The spatial representation of numerical and non-numerical ordered sequences: Insights from a random generation task

It is widely believed that numbers are spatially represented from left to right on the mental number line. Whether this spatial format of representation is specific to numbers or is shared by non-numerical ordered sequences remains controversial. When healthy participants are asked to randomly generate digits they show a systematic small-number bias that has been interpreted in terms of “pseudoneglect in number space”. Here we used a random generation task to compare numerical and non-numerical order. Participants performed the task at three different pacing rates and with three types of stimuli (numbers, letters, and months). In addition to a small-number bias for numbers, we observed a bias towards “early” items for letters and no bias for months. The spatial biases for numbers and letters were rate independent and similar in size, but they did not correlate across participants. Moreover, letter generation was qualified by a systematic forward direction along the sequence, suggesting that the ordinal dimension was more salient for letters than for numbers in a task that did not require its explicit processing. The dissociation between numerical and non-numerical orders is consistent with electrophysiological and neuroimaging studies and suggests that they rely on at least partially different mechanisms.     

The associations between space and order in numerical and non-numerical sequences

Spatial-numerical associations have been found across different studies, yet the basis for these associations remains debated. The current study employed an order judgment task to adjudicate between two competing accounts of such associations, namely the Mental Number Line (MNL) and Working Memory (WM) models. On this task, participants judged whether number pairs were in ascending or descending order. Whereas the MNL model predicts that ascending and descending orders should map onto opposite sides of space, the WM model predicts no such mapping. Moreover, we compared the spatial-order mapping for numerical and non-numerical sequences because the WM model predicts no difference in mapping. Across two experiments, we found consistent spatial mappings for numerical order along both horizontal and vertical axes, consistent with a MNL model. In contrast, we found no consistent mappings for letter sequences. These findings are discussed in the context of conflicting extant data related to these two models.     

Children's equivalence judgments: Crossmapping effects

Preschoolers made numerical comparisons between sets with varying degrees of shared surface similarity. When surface similarity was pitted against numerical equivalence (i.e., crossmapping), children made fewer number matches than when surface similarity was neutral (i.e, all sets contained the same objects). Only children who understood the number words for the target sets performed above chance in the crossmapping condition. These findings are consistent with previous research on children's non-numerical comparisons (e.g., [Rattermann, M. J., & Gentner, D. (1998). The effect of language on similarity: The use of relational labels improves young children's performance in a mapping task. In K. Holyoak, D. Gentner, & B. Kokinov (Eds.), Advances in analogy research: Integration of theory and data from cognitive, computational, and neural sciences (pp. 274–282). Sofia: New Bulgarian University; Smith, L. B. (1993). The concept of same. In H. W. Reese (Ed.), Advances in child development and behavior, Vol. 24 (pp. 215–252). New York: Academic Press]) and suggest that the same mechanisms may underlie numerical development.     

Seven-month-olds detect ordinal numerical relationships within temporal sequences

Previous evidence has shown that 11-month-olds represent ordinal relations between purely numerical values, whereas younger infants require a confluence of numerical and non-numerical cues. In this study, we show that when multiple featural cues (i.e., color and shape) are provided, 7-month-olds detect reversals in the ordinal direction of numerical sequences relying solely on number when non-numerical quantitative cues are controlled. These results provide evidence for the influence of featural information and multiple cue integration on infants’ proneness to detect ordinal numerical information.     

Numerical and non-numerical ordinality processing in children with and without developmental dyscalculia: Evidence from fMRI

Ordinality is – beyond numerical magnitude (i.e., quantity) – an important characteristic of the number system. There is converging empirical evidence that (intra)parietal brain regions mediate number magnitude processing. Furthermore, recent findings suggest that the human intraparietal sulcus (IPS) supports magnitude and ordinality in a domain-general way. However, the latter findings are derived from adult studies and with respect to children (i.e., developing brain systems) both the neural correlates of ordinality processing and the precise role of the IPS (domain-general vs. domain-specific) in ordinality processing are thus far unknown. The present study aims at filling this gap by employing functional magnetic resonance imaging (fMRI) to investigate numerical and non-numerical ordinality knowledge in children with and without developmental dyscalculia. In children (without DD) processing of numerical and non-numerical ordinality alike is supported by (intra)parietal cortex, thus extending the notion of a domain-general (intra)parietal cortex to developing brain systems. Moreover, activation extents in response to numerical ordinality processing differ significantly between children with and without dyscalculia in inferior parietal regions (supramarginal gyrus and IPS).     

