Numerical values modulate size perception

The link between various codes of magnitude and their interactions has been studied extensively for many years. In the current study, we examined how the physical and numerical magnitudes of digits are mapped into a combined mental representation. In two psychophysical experiments, participants reported the physically larger digit among two digits. In the identical condition, participants compared digits of an identical value (e.g., "2" and "2"); in the different condition, participants compared digits of distinct numerical values (i.e., "2" and "5"). As anticipated, participants overestimated the physical size of a numerically larger digit and underestimated the physical size of a numerically smaller digit. Our results extend the shared-representation account of physical and numerical magnitudes.

Endogenous attention modulates automaticity of number processing

Numerals (i.e., symbolic representations of numerical magnitude) are widespread in our environment and are fundamental to many decisions we make. It has been suggested that the processing of numerical magnitude is automatic. Various robust psychological effects, such as the distance effect (Moyer & Landauer, Nature, 215 (5109), 1519-1520, 1967) and the physical size-congruity effect (SiCE; Henik & Tzelgov, Memory & Cognition, 10 (4), 389-395, 1982), support this claim. Importantly, these effects demonstrate that the processing of numerical magnitude occurs unintentionally and while irrelevant to the task. These aspects often serve as criteria to assess the automatic nature of mental processes. However, evidence for the involvement of attention in the processing of magnitude of numerals somewhat subverts the automaticity account that was originally put forward. To reconcile between evidence in support of the automaticity account and evidence that compromises this account, we drew on another account of automaticity. This account distinguishes between strongly automatic and partly automatic mental processes based on their susceptibility to attentional influences. In the current study, we manipulated endogenous attention while participants completed numerical and physical comparisons of numerals, separately. We observed modulations of the SiCE for physical comparisons but not for numerical comparisons of numerals. That is, the processing of numerical magnitude when irrelevant was subjected to attentional influences, but the processing of their physical magnitude (i.e., size) was not. Therefore, we concluded that processing the numerical magnitude is partly automatic, whereas processing their physical magnitude is strongly automatic.

Do numbers make us handy? Behavioral and electrophysiological evidence for number-hand congruency effect

Finger counting facilitates numerical representations and mathematical processing. The current study investigated the association between finger counting habits and number processing by employing behavioral and electrophysiological measures. We explored whether small and large numerical primes influence the recognition of embodied target hand stimuli. Twenty-four right-handed participants that were grouped into right-starters (n = 13) and left-starters (n = 11) for finger counting performed a hand recognition task that consisted of numerical magnitudes as prime and hand recognition as targets. Based on the finger counting habits, congruent (i.e., left-starters: small number/left hand or large number/right hand; right-starters: small number/right hand or large number/left hand) and incongruent (i.e., left-starters: large number/left hand or small number/right hand; right-starters: large number/right hand or small number/left hand) conditions were presented to the participants. The participants were required to indicate whether the targets were left or right hand by simply pressing the left or the right key, respectively. Results indicated faster reaction times (RTs) for congruent as opposed to incongruent trials for all participants. The mean amplitude of the centro-parietal P300 component was significantly increased for the incongruent compared to congruent condition, indicating increased mental effort. Also, analysis of the latency of the P300 in terms of congruency effect in all participants revealed significant results. These combined results provide behavioral and electrophysiological evidence indicating the embodied nature of numbers. The results are interpreted in light of the general findings related to the P300 component. This research supports the association of number-hand representations and corroborates the idea of embodied numerosity.

The co-occurrence of Attention-Deficit/Hyperactivity Disorder and mathematical difficulties: An investigation of the role of basic numerical skills

BACKGROUND

Attention-deficit/hyperactivity disorder (ADHD) and dyscalculia, also called mathematics disorder, frequently co-occur, yet the etiology of this comorbidity is poorly understood.

AIMS

This study investigated whether impairments in the understanding of numbers and magnitudes (basic numerical skills) are a unique risk factor for mathematical difficulties (MD) or a shared risk factor that could help to explain the association between ADHD and MD.

