从数量表征到数表征：具身认知视角下的人类数能力获得

数能力是数学认知的基本成分。与动物所具有的基本数能力不同,人类不仅具备数量表征能力,更重要的是还拥有对数概念进行表征的数表征能力。虽然具身认知与离身认知都对数概念的表征问题进行了解释,但二者却存在明显理论分歧。具身认知观点主要从具身数量表征和数能力发展的具身认知机制两方面为人类独特数能力的获得提供了理论支撑及实证证据。这启示人们需要重视具身学习在数能力形成实践中的关键作用,重视具身数量表征在数学教学中的作用,仍需进一步揭示其内在的心理和神经基础。     

Comparing Data Sets: Implicit Summaries of the Statistical Properties of Number Sets

Comparing datasets, that is, sets of numbers in context, is a critical skill in higher order cognition. Although much is known about how people compare single numbers, little is known about how number sets are represented and compared. We investigated how subjects compared datasets that varied in their statistical properties, including ratio of means, coefficient of variation, and number of observations, by measuring eye fixations, accuracy, and confidence when assessing differences between number sets. Results indicated that participants implicitly create and compare approximate summary values that include information about mean and variance, with no evidence of explicit calculation. Accuracy and confidence increased, while the number of fixations decreased as sets became more distinct (i.e., as mean ratios increase and variance decreases), demonstrating that the statistical properties of datasets were highly related to comparisons. The discussion includes a model proposing how reasoners summarize and compare datasets within the architecture for approximate number representation.     

Mental representations of magnitude and order: A dissociation by sensorimotor learning

Numbers and spatially directed actions share cognitive representations. This assertion is derived from studies that have demonstrated that the processing of small- and large-magnitude numbers facilitates motor behaviors that are directed to the left and right, respectively. However, little is known about the role of sensorimotor learning for such number–action associations. In this study, we show that sensorimotor learning in a serial reaction-time task can modify the associations between number magnitudes and spatially directed movements. Experiments 1 and 3 revealed that this effect is present only for the learned sequence and does not transfer to a novel unpracticed sequence. Experiments 2 and 4 showed that the modification of stimulus–action associations by sensorimotor learning does not occur for other sets of ordered stimuli such as letters of the alphabet. These results strongly suggest that numbers and actions share a common magnitude representation that differs from the common order representation shared by letters and spatially directed actions. Only the magnitude representation, but not the order representation, can be modified episodically by sensorimotor learning.     

Why is number word learning hard? Evidence from bilingual learners

Young children typically take between 18 months and 2 years to learn the meanings of number words. In the present study, we investigated this developmental trajectory in bilingual preschoolers to examine the relative contributions of two factors in number word learning: (1) the construction of numerical concepts, and (2) the mapping of language specific words onto these concepts. We found that children learn the meanings of small number words (i.e., one, two, and three) independently in each language, indicating that observed delays in learning these words are attributable to difficulties in mapping words to concepts. In contrast, children generally learned to accurately count larger sets (i.e., five or greater) simultaneously in their two languages, suggesting that the difficulty in learning to count is not tied to a specific language. We also replicated previous studies that found that children learn the counting procedure before they learn its logic – i.e., that for any natural number, n, the successor of n in the count list denotes the cardinality n + 1. Consistent with past studies, we found that children’s knowledge of successors is first acquired incrementally. In bilinguals, we found that this knowledge exhibits item-specific transfer between languages, suggesting that the logic of the positive integers may not be stored in a language-specific format. We conclude that delays in learning the meanings of small number words are mainly due to language-specific processes of mapping words to concepts, whereas the logic and procedures of counting appear to be learned in a format that is independent of a particular language and thus transfers rapidly from one language to the other in development.     

Agreement and movement: a syntactic analysis of attraction

This paper links experimental psycholinguistics and theoretical syntax in the study of subject-verb agreement. Three experiments of elicited spoken production making use of specific characteristics of Italian and French are presented. They manipulate and examine its impact on the occurrence of 'attraction' errors (i.e. incorrect agreement with a word that is not the subject of the sentence). Experiment 1 (in Italian) shows that subject modifiers do not trigger attraction errors in free inverted VS (Verb Subject) structures, although attraction was found in VS interrogatives in English (Vigliocco, G., & Nicol, J. (1998). Separating hierarchical relations and word order in language production. Is proximity concord syntactic or linear? Cognition, 13-29) In Experiment 2 (in French), we report stronger attraction with preverbal clitic object pronouns than with subject modifiers. Experiment 3 (in French) shows that displaced direct objects in the cleft construction trigger attraction effects, in spite of the fact that the object does not intervene between the subject and the verb in the surface word order (OSV). Moreover, attraction is stronger in structures with subject-verb inversion (...). These observations are shown to be naturally interpretable through the tools of formal syntax, as elaborated within the Principles and Parameters/Minimalist tradition. Three important constructs are discussed: (1) the hierarchical representation of the sentence during syntactic construction, and the role of intermediate positions by which words transit when they move; (2) the role of specific hierarchical (c-command) but also linear (precedence) relations; and (3) the possibility that agreement involves two functionally distinct components. A gradient of computational complexity in agreement is presented which relates empirical evidence to these theoretical constructs.     

