Time counts: Bidirectional interaction between time and numbers in human adults

Number is known for influencing time processing, but to what extent time influences number in human adults is unclear. We investigated possible bidirectional interactions (number on time and time on number) using a novel Stroop-like task; participants compared numbers or temporal durations in congruent (larger number presented for longer duration) or incongruent conditions (smaller number presented for longer duration). Time and number tasks were presented in different blocks (Experiment 1) or within the same block of trials with task instructions provided at the offset of the stimuli (Experiment 2). Analyses of response times (RTs) and their distribution revealed that number affected time from early RTs, and time affected number at late RTs – an asymmetry observed only when time and number tasks were presented in separate blocks. Thus, carefully chosen tasks and appropriate data analysis can reveal bidirectionality between time and number, consistent with shared magnitude or decision mechanisms.     

Tapping into the interplay of lexical and number knowledge using fast mapping: A longitudinal eye-tracking study with two-year-olds

Language skills and mathematical competencies are argued to influence each other during development. While a relation between the development of vocabulary size and mathematical skills is already documented in the literature, this study further examines how children’s ability to map a novel word to an unknown object as well as their ability to retain this word from memory may be related to their knowledge of number words. Twenty-five children were tested longitudinally (at 30 and at 36 months of age) using an eye-tracking-based fast mapping task, the Give-a-Number task, and standardized measures of vocabulary. The results reveal that children’s ability to create and retain a mental representation of a novel word was related to number knowledge at 30 months, but not at 36 months while vocabulary size correlated with number knowledge only at 36 months. These results show that even specific mapping processes are initially related to the acquisition of number words and they speak for a parallelism between the development of lexical and number-concept knowledge despite their semantic and syntactic differences.     

Expectations about numerical events in four lemur species (Eulemur fulvus, Eulemur mongoz, Lemur catta and Varecia rubra)

Although much is known about how some primates—in particular, monkeys and apes—represent and enumerate different numbers of objects, very little is known about the numerical abilities of prosimian primates. Here, we explore how four lemur species (Eulemur fulvus, E. mongoz, Lemur catta, and Varecia rubra) represent small numbers of objects. Specifically, we presented lemurs with three expectancy violation looking time experiments aimed at exploring their expectations about a simple 1+1 addition event. In these experiments, we presented subjects with displays in which two lemons were sequentially added behind an occluder and then measured subjects duration of looking to expected and unexpected outcomes. In experiment 1, subjects looked reliably longer at an unexpected outcome of only one object than at an expected outcome of two objects. Similarly, subjects in experiment 2 looked reliably longer at an unexpected outcome of three objects than at an expected outcome of two objects. In experiment 3, subjects looked reliably longer at an unexpected outcome of one object twice the size of the original than at an expected outcome of two objects of the original size. These results suggest that some prosimian primates understand the outcome of simple arithmetic operations. These results are discussed in light of similar findings in human infants and other adult primates.This revised version was published online in March 2005 with corrections to Fig. 6.     

Time and number discrimination in a bisection task with a sequence of stimuli: a developmental approach

Children, aged 5 and 8 years, and adults were tested in a bisection task with a sequence of stimuli in which time and number co-varied. In a counting and a non-counting condition, they were instructed either to process the duration of this sequence while ignoring the number of stimuli (temporal bisection), or to process the number of stimuli while ignoring the duration (numerical bisection). In the temporal bisection task, number interfered with the 5-year-olds' temporal performance, indicating that young children did not process time and number independently in a sequence of stimuli when they had to attend to duration. However, number interference decreased both with age and counting strategy. In contrast, in the numerical bisection task, duration did not interfere with numerical discrimination for any age group.     

Inhibitory control in number-conservation and class-inclusion tasks: A neo-Piagetian inter-task priming study

We investigated whether success in number-conservation and class-inclusion tasks relies on a general ability to inhibit misleading strategies. Two groups of 10-year-olds performed inter-task priming between computerized versions of class-inclusion and number-conservation tasks (Experiment 1). In one group, the class-inclusion task served as a prime and the number-conservation task as an assessment probe and vice versa in the other group. Response times were shorter in the number-conservation task when performed after the class-inclusion task (and vice versa in the other group) than in control prime-probe sequences in which the primes did not require the inhibition of a misleading strategy. Experiment 2 showed that these inter-task priming effects did not simply reflect that class inclusion and number conservation rely on the reversibility of concrete operations as Piaget would have hypothesized. Taken together the results are consistent with the neo-Piagetian assumption that cognitive development is rooted in both the acquisition of knowledge of incremental complexity and the ability to resist (inhibit) previous knowledge. Critically, the present finding suggests that the inhibitory ability is not domain- or strategy-specific.     

Children's representation of symbolic and nonsymbolic magnitude examined with the priming paradigm

How people process and represent magnitude has often been studied using number comparison tasks. From the results of these tasks, a comparison distance effect (CDE) is generated, showing that it is easier to discriminate two numbers that are numerically further apart (e.g., 2 and 8) compared with numerically closer numbers (e.g., 6 and 8). However, it has been suggested that the CDE reflects decisional processes rather than magnitude representation. In this study, therefore, we investigated the development of symbolic and nonsymbolic number processes in kindergartners and first, second, and sixth graders using the priming paradigm. This task has been shown to measure magnitude and not decisional processes. Our findings revealed that a priming distance effect (PDE) is already present in kindergartners and that it remains stable across development. This suggests that formal schooling does not affect magnitude representation. No differences were found between the symbolic and nonsymbolic PDE, indicating that both notations are processed with comparable precision. Finally, a poorer performance on a standardized mathematics test seemed to be associated with a smaller PDE for both notations, possibly suggesting that children with lower mathematics scores have a less precise coding of magnitude. This supports the defective number module hypothesis, which assumes an impairment of number sense.     

Are historic years understood as numbers or events? An fMRI study of numbers with semantic associations

While numbers generally cue processing of quantity or order, they can also contain semantic information, as in the case of historic years (e.g., “1492” calls forth associations of Columbus sailing the ocean blue). Whether these dates are processed as quantities or events may depend on the context in which they occur. We examined such “ambiguous number” processing in two different contexts using a paired-comparison task, recording both behavioral responses and brain activity. Participants were either asked to think of all items as numbers and to choose the larger number, or were told to treat the comparators as events and to choose the later event. Behaviorally, all events showed a normal distance effect, establishing that they may be understood and compared in an ordinal sequence. Functional magnetic resonance imaging (fMRI) demonstrated significant differences between years when treated as numbers versus as events. Dates in both contexts shared activity in parietal lobe regions previously implicated in number processing. Dates as numbers showed no extra-numeric activity, while dates thought of as events evoked activity in temporal semantic processing and frontal semantic retrieval areas. These differences suggest that extra-numeric information may be easily accessed and incorporated during processing when supported by even a weak context. This work supports previous studies showing a dissociation between quantity and meaning, and illustrates the brain areas involved in these different aspects.     

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.