Multiple spatial representations of number: evidence for co-existing compressive and linear scales

Differential roles of brain oscillations in numerical processing: evidence from resting-state EEG and mental number line

Recent works point to the importance of emotions in special-numerical associations. There remains a notable gap in understanding the electrophysiological underpinnings of such associations. Exploring resting-state (rs) EEG, particularly in frontal regions, could elucidate emotional aspects, while other EEG measures might offer insights into the cognitive dimensions correlating with behavioral performance. The present work investigated the relationship between rs-EEG measures (emotional and cognitive traits) and performance in the mental number line (MNL). EEG activity in theta (3-7 Hz), alpha (8-12 Hz, further subdivided into low-alpha and high-alpha), sensorimotor rhythm (SMR, 13-15 Hz), beta (16-25 Hz), and high-beta/gamma (28-40 Hz) bands was assessed. 76 university students participated in the study, undergoing EEG recordings at rest before engaging in a computerized number-to-position (CNP) task. Analysis revealed significant associations between frontal asymmetry, specific EEG frequencies, and MNL performance metrics (i.e., mean direction bias, mean absolute error, and mean reaction time). Notably, theta and beta asymmetries correlated with direction bias, while alpha peak frequency (APF) and beta activity related to absolute errors in numerical estimation. Moreover, the study identified significant correlations between relative amplitude indices (i.e., theta/beta ratio, theta/SMR ratio) and both absolute errors and reaction times (RTs). Our findings offer novel insights into the emotional and cognitive aspects of EEG patterns and their links to MNL performance.

More linear than log? Non-symbolic number-line estimation in 3- to 5-year-old children

The number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties (e.g., equidistance, ordinality) can easily be transferred into a spatial format of an oriented number line. This triggers the question on basic characteristics of number line estimation before children get fully familiar with the symbolic number system, i.e., when they mostly rely on approximate system for non-symbolic quantities. In our three studies, we examine therefore how preschool children (3–5-years old) estimate position of non-symbolic quantities on a line, and how this estimation is related to the developing symbolic number knowledge and cultural (left-to-right) directionality. The children were tested with the Give-a-number task, then they performed a computerized number-line task. In Experiment 1, lines bounded with sets of 1 and 20 elements going left-to-right or right-to-left were used. Even in the least numerically competent group, the linear model better fit the estimates than the logarithmic or cyclic power models. The line direction was irrelevant. In Experiment 2, a 1–9 left-to-right oriented line was used. Advantage of linear model was found at group level, and variance of estimates correlated with tested numerosities. In Experiment 3, a position-to-number procedure again revealed the advantage of the linear model, although the strategy of selecting an option more similar to the closer end of the line was prevalent. The precision of estimation increased with the mastery of counting principles in all three experiments. These results contradict the hypothesis of the log-to-linear shift in development of basic numerical representation, rather supporting the linear model with scalar variance. However, the important question remains whether the number-line task captures the nature of the basic numerical representation, or rather the strategies of mapping that representation to an external space.

Morningness-Eveningness Preference, Time Perspective, and Passage of Time Judgments

Recent studies have shown that making accurate passage of time judgments (POTJs) for long-time intervals is an important cognitive ability. Different temporal domains, such as circadian typology (biological time) and time perspective (psychological time), could have an effect on subjective POTJs, but few studies have investigated the reciprocal influences among these temporal domains. The present study is the first systematic attempt to fill this gap. A sample of 222 participants (53.20% females; 19-60 years) filled in the Zimbardo Time Perspective Inventory for the measurement of time perspective, the reduced version of the Morningness-Eveningness Questionnaire (rMEQ) for chronotypes, and an ad-hoc questionnaire assessing sleep habits during weekdays and the weekend (for social jetlag). The POTJ was measured using a modified version of a pictorial timeline presented at five different moments. Also, participants judged how different temporal expressions were related to the past, present, and future along a 7-point Likert scale. After confirming the association between eveningness and present-hedonism orientation and morningness and future-orientation, we found that evening-types produced higher scores for future expressions. The subjective POTJ expressed in minutes was predicted by Deviation from Balanced Time (DBTP), present-fatalism orientation, and social jetlag. Finally, the rMEQ score, past-positive orientation, and DBTP predicted the difference between subjective and objective POT. The results are discussed offering an explanation in terms of the interconnections between circadian typology, individual time perspective, and the sense of the POT, suggesting the multicomponent nature of the concept of time.

