Quantity Judgments Research Articles

A rapid, objective and implicit measure of visual quantity discrimination

There is evidence that accurate and rapid judgments of visual quantities form an essential component of human mathematical ability. However, explicit behavioural discrimination measures of visual quantities are readily contaminated both by variations in low-level physical parameters and higher order cognitive factors, while implicit measures often lack objectivity and sensitivity at the individual participant level. Here, with electrophysiological frequency tagging, we show discrimination differences between briefly presented visual quantities as low as a ratio of 1.4 (i.e., 14 vs. 10 elements). From this threshold, the neural discrimination response increases with parametrically increasing differences in ratio between visual quantities. Inter-individual variability in magnitude of the EEG response at this population threshold ratio predicts behavioural performance at an independent number comparison task. Overall, these findings indicate that visual quantities are perceptually discriminated automatically and rapidly (i.e., at a glance) within the occipital cortex. Given its high sensitivity, this paradigm could provide an implicit diagnostic neural marker of this process suitable for a wide range of fundamental and clinical applications.

A Design of Underground Communication Network

In order to meet the communication needs of underground space, it is necessary to build an underground communication network. Taking the data of a region as an example, considering the construction cost, the communication expense, equipment and lines carrying capacity and the characteristics of underground communication, the model of the problem was analyzed. By using the iterative judgment of position and quantity of information flow, k-means clustering algorithm, genetic algorithm and other methods, we established LANs and high-level nodes, as well as the trunk line and the branch line. The process of designing the underground network took into account many factors. And the final results meet the actual requirements and the common sense. It provides a reference for similar problems in the future.

Study into long-term memory of a complex learning task in Shetland ponies (Equus caballus)

Three Shetland ponies were tested in memorizing a quantity discrimination task with a computer based learning device. One year before, they had learned to relate a quantity of geometric symbols to another on a computer flat screen presented in a matching to sample design. All three of the ponies solved the task to discriminate between the quantities 4 vs. 5. After a delay of one year, animals were re-tested with an identical test procedure with a quantity discrimination of 3 vs. 2 and the original matching to sample task. None of the animals could recall the discrimination task, while the instrumental task of the test procedure was instantly performed correctly. Considering the evidence on multiple memory systems, we suggest that learning of the motor sequence of the test procedure and learning of the higher-order cognition task involved different parts of the brain and were possibly also encoded, stored and recalled in different ways. The ponies were not involved in any other training, learning tests or activities in the interim. Thus, we suggest that continuous learning and training stimuli might enhance the retention of a complex learning task like the quantity judgment, based on a matching to sample arrangement. These results give implications for further research in the field of learning, memory and training of the horse.

Dealing with interference: Chimpanzees respond to conflicting cues in a food-choice memory task.

Interference effects emerge when responding on the basis of task-relevant features is directly pitted against task-irrelevant cues that could lead to errors. To study potential interference effects in a food-choice memory test, 3 chimpanzees were presented with conflicting information in a magnitude judgment task. In Experiment 1, chimpanzees were presented with an ordinal series of colored containers that they sequenced on the basis of relative preference for the different foods that were consistently hidden under the containers. Chimpanzees also were presented with a relative quantity judgment task in which they saw identical containers cover different amounts of a consistent food type. Then, the ordinal and quantity tasks were combined such that the colored containers from the ordinal task were used as covers for the consistent food type from the quantity task. This created instances of congruency (e.g., a highly preferred colored container placed over the largest food quantity) and incongruency (e.g., a highly preferred colored container placed over a small food quantity) between task-relevant and task-irrelevant features. Interference effects were evident when chimpanzees responded on the basis of task-irrelevant features (i.e., container value) rather than task-relevant features (i.e., food quantity), sometimes leading to suboptimal responses in incongruent trial types. Chimpanzees also demonstrated some evidence of the cognitive control needed to inhibit responding based solely on the learned values of the containers. (PsycINFO Database Record

