Discrimination of small quantities by fish (redtail splitfin, Xenotoca eiseni)

Discrimination of quantity has been argued to rely on two non-verbal representational systems: an object file system (OFS) for representing small values (≤3–4) and an analog magnitude system (AMS) for representing large magnitudes (>4). Infants’ ability to discriminate 1 versus 2, 1 versus 3, 2 versus 3, but not 1 versus 4 or 2 versus 4 seems to prove the independence of such systems. Here, we show that redtail splitfin fish (Xenotoca eiseni) performed relative quantity estimations preferring to approach the location previously occupied by the larger in number between two groups of conspecifics (no longer visible at test) in sets of 1 versus 2 and 2 versus 3 items, but failed at 3 versus 4 items, thus showing the same set-size limit as infants for discrimination of small quantities. However, when tested with quantities that spanned the boundary of the two systems, that is, 1 versus 4 and 2 versus 4, fish succeeded. These results thus point to either the use of continuous physical variables and/or the use of the AMS also for small numerousness in fish in this task.     

Cross-domain transfer of quantitative discriminations: Is it all a matter of proportion?

Meck and Church (1983) estimated a 5:1 scale factor relating the mental magnitudes representing number to the mental magnitudes representing duration. We repeated their experiment with human subjects. We obtained transfer regardless of the objective scaling between the ranges; a 5:1 scaling for number versus duration (measured in seconds) was not necessary. We obtained transfer even when the proportions between the endpoints of the number range were different. We conclude that, at least in human subjects, transfer from a discrimination based on continuous quantity (duration) to a discrimination based on discrete quantity (number) is mediated by the cross-domain comparability of withindomain proportions. The results of our second and third experiments also suggest that the subjects compare a probe with a criterion determined by the range of stimuli tested rather than by trial-specific referents, in accordance with the pseudologistic model of Killeen, Fetterman, and Bizo (1997).     

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.     

Generalising principles in spite of procedural differences: Children's understanding of division

Between ages 5 and 7, children are known to be quite good at sharing discrete quantities but very bad at sharing continuous quantities. Our aim was to find whether they can transfer their understanding of logical relations from discrete to continuous quantities though the procedures used in sharing these quantities are markedly different.Two samples of 5- to 7-year-olds participated in two studies. In the first study, the items involved partitive division; in the second, quotitive division tasks. In both studies, the children solved tasks with discrete and continuous quantities.Performance varied significantly across age level and logical principle (equivalence between different rounds of sharing versus inverse relation between the divisor and the quotient) but not across type of quantity (discrete versus continuous). There was a very strong relation between performance across type of quantity. We conclude that children can generalise reasoning principles in division across type of quantity in spite of the difference in sharing procedures.     

Non-symbolic arithmetic in adults and young children

Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on non-symbolic numerosities. Previous research has demonstrated that human adults, human infants, and non-human animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these non-symbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on non-numerical continuous quantities, modality-specific quantity information, the adoption of alternative non-arithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.     

The Development of Proportional Reasoning: Effect of Continuous Versus Discrete Quantities

This study examines the development of children's ability to reason about proportions that involve either discrete entities or continuous amounts. Six-, 8- and 10-year olds were presented with a proportional reasoning task in the context of a game involving probability. Although all age groups failed when proportions involved discrete quantities, even the youngest age group showed some success when proportions involved continuous quantities. These findings indicate that quantity type strongly affects children's ability to make judgments of proportion. Children's greater success in judging proportions involving continuous quantities appears to be related to their use of different strategies in the presence of countable versus noncountable entities. In two discrete conditions, children—particularly 8- and 10-year-olds—adopted an erroneous counting strategy, considering the number of target elements but not the relation between target and nontarget elements, either in terms of number or amount. In contrast, in the continuous condition, when it was not possible to count, children may have relied on an early developing ability to code the relative amounts of target and nontarget regions.     

One, two, three, four, or is there something more? Numerical discrimination in day-old domestic chicks

Human adults master sophisticated, abstract numerical calculations that are mostly based on symbolic language and thus inimitably human. Humans may nonetheless share a subset of non-verbal numerical skills, available soon after birth and considered the evolutionary foundation of more complex numerical reasoning, with other animals. These skills are thought to be based on the two systems: the object file system which processes small values (<3) and the analogue magnitude system which processes large magnitudes (>4). Infants’ ability to discriminate 1 vs. 2, 1 vs. 3, 2 vs. 3, but not 1 vs. 4, seems to indicate that the two systems are independent, implying that the conception of a continuous number processing system is based on precursors that appear to be interrupted at or about the number four. The findings from the study being presented here indicating that chicks are able to make a series of discriminations regarding that borderline number (1 vs. 4, 1 vs. 5, 2 vs. 4) support the hypothesis that there is continuity in the number system which processes both small and large numerousness.     

