Quantity Judgments Research Articles

Quantity without numbers and numbers without quantity in the parietal cortex

A dominant view in numerical cognition is that processing the quantity indicated by numbers (e.g. deciding the larger between two numbers such as ‘12.07’ or ‘15.02’) relies on the intraparietal regions (IPS) of the cerebral cortex. However, it remains unclear whether the IPS could play a more general role in numerical cognition, for example in (1) quantity processing even with non-numerical stimuli (e.g. choosing the larger of ‘bikini’ and ‘coat’); and/or (2) conceptual tasks involving numbers beyond those requiring quantity processing (e.g. attributing a summer date to either ‘12.07’ or ‘15.02’).In this study we applied fMRI-guided TMS to the left and right IPS, while independently manipulating stimulus and task. Our results showed that IPS involvement in numerical cognition is neither stimulus-specific nor specific for conceptual tasks. Thus, quantity judgments with numerical and non-numerical stimuli were equally affected by IPS-TMS, as well as a number conceptual task not requiring quantity comparisons. However, IPS-TMS showed no impairment for perceptual decisions on numbers without any conceptual processing (i.e. colour judgment), nor for conceptual decisions that did not involve quantity or number stimuli (e.g. summer object: ‘bikini’ or ‘coat’?). These results are consistent with proposals that the parietal areas are engaged in the conceptual representation of numbers but they challenge the most common view that number processing is so automatic that the simple presentation of numbers activates the IPS and a sense of magnitude. Rather, our results show that the IPS is only necessary when conceptual operations need to be explicitly oriented to numerical concepts.

Judgments of discrete and continuous quantity: An illusory Stroop effect

Evidence from human cognitive neuroscience, animal neurophysiology, and behavioral research demonstrates that human adults, infants, and children share a common nonverbal quantity processing system with nonhuman animals. This system appears to represent both discrete and continuous quantity, but the proper characterization of the relationship between judgments of discrete and continuous quantity remains controversial. Some researchers have suggested that both continuous and discrete quantity may be automatically extracted from a scene and represented internally, and that competition between these representations leads to Stroop interference. Here, four experiments provide evidence for a different explanation of adults’ performance on the types of tasks that have been said to demonstrate Stroop interference between representations of discrete and continuous quantity. Our well-established tendency to underestimate individual two-dimensional areas can provide an alternative explanation (introduced here as the “illusory-Stroop” hypothesis). Though these experiments were constructed like Stroop tasks, and they produce patterns of performance that initially appear consistent with Stroop interference, Stroop interference effects are not involved. Implications for models of the construction of cumulative area representations and for theories of discrete and continuous quantity processing in large sets are discussed.

Relative quantity judgment by Asian elephants (Elephas maximus)

This study investigated whether Asian elephants can make relative quantity judgment (RQJ), a dichotomous judgment of unequal quantities ordered in magnitude. In Experiment 1, elephants were simultaneously shown two baskets with differing quantities of bait (up to 6 items). In Experiment 2, elephants were sequentially presented with baits, which could not be seen by elephants in their total quantities. The task of elephants was to choose the larger quantity in both experiments. Results showed that the elephants chose the larger quantity with significantly greater frequency. Interestingly, the elephants did not exhibit disparity or magnitude effects, in which performance declines with a smaller difference between quantities in a two-choice task, or the total quantity increases, respectively. These findings appear to be inconsistent with the previous reports of RQJ in other animals, suggesting that elephants may be using a different mechanism to compare and represent quantities than previously suggested for other species.