The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities?

Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic "Where was the number?" (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children's SNAs on the WTN task and math ability, children's SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability.

Related but different: Examining pseudoneglect in audition, touch and vision

Although researchers have consistently demonstrated a leftward attentional bias in visual and representational (e.g. tactile/mental number line) line bisection tasks, the results from audition have been mixed. Differences in methodology between auditory and visual bisection tasks, especially with regards to the location of stimuli of peripersonal versus extrapersonal space, have also meant that researchers have not been able to compare performance in visual, tactile and auditory line bisection directly. In this research, 39 neurologically typical individuals participated in standard visual and tactile line bisection tasks, together with a newly developed auditory line bisection task. Results demonstrated significant leftward bisection biases across all three modalities. Hence, we demonstrate auditory pseudoneglect in peripersonal space for the first time. Tactile and auditory line bisections showed a relatively small but statistically reliable correlation, but neither task correlated with visual line bisection. This suggests that the processes underlying auditory line bisection are not synonymous to those involved in visual perceptual bisection, and further we argue that this bias may be related to representational pseudoneglect.

Representations of numerical sequences and the concept of middle in preschoolers

The present study concerns preschoolers' understanding of the middle concept as it applies to numerical sequences. Previous research using implicit psychophysical assessment suggests that the numerical midpoint is embedded within numerical representations by 4 years of age. Here, we examined 3- to 5-year-olds' ability to identify the midpoint value in triplets of non-symbolic numbers when explicitly probed to do so. We found that whereas 4- and 5-year-olds were capable of explicit access to numerical midpoint values and showed ratio-dependent performance, a signature of the approximate number system (ANS), 3-year-olds performed at chance. Children's difficulty in identifying numerical midpoint values was not due to comparing multiple arrays, nor was it entirely due to a spatial association with the word "middle" used in the task. We speculate that explicit access to numerical midpoint values may be jointly supported by endogenous control of attentional mechanisms and the development of a mental number line.

Representational pseudoneglect: a review

Pseudoneglect, the tendency to be biased towards the left-hand side of space, is a robust and consistent behavioural observation best demonstrated on the task of visuospatial line bisection, where participants are asked to centrally bisect visually presented horizontal lines at the perceived centre. A number of studies have revealed that a representational form of pseudoneglect exists, occurring when participants are asked to either mentally represent a stimulus or explore a stimulus using touch in the complete absence of direct visuospatial processing. Despite the growing number of studies that have demonstrated representational pseudoneglect there exists no current and comprehensive review of these findings and no discussion of a theoretical framework into which these findings may fall. An important gap in the current representational pseudoneglect literature is a discussion of the developmental trajectory of the bias. The focus of the current review is to outline studies that have observed representational pseudoneglect in healthy participants, consider a theoretical framework for these observations, and address the impact of lifespan factors such as cognitive ageing on the phenomenon.

Elementary school children's attentional biases in physical and numerical space

Numbers are conceptualized spatially along a horizontal mental line. This view is supported by mounting evidence from healthy adults and patients with unilateral spatial neglect. Little is known about children's representation of numbers with respect to space. This study investigated elementary school children's directional biases in physical and numerical space to better understand the relation between space and number. We also examined the nature of spatial organization in numerical space. In two separate tasks, children (n=57) were asked to bisect a physical line and verbally estimate the midpoint of number pairs. In general, results indicated leftward biases in both tasks, but the degree of deviation did not correlate between the tasks. In the number bisection task, leftward bias (underestimating the midpoint) increased as a function of numerical magnitude and interval between number pairs. In contrast, a rightward deviation was found for smaller number pairs. These findings suggest that different underlying spatial attentional mechanisms might be directed in physical and numerical space in young school children, which would be integrated in adulthood.

