Beyond the mental number line: A neural network model of number–space interactions

It is commonly assumed that there is an interaction between the representations of number and space (e.g., Dehaene et al., 1993, Walsh, 2003), typically ascribed to a mental number line. The exact nature of this interaction has remained elusive, however. Here we propose that spatial aspects are not inherent to number representations, but that instead spatial and numerical representations are separate. However, cultural factors establish ties between them. By extending earlier models (Gevers et al., 2006, Verguts et al., 2005) based on this hypothesis, the authors present computer simulations showing that a model incorporating this idea can account for data from a series of studies. These results suggest that number–space interactions are emergent properties resulting from the interaction between different brain areas.     

Choosing the number of clusters in Κ-means clustering

Steinley (2007) provided a lower bound for the sum-of-squares error criterion function used in K-means clustering. In this article, on the basis of the lower bound, the authors propose a method to distinguish between 1 cluster (i.e., a single distribution) versus more than 1 cluster. Additionally, conditional on indicating there are multiple clusters, the procedure is extended to determine the number of clusters. Through a series of simulations, the proposed methodology is shown to outperform several other commonly used procedures for determining both the presence of clusters and their number.     

Is the number of trials a primary determinant of conditioned responding?

Acquisition of conditioned responding is thought to be determined by the number of pairings of a conditioned stimulus (CS) and an unconditioned stimulus (US). However, it is possible that acquisition is primarily determined not by the number of trials but rather by quantities that often correlate with the number of trials, such as cumulative intertrial interval (ITI) and the number of sessions. Four experiments examined whether the number of trials has an effect on acquisition of conditioned responding, once cumulative ITI and number of sessions are equated. Results of the experiments with rats and mice favor the hypothesis that over an eightfold range, variation in number of CS-US pairings has little effect. It is suggested that learning curves might more accurately be plotted across cumulative ITI or number of sessions, and not across number of trials. Results pose a challenge to trial-centered accounts of conditioning, as demonstrated by simulations of the Rescorla-Wagner model, a simplified version of Wagner's standard operating procedure model (SOP), and Stout & Miller's sometimes competing retrieval model (SOCR). A time-centered account, rate estimation theory (RET), predicts the main finding but has trouble with other aspects of the learning process more easily accommodated by trial-centered models.     

The influence of the number of toys in the environment on toddlers’ play

We tested the hypothesis that an environment with fewer toys will lead to higher quality of play for toddlers. Each participant (n = 36) engaged in supervised, individual free play sessions under two conditions: Four Toy and Sixteen Toy. With fewer toys, participants had fewer incidences of toy play, longer durations of toy play, and played with toys in a greater variety of ways (Z = −4.448, p < 0.001, r = −0.524; Z = 2.828, p = 0.005, r = 0.333; and Z = 4.676, p < 0.001, r = 0.55, respectively). This suggests that when provided with fewer toys in the environment, toddlers engage in longer periods of play with a single toy, allowing better focus to explore and play more creatively. This can be offered as a recommendation in many natural environments to support children’s development and promote healthy play.     

Infants’ listening in multitalker environments: Effect of the number of background talkers

Infants are often spoken to in the presence of background sounds, including speech from other talkers. In the present study, we compared 5- and 8.5-month-olds’ abilities to recognize their own names in the context of three different types of background speech: that of a single talker, multitalker babble, and that of a single talker played backward. Infants recognized their names at a 10-dB signal-to-noise ratio in the multiple-voice condition but not in the single-voice (nonreversed) condition, a pattern opposite to that of typical adult performance. Infants similarly failed to recognize their names when the background talker’s voice was reversed&#x2014;that is, unintelligible, but with speech-like acoustic properties. These data suggest that infants may have difficulty segregating the components of different speech streams when those streams are acoustically too similar. Alternatively, infants’ attention may be drawn to the time-varying acoustic properties associated with a single talker’s speech, causing difficulties when a single talker is the competing sound.     

Are number gestures easier than number words for preschoolers?

Some researchers have argued that children's earliest symbols are based on their sensorimotor experience and that arbitrary symbol-referent mapping poses a challenge for them. If so, exposure to iconic symbols (such as one-finger-for-one-object manual gestures) might help children in a difficult domain such as number. We assessed 44 preschoolers’ processing of number gestures and number words on two tasks: give-a-number and how-many. Children were generally more accurate when mapping number words than number gestures onto a number of toys, suggesting that the iconicity of number gestures does not help children map symbols to numbers. We argue that children learn number gestures as arbitrary symbols and do not take advantage of their iconicity.     