Who has more? The influence of linguistic form on quantity judgments

Quantity judgment tasks have been increasingly used within and across languages as a diagnostic for noun semantics. Overwhelmingly, results show that notionally atomic nouns (Who has more cats?) are counted, while notionally non-atomic nouns (Who has more milk?) are measured by volume. There are two primary outliers to the strict atomicity-tracking pattern. First, some nouns, like furniture, show primarily cardinality-based results in some studies, indicating atomicity, but nevertheless show systematic non-cardinality judgments in other studies, with comparison based instead on value/utility. Second, it has been reported that speakers of the Amazonian language Yudja favor cardinality-based quantity comparison for all nouns regardless of notional atomicity. In the current study, we show that both of these patterns arise in naïve English speakers in the absence of clear linguistic cues to atomicity, and suggest that the absence or mis-diagnosis of linguistic cues may be behind the reported outliers to atomicity-tracking.

How Illusory Is the Solitaire Illusion? Assessing the Degree of Misperception of Numerosity in Adult Humans

The Solitaire illusion occurs when the spatial arrangement of items influences the subjective estimation of their quantity. Unlike other illusory phenomena frequently reported in humans and often also in non-human animals, evidence of the Solitaire illusion in species other than humans remains weak. However, before concluding that this perceptual bias affects quantity judgments differently in human and non-human animals, further investigations on the strength of the Solitaire illusion is required. To date, no study has assessed the exact misperception of numerosity generated by the Solitaire arrangement, and the possibility exists that the numerical effects generated by the illusion are too subtle to be detected by non-human animals. The present study investigated the strength of this illusion in adult humans. In a relative numerosity task, participants were required to select which array contained more blue items in the presence of two arrays made of identical blue and yellow items. Participants perceived the Solitaire illusion as predicted, overestimating the Solitaire array with centrally clustered blue items as more numerous than the Solitaire array with blue items on the perimeter. Their performance in the presence of the Solitaire array was similar to that observed in control trials with numerical ratios larger than 0.67, suggesting that the illusory array produces a substantial overestimation of the number of blue items in one array relative to the other. This aspect was more directly investigated in a numerosity identification task in which participants were required to estimate the number of blue items when single arrays were presented one at a time. In the presence of the Solitaire array, participants slightly overestimated the number of items when they were centrally located while they underestimated the number of items when those items were located on the perimeter. Items located on the perimeter were perceived to be 76% as numerous as centrally located items. The magnitude of misperception of numerosity reported here may represent a useful tool to help to understand whether non-human animals have different perceptual mechanisms or, instead, do not display adequate numerical abilities to spot the illusory difference generated in the Solitaire array.

Judging crowds’ size by ear and by eye in virtual reality

Judging the size of a group of people is an everyday task, on which many decisions are based. In the present study, we investigated whether judgment of size of different groups of people depended on whether they were presented through the auditory channel, through the visual channel, or through both auditory and visual channels. Groups of humanoids of different sizes (from 8 to 128) were presented within a virtual environment to healthy participants. They had to judge whether there was a lot of people in each group and rate their discomfort in relation to the stimuli with Subjective Units of Distress. Our groups of 96 and 128 virtual humans were judged as crowds regardless of their sensory presentation. The sensory presentation influenced participants’ judgment of virtual human group size ranging from 8 to 48. Moreover, while the quantity judgments in the auditory condition increased linearly with the group size, participants judged the quantity of people in a logarithmic manner in the two other sensory conditions. These results suggest that quantity judgment based on auditory information in a realistic context may often involve implicit arithmetic. Even though our participants were not phobic of crowds, our findings are of interest for the field of virtual reality-based therapy for diverse disorders because they indicate that quantity judgment can potentially be altered in a sensory-specific manner in patients with fear of crowds.

Counting with Colours? Effect of Colours on the Numerical Abilities of House Crows (Corvus splendens) and Common Myna (Acridotheres tristis).

