Arabic numbers—whether numerals are mapped directly to the quantities they represent or instead if numerals are mapped to quantities indirectly via a direct mapping to number words. We found support for the latter suggesting that children may first map numerals to number words (another symbolic representation) and only through this mapping are numerals subsequently tied to the quantities they represent. In addition, unlike both mappings involving quantity, the mapping between the 2 symbolic representations of number (numerals and number words) was not set-size-dependent, therefore providing further evidence that children may map symbols to other symbols in the absence of a quantity referent. Together, the results provide new insight into the important processes involved in how children acquire an understanding of symbolic representations of number.     

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.     

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.     

Use of number by crows: investigation by matching and oddity learning

Hooded crows were trained in two-alternative simultaneous matching and oddity tasks with stimulus sets of three different categories: color (black and white), shape (Arabic Numerals 1 and 2, which were used as visual shapes only), and number of elements (arrays of one and two items). These three sets were used for training successively and repeatedly; the stimulus set was changed to the next one after the criterion (80% correct or better over 30 consecutive trials) was reached with the previous one. Training was continued until the criterion could be reached within the first 30 to 50 trials for each of the three training sets. During partial transfer tests, familiar stimuli (numerals and arrays in the range from 1 to 2) were paired with novel ones (numerals and arrays in the range from 3 to 4). At the final stage of testing only novel stimuli were presented (numerals and arrays in the range from 5 to 8). Four of 6 birds were able to transfer in these tests, and their performance was significantly above chance. Moreover, performance of the birds on the array stimuli did not differ from their performance on the color or shape stimuli. They were capable of recognizing the number of elements in arrays and comparing the stimuli by this attribute. It was concluded that crows were able to apply the matching (or oddity) concept to stimuli of numerical category.     

Further evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus)

A Grey parrot (Psittacus erithacus), able to quantify sets of eight or fewer items (including heterogeneous subsets), to sum two sequentially presented sets of 0–6 items (up to 6), and to identify and serially order Arabic numerals (1–8), all by using English labels (Pepperberg in J Comp Psychol 108:36–44, 1994; J Comp Psychol 120:1–11, 2006a; J Comp Psychol 120:205–216, 2006b; Pepperberg and Carey submitted), was tested on addition of two Arabic numerals or three sequentially presented collections (e.g., of variously sized jelly beans or nuts). He was, without explicit training and in the absence of the previously viewed addends, asked, “How many total?” and required to answer with a vocal English number label. In a few trials on the Arabic numeral addition, he was also shown variously colored Arabic numerals while the addends were hidden and asked “What color number (is the) total?” Although his death precluded testing on all possible arrays, his accuracy was statistically significant and suggested addition abilities comparable with those of nonhuman primates.     

Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude ( Butterworth, 1999 , 2005 ; A. J. Wilson & Dehaene, 2007 ) and/or to access that number magnitude representation from numerical symbols ( Rousselle & No?l, 2007 ). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.     

Effect of task, selection set, and dispersion of attention on visual identification time

Schioldborg, P., Paus, E. & Myhre, G. Effect of task, selection set, and dispersion of attention on visual identification time. Scand. J. Psychol., 1973, 14, 195–198. Letters and digits were presented in rows of one, two, or three, and the time required for identifying their position, colour, class, or names recorded for three subjects. The identification tasks were performed under three different conditions of preparatory set, making use of position, colour, and class as selecting attributes. (1) Under each selection set, only the identification of letters and digits as such required more time with increasing number of items, indicating limiting conditions for a "constancy" model of attention. (2) For all identification tasks, selection procured by class yielded longer identification time than selection by position or colour, suggesting different levels of complexity in information processing. (3) Identification of class under a position set, or position under a class set, required less time than the corresponding combinations with colour in place of position, suggesting that analysis of form and position relate basically to the same processes.     

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.     

Automatic quantity processing in 5-year olds and adults

In this study adults performed numerical and physical size judgments on a symbolic (Arabic numerals) and non-symbolic (groups of dots) size congruity task. The outcomes would reveal whether a size congruity effect (SCE) can be obtained irrespective of notation. Subsequently, 5-year-old children performed a physical size judgment on both tasks. The outcomes will give a better insight in the ability of 5-year-olds to automatically process symbolic and non-symbolic numerosities. Adult performance on the symbolic and non-symbolic size congruity tasks revealed a SCE for numerical and physical size judgments, indicating that the non-symbolic size congruity task is a valid indicator for automatic processing of non-symbolic numerosities. Physical size judgments on both tasks by children revealed a SCE only for non-symbolic notation, indicating that the lack of a symbolic SCE is not related to the mathematical or cognitive abilities required for the task but instead to an immature association between the number symbol and its meaning.     