Quantity judgments of sequentially presented food items by capuchin monkeys (Cebus apella)

Recent assessments have shown that capuchin monkeys, like chimpanzees and other Old World primate species, are sensitive to quantitative differences between sets of visible stimuli. In the present study, we examined capuchins' performance in a more sophisticated quantity judgment task that required the ability to form representations of food quantities while viewing the quantities only one piece at a time. In three experiments, we presented monkeys with the choice between two sets of discrete homogeneous food items and allowed the monkeys to consume the set of their choice. In Experiments 1 and 2, monkeys compared an entirely visible food set to a second set, presented item-by-item into an opaque container. All monkeys exhibited high accuracy in choosing the larger set, even when the entirely visible set was presented last, preventing the use of one-to-one item correspondence to compare quantities. In Experiment 3, monkeys compared two sets that were each presented item-by-item into opaque containers, but at different rates to control for temporal cues. Some monkeys performed well in this experiment, though others exhibited near-chance performance, suggesting that this species' ability to form representations of food quantities may be limited compared to previously tested species such as chimpanzees. Overall, these findings support the analog magnitude model of quantity representation as an explanation for capuchin monkeys' quantification of sequentially presented food items.

Quantity without numbers and numbers without quantity in the parietal cortex

A dominant view in numerical cognition is that processing the quantity indicated by numbers (e.g. deciding the larger between two numbers such as ‘12.07’ or ‘15.02’) relies on the intraparietal regions (IPS) of the cerebral cortex. However, it remains unclear whether the IPS could play a more general role in numerical cognition, for example in (1) quantity processing even with non-numerical stimuli (e.g. choosing the larger of ‘bikini’ and ‘coat’); and/or (2) conceptual tasks involving numbers beyond those requiring quantity processing (e.g. attributing a summer date to either ‘12.07’ or ‘15.02’). In this study we applied fMRI-guided TMS to the left and right IPS, while independently manipulating stimulus and task. Our results showed that IPS involvement in numerical cognition is neither stimulus-specific nor specific for conceptual tasks. Thus, quantity judgments with numerical and non-numerical stimuli were equally affected by IPS-TMS, as well as a number conceptual task not requiring quantity comparisons. However, IPS-TMS showed no impairment for perceptual decisions on numbers without any conceptual processing (i.e. colour judgment), nor for conceptual decisions that did not involve quantity or number stimuli (e.g. summer object: ‘bikini’ or ‘coat’?). These results are consistent with proposals that the parietal areas are engaged in the conceptual representation of numbers but they challenge the most common view that number processing is so automatic that the simple presentation of numbers activates the IPS and a sense of magnitude. Rather, our results show that the IPS is only necessary when conceptual operations need to be explicitly oriented to numerical concepts.

The effect of emergent features on judgments of quantity in configural and separable displays.

Two experiments investigated effects of emergent features on perceptual judgments of comparative magnitude in three diagrammatic representations: kiviat charts, bar graphs, and line graphs. Experiment 1 required participants to compare individual values; whereas in Experiment 2 participants had to integrate several values to produce a global comparison. In Experiment 1, emergent features of the diagrams resulted in significant distortions of magnitude judgments, each related to a common geometric illusion. Emergent features are also widely believed to underlie the general superiority of configural displays, such as kiviat charts, for tasks requiring the integration of information. Experiment 2 tested the extent of this benefit using diagrams with a wide range of values. Contrary to the results of previous studies, the configural display produced the poorest performance compared to the more separable displays. Moreover, the pattern of responses suggests that kiviat users switched from an integration strategy to a sequential one depending on the shape of the diagram. The experiments demonstrate the powerful interaction between emergent visual properties and cognition and reveal limits to the benefits of configural displays for integration tasks.

Neuronal population coding of continuous and discrete quantity in the primate posterior parietal cortex.

Quantitative knowledge guides vital decisions in the life of animals and humans alike. The posterior parietal cortex in primates has been implicated in representing abstract quantity, both continuous (extent) and discrete (number of items), supporting the idea of a putative generalized magnitude system in this brain area. Whether or not single neurons encode different types of quantity, or how quantitative information is represented in the neuronal responses, however, is unknown. We show that length and numerosity are encoded by functionally overlapping groups of parietal neurons. Using a statistical classifier, we found that the activity of populations of quantity-selective neurons contained accurate information about continuous and discrete quantity. Unexpectedly, even neurons that were nonselective according to classical spike-count measures conveyed robust categorical information that predicted the monkeys' quantity judgments. Thus, different information-carrying processes of partly intermingled neuronal networks in the parietal lobe seem to encode various forms of abstract quantity.