F6,30 = 1.748, p = 0.144; and green: F6,30= 1.085, p = 0.394). Common myna, however, showed a strong influence of red colour in their quantity judgment ( F6,30 = 2.922, p = 0.023) as they succeeded in choosing the largest amount of food between two cups, but not when offered food using green coloured mealworms ( F6,30 = 1.183, p = 0.342). In the next experiment, we hypothesised that both house crow and common myna will prefer red coloured food items over green coloured food items, when factors such as the amount of food is equal. We chose to test red and green colours because both colours play an important role in most avian food selections. Results showed that there were no significant differences in the selection of red or green coloured mealworms for both house crows ( F6,30 = 2.310, p = 0.06) and common myna ( F6,30 = 0.823, p = 0.561).

An exploratory study of calculation of appropriate return days for securing the objectivity of judgment of wagon quantity required

An exploratory study of calculation of appropriate return days for securing the objectivity of judgment of wagon quantity required

Capuchin monkeys (Cebus apella) treat small and large numbers of items similarly during a relative quantity judgment task.

A key issue in understanding the evolutionary and developmental emergence of numerical cognition is to learn what mechanism(s) support perception and representation of quantitative information. Two such systems have been proposed, one for dealing with approximate representation of sets of items across an extended numerical range and another for highly precise representation of only small numbers of items. Evidence for the first system is abundant across species and in many tests with human adults and children, whereas the second system is primarily evident in research with children and in some tests with non-human animals. A recent paper (Choo & Franconeri, Psychonomic Bulletin & Review, 21, 93-99, 2014) with adult humans also reported "superprecise" representation of small sets of items in comparison to large sets of items, which would provide more support for the presence of a second system in human adults. We first presented capuchin monkeys with a test similar to that of Choo and Franconeri in which small or large sets with the same ratios had to be discriminated. We then presented the same monkeys with an expanded range of comparisons in the small number range (all comparisons of 1-9 items) and the large number range (all comparisons of 10-90 items in 10-item increments). Capuchin monkeys showed no increased precision for small over large sets in making these discriminations in either experiment. These data indicate a difference in the performance of monkeys to that of adult humans, and specifically that monkeys do not show improved discrimination performance for small sets relative to large sets when the relative numerical differences are held constant.

Grammatical and pragmatic properties of the DP in children with Specific Language Impairment (SLI) and in children with High Functioning Autism (HFA)

This study investigates whether grammar and pragmatics are separate linguistic components or not, and whether children with SLI and children with HFA have overlapping or distinct linguistic profiles. We examine two DP-related phenomena: the mass-count distinction (grammatical) and the choice for a definite/indefinite article (pragmatic). We tested 27 children with HFA aged 6–14, age and gender matched to 27 children with SLI, and 27 TD controls on a Quantity Judgment Task (mass-count) and an Elicited Production Task (article choice). Our results show that pragmatics can be impaired independently from grammar (in HFA) and vice versa (in SLI), providing evidence for a modular view of grammar and pragmatics, and against an overlap in the profiles of SLI and HFA.

The elusive illusion: Do children (Homo sapiens) and capuchin monkeys (Cebus apella) see the Solitaire illusion?

One approach to gaining a better understanding of how we perceive the world is to assess the errors that human and nonhuman animals make in perceptual processing. Developmental and comparative perspectives can contribute to identifying the mechanisms that underlie systematic perceptual errors often referred to as perceptual illusions. In the visual domain, some illusions appear to remain constant across the lifespan, whereas others change with age. From a comparative perspective, many of the illusions observed in humans appear to be shared with nonhuman primates. Numerosity illusions are a subset of visual illusions and occur when the spatial arrangement of stimuli within a set influences the perception of quantity. Previous research has found one such illusion that readily occurs in human adults, the Solitaire illusion. This illusion appears to be less robust in two monkey species, rhesus macaques and capuchin monkeys. We attempted to clarify the ontogeny of this illusion from a developmental and comparative perspective by testing human children and task-naïve capuchin monkeys in a computerized quantity judgment task. The overall performance of the monkeys suggested that they perceived the numerosity illusion, although there were large differences among individuals. Younger children performed similarly to the monkeys, whereas older children more consistently perceived the illusion. These findings suggest that human-unique perceptual experiences with the world might play an important role in the emergence of the Solitaire illusion in human adults, although other factors also may contribute.