Grey parrot number acquisition: The inference of cardinal value from ordinal position on the numeral list

A Grey parrot (Psittacus erithacus) had previously been taught to use English count words ("one" through "sih" [six]) to label sets of one to six individual items (Pepperberg, 1994). He had also been taught to use the same count words to label the Arabic numerals 1 through 6. Without training, he inferred the relationship between the Arabic numerals and the sets of objects (Pepperberg, 2006b). In the present study, he was then trained to label vocally the Arabic numerals 7 and 8 ("sih-none", "eight", respectively) and to order these Arabic numerals with respect to the numeral 6. He subsequently inferred the ordinality of 7 and 8 with respect to the smaller numerals and he inferred use of the appropriate label for the cardinal values of seven and eight items. These data suggest that he constructed the cardinal meanings of "seven" ("sih-none") and "eight" from his knowledge of the cardinal meanings of one through six, together with the place of "seven" ("sih-none") and "eight" in the ordered count list.     

Verbal interference with perceptual classification: The effect of semantic structure

In a task of the same form as the standard Stroop test, the relevant attribute was ellipse size and the required responses were the numbers 1 through 6 assigned to each of the ellipses in order of increasing size. The irrelevant attribute consisted of either alphabet letters or the numerical symbols 1 through 6 displayed in the center of each ellipse. The numerals produced more interference with the classification of the relevant attribute than the alphabet letters, supporting Klein’s (1964) results. In addition, the interference due to the irrelevant numerical symbols increased as the distance between the values of the relevant and irrelevant attributes was decreased. Since “distance” is a structural property of the number system, this indicated that the competing response tendencies aroused by the irrelevant numericals involved the semantic structure for numbers. The same results were obtained when numerical quantity, rather than ellipse size, was the relevant attribute.     

"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     

Beyond quantity: Individual differences in working memory and the ordinal understanding of numerical symbols

In two different contexts, we examined the hypothesis that individual differences in working memory (WM) capacity are related to the tendency to infer complex, ordinal relationships between numerical symbols. In Experiment 1, we assessed whether this tendency arises in a learning context that involves mapping novel symbols to quantities by training adult participants to associate dot-quantities with novel symbols, the overall relative order of which had to be inferred. Performance was best for participants who were higher in WM capacity (HWMs). HWMs also learned ordinal information about the symbols that lower WM individuals (LWMs) did not. In Experiment 2, we examined whether WM relates to performance when participants are explicitly instructed to make numerical order judgments about highly enculturated numerical symbols by having participants indicate whether sets of three Arabic numerals were in increasing order. All participants responded faster when sequential sets (3-4-5) were in order than when they were not. However, only HWMs responded faster when non-sequential, patterned sets (1-3-5) were in order, suggesting they were accessing ordinal associations that LWMs were not. Taken together, these experiments indicate that WM capacity plays a key role in extending symbolic number representations beyond their quantity referents to include symbol-symbol ordinal associations, both in a learning context and in terms of explicitly accessing ordinal relationships in highly enculturated stimuli.     

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).     

Negative numbers are generated in the mind

The goal of the present study was to disentangle two possible representations of negative numbers--the holistic representation, where absolute magnitude is integrated with polarity; and the components representation, where absolute magnitude is stored separately from polarity. Participants' performance was examined in two tasks involving numbers from--100 to 100. In the numerical comparison task, participants had to decide which number of a pair was numerically larger/smaller. In the number line task, participants were presented with a spatial number line on which they had to place a number. The results of both tasks support the components representation of negative numbers. The findings suggest that processing of negative numbers does not involve retrieval of their meaning from memory, but rather the integration of the polarity sign with the digits' magnitudes.     

Pragmatic inference, not semantic competence, guides 3-year-olds' interpretation of unknown number words

Before children learn the specific meanings of numerals like six, do they know that they represent precise quantities? Previous studies have reported conflicting evidence and have found that children expect numerals to label precise quantities in some tasks but not in others (Condry & Spelke, 2008; Sarnecka & Gelman, 2004). In this article, we present evidence that some of children's apparent successes are best explained not by domain-specific semantic understanding of number but instead by language-general pragmatic abilities. In Experiment 1, we replicated the findings of the previous studies in a within-subject design. When 3-year-olds saw a set labeled with a number (e.g., five) and an item was added, they preferred a new label (six) over the old one, as though they believed that number words have precise meanings. However, when 1 of 2 sets was labeled (e.g., as five) and children were asked to find the same quantity (five) or a new quantity (six), they performed identically whether the original set was changed in quantity or merely rearranged. Thus, when 2 numerals were offered as alternative labels for 1 set, children behaved as though they had precise meanings, whereas when they were asked to determine which of 2 sets a single numeral referred to, they did not. In Experiment 2, children were tested using similar methods but with novel nouns and objects that were transformed, instead of sets. Children showed the identical pattern of results despite lacking meanings for these words, suggesting that their judgments for numerals may not have relied on semantic knowledge that numerals have precise meanings. We propose that children's behavior can be explained by the use of domain-general pragmatic inference and does not require positing domain-specific numerical knowledge. (PsycINFO Database Record (c) 2013 APA, all rights reserved).