Summation and quantity judgments of sequentially presented sets by capuchin monkeys (Cebus apella)

Capuchin monkeys (Cebus apella) were presented with two sets of food items, identical in food type but differing in number. Animals selected one set and were permitted to consume their choice. Set sizes ranged from 1 to 6 items. In experiment 1, each set was uncovered and recovered before a response was made, and the monkeys selected the larger set at high levels. Experiment 2 presented sets that had both visible and nonvisible food items in them at the time of the response, thus requiring the monkeys to sum the total amount of food that was available. The monkeys again selected the larger set with no decrement in performance. Overall, the data indicate that capuchins, like other more extensively studied primate species in this area of research, are responsive to quantitative differences between sets. Capuchins succeed in making these quantity judgments when sets are nonvisible at choice time and when summation of items must be performed, thus demonstrating coordination of quantification skills and memory. Capuchins also inhibit responses to visible food items when those items are only part of an overall smaller quantity of food compared with a completely nonvisible set.

Events and the ontology of individuals: Verbs as a source of individuating mass and count nouns

What does mass–count syntax contribute to the interpretation of noun phrases (NPs), and how much of NP meaning is contributed by lexical items alone? Many have argued that count syntax specifies reference to countable individuals (e.g., cats) while mass syntax specifies reference to unindividuated entities (e.g., water). We evaluated this claim using the quantity judgment method, and tested the interpretation of words used in mass and count syntax that described either protracted, “durative” events (e.g., mass: some dancing; count: a dance), or instantaneous, “punctual” events (e.g., mass: some jumping; count: a jump). For durative words, participants judged, for example, that six brief dances are more dances but less dancing than two long dances, thus showing a significant difference in their interpretation of the count and mass usages. However, for punctual words, participants judged, for example, that six small jumps are both more jumps and more jumping than two long jumps, resulting in no difference due to mass–count syntax. Further, when asked which dimensions are important for comparing quantities of durative and punctual events, participants ranked number as first in importance for durative and punctual words presented in count syntax, but also for punctual words presented in mass syntax. These results indicate that names for punctual events individuate when used in either mass or count syntax, and thus provide evidence against the idea that mass syntax forces an unindividuated construal. They also indicate that event punctuality as encoded by verbs is importantly linked to the individuation of NPs, and may access a common underlying ontology of individuals.

Rhesus monkeys (Macaca mulatta) succeed on a computerized test designed to assess conservation of discrete quantity

Conservation of quantity occurs through recognition that changes in the physical arrangement of a set of items do not change the quantity of items in that set. Rhesus monkeys (Macaca mulatta) were presented with a computerized quantity judgment task. Monkeys were rewarded for selecting the greater quantity of items in one of two horizontal arrays of items on the screen. On some trials, after a correct selection, no reward was given but one of the arrays was manipulated. In some cases, this manipulation involved moving items closer together or farther apart to change the physical arrangement of the array without changing the quantity of items in the array. In other cases, additional items were added to the initially smaller array so that it became quantitatively larger. Monkeys then made another selection from the two rows of items. Monkeys were sensitive to these manipulations, changing their selections when the number of items in the rows changed but not when the arrangement only was changed. Therefore, monkeys responded on the basis of the quantity of items, and they were not distracted by non-quantitative manipulations of the sets.

Children's Early Understanding of Mass-Count Syntax: Individuation, Lexical Content, and the Number Asymmetry Hypothesis

How does mass-count syntax affect word meaning? Many theorists have proposed that count nouns denote individuals, whereas mass nouns do not (Bloom, 1999; Gordon, 1985; Link, 1983), a proposal that is supported by prototypical examples of each (table, water). However, studies of quantity judgments in 4-year-olds and adults demonstrate that some mass nouns (furniture) do denote individuals (Barner & Snedeker, 2005). This is problematic for bootstrapping theories that posit one-to-one syntax-semantics mappings (individual " count; nonindividual " mass; Bloom, 1999), unless mass nouns that denote individuals are late-learned exceptions to mappings. This article investigates this possibility in 3-year-olds and adults using 2 methods: word extension and quantity judgment. Both methods indicate that novel mass nouns can denote individuals in both age groups, and thus fail to support simplified syntax-semantics mappings. Also, differences between word extension and quantity judgment raise the possibility that the tasks measure different underlying knowledge.

