儿童数字估计的表征模式与发展

探讨中国儿童数字估计的表征模式与发展趋势。包括两个实验,均采用数字线估计任务,实验一以92名幼儿园、一年级及二年级儿童为被试,考察其在0~100范围的数字估计,结果显示,幼儿园儿童在数字估计更多地采用对数表征,而一二年级的儿童在数字估计中更多地采用线性表征;实验二以86名一、三、五年级儿童为被试,考察其在0~1000范围的数字估计,结果显示,一年级儿童有一半采用对数表征,另一半采用线性表征,而三五年级儿童大多采用线性表征。中国儿童的数字估计表现出与美国儿童相同的发展模式,都是由不精确的对数表征逐步向精确的线性表征发展;人的数表征有多种形式,即使在同一年龄阶段,也会因任务难度的不同而选择不同的表征模式。中国儿童精确数字估计能力的出现要早于美国儿童。     

Physical similarity (and not quantity representation) drives perceptual comparison of numbers: evidence from two Indian notations

Numerical quantity seems to affect the response in any task that involves numbers, even in tasks that do not demand access to quantity (e.g., perceptual tasks). That is, readers seem to activate quantity representations upon the mere presentation of integers. One important piece of evidence in favor of this view comes from the finding of a distance effect in perceptual tasks: When one compares two numbers, response times (RTs) are a function of the numerical distance between them. However, recent studies have suggested that the physical similarity between Arabic numbers is strongly correlated with their numerical distance, and that the former could be a better predictor of RT data in perceptual tasks in which magnitude processing is not required (Cohen, 2009a). The present study explored the Persian and Arabic versions of Indian numbers (Exps. 1 and 2, respectively). Na?ve participants (speakers of Spanish) and users of these notations (Pakistanis and Jordanians) participated in a physical same–different matching task. The RTs of users of the Indian notations were regressed on perceptual similarity (estimated from the Spanish participants’ RTs) and numerical distance. The results showed that, regardless of the degree of correlation between the perceptual similarity function and the numerical distance function, the critical predictor for RTs was perceptual similarity. Thus, participants do not automatically activate Indian integers’ quantity representations, at least not when these numbers are presented in simple perceptual tasks.     

Counting to Infinity: Does Learning the Syntax of the Count List Predict Knowledge That Numbers Are Infinite?

By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: Every natural number, n, has a successor, n + 1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language-specific counting routines (e.g., the rules in English that represent base-10 structure). We tested 4- and 5-year-old children’s knowledge of counting with three tasks, which we then related to (a) children’s belief that 1 can always be added to any number (the successor function) and (b) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge was not directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as 4 years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end.     

Specific temporal-lobe signs and enhanced delayed cross-modal matching performance

Structural damage to the amygdala severely retards delayed cross-modal (tactile-to-sight) matching in primates. Conversely, we hypothesized that people who display signs suggestive of specific temporal lobe lability should show enhanced delayed cross-modal matching performance. The hypothesis was supported. In a single experiment involving 25 subjects, significant negative correlations obtained between the numbers of errors on the cross-modal matching task and numbers of affirmative responses within clusters of items that contained themes of meaningfulness, religious beliefs or ictal, complex partial epileptic (limbic) states. On the other hand, the numbers of errors were not significantly correlated with either clusters of control items or items that are presumed to reflect the function of other temporal-lobe structures. Both matching and questionnaire data were collected under double-blind conditions.     

Capacity limitations and representational shifts in spatial short-term memory

Performance was examined in a task requiring the reconstruction of spatial locations. Previous research suggests that it may be necessary to differentiate between memory for smaller and larger numbers of locations (Postma & DeHaan, 1996), at least when locations are presented simultaneously (Igel & Harvey, 1991). Detailed analyses of the characteristics of performance showed that such a differentiation might also be required for sequential presentation. Furthermore the slope of the function relating each successive response to accuracy was greater with 3 than with 6, 8, or 10 locations that did not differ. Participants also reconstructed the arrays as being more proximal than in fact they were; sequential presentation eliminated this distortion when there were three but not when there were more than three locations. These results support the idea that very small numbers of locations are remembered using a specific form of representation, which is unavailable to larger numbers of locations.     