METHODS AND PROCEDURES

Basic numerical skills were assessed with eight subtests in children (age 6-10 years, N = 86) with clinically significant ADHD symptoms and/or MD and typically developing children (control group). This double dissociation design allowed to test for main and interaction effects of ADHD and MD using both classical and Bayesian analysis of variance (ANOVA).

OUTCOMES AND RESULTS

Children with MD were impaired in transcoding, complex number and magnitude comparison, and arithmetic fact retrieval. They were not impaired in tasks assessing core markers of numeracy, which might be explained by the sample including children with mathematical difficulties instead of a diagnosed dyscalculia. ADHD was not associated with deficits in any of the tasks. The evidence for an additive combination of cognitive profiles was weak.

CONCLUSIONS AND IMPLICATIONS

Impairments in basic numerical skills are uniquely associated with MD and do not represent a shared risk factor for ADHD symptoms and MD.

Non-symbolic and symbolic number lines are dissociated

People use mental number lines for both symbolic numerals and numerosity, but little is known about how these two mental number lines are related. The current study investigated the association in effect size, directionality of the mental number line, and development between symbolic and non-symbolic mental number lines to determine if they were related to or independent from each other. We collected data from numerosity- and digit-matching tasks that used the following numbers: 11, 14, 17, 20, 23, 26, and 29. Tasks were performed by college undergraduates and the fifth-grade primary school students. The results showed that none of the effects for non-symbolic numbers was related to any of the effects for symbolic numbers, and vice versa, in both adults and children. Another notable finding was that the correlation between the SNARC (spatial-numerical association of response code) effect size and mathematical ability was negative in the adult group. These results are consistent with the dissociated processes hypothesis and suggest that mental number lines are notation-dependent. As shown by the SNARC effect, the mental number line might result in interference in the current task by an automatically activated spatial notation-dependent representation of numbers.

How does number magnitude influence temporal and spatial parameters of eye movements?

The influence of numerical processing on individuals' behavior is now well documented. The spatial representation of numbers on a left-to-right mental line (i.e., SNARC effect) has been shown to have sensorimotor consequences, the majority of studies being mainly concerned with its impact on the response times. Its impact on the motor programming stage remains less documented, although swiping movement amplitudes have recently been shown to be modulated by number magnitude. Regarding saccadic eye movements, the few available studies have not provided clear-cut conclusions. They showed that spatial-numerical associations modulated ocular drifts, but not the amplitude of memory-guided saccades. Because these studies held saccadic coordinates constant, which might have masked potential numerical effects, we examined whether spontaneous saccadic eye movements (with no saccadic target) could reflect numerical effects. Participants were asked to look either to the left or to the right side of an empty screen to estimate the magnitude (< or > 5) of a centrally presented digit. Latency data confirmed the presence of the classical SNARC and distance effects. More critically, saccade amplitude reflected a numerical effect: participants' saccades were longer for digits far from the standard (1 and 9) and were shorter for digits close to it (4 and 6). Our results suggest that beyond response times, kinematic parameters also offer valuable information for the understanding of the link between numerical cognition and motor programming.

Impact of stimulus format and reward value on quantity discrimination in capuchin and squirrel monkeys

Quantity discrimination abilities are seen in a diverse range of species with similarities in performance patterns, suggesting common underlying cognitive mechanisms. However, methodological factors that impact performance make it difficult to draw broad phylogenetic comparisons of numerical cognition across studies. For example, some Old World monkeys selected a higher quantity stimulus more frequently when choosing between inedible (pebbles) than edible (food) stimuli. In Experiment 1 we presented brown capuchin (Cebus [Sapajus] paella) and squirrel monkeys (Saimiri sciureus) with the same two-choice quantity discrimination task in three different stimulus conditions: edible, inedible, and edible replaced (in which choice stimuli were food items that stood in for the same quantity of food items that were given as a reward). Unlike Old World monkeys, capuchins selected the higher quantity stimulus more in the edible condition and squirrel monkeys showed generally poor performance across all stimulus types. Performance patterns suggested that differences in subjective reward value might motivate differences in choice behavior between and within species. In Experiment 2 we manipulated the subjective reinforcement value of the reward by varying reward type and delay to reinforcement and found that delay to reinforcement had no impact on choice behavior, while increasing the value of the reward significantly improved performance by both species. The results of this study indicate that species presented with identical tasks may respond differently to methodological factors such as stimulus and reward types, resulting in significant differences in choice behavior that may lead to spurious suggestions of species differences in cognitive abilities.