Rightward biases in free-viewing visual bisection tasks: Implications for leftward responses biases on similar tasks

Neurologically normal individuals exhibit strong leftward response biases during free-viewing perceptual judgments of brightness, quantity, and size. When participants view two mirror-reversed objects and they are forced to choose which object appears darker, more numerous, or larger, the stimulus with the relevant feature on the left side is chosen 60–75% of the time. This effect could be influenced by inaccurate judgments of the true centre-point of the objects being compared. In order to test this possibility, 10 participants completed three visual bisection tasks on stimuli known to elicit strong leftward response biases. Participants were monitored using a remote eye-tracking device and instructed to stare at the subjective midpoint of objects presented on a computer screen. Although it was predicted that bisection errors would deviate to the left of centre (as is the case in the line bisection literature), the opposite effect was found. Significant rightward bisection errors were evident on two of the three tasks, and the leftward biases seen during forced-choice tasks could be the result of misjudgments to the right of centre on these same tasks.

Quantity judgments and individuation: evidence that mass nouns count

Three experiments explored the semantics of the mass-count distinction in young children and adults. In Experiments 1 and 2, the quantity judgments of participants provided evidence that some mass nouns refer to individuals, as such. Participants judged one large portion of stuff to be “more” than three tiny portions for substance-mass nouns (e.g. mustard, ketchup), but chose according to number for count nouns (e.g. shoes, candles) and object-mass nouns (e.g. furniture, jewelry). These results suggest that some mass nouns quantify over individuals, and that therefore reference to individuals does not distinguish count nouns from mass nouns. Thus, Experiments 1 and 2 failed to support the hypothesis that there exist one-to-one mappings between mass-count syntax and semantics for either adults or young children. In Experiment 3, it was found that for mass-count flexible terms (e.g. string, stone) participants based quantity judgments on number when the terms were used with count syntax, but on total amount of stuff when used with mass syntax. Apparently, the presence of discrete physical objects in a scene (e.g. stones) is not sufficient to permit quantity judgments based on number. It is proposed that object-mass nouns (e.g. furniture) can be used to refer to individuals due to lexically specified grammatical features that normally occur in count syntax. Also, we suggest that children learning language parse words that refer to individuals as count nouns unless given morpho-syntactic and referential evidence to the contrary, in which case object-mass nouns are acquired.

The Effect of Package Shape on Consumers' Judgments of Product Volume: Attention as a Mental Contaminant

A series of experiments examined how a container’s shape can bias judgments of product quantity. Packages that have shapes that are perceived as attracting more attention are also perceived to contain a greater volume of a product than same‐sized packages that attract less attention. The disparity in attention leads to “mental contamination” of the volume judgment. The bias holds for different sets of containers, for containers placed in different contexts, and for containers with contents varying in desirability. Habituation to an unusual container that attracts attention can reduce the effect, as can viewing containers with a disliked content.

Structure mapping in spatial reasoning

Four experiments examined whether spatial reasoning about non-spatial concepts is based on mapping concepts to space according to similarities in relational structure. Six- and 7-year-old children without any prior graphing experience were asked to reason with graph-like diagrams. In Experiments 1 and 2, children were taught to map time and quantity to vertical and horizontal lines, and then were asked to judge the relative value of a second-order variable (rate) or a first-order variable (quantity) represented in a function line. Children’s judgments indicated that they mapped concepts to space by aligning similar relational structures: quantity judgments corresponded to line height, and rate judgments corresponded to line slope. These correspondences entail a mapping of first-order concepts to first-order spatial dimensions and second-order concepts to second-order spatial dimensions. Experiments 3 and 4 investigated the role of context in establishing relational structure. Children were taught to map age and rate (Experiment 3) or age and size (Experiment 4) to vertical and horizontal lines, and were then asked to judge the rate or the size represented by a function line. In this context, both rate and size were first-order variables, and children’s judgments corresponded to line height, also a first-order variable. The results indicate that spatial reasoning involves a structure-sensitive mapping between concepts and space.