Individual differences in algebraic cognition: Relation to the approximate number and semantic memory systems

The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial–numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities.

Quantity judgments in bilingual speakers (Yudja/Brazilian Portuguese)

This paper contributes to investigations on language-specific encoding of the count mass distinction. Two quantity judgment studies tested adult bilingual speakers in two languages that encode the count-mass distinction differently: Yudja and Brazilian Portuguese. In Yudja all nouns have count denotations. That is, all nouns (including notional mass nouns like ‘water’) can be directly combined with numerals and with count quantifiers. Conversely, in Brazilian Portuguese count and mass nouns are grammaticalized in different ways. For example, only count nouns can be directly combined with numerals. In two studies with 20 Yudja bilingual adults we investigate whether the performance of Yudja speakers in quantity judgment tasks varied depending on the language that they were operating. ************************************************************************************************************************************************************************************************************ Julgamentos de quantidade em falantes bilingues (Yudja/Portugues Brasileiro) Este artigo contribui para o debate sobre a codificacao gramatical da distincao contavel-massivo. Dois julgamentos de quantidade testaram adultos bilingues em duas linguas que codificam a distincao contavel-massivo de formas diferentes: Yudja e o portugues brasileiro. Em Yudja todos os nomes tem uma denotacao contavel. Isto e, todos os nomes (incluindo nomes nocionalmente massivos como agua) podem ser diretamente combinados a numerais e a quantificadores contaveis. Por outro lado, no portugues brasileiro, nomes contaveis e massivos sao gramaticalmente distintos. Por exemplo, somente nomes contaveis podem ser diretamente combinados a numerais. Em dois estudos experimentais com 20 falantes bilingues da lingua Yudja, nos investigamos se a performance dos falantes Yudja bilingues em tarefas de julgamento de quantidade variavam de acordo com a lingua na qual eles eramtestados. Palavras-chave: Contavel-massivo; Semântica; Aquisicao; Bilinguismo; Tarefa de julgamento de quantidade

All notional mass nouns are count nouns in Yudja

<p>This paper investigates the linguistic expression of individuation and counting in Yudja (Juruna family), a Tupi language spoken in Brazil. Relying on the principles of mereotopology (Casati and Varzi 1999, Varzi 2007), the main claim of this paper is that in Yudja all nouns can be used as count nouns. That is, in Yudja maximal self-connected concrete portions of a kind can be considered as atoms and can be counted. This claim is based on two fundamental properties of Yudja. First, all notional mass nouns can be directly combined with numerals. Second, the results of quantity judgments studies with Yudja children and adults suggest that all nouns can be directly combined with count-quantifiers and that count-quantifiers are necessarily interpreted as referring to the number of concrete portions. These properties together suggest that all nouns in Yudja are interpreted as count nouns.</p>

Brazilian Bare Phrases and Referentiality: Evidences from an Experiment

This paper experimentally investigates the denotation of the bare singulars (BS) and bare plural noun phrases (BP) in Brazilian Portuguese (BrP). The first section reviews two theories concerning the semantics of the bare nouns in BrP: the count theory according to which bare nouns are countable (plural sums); and the mass theory, which proposes that there is a difference between these bare nouns, since only the BP is a count noun. The second section presents the experiment. It explores the relation between the semantics of BSs and BPs in a mass context. 64 participants were asked to perform quantity judgments (on number, volume, both number and volume or none) in a comprehension task. The task relied on visual and hearing information. The results show that BSs allow preferentially comparison by volume scales, though they accept the count reading. The presence of the plural morpheme ([-s]) only allows quantity judgments based on number. Thus, the experiment failed to support the count hypothesis that BSs are number neutral (SCHMITT; MUNN, 1999; MUNN; SCHMITT, 2005; MÜLLER, 2002), and corroborates Pires de Oliveira & Rothstein's (2011) mass hypothesis. Relying on Rothstein & Pires de Oliveira (in press), we propose that the morpho-syntactic plural mark imposes counting and that the cardinal reading of the BS is derived from measuring.