Comparisons of intervals between subjective numbers

Fifty Ss compared the subjective magnitudes of adjacent, objectively equal intervals between numbers by the method of triads. The numbers investigated were the integers from 1 to 10. For every comparison, the interval between the larger integers was more frequently judged closer, which supported previous findings that, for numbers used in magnitude estimation, subjective number is a negatively accelerated function of objective number. The nature of the psychological and physical variables in a number psychophysical function was discussed.     

Number words and reference to numbers

A realist view of numbers often rests on the following thesis: statements like ‘The number of moons of Jupiter is four’ are identity statements in which the copula is flanked by singular terms whose semantic function consists in referring to a number (henceforth: Identity). On the basis of Identity the realists argue that the assertive use of such statements commits us to numbers. Recently, some anti-realists have disputed this argument. According to them, Identity is false, and, thus, we may deny that the relevant statements commit us to numbers. The present paper argues that the correct linguistic analysis of the relevant number statements supports the anti-realist view that Identity is false. However, as will further be shown, pace the anti-realist, this analysis does not establish that such statements do not commit us to numbers after all.     

Single-digit Arabic numbers do not automatically activate magnitude representations in adults or in children: Evidence from the symbolic same–different task

We investigated whether the mere presentation of single-digit Arabic numbers activates their magnitude representations using a visually-presented symbolic same–different task for 20 adults and 15 children. Participants saw two single-digit Arabic numbers on a screen and judged whether the numbers were the same or different. We examined whether reaction time in this task was primarily driven by (objective or subjective) perceptual similarity, or by the numerical difference between the two digits. We reasoned that, if Arabic numbers automatically activate magnitude representations, a numerical function would best predict reaction time; but if Arabic numbers do not automatically activate magnitude representations, a perceptual function would best predict reaction time. Linear regressions revealed that a perceptual function, specifically, subjective visual similarity, was the best and only significant predictor of reaction time in adults and in children. These data strongly suggest that, in this task, single-digit Arabic numbers do not necessarily automatically activate magnitude representations in adults or in children. As the first study to date to explicitly study the developmental importance of perceptual factors in the symbolic same–different task, we found no significant differences between adults and children in their reliance on perceptual information in this task. Based on our findings, we propose that visual properties may play a key role in symbolic number judgements.     

Mental number line disruption in a right-neglect patient after a left-hemisphere stroke

A right-neglect patient with focal left-hemisphere damage to the posterior superior parietal lobe was assessed for numerical knowledge and tested on the bisection of numerical intervals and visual lines. The semantic and verbal knowledge of numbers was preserved, whereas the performance in numerical tasks that strongly emphasize the visuo-spatial layout of numbers (e.g. number bisection) was impaired. The behavioral pattern of error in the two bisection tasks mirrored the one previously described in left-neglect patients. In other words, our patient misplaced the subjective midpoint (numerical or visual) to the left as function of the interval size. These data, paired with the patient's lesion site are strictly consistent with the tripartite organization of number-related processes in the parietal lobes as proposed by Dehaene and colleagues. According to these authors, the posterior superior parietal lobe on both hemispheres underpins the attentional orientation on the putative mental number line, the horizontal segment of the intraparietal sulcus is bilaterally related to the semantic of the numerical domain, whereas the left angular gyrus subserves the verbal knowledge of numbers. In summary, our results suggest that the processes involved in the navigation along the mental number line, which are related to the parietal mechanisms for spatial attention, and the processes involved in the semantic and verbal knowledge of numbers, are dissociable.     