Anticipatory response control in motor sequence learning: evidence from stimulus-response compatibility

Three experiments using a serial four-choice reaction-time (RT) task explored the interaction of sequence learning and stimulus-based response conflict. In Experiment 1, the spatial stimulus-response (S-R) mapping was manipulated between participants. Incompatible S-R mappings produced much higher RTs than the compatible mapping, but sequence learning decreased this S-R compatibility effect. In Experiment 2, the spatial stimulus feature was made task-irrelevant by assigning responses to symbols that were presented at unpredictable locations. The data indicated a Simon effect (i.e., increased RT when irrelevant stimulus location is spatially incompatible with response location) that was reduced by sequence learning. However, this effect was observed only in participants who developed an explicit sequence representation. Experiment 3 replicated this learning-based modulation of the Simon effect using explicit sequence-learning instructions. Taken together, the data support the notion that explicit sequence learning can lead to motor 'chunking', so that pre-planned response sequences are shielded from conflicting stimulus information.     

Dissociating cue-related and task-related processes in task inhibition: evidence from using a 2:1 cue-to-task mapping.

Performance of task sequences is assumed to rely on activation and inhibition of tasks. An empirical marker of task inhibition is the so-called n-2 repetition cost, which is assessed by comparing performance in trial n-2 task repetitions (i.e., ABA) with that in n-2 task switches (i.e., CBA). Current theoretical accounts assume that inhibition acts on the level of task representations (i.e., task sets). However, another potential target of task inhibition could be the representation of the task cue. To decide between these two alternatives, the authors used a 2:1 cue-to-task mapping design. They found significant n-2 task repetition costs both with n-2 cue repetitions and n-2 cue switches. These costs were about equal (Experiment 1), and this data pattern was found for both short and long cuing intervals (Experiment 2). Together, the data suggest that task inhibition acts on task sets and not on cue representations.     

Mapping Among Number Words,Numerals, and Nonsymbolic Quantities in Preschoolers

In mathematically literate societies, numerical information is represented in 3 distinct codes: a verbal code (i.e., number words); a digital, symbolic code (e.g., Arabic numerals); and an analogical code (i.e., quantities; Dehaene, 1992). To communicate effectively using these numerical codes, our understanding of number must involve an understanding of each representation as well as how they map to other representations. In the current study, we looked at 3- and 4-year-old children’s understanding of Arabic numerals in relation to both quantities and number words. The results suggest that the mapping between quantities and numerals is more difficult than the mapping between numerals and number words and between number words and quantities. Thus, we compared 2 competing models designed to investigate how children represent the meanings of Arabic numbers—whether numerals are mapped directly to the quantities they represent or instead if numerals are mapped to quantities indirectly via a direct mapping to number words. We found support for the latter suggesting that children may first map numerals to number words (another symbolic representation) and only through this mapping are numerals subsequently tied to the quantities they represent. In addition, unlike both mappings involving quantity, the mapping between the 2 symbolic representations of number (numerals and number words) was not set-size-dependent, therefore providing further evidence that children may map symbols to other symbols in the absence of a quantity referent. Together, the results provide new insight into the important processes involved in how children acquire an understanding of symbolic representations of number.     

Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude ( Butterworth, 1999 , 2005 ; A. J. Wilson & Dehaene, 2007 ) and/or to access that number magnitude representation from numerical symbols ( Rousselle & No?l, 2007 ). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.     