Role of mental representations in quantity judgments by jackdaws (Corvus monedula).

The representation of quantity by the preverbal or nonverbal mind is a question of considerable interest in the study of cognition, as it should be generally adaptive to most animals to be able to distinguish quantity. We already know that some primate species and human infants represent and enumerate objects in similar ways. Considerable data also exist concerning such abilities in birds. Our aim in this study has been to find out whether jackdaws (Corvus monedula) are capable of performing relative quantity judgments based on mental representations, and if so, what are the limiting factors to their abilities. In our setting the birds were required to make a choice between two visibly and sequentially placed set of food items which, at the moment of choice were not visible to the subjects. We investigated all the number combinations between 1 and 5. Our results show that jackdaws are able to perform relative quantity judgments successfully, even when temporal cues are controlled for, whereas their performance declines in the direction of larger set size (numerical size effect), and when the difference between the two arrays decreases (numerical distance and ratio effect). These signatures are usually interpreted as evidence for the "accumulator" model of mental representation of quantity. Our control results suggest that jackdaws do not use temporal cues, but may well use total volume as basis for discrimination, perhaps among other attributes (choice may be based on multiple cues).

Chimpanzees sometimes see fuller as better: Judgments of food quantities based on container size and fullness

The context in which food is presented can alter quantity judgments leading to sub-optimal choice behavior. Humans often over-estimate food quantity on the basis of how food is presented. Food appears larger if plated on smaller dishes than larger dishes and liquid volumes appear larger in taller cups than shorter cups. Moreover, smaller but fuller containers are preferred in comparison to larger, but less full containers with a truly larger quantity. Here, we assessed whether similar phenomena occur in chimpanzees. Four chimpanzees chose between two amounts of food presented in different sized containers, a large (2 oz.) and small (1 oz.) cup. When different quantities were presented in the same-sized cups or when the small cup contained the larger quantity, chimpanzees were highly accurate in choosing the larger food amount. However, when different-sized cups contained the same amount of food or the smaller cup contained the smaller amount of food (but looked relatively fuller), the chimpanzees often showed a bias to select the smaller but fuller cup. These findings contribute to our understanding of how quantity estimation and portion judgment is impacted by the surrounding context in which it is presented.

Response Category Width as a Psychophysical Manifestation of Construal Level and Distance

The present research suggests that people adjust their mental response scales to an object's distance and construal level. People make use of wider response categories when they judge distant and abstract as compared with close and concrete stimuli. Across five experiments, participants worked on visual and verbal estimation problems (e.g., length or quantity judgments). Answers were provided in interval format, and differences between minimal and maximal estimates served as a measure of response category width. When target objects were framed as spatially distant rather than close (Studies 1 and 3), as unlikely rather than likely (Study 2), and as abstract rather than concrete (Study 4), category widths increased. Similarly, priming a high-level rather than a low-level mindset yielded wider interval estimates (Study 5). The general discussion highlights the usefulness of category width as a basic measure of construal level and as a theoretical link between various branches of construal-level theory.

Quantity estimation and comparison in western lowland gorillas (Gorilla gorilla gorilla)

We investigated the quantity judgment abilities of two adult male western lowland gorillas (Gorilla gorilla gorilla) by presenting discrimination tasks on a touch-screen computer. Both gorillas chose the larger quantity of two arrays of dot stimuli. On some trials, the relative number of dots was congruent with the relative total area of the two arrays. On other trials, number of dots was incongruent with area. The gorillas were first tested with static dots, then with dots that moved within the arrays, and finally on a task where they were required to discriminate numerically larger subsets within arrays of moving dots. Both gorillas achieved above-chance performance on both congruent and incongruent trials with all tasks, indicating that they were able to use number as a cue even though ratio of number and area significantly controlled responding, suggesting that number was not the only relevant dimension that the gorillas used. The pattern of performance was similar to that found previously with monkeys and chimpanzees but had not previously been demonstrated in gorillas within a computerized test format, and with these kinds of visual stimuli.