Memory comparison time for complex digit stimuli: The effect of number length

Woodworth (1938) reported that naming latency increased linearly with the number of digits per number (number length). In the present study, the Sternberg memory scanning paradigm was utilized to investigate this effect. It was found that the slope of the memory scanning function increased linearly with number length: memory scanning time was 40 msec for one-digit numbers, 70 msec for two-digit numbers and 101 msec for 3 digit numbers. The intercepts of the memory scanning functions did not differ for the three types of numbers. Thus the increase in latency may be due to the memory comparison stage of processing. The data suggest that a memory comparison operation occurs for each digit position of the complex memory items composed of more than one digit.     

Trials‐to‐criterion latent inhibition in humans as a function of stimulus pre‐exposure and positive‐schizotypy

Latent inhibition (LI) is a phenomenon during which non‐reinforced pre‐exposure to a stimulus retards later learning of associations with that same stimulus. It has been suggested that LI is a positive function of the amount of stimulus pre‐exposure (PE) and that with very small amounts of PE, facilitation rather than inhibition will occur—particularly in high positive‐schizotypes. Although LI has been demonstrated as a function of the amount of pre‐exposure in animals, human findings have not proved to be so uniform or consistent. The primary objective of the present study was to establish LI as a function of numbers of pre‐exposure on visual and auditory trials‐to‐criterion tasks, with a secondary objective to establish latent facilitation (LF) with very low numbers of pre‐exposure in high positive‐schizotypes. Results revealed a uniform pattern of learning across pre‐exposure conditions, including latent facilitation, on the visual, but not the auditory task. LF was also observed in the high, but not low, scorers in positive‐schizotypy with very low numbers of pre‐exposure on the visual task.     

Two subjective scales of number

The way in which the apparent magnitude of numbers grows with their absolute magnitude was measured with a modified version of the direct technique Marks and Slawson (1966) used to determine the psychophysical exponent for loudness. This modified technique required subjects to estimate how evenly and randomly a sequence of integers appeared to sample the numerical continuum. The results indicate that the apparent magnitude of numbers increases with a decelerated power function of their arithmetic magnitude when a series samples from an open-ended range. However, when an upper boundary of the range is specified, the subjective scale seems to be linear. Random productions of numbers parallel the results found with judgments of presented sequences. The two scales of number provide the basis for an interpretation of the difference between magnitude and category scales: that subjects use numbers differently when the response scale is open-ended Imagnitude estimation than when it has a fixed upper limit tcategory scale. Given the assumption that subjects use numbers in this way in the two tasks, the qualitative relation between magnitude and category scales is predicted.     

Without mercy: the immediate impact of group size on lynch mob atrocity

Two independent research traditions have focused on social contributions to lynching. The sociological power threat hypothesis has argued that lynching atrocity will increase as a function of the relative number of African Americans. The psychological self-attention theory has argued that lynching atrocity will increase as a function of the relative number of mob members. Two series of analyses (one using newspaper reports and the second using photographic records) using different and nonoverlapping samples of lynching events rendered a consistent pattern of results: Lynch mob atrocity did not increase as a function of the relative numbers of African Americans in the county population but it did increase as a function of the relative numbers of mob members in the lynch mob. Discussion considers the implications of these results.     

A test of a two-stage model of magnitude judgment

It has been suggested that the power law J = aⁿ, describing the relationship between numerical magnitude judgments and physical magnitudes, confounds a sensory or input function with an output function flawing to do with O’s use of numbers. Judged magnitudes of differences between stimuli offer some opportunity for separating these functions. We obtained magnitude judgments of differences between paired weights, as well as magnitude judgments of the weights making up the pairs. From the former we calculated simultaneously an input exponent and an output exponent, working upon Attneave’s assumption that both transformations are describable as power functions. The inferred input and output functions, in combination, closely predict the judgments of individual weights by the same Os. Although pooled data (geometric means of judgments) conform fairly well to a linear output function, individual data do not; i.e., individual Os deviate quite significantly fromlinearity and from one another in their use of numbers. Individual values of the inferred sensory exponent, k, show significantly better uniformity over Os than do values of the phenotypica! magnitude exponent previously found to describe interval judgments of weight.     

