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.     

What do infants know about continuous quantity?

We investigated infants' sensitivity to amount of continuous quantity and to change in amount of continuous quantity. Using a habituation procedure, Experiment 1 examined whether 6-month-old infants can distinguish between different amounts of liquid in a container. Infants looked significantly longer at a novel quantity than at the familiar quantity. Using a violation-of-expectation paradigm, Experiment 2 examined whether 9-month-old infants expect a change in amount when liquid is added to a hidden container which is already one-fourth full of liquid. Infants looked significantly longer at the impossible event than at the possible event. These findings indicate that infants are sensitive to amount, calling into question claims that infants have a quantitative mechanism which is exclusive to number.     

Non-verbal numerical cognition: from reals to integers

Data on numerical processing by verbal (human) and non-verbal (animal and human) subjects are integrated by the hypothesis that a non-verbal counting process represents discrete (countable) quantities by means of magnitudes with scalar variability. These appear to be identical to the magnitudes that represent continuous (uncountable) quantities such as duration. The magnitudes representing countable quantity are generated by a discrete incrementing process, which defines next magnitudes and yields a discrete ordering. In the case of continuous quantities, the continuous accumulation process does not define next magnitudes, so the ordering is also continuous ('dense'). The magnitudes representing both countable and uncountable quantity are arithmetically combined in, for example, the computation of the income to be expected from a foraging patch. Thus, on the hypothesis presented here, the primitive machinery for arithmetic processing works with real numbers (magnitudes).     

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.     

Judgement of discrete and continuous quantity in adults: number counts!

Three experiments involving a Stroop-like paradigm were conducted. In Experiment 1, adults received a number comparison task in which large sets of dots, orthogonally varying along a discrete dimension (number of dots) and a continuous dimension (cumulative area), were presented. Incongruent trials were processed more slowly and with less accuracy than congruent trials, suggesting that continuous dimensions such as cumulative area are automatically processed and integrated during a discrete quantity judgement task. Experiment 2, in which adults were asked to perform area comparison on the same stimuli, revealed the reciprocal interference from number on the continuous quantity judgements. Experiment 3, in which participants received both the number and area comparison tasks, confirmed the results of Experiments 1 and 2. Contrasting with earlier statements, the results support the view that number acts as a more salient cue than continuous dimensions in adults. Furthermore, the individual predisposition to automatically access approximate number representations was found to correlate significantly with adults' exact arithmetical skills.     

Judgement of discrete and continuous quantity in adults: Number counts!

Three experiments involving a Stroop-like paradigm were conducted. In Experiment 1, adults received a number comparison task in which large sets of dots, orthogonally varying along a discrete dimension (number of dots) and a continuous dimension (cumulative area), were presented. Incongruent trials were processed more slowly and with less accuracy than congruent trials, suggesting that continuous dimensions such as cumulative area are automatically processed and integrated during a discrete quantity judgement task. Experiment 2, in which adults were asked to perform area comparison on the same stimuli, revealed the reciprocal interference from number on the continuous quantity judgements. Experiment 3, in which participants received both the number and area comparison tasks, confirmed the results of Experiments 1 and 2. Contrasting with earlier statements, the results support the view that number acts as a more salient cue than continuous dimensions in adults. Furthermore, the individual predisposition to automatically access approximate number representations was found to correlate significantly with adults' exact arithmetical skills.     

Relation of cognitive operational performance to comprehension and production of dimensional adjectives

This study compared normal developing children, aged 34 to 51 mo., on comprehension and production of relative dimensional adjectives using object manipulations in a close elicitation procedure and on Piagetian operational tests of conservation of continuous quantity, length, reversibility and seriation. Analysis indicated a significant difference on the expressive language performance of the transitional and the concrete operational children over the preoperational children, but no significant differences occurred between the first two groups. Children who performed better on seriation were significantly better on expressive language performance. Children classified as operational for length performed better on all language measures than those classified as nonoperational. Reversibility and conservation of a continuous quantity did not differentiate children.     

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.     

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.     