Chimpanzees (Pan troglodytes) accurately compare poured liquid quantities

Although many studies have shown that nonhuman animals can choose the larger of two discrete quantities of items, less emphasis has been given to discrimination of continuous quantity. These studies are necessary to discern the similarities and differences in discrimination performance as a function of the type of quantities that are compared. Chimpanzees made judgments between continuous quantities (liquids) in a series of three experiments. In the first experiment, chimpanzees first chose between two clear containers holding differing amounts of juice. Next, they watched as two liquid quantities were dispensed from opaque syringes held above opaque containers. In the second experiment, one liquid amount was presented by pouring it into an opaque container from an opaque syringe, whereas the other quantity was visible the entire time in a clear container. In the third experiment, the heights at which the opaque syringes were held above opaque containers differed for each set, so that sometimes sets with smaller amounts of juice were dropped from a greater height, providing a possible visual illusion as to the total amount. Chimpanzees succeeded in all tasks and showed many similarities in their continuous quantity estimation to how they performed previously in similar tasks with discrete quantities (for example, performance was constrained by the ratio between sets). Chimpanzees could compare visible sets to nonvisible sets, and they were not distracted by perceptual illusions created through various presentation styles that were not relevant to the actual amount of juice dispensed. This performance demonstrated a similarity in the quantitative discrimination skills of chimpanzees for continuous quantities that matches that previously shown for discrete quantities.     

Erratum to: Chimpanzees (Pan troglodytes) accurately compare poured liquid quantities

Although many studies have shown that nonhuman animals can choose the larger of two discrete quantities of items, less emphasis has been given to discrimination of continuous quantity. These studies are necessary to discern the similarities and differences in discrimination performance as a function of the type of quantities that are compared. Chimpanzees made judgments between continuous quantities (liquids) in a series of three experiments. In the first experiment, chimpanzees first chose between two clear containers holding differing amounts of juice. Next, they watched as two liquid quantities were dispensed from opaque syringes held above opaque containers. In the second experiment, one liquid amount was presented by pouring it into an opaque container from an opaque syringe, whereas the other quantity was visible the entire time in a clear container. In the third experiment, the heights at which the opaque syringes were held above opaque containers differed for each set, so that sometimes sets with smaller amounts of juice were dropped from a greater height, providing a possible visual illusion as to the total amount. Chimpanzees succeeded in all tasks and showed many similarities in their continuous quantity estimation to how they performed previously in similar tasks with discrete quantities (for example, performance was constrained by the ratio between sets). Chimpanzees could compare visible sets to nonvisible sets, and they were not distracted by perceptual illusions created through various presentation styles that were not relevant to the actual amount of juice dispensed. This performance demonstrated a similarity in the quantitative discrimination skills of chimpanzees for continuous quantities that matches that previously shown for discrete quantities.     

Comparing performance in discrete and continuous comparison tasks

The approximate number system (ANS) theory suggests that all magnitudes, discrete (i.e., number of items) or continuous (i.e., size, density, etc.), are processed by a shared system and comply with Weber's law. The current study reexamined this notion by comparing performance in discrete (comparing numerosities of dot arrays) and continuous (comparisons of area of squares) tasks. We found that: (a) threshold of discrimination was higher for continuous than for discrete comparisons; (b) while performance in the discrete task complied with Weber's law, performance in the continuous task violated it; and (c) performance in the discrete task was influenced by continuous properties (e.g., dot density, dot cumulative area) of the dot array that were not predictive of numerosities or task relevant. Therefore, we propose that the magnitude processing system (MPS) is actually divided into separate (yet interactive) systems for discrete and continuous magnitude processing. Further subdivisions are discussed. We argue that cooperation between these systems results in a holistic comparison of magnitudes, one that takes into account continuous properties in addition to numerosities. Considering the MPS as two systems opens the door to new and important questions that shed light on both normal and impaired development of the numerical system.     