The relation between space and math: developmental and educational implications

There is a well-known relation between spatial ability and mathematics dating back to the work of early twentieth century factor analysts. This connection is a ripe opportunity for educators, who might use spatial training to improve math learning. However, a closer look at the literature reveals gaps that impede direct application. The primary problem is that although this relation is well established in older children and adults, its emergence in early development and subsequent developmental interactions are not well documented. Moreover, there is a need for more mechanistic explanations that might be leveraged to improve math education. In this chapter, we attempt to address these issues by reviewing the existing literature to identify instances where answers are available and others where further research is needed.

Approaching stimuli bias attention in numerical space

Increasing evidence suggests that common mechanisms underlie the direction of attention in physical space and numerical space, along the mental number line. The small leftward bias (pseudoneglect) found on paper-and-pencil line bisection is also observed when participants 'bisect' number pairs, estimating (without calculating) the number midway between two others. Here we investigated the effect of stimulus motion on attention in numerical space. A two-frame apparent motion paradigm manipulating stimulus size was used to produce the impression that pairs of numbers were approaching (size increase from first to second frame), receding (size decrease), or not moving (no size change). The magnitude of pseudoneglect increased for approaching numbers, even when the final stimulus size was held constant. This result is consistent with previous findings that pseudoneglect in numerical space (as in physical space) increases as stimuli are brought closer to the participant. It also suggests that the perception of stimulus motion modulates attention over the mental number line and provides further support for a connection between the neural representations of physical space and number.

Common spatial organization of number and emotional expression: a mental magnitude line

Converging behavioral and neural evidence suggests that numerical representations are mentally organized in left-to-right orientation. Here we show that this format of spatial organization extends to emotional expression. In Experiment 1, right-side responses became increasingly faster as number (represented by Arabic numerals) or happiness (depicted in facial stimuli) increased, for judgments completely unrelated to magnitude. Additional experiments suggest that magnitude (i.e., more/less relations), not valence (i.e., positive/negative), underlies left-to-right orientation of emotional expression (Experiment 2), and that this orientation accommodates to the context-relevant emotion (e.g., happier faces are more rightward when judged on happiness, but more leftward when judged on angriness; Experiment 3). These findings show that people automatically extract magnitude from a variety of stimuli, representing such information in common left-to-right format, perhaps reflecting a mental magnitude line. We suggest that number is but one dimension in a hyper-general representational system uniting disparate dimensions of magnitude and likely subserved by common neural mechanisms in posterior parietal cortex.

Number line compression and the illusory perception of random numbers

Developmental studies indicate that children initially possess a compressed intuition of numerical distances, in which larger numbers are less discriminable than small ones. Education then "linearizes" this responding until by about age eight, children become able to map symbolic numerals onto a linear spatial scale. However, this illusion of compression of symbolic numerals may still exist in a dormant form in human adults and may be observed in appropriate experimental contexts. To investigate this issue, we asked adult participants to rate whether a random sequence of numbers contained too many small numbers or too many large ones. Participants exhibited a large bias, judging as random a geometric series that actually oversampled small numbers, consistent with a compression of large numbers. This illusion resisted training on a number-space mapping task, even though performance was linear on this task. While the illusion was moderately reduced by explicit exposure to linear sequences, responding was still significantly compressed. Thus, the illusion of compression is robust in this task, but linear and compressed responding can be exhibited in the same participants depending on the experimental context.

Bisecting the mental number line in near and far space

Much evidence suggests that common posterior parietal mechanisms underlie the orientation of attention in physical space and along the mental number line. For example, the small leftward bias (pseudoneglect) found in paper-and-pencil line bisection is also found when participants "bisect" number pairs, estimating (without calculating) the number midway between two others. For bisection of physical lines, pseudoneglect has been found to shift rightward as lines are moved from near space (immediately surrounding the body) to far space. We investigated whether the presentation of stimuli in near or far space also modulated spatial attention for the mental number line. Participants bisected physical lines or number pairs presented at four distances (60, 120, 180, 240cm). Clear rightward shifts in bias were observed for both tasks. Furthermore, the rate at which this shift occurred in the two tasks, as measured by least-squares regression slopes, was significantly correlated across participants, suggesting that the transition from near to far distances induced a common modulation of lateral attention in physical and numerical space. These results demonstrate a tight coupling between number and physical space, and show that even such prototypically abstract concepts as number are modulated by our on-line interactions with the world.