Place‐value understanding in number line estimation predicts future arithmetic performance

In multi‐digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place‐value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place‐value understanding in first grade. We hypothesized that estimating the position of two‐digit numbers requires place‐value understanding. Therefore, we fitted a linear function to children's estimates of two‐digit numbers and considered the resulting slope as a measure of children's place‐value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place‐value understanding. Additionally, the slope for two‐digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first‐graders' place‐value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place‐value understanding) may drive the association between number line estimation and arithmetic performance.     

Fractions but not negative numbers are represented on the mental number line

The present study is the first to directly compare numerical representations of positive numbers, negative numbers and unit fractions. The results show that negative numbers and unit fractions were not represented in the same way. Distance effects were found when positive numbers were compared with fractions but not when they were compared with negative numbers, thus suggesting that unit fractions but not negative numbers were represented on the number line with positive numbers. As indicated by the semantic congruity effect, negative numbers were perceived to be small, positive numbers were perceived as large, while unit fractions were perceived neither as large nor small. Comparisons between negative numbers were faster than between unit fractions, possibly due to the smaller differences between the holistic magnitudes of the unit fractions. Finally, comparing unit fractions to 1 was faster than comparing them to 0, consistent with the idea that unit fractions are perceived as entities smaller than 1 (Kallai & Tzelgov, 2009). The results are consistent with the idea of a mental division between numbers that represent a quantity (positive numbers and unit fractions) and those that do not (negative numbers).     

Extracting thresholds from noisy psychophysical data.

Psychophysical studies with infants or with patients often are unable to use pilot data, training, or large numbers of trials. To evaluate threshold estimates under these conditions, computer simulations of experiments with small numbers of trials were performed by using psychometric functions based on a model of two types of noise: stimulus-related noise (affecting slope) and extraneous noise (affecting upper asymptote). Threshold estimates were biased and imprecise when extraneous noise was high, as were the estimates of extraneous noise. Strategies were developed for rejecting data sets as too noisy for unbiased and precise threshold estimation; these strategies were most successful when extraneous noise was low for most of the data sets. An analysis of 1,026 data sets from visual function tests of infants and toddlers showed that extraneous noise is often considerable, that experimental paradigms can be developed that minimize extraneous noise, and that data analysis that does not consider the effects of extraneous noise may underestimate test-retest reliability and overestimate interocular differences.     

Effect of reference number on magnitude estimation

The psychophysical function obtained by the method of magnitude estimation was influenced by the reference number (modulus) assigned to a “standard” line and the position of this standard in the range of comparison stimuli. Data from two experiments with judgments of apparent length of lines show how both variables systematically affect the slope of the power function. AllowingO to choose his own reference numbers, even though these numbers varied among as, tended to produce less variability in slope than ifE imposed fixed reference numbers forO to use.     

To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

Recent accounts of number word learning posit that when children learn to accurately count sets (i.e., become “cardinal principle” or “CP” knowers), they have a conceptual insight about how the count list implements the successor function – i.e., that every natural number n has a successor defined as n + 1 (Carey, 2004, 2009; Sarnecka & Carey, 2008). However, recent studies suggest that knowledge of the successor function emerges sometime after children learn to accurately count, though it remains unknown when this occurs, and what causes this developmental transition. We tested knowledge of the successor function in 100 children aged 4 through 7 and asked how age and counting ability are related to: (1) children’s ability to infer the successors of all numbers in their count list and (2) knowledge that all numbers have a successor. We found that children do not acquire these two facets of the successor function until they are about 5½ or 6 years of age – roughly 2 years after they learn to accurately count sets and become CP-knowers. These findings show that acquisition of the successor function is highly protracted, providing the strongest evidence yet that it cannot drive the cardinal principle induction. We suggest that counting experience, as well as knowledge of recursive counting structures, may instead drive the learning of the successor function.