Size matters: impact of item size and quantity on array choice by chimpanzees (Pan troglodytes).

The authors previously reported that chimpanzees (Pan troglodytes) showed a striking bias to select the larger of 2 candy arrays, despite a reversed reward contingency in which the animals received the smaller, nonselected array as a reward, except when Arabic numerals were used as stimuli. A perceptual or incentive-based interference occurred that was overcome by symbolic stimuli. The authors of the present study examined the impact of element size in choice arrays, using 1 to 5 large and small candies. Five test-sophisticated chimpanzees selected an array from the 2 presented during each trial. Their responses were not optimal, as animals generally selected arrays with larger total mass; thus, they received the smaller remaining array as a reward. When choice stimuli differed in size and quantity, element size was more heavily weighted, although choices reflected total candy mass. These results replicate previous findings showing chimpanzees' difficulties with quantity judgments under reverse reward contingencies and also show that individual item size exerts a more powerful interference effect.

Spontaneous use of magnitude discrimination and ordination by the orangutan (Pongo pygmaeus).

The ability to discriminate quantity is descriptive of general cognitive ability. In this study, the authors presented 2 orangutans (Pongo pygmaeus) with a quantity judgment task. Each trial consisted of 2 choices, ranging from 1 to 6 food items in each. The orangutan chose 1 of the quantities, which was removed, and the remaining array was given as a reward. In contrast to chimpanzees previously tested on the same task (S. T. Boysen & G. G. Berntson, 1995; S. T. Boysen, G. G. Berntson, M. B. Hannan, & J. T. Cacioppo, 1996; S. T. Boysen, K. L. Mukobi, & G. G. Berntson, 1999), the orangutans optimized their performance. Orangutans, therefore, attend to differences in magnitude and can spontaneously use ordinality. Results also suggest a cognitive difference between chimpanzees and orangutans.

Effects of group size and procedural influence on consensual judgments of quantity: The examples of damage awards and mock civil juries.

Six-person mock civil juries awarded significantly larger amounts for damages than did 12-person juries, and individuals preferred even larger average awards. A reversal of the ''deep-pockets bias'' observed earlier, an explanation involving temporal fluctuation in normative standards, during the rime interval between the studies, was supported by independent data showing temporal trends in actual civil trial awards. A computational model of consensus that assumed a strong majority of those members with the most similar (closest) personal Preferences decided on the median of their preferences accurately predicted award magnitude. Computer simulations explored the effects of critical faction size (majority, etc.) and location within the group, features that might in turn depend on task environment, cultural dynamics, and social context.

Effects of group size and procedural influence on consensual judgments of quantity: The examples of damage awards and mock civil juries.

Effects of group size and procedural influence on consensual judgments of quantity: The examples of damage awards and mock civil juries.

Split decisions: Problems in the interpretation of results from commissurotomized subjects

It has been widely accepted that there is little if any interhemispheric transfer of perceptual information in subjects following section of the forebrain commissures, although there may be diffuse transfer of emotional and connotative information, and an ability to direct spatial attention between hemispheres. However Sergent has recently shown that split-brained subjects can make rapid and accurate perceptual judgments based on visual stimuli presented simultaneously to the two visual fields. These include judgments of alignment, judgments of relative quantity, and higher-order judgments such as lexical decisions about letter strings straddling the visual fields. However it is premature to conclude that these tasks are accomplished by subcortical transfer of perceptual information; at least some of them can be explained in terms of strategies based on information available to just one hemisphere, or on the transfer of rudimentary information, perhaps accomplished through cross-cueing.