Neural signatures of number processing in human infants: evidence for two core systems underlying numerical cognition

Behavioral research suggests two cognitive systems are at the foundations of numerical thinking: one for representing 1-3 objects in parallel and one for representing and comparing large, approximate numerical magnitudes. We tested for dissociable neural signatures of these systems in preverbal infants, by recording event-related potentials (ERPs) as 6-7.5 month-old infants (n = 32) viewed dot arrays containing either small (1-3) or large (8-32) sets of objects in a number alternation paradigm. If small and large numbers are represented by the same neural system, then the brain response to the arrays should scale with ratio for both number ranges, a behavioral and brain signature of the approximate numerical magnitude system obtained in animals and in human adults. Contrary to this prediction, a mid-latency positivity (P500) over parietal scalp sites was modulated by the ratio between successive large, but not small, numbers. Conversely, an earlier peaking positivity (P400) over occipital-temporal sites was modulated by the absolute cardinal value of small, but not large, numbers. These results provide evidence for two early developing systems of non-verbal numerical cognition: one that responds to small quantities as individual objects and a second that responds to large quantities as approximate numerical values. These brain signatures are functionally similar to those observed in previous studies of non-symbolic number with adults, suggesting that this dissociation may persist over vast differences in experience and formal training in mathematics.     

Shared system for ordering small and large numbers in monkeys and humans

There is increasing evidence that animals share with adult humans and perhaps human infants a system for representing objective number as psychological magnitudes that are an analogue of the quantities they represent. Here we show that rhesus monkeys can extend a numerical rule learned with the values 1 through 9 to the values 10, 15, 20, and 30, which suggests that there is no upper limit on a monkey's numerical capacity. Instead, throughout the numerical range tested, both accuracy and latency in ordering two numerical values were systematically controlled by the ratio of the values compared. In a second experiment, we directly compared humans' and monkeys' performance in the same ordinal comparison task. The qualitative and quantitative similarity in their performance provides the strongest evidence to date of a single nonverbal, evolutionarily primitive mechanism for representing and comparing numerical values.     

The logic of interpreting evidence of developmental ordering: strong inference and categorical measures

Developmental ordering is a fundamental prediction of developmental theories and a central issue in developmental research. However, logically sound evidence of developmental ordering is difficult to obtain. This article analyzes the logical basis of testing developmental order hypotheses with categorical measures. Depending on whether saltatory (i.e., discrete) or continuous developmental changes are being assessed, the observed relationship between categorical measures yields very different types of information about developmental ordering. When change is continuous, the relationship between the measures does not confirm any one ordering hypothesis, but rather, disconfirms one or more hypotheses. Whether an underlying variable undergoes saltatory or continuous development has long been recognized as an important theoretical issue, but its impact on the interpretation of developmental ordering has not previously been explicated.     

Verbal interference suppresses exact numerical representation

Language for number is an important case study of the relationship between language and cognition because the mechanisms of non-verbal numerical cognition are well-understood. When the Pirahã (an Amazonian hunter-gatherer tribe who have no exact number words) are tested in non-verbal numerical tasks, they are able to perform one-to-one matching tasks but make errors in more difficult tasks. Their pattern of errors suggests that they are using analog magnitude estimation, an evolutionarily- and developmentally-conserved mechanism for estimating quantities. Here we show that English-speaking participants rely on the same mechanisms when verbal number representations are unavailable due to verbal interference. Followup experiments demonstrate that the effects of verbal interference are primarily manifest during encoding of quantity information, and—using a new procedure for matching difficulty of interference tasks for individual participants—that the effects are restricted to verbal interference. These results are consistent with the hypothesis that number words are used online to encode, store, and manipulate numerical information. This linguistic strategy complements, rather than altering or replacing, non-verbal representations.     

The relative salience of discrete and continuous quantity in young infants

Whether human infants spontaneously represent number remains contentious. Clearfield & Mix (1999 ) and Feigenson, Carey & Spelke (2002 ) put forth evidence that when presented with small sets of 1–3 items infants may preferentially attend to continuous properties of stimuli rather than to number, and these results have been interpreted as evidence that infants may not have numerical competence. Here we present three experiments that test the hypothesis that infants prefer to represent continuous variables over number. In Experiment 1, we attempt to replicate the Clearfield & Mix study with a larger sample of infants. Although we replicated their finding that infants attend to changes in contour length, infants in our study attended to number and perimeter/area simultaneously. In Experiments 2 and 3, we pit number against continuous extent for exclusively large sets (Experiment 2) and for small and large sets combined (Experiment 3). In all three experiments, infants noticed the change in number, suggesting that representing discrete quantity is not a last resort for human infants. These results should temper the conclusion that infants find continuous properties more salient than number and instead suggest that number is spontaneously represented by young infants, even when other cues are available.     