Quantifier comprehension in corticobasal degeneration

In this study, we investigated patients with focal neurodegenerative diseases to examine a formal linguistic distinction between classes of generalized quantifiers, like "some X" and "less than half of X." Our model of quantifier comprehension proposes that number knowledge is required to understand both first-order and higher-order quantifiers. The present results demonstrate that corticobasal degeneration (CBD) patients, who have number knowledge impairments but little evidence for a deficit understanding other aspects of language, are impaired in their comprehension of quantifiers relative to healthy seniors, Alzheimer's disease (AD) and frontotemporal dementia (FTD) patients [F(3,77)=4.98; p<.005]. Moreover, our model attempts to honor a distinction in complexity between classes of quantifiers such that working memory is required to comprehend higher-order quantifiers. Our results support this distinction by demonstrating that FTD and AD patients, who have working memory limitations, have greater difficulty understanding higher-order quantifiers relative to first-order quantifiers [F(1,77)=124.29; p<.001]. An important implication of these findings is that the meaning of generalized quantifiers appears to involve two dissociable components, number knowledge and working memory, which are supported by distinct brain regions.     

The cognitive profile of Chinese children with mathematics difficulties

This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children’s mathematics learning difficulties. Children with mathematics difficulties (MD) of two age groups (7-8 and 9-11 years) were compared with age-matched typically achieving children. For both age groups, children with MD performed significantly worse than their age-matched controls on all of the domain-specific and domain-general measures. Further analyses revealed that the MD children with literacy difficulties (MD/RD group) performed the worst on all of the measures, whereas the MD-only group was significantly outperformed by the controls on the four domain-specific measures and verbal working memory. Stepwise discriminant analyses showed that both number fact retrieval and place value concept were significant factors differentiating the MD and non-MD children. To conclude, deficits in domain-specific skills, especially those of number fact retrieval and place value understanding, characterize the profile of Chinese children with MD.     

“Counting” serially presented stimuli by human and nonhuman primates and pigeons

Much of Stewart Hulse's career was spent analyzing how animals can extract patterned information from sequences of stimuli. Yet an additional form of information contained in a sequence may be the number of times different elements occurred. Experiments that required numerical discrimination between different stimulus items presented in sequence are analyzed for primates and pigeons. It is shown that a model based on magnitude discrimination can account well for data from human and nonhuman primate experiments. Similar experiments carried out with pigeons showed a strong recency effect not found with primates. The pigeon data are modeled successfully, however, by assuming that representations of events decay as other events are presented.     

Leibniz's Concept of Substance and his Reception of John Calvin's Doctrine of the Eucharist

Leibniz saw the question of the eucharist as a crucial stumbling block to the agreement between Lutherans and Calvinists. Mandated together with Daniel Ernst Jablonsky to prepare working documents for the negotiations between Hanover and Brandenburg in 1697, Leibniz carefully read through the Calvinist Confessions of faith and the works of Calvin in their 1671 edition. He made an extensive collection of excerpts from the Confessions of faith and from Calvin's Institutes all intended to show that Calvinists admitted the substantial presence of Christ's body in the eucharist. (This collection of excerpts is analysed here for the first time and compared with another little-known document, the Unvorgreiffliches Bedencken). L. had argued previously in 1691/92 that, contrary to the assertions of Pellisson-Fontanier, his own conception of substance and of Christ's presence in the eucharist was completely different from Calvin's. However, by 1697, it was clear to Leibniz that Calvin's concept of substance, which was broadly speaking Aristotelian, was never defined clearly by the reformer, and could be made to coincide with Leibniz's own notion of substance as force rather than substance in its dimensional sense. At the same time L. dissociated Ubiquitarianism (doctrine characteristic of late sixteenth century Lutheranism, which defended the dimensional presence of Christ's body in heaven and in the eucharist, by arguing that Christ in his divine nature could cause his physical body to be present in several places at the same time) from Lutheranism. He also drove a wedge between the doctrines of Zwingli and Calvin. L. thus attempted to find religious union on a common ontology and he might well have succeeded if it were not for complex political circumstances, which ultimately caused the failure of the negotiations.     

How do we convert a number into a finger trajectory?

How do we understand two-digit numbers such as 42? Models of multi-digit number comprehension differ widely. Some postulate that the decades and units digits are processed separately and possibly serially. Others hypothesize a holistic process which maps the entire 2-digit string onto a magnitude, represented as a position on a number line. In educated adults, the number line is thought to be linear, but the “number sense” hypothesis proposes that a logarithmic scale underlies our intuitions of number size, and that this compressive representation may still be dormant in the adult brain. We investigated these issues by asking adults to point to the location of two-digit numbers on a number line while their finger location was continuously monitored. Finger trajectories revealed a linear scale, yet with a transient logarithmic effect suggesting the activation of a compressive and holistic quantity representation. Units and decades digits were processed in parallel, without any difference in left-to-right vs. right-to-left readers. The late part of the trajectory was influenced by spatial reference points placed at the left end, middle, and right end of the line. Altogether, finger trajectory analysis provides a precise cognitive decomposition of the sequence of stages used in converting a number to a quantity and then a position.