NoA’s ark: Influence of the number of associates in visual word recognition

The main aim of this study was to explore the extent to which the number of associates of a word (NoA) influences lexical access, in four tasks that focus on different processes of visual word recognition: lexical decision, reading aloud, progressive demasking, and online sentence reading. Results consistently showed that words with a dense associative neighborhood (high-NoA words) were processed faster than words with a sparse neighborhood (low-NoA words), extending previous findings from English lexical decision and categorization experiments. These results are interpreted in terms of the higher degree of semantic richness of high-NoA words as compared with low-NoA words. 2008 Psychonomic Society, Inc. Author Note     

Reconceptualizing the origins of number knowledge: A “non-numerical” account

This paper presents a new conceptualization of the origins of numerical competence in humans. I first examine the existing claim that infants are innately provided with a system of specifically numerical knowledge, consisting of both cardinal and ordinal concepts. I suggest instead that the observed behaviors require only simple perceptual discriminations based on domain-independent competencies. At most, these involve the formal equivalent of cardinal information. Finally, I present a “non-numerical” account that characterizes infants competencies with regard to numerosity as emerging primarily from some general characteristics of the human perception and attention system.     

Approximate number sense,symbolic number processing,or number–space mappings: What underlies mathematics achievement?

In this study, the performance of typically developing 6- to 8-year-old children on an approximate number discrimination task, a symbolic comparison task, and a symbolic and nonsymbolic number line estimation task was examined. For the first time, children’s performances on these basic cognitive number processing tasks were explicitly contrasted to investigate which of them is the best predictor of their future mathematical abilities. Math achievement was measured with a timed arithmetic test and with a general curriculum-based math test to address the additional question of whether the predictive association between the basic numerical abilities and mathematics achievement is dependent on which math test is used. Results revealed that performance on both mathematics achievement tests was best predicted by how well children compared digits. In addition, an association between performance on the symbolic number line estimation task and math achievement scores for the general curriculum-based math test measuring a broader spectrum of skills was found. Together, these results emphasize the importance of learning experiences with symbols for later math abilities.     

The impact of inhibition capacities and age on number–space associations

Numerical and spatial representations are tightly linked, i.e., when doing a binary classification judgment on Arabic digits, participants are faster to respond with their left/right hand to small/large numbers, respectively (Spatial-Numerical Association of Response Codes, SNARC effect, Dehaene et al. in J Exp Psychol Gen 122:371–396, 1993). To understand the underlying mechanisms of the well-established SNARC effect, it seems essential to explore the considerable inter-individual variability characterizing it. The present study assesses the respective roles of inhibition, age, working memory (WM) and response speed. Whereas these non-numerical factors have been proposed as potentially important factors to explain individual differences in SNARC effects, none (except response speed) has so far been explored directly. Confirming our hypotheses, the results show that the SNARC effect was stronger in participants that had weaker inhibition abilities (as assessed by the Stroop task), were relatively older and had longer response times. Interestingly, whereas a significant part of the age influence was mediated by cognitive inhibition, age also directly impacted the SNARC effect. Similarly, cognitive inhibition abilities explained inter-individual variability in number–space associations over and above the factors age, WM capacity and response speed. Taken together our results provide new insights into the nature of number–space associations by describing how these are influenced by the non-numerical factors age and inhibition.     

Heterogeneity of borderline personality disorder: do the number of criteria met make a difference?

Many studies have compared the demographic and clinical characteristics of patients with and without borderline personality disorder (BPD), but there is limited knowledge on differences within the population of borderline patients. One potential index of heterogeneity is disorder severity. In the present report from the Rhode Island Methods to Improve Diagnostic Assessment and Services (MIDAS) project, we examined whether the severity of borderline personality disorder, as measured by the number of criteria present, is associated with co-morbidity of Axis I and Axis II diagnoses, as well as demographic factors and psychosocial functioning. Two thousand three hundred psychiatric outpatients were interviewed with the Structured Interview for DSM-IV Personality (SIDP-IV). Approximately ten percent (n = 237) of the patients were diagnosed with BPD, and they were divided into four groups based on the number of DSM-IV criteria met, 5 (n = 89), 6 (n = 70), 7 (n = 46), and 8 or 9 (n = 32). There were greater rates of drug use disorders and comorbid Axis II disorders, as well as a greater number of suicidal gestures, in patients meeting seven or more BPD criteria. There were no significant differences between the groups in the number and specific rates of other co-morbid Axis I disorders, other measures of psychosocial functioning, or demographic correlates. Sub-typing of borderline patients by the number of criteria met provides a limited explanation for heterogeneity within BPD patients.     