Dissociation between magnitude comparison and relation identification across different formats for rational numbers

ABSTRACT

The present study examined whether a dissociation among formats for rational numbers (fractions, decimals, and percentages) can be obtained in tasks that require comparing a number to a non-symbolic quantity (discrete or else continuous). In Experiment 1, college students saw a discrete or else continuous image followed by a rational number, and had to decide which was numerically larger. In Experiment 2, participants saw the same displays but had to make a judgment about the type of ratio represented by the number. The magnitude task was performed more quickly using decimals (for both quantity types), whereas the relation task was performed more accurately with fractions (but only when the image showed discrete entities). The pattern observed for percentages was very similar to that for decimals. A dissociation between magnitude comparison and relational processing with rational numbers can be obtained when a symbolic number must be compared to a non-symbolic display.     

Development of proportional reasoning: where young children go wrong

Previous studies have found that children have difficulty solving proportional reasoning problems involving discrete units until 10 to 12 years of age, but can solve parallel problems involving continuous quantities by 6 years of age. The present studies examine where children go wrong in processing proportions that involve discrete quantities. A computerized proportional equivalence choice task was administered to kindergartners through 4th-graders in Study 1, and to 1st- and 3rd-graders in Study 2. Both studies involved 4 between-subjects conditions that were formed by pairing continuous and discrete target proportions with continuous and discrete choice alternatives. In Study 1, target and choice alternatives were presented simultaneously; in Study 2, target and choice alternatives were presented sequentially. In both studies, children performed significantly worse when both the target and choice alternatives were represented with discrete quantities than when either or both of the proportions involved continuous quantities. Taken together, these findings indicate that children go astray on proportional reasoning problems involving discrete units only when a numerical match is possible, suggesting that their difficulty is due to an overextension of numerical equivalence concepts to proportional equivalence problems.     

Piagetian conservation of discrete quantities in bonobos (Pan paniscus), chimpanzees (Pan troglodytes), and orangutans (Pongo pygmaeus)

This study investigated whether physical discreteness helps apes to understand the concept of Piagetian conservation (i.e. the invariance of quantities). Subjects were four bonobos, three chimpanzees, and five orangutans. Apes were tested on their ability to conserve discrete/continuous quantities in an over-conservation procedure in which two unequal quantities of edible rewards underwent various transformations in front of subjects. Subjects were examined to determine whether they could track the larger quantity of reward after the transformation. Comparison between the two types of conservation revealed that tests with bonobos supported the discreteness hypothesis. Bonobos, but neither chimpanzees nor orangutans, performed significantly better with discrete quantities than with continuous ones. The results suggest that at least bonobos could benefit from the discreteness of stimuli in their acquisition of conservation skills.     

Putting the elephant back in the herd: elephant relative quantity judgments match those of other species

The ability to discriminate between quantities has been observed in many species. Typically, when an animal is given a choice between two sets of food, accurate performance (i.e., choosing the larger amount) decreases as the ratio between two quantities increases. A recent study reported that elephants did not exhibit ratio effects, suggesting that elephants may process quantitative information in a qualitatively different way from all other nonhuman species that have been tested (Irie-Sugimoto et al. in Anim Cogn 12:193-199, 2009). However, the results of this study were confounded by several methodological issues. We tested two African elephants (Loxodonta africana) to more thoroughly investigate relative quantity judgment in this species. In contrast to the previous study, we found evidence of ratio effects for visible and nonvisible sequentially presented sets of food. Thus, elephants appear to represent and compare quantities in much the same way as other species, including humans when they are prevented from counting. Performance supports an accumulator model in which quantities are represented as analog magnitudes. Furthermore, we found no effect of absolute magnitude on performance, providing support against an object-file model explanation of quantity judgment.     

Children’s multiplicative transformations of discrete and continuous quantities

Recent studies have documented an evolutionarily primitive, early emerging cognitive system for the mental representation of numerical quantity (the analog magnitude system). Studies with nonhuman primates, human infants, and preschoolers have shown this system to support computations of numerical ordering, addition, and subtraction involving whole number concepts prior to arithmetic training. Here we report evidence that this system supports children’s predictions about the outcomes of halving and perhaps also doubling transformations. A total of 138 kindergartners and first graders were asked to reason about the quantity resulting from the doubling or halving of an initial numerosity (of a set of dots) or an initial length (of a bar). Controls for dot size, total dot area, and dot density ensured that children were responding to the number of dots in the arrays. Prior to formal instruction in symbolic multiplication, division, or rational number, halving (and perhaps doubling) computations appear to be deployed over discrete and possibly continuous quantities. The ability to apply simple multiplicative transformations to analog magnitude representations of quantity may form a part of the toolkit that children use to construct later concepts of rational number.