Chunking in task sequences modulates task inhibition

In a study of the formation of representations of task sequences and its influence on task inhibition, participants first performed tasks in a predictable sequence (e.g., ABACBC) and then performed the tasks in a random sequence. Half of the participants were explicitly instructed about the predictable sequence, whereas the other participants did not receive these instructions. Task-sequence learning was inferred from shorter reaction times (RTs) in predictable relative to random sequences. Persisting inhibition of competing tasks was indicated by increased RTs in n- 2 task repetitions (e.g., ABA) compared with n- 2 nonrepetitions (e.g., CBA). The results show task-sequence learning for both groups. However, task inhibition was reduced in predictable relative to random sequences among instructed-learning participants who formed an explicit representation of the task sequence, whereas sequence learning and task inhibition were independent in the noninstructed group. We hypothesize that the explicit instructions led to chunking of the task sequence, and that n- 2 repetitions served as chunk points (ABA-CBC), so that within-chunk facilitation modulated the inhibition effect.     

表面相似性与共享标签知识对智障儿童数量表征的影响

采用给数取物任务和数量比较任务,考察表面相似性与共享标签知识对96名7～16岁智障儿童数量表征的影响。研究结果表明：（1）智障儿童数量表征能力随着年龄增长而提高,11～13岁和14～16岁智障儿童完成数量比较任务的正确率显著高于7～10岁;（2）智障儿童在高表面相似物体下完成数量比较任务的正确率显著高于低表面相似物体下的正确率;（3）擅长使用数字标签的智障儿童,在数量比较任务的表现显著优于不擅长组。     

Temporal order judgment reveals how number magnitude affects visuospatial attention

The existence of spatial components in the mental representation of number magnitude has raised the question regarding the relation between numbers and spatial attention. We present six experiments in which this relation was examined using a temporal order judgment task to index attentional allocation. Results demonstrate that one important consequence of numerical processing is the automatic allocation of spatial attention, which in turn affects the perception of the temporal order of visual events. Given equal onset time, left-side stimuli are perceived to occur before right-side stimuli when a small number (1, 2) is processed, whereas right-side stimuli are perceived to occur before left-side stimuli when a larger number (8, 9) is processed. In addition, we show that this attentional effect is specific to quantity processing and does not generalize to non-numerical ordinal sequences.     

Eye gaze patterns reveal how we reason about fractions

ABSTRACT

Fractions are defined by numerical relationships, and comparing two fractions’ magnitudes requires within-fraction (holistic) and/or between-fraction (componential) relational comparisons. To better understand how individuals spontaneously reason about fractions, we collected eye-tracking data while they performed a fraction comparison task with conditions that promoted or obstructed different types of comparisons. We found evidence for both componential and holistic processing in this mixed-pairs task, consistent with the hybrid theory of fraction representation. Additionally, making within-fraction eye movements on trials that promoted a between-fraction comparison strategy was associated with slower responses. Finally, participants who performed better on a non-numerical test of reasoning took longer to respond to the most difficult fraction trials, which suggests that those who had greater facility with non-numerical reasoning attended more to numerical relationships. These findings extend prior research and support the continued investigation into the mechanistic links between numerical and non-numerical reasoning.     

Nonverbal estimation during numerosity judgements by adult humans

On an automated task, humans selected the larger of two sets of items, each created through the one-by-one addition of items. Participants repeated the alphabet out loud during trials so that they could not count the items. This manipulation disrupted counting without producing major effects on other cognitive capacities such as memory or attention, and performance of this experimental group was poorer than that of participants who counted the items. In Experiment 2, the size of individual items was varied, and performance remained stable when the larger numerical set contained a smaller total amount than the smaller numerical set (i.e., participants used numerical rather than nonnumerical quantity cues in making judgements). In Experiment 3, reports of the number of items in a single set showed scalar variability as accuracy decreased, and variability in responses increased with increases in true set size. These data indicate a mechanism for the approximate representation of numerosity in adult humans that might be shared with nonhuman animals.     