Familiarity increases the number of remembered Pokémon in visual short-term memory

Long-term memory (LTM) can influence many aspects of short-term memory (STM), including increased STM span. However, it is unclear whether LTM enhances the quantitative or qualitative aspect of STM. That is, do we retain a larger number of representations or more precise representations in STM for familiar stimuli than unfamiliar stimuli? This study took advantage of participants’ prior rich multimedia experience with Pokémon, without investing on laboratory training to examine how prior LTM influenced visual STM. In a Pokémon visual STM change detection task, participants remembered more first-generation Pokémon characters that they were more familiar with than recent-generation Pokémon characters that they were less familiar with. No significant difference in memory quality was found when quantitative and qualitative effects of LTM were isolated using receiver operating characteristic (ROC) analyses. Critically, these effects were absent in participants who were unfamiliar with first-generation Pokémon. Furthermore, several alternative interpretations were ruled out, including general video-gaming experience, subjective Pokémon preference, and verbal encoding. Together, these results demonstrated a strong link between prior stimulus familiarity in LTM and visual STM storage capacity.     

Disruption of writing by background speech: Does sound source location and number of voices matter?

It is not unusual that people have to write in an environment where background speech is present. Background speech can vary in both speech intelligibility and location of the sound source. Earlier research has shown disruptive effects of background speech on writing performance. To expand and reinforce this knowledge, the present study investigated the role of number of voices and sound source location in the relation between background speech and writing performance. Participants wrote texts in quiet or in background speech consisting of one or seven voices talking simultaneously located in front of or behind them. Overall, one voice was more disruptive than seven voices talking simultaneously. Self‐reports showed that sound from the front was more disruptive compared with sound from behind. Results are in line with theory of interference‐by‐process, attentional capture, and the cross‐modal theory of attention. The relevance of the results for open‐office environments is discussed.     

Is Dunbar's number up?

This article is a commentary on ‘Relationships and the social brain: Integrating psychological and evolutionary perspectives’ ( Sutcliffe, Dunbar, Binder, & Arrow, 2012 ).     

The number–weight illusion

Psychonomic Bulletin & Review - When objects are manually lifted to compare their weight, then smaller objects are judged to be heavier than larger objects of the same physical weights: the...     

Multiple cues for quantification in infancy: is number one of them?

A review and synthesis of the literature on quantification in infancy and early childhood is provided. In most current conceptualizations, early quantification is assumed to be number based. However, the extant literature provides no clear-cut evidence that infants use number to perform quantitative tasks. Instead, new research suggests that quantification is initially based on nonnumerical cues, such as area and contour length, whether or not a task involves discrete items. The authors discuss the implications of these findings with respect to early quantification and its relation to later numerical development.     

The role of attention in subitizing: Is the magical number 1?

Subitizing, the fast and accurate enumeration of up to about 3 or 4 objects, has often been thought to be dependent on limited-capacity preattentive mechanisms. We used an attentional blink paradigm to investigate the extent to which subitizing requires attentional resources. On each trial, subjects identified a target letter in an RSVP stream and then enumerated dots presented in the stream that were either simultaneous with the target letter or followed it by up to 400 ms. For numerosities from 2 to 9, evidence of an attentional blink was observed; only enumeration of 0 or 1 elements was independent of lag. Thus, even enumeration of 2–3 objects, which is within the traditional subitizing range, appears to require attentional resources. The relation of this work to studies on the attentional requirements of detecting a unique item among distractors, a supposedly preattentive discrimination, is briefly discussed.     

Stimulus range,number of categories,and the “virtual” exponent

Three different stimulus modalities (line length, number, and sound pressure) were judged by magnitude scaling techniques and by 7-, 15-, 31-, and 75-point category scales. All of the 40 subjects were given the same number stimuli, but two different sound-pressure ranges were presented (each to 20 subjects) and four different line-length ranges were presented (each to 10 subjects). Analyses of lack of fit for various simple functions were performed to determine bestfitting functions. The simple power function was often found to be an adequate fit to the data for all the response modalities used, although all of the response modalities were sensitive to changes in stimulus range. For simple power functions, the category-scale exponent was a function of both the range of stimuli and the number of categories provided. Category scales did not always produce exponents smaller than those obtained with magnitude estimation, which calls into question the concept of a virtual exponent for category scales.     

Evolved sex differences in the number of partners desired? The long and the short of it

Do men seek more short-term mates than women? Buss and Schmitt (1993) showed a pattern of mean difference in the ideal number of sexual partners men and women desired over various time frames. We replicated these mean sex differences (e.g., ideal number over the next 30 years: Ms = 7.69 and 2.78 for men and women, respectively), but in both data sets the sampling distributions were highly skewed. In Study 1, we found few sex differences in medians across time frames (e.g., ideal number over the next 30 years: Mdn = 1 for both men and women). In Study 2, most college men (98.9%) and women (99.2%) said they wanted to settle down with one mutually exclusive sexual partner at some point in their life, ideally within the next 5 years. Neither medians in number of partners desired overall before settling down (replicating Study 1) nor medians in short-term partners desired before settling down (Mdn = 0) differed significantly by gender. Rather, men and women concurred: Short-term mating is not what humans typically seek.