空间量化的心理表征

空间量化(spacial quantification)是空间知觉的基础, 是对特定空间性质的表达。离散量(discrete magnitude)与连续量(continuous quantity)分别反映了空间分立和连续的性质, 二者有着相似的行为效应, 在神经表达上也有部分重叠, 这些证据暗示了二者可能有共同的表征机制——模拟表征(analog magnitude representation)。数量空间映射(number–space mappings)提供了数量与空间关系的直接证据。但空间量化的研究中还有许多未解之谜, 如：空间量化的动态表征、量化机制的普遍性、参照点问题、复杂和多维空间的量化等。在具身认知(embodied cognition)的框架下, 空间量化的心理表征研究将对空间的性质做出更深刻的回答。     

类比数量表征的线索:离散量还是连续量

运用Stroop研究范式,对30名成人被试的类比数量表征线索进行了研究。结果发现:在类比数量的小数和大数段上,不存在表征线索的分化;在个数任务中不出现Stroop效应,但被试在强不一致条件下的错误率显著高于强一致条件下的错误率;在累积面积任务中出现Stroop效应,且在弱不一致、强不一致条件下的反应时、错误率都显著高于强一致、弱一致条件下的反应时、错误率。以上结果表明被试在进行类比数量表征时,可以同时提取离散量线索和连续量线索,但是对离散量线索的抑制要难于对连续量线索的抑制。     

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.     

3～6岁儿童空间非标准测量认知加工线索研究

本研究选取北京市3所幼儿园108名3~6岁儿童为被试,采用个别测查法对儿童进行非标准空间测量时所依据的认知加工线索进行研究,结果表明:1、3~6岁儿童在一维空间测量中从两个端点和端点之间的连续空间提取信息进行量的判断,当从端点和端点之间的连续空间提取的线索不一致时,多数儿童不能对其进行整合;2、3~6岁儿童二维空间测量主要遵循一维规则,并且一维规则适用于各种图形;3、儿童根据空间量中显著特征进行量的判断,儿童对作为判断依据的显著特征的选择受各维度对比度影响。     

A disconfirmation of the quantitative identity-quantitative equivalence sequence

Sixty-four kindergarten children received tests of quantitative identity and quantitative equivalence for the conservations of number and continuous quantity. Two types of identity trials were included: a standard version using a single stimulus, and a modified version which paralleled the equivalence task in its use of two stimuli. In addition, half of the children were asked two questions on each trial (one preceding and one following the transformation), whereas half were asked only the post-transformation question. Neither the number of stimuli used nor the number of questions asked had any effect on performance. In contrast to some previous reports, tests of quantitative identity were no easier than tests of quantitative equivalence. It was concluded that the identity-equivalence décalage, if it exists at all, is less important than previous authors have claimed.     

Weighing evils: the C. S. Lewis approach

It is often argued that the great quantity of evil in our world makes God’s existence less likely than a lesser quantity would, and this, presumably, because the probability that some evils are gratuitous increases as the overall quantity of evil increases. Often, an additive approach to quantifying evil is employed in such arguments. In this paper, we examine C. S. Lewis’ objection to the additive approach, arguing that although he is correct to reject this approach, there is a sense in which he underestimates the quantity of pain. However, the quantity of pain in that sense does not significantly increase the probability that some pain is gratuitous. Therefore, the quantitative argument likely fails.     

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.     

Task differences on conservation and transitivity problems

Task differences were assessed with 120 first graders in a 2 × 3 × 3 factorial design combining Socioeconomic level (middle vs lower), Concepts (conservation of length, transitivity of length, and conservation of continuous quantity), and Tasks (three tasks for each concept selected on the basis of frequency of usage). As predicted, task by concept interactions were found on all four task response measures. This finding is consistent with Piaget's concept of horizontal decalage and with Flavell and Wohlwill's competence-performance model, which proposes that intermediate phases of stage transition are characterized by considerable intertask differences and initial and final phases by intertask consistency in performance. The importance of investigating the effects of task variables on multiple dependent measures at various age levels and points in time was emphasized. Concept, task, and socioeconomic class differences were also found on some of the measures.     

Shorter articles and notes conservation of quantity in children: The effect of vocabulary and participation

In this pilot experiment some new tests for conservation of quantity were devised. The experimental group given these tests was able to make correct judgements, while a control group, matched for age and intelligence, failed to show conservation of quantity in the standard tests used by Piaget. The difference between the tests was analysed, and some factors emerged which, it is suggested, can serve to facilitate the child's performance.     

Rhesus monkeys (<Emphasis Type="Italic">Macaca mulatta</Emphasis>) 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.Electronic Supplementary Material Supplementary material is available for this article at