Nonverbal estimation during numerosity judgements by adult humans

On an automated task, humans selected the larger of two sets of items, each created through the one-by-one addition of items. Participants repeated the alphabet out loud during trials so that they could not count the items. This manipulation disrupted counting without producing major effects on other cognitive capacities such as memory or attention, and performance of this experimental group was poorer than that of participants who counted the items. In Experiment 2, the size of individual items was varied, and performance remained stable when the larger numerical set contained a smaller total amount than the smaller numerical set (i.e., participants used numerical rather than nonnumerical quantity cues in making judgements). In Experiment 3, reports of the number of items in a single set showed scalar variability as accuracy decreased, and variability in responses increased with increases in true set size. These data indicate a mechanism for the approximate representation of numerosity in adult humans that might be shared with nonhuman animals.     

Coding strategies in number space: memory requirements influence spatial-numerical associations

The tendency to respond faster with the left hand to relatively small numbers and faster with the right hand to relatively large numbers (spatial numerical association of response codes, SNARC effect) has been interpreted as an automatic association of spatial and numerical information. We investigated in two experiments the impact of task-irrelevant memory representations on this effect. Participants memorized three Arabic digits describing a left-to-right ascending number sequence (e.g., 3-4-5), a descending sequence (e.g., 5-4-3), or a disordered sequence (e.g., 5-3-4) and indicated afterwards the parity status of a centrally presented digit (i.e., 1, 2, 8, or 9) with a left/right keypress response. As indicated by the reaction times, the SNARC effect in the parity task was mediated by the coding requirements of the memory tasks. That is, a SNARC effect was only present after memorizing ascending or disordered number sequences but disappeared after processing descending sequences. Interestingly, the effects of the second task were only present if all sequences within one experimental block had the same type of order. Taken together, our findings are inconsistent with the idea that spatial-numerical associations are the result of an automatic and obligatory cognitive process but do suggest that coding strategies might be responsible for the cognitive link between numbers and space.     

Verbal representation in task order control: An examination with transition and task cues in random task switching

Recent task-switching studies in which a predictable task sequence has been used have indicated that verbal representation contributes to the control of task order information. The present study focused on the role of verbal representation in sequential task decisions, which are an important part of task order control, and examined the effects of articulatory suppression in a random-task-cuing paradigm with two different types of cues presented just before the presentation of a stimulus: a transition cue and a task cue. The former cue provided information only about switching or repeating the task, whereas the latter was associated directly with the identity of the task (i.e., indicating a parity or a magnitude task). In Experiment 1, in which transition cues guided task sequences, articulatory suppression impaired performance in both repetition and switch trials, thereby increasing the mixing costs. In Experiment 2, in which a task cue, rather than a transition cue, was presented to examine the influence of a cue-decoding process, articulatory suppression had no specific effect on task performance. Experiment 3, in which the transition cue and the task cue were randomly presented in the same block to equalize the memory load and task strategy for the two types of cues, confirmed that articulatory suppression significantly increased the mixing costs only in transition cue trials. The results from the three experiments indicated that the use of verbal representation is effective in sequential task decision—that is, in selecting a task set on the basis of transient task order information in both repetition and switch trials.     

Representation and retention of three-event sequences in pigeons

The present experiments studied a three-event delayed sequence-discrimination (DSD) task: one arrangement (order) of two stimuli (red and yellow overhead lights) taken three in succession (e.g., red, yellow, red) was the positive sequence and the remaining seven arrangements were the negative sequences for responding and reward during the subsequent test stimulus. In Experiment 1, the final stimulus (recency) and the order of stimuli in the positive sequence controlled acquisition of discrimination. In Experiment 2, increasing the duration of memory intervals between stimuli reduced the discriminability of those negative sequences identical to the positive sequence after the delay. Three-event DSD performance in Experiments 1 and 2 was similar to two-event DSD performance in comparable published experiments. Models developed to explain pigeon performance in two-event DSD were extended to the three-event task. Results from both two- and three-event versions of the DSD task falsified a noncumulative model and several cumulative integration models (i.e., adding, averging, and some multiplying models), but corroborated one cumulative, multiplying model.     

The representation of negative numbers: Exploring the effects of mode of processing and notation

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.