Sensorimotor lateralization scaffolds cognitive specialization

In this chapter, we review hemispheric differences for sensorimotor function and cognitive abilities. Specifically, we examine the left-hemisphere specialization for visuomotor control and its interplay with language, executive function, and musical training. Similarly, we discuss right-hemisphere lateralization for haptic processing and its relationship to spatial and numerical processing. We propose that cerebral lateralization for sensorimotor functions served as a foundation for the development of higher cognitive abilities and their hemispheric functional specialization. We further suggest that sensorimotor and cognitive functions are inextricably linked. Based on the studies discussed in this chapter our view is that sensorimotor control serves as a loom upon which the fibers of language, executive function, spatial, and numerical processing are woven together to create the fabric of cognition.

The neural bases of price estimation: Effects of size and precision of the estimate

People are often confronted with the need of estimating the market price of goods. An important question is how people estimate prices, given the variability of products and prices available. Using event-related fMRI, we investigated how numerical processing modulates the neural bases of retail price estimation by focusing on two numerical dimensions: the size and precision of the estimates. Participants were presented with several product labels and made market price estimates for those products. Measures of product buying frequency and market price variability were also collected. The estimation of higher prices required longer response times, was associated with greater variation in responses across participants, and correlated with increasing medial and lateral prefrontal cortex (PFC) activity. Moreover, price estimates followed Weber's law, a hallmark feature of numerical processing. Increasing accuracy in price estimation, indexed by decreasing Weber fraction, engaged the intraparietal sulcus (IPS), a critical region in numerical processing. Our findings provide evidence for distinguishable neural mechanisms associated with the size and the precision of price estimates.

Abstract numerical discrimination learning in rats

In this study, we examined rats' discrimination learning of the numerical ordering positions of objects. In Experiments 1 and 2, five out of seven rats successfully learned to respond to the third of six identical objects in a row and showed reliable transfer of this discrimination to novel stimuli after being trained with three different training stimuli. In Experiment 3, the three rats from Experiment 2 continued to be trained to respond to the third object in an object array, which included an odd object that needed to be excluded when identifying the target third object. All three rats acquired this selective-counting task of specific stimuli, and two rats showed reliable transfer of this selective-counting performance to test sets of novel stimuli. In Experiment 4, the three rats from Experiment 3 quickly learned to respond to the third stimulus in object rows consisting of either six identical or six different objects. These results offer strong evidence for abstract numerical discrimination learning in rats.

Conceptual size representation in ventral visual cortex

Recent findings suggest that visual objects may be mapped along the ventral occipitotemporal cortex according to their real-world size (Konkle and Oliva, 2012). It has been argued that such mapping does not reflect an abstract, conceptual size representation, but rather the visual or functional properties associated with small versus big real-world objects. To determine whether a more abstract conceptual size representation may affect visual cortical activation we used meaningless geometrical shapes, devoid of semantic or functional associations, which were associated with specific size representations by virtue of extensive training. Following training, participants underwent functional magnetic resonance imaging (fMRI) scanning while performing a conceptual size comparison task on the geometrical shapes. In addition, a size comparison task was conducted for numeral digits denoting small and big numbers. A region-of-interest analysis revealed larger blood oxygenation level dependent (BOLD) responses for conceptually 'big' than for conceptually 'small' shapes, as well as for big versus small numbers, within medial (parahippocampal place area, PPA) and lateral (occipital place area, OPA) place-selective regions. Processing of the 'big' visual shapes further elicited enhanced activation in early visual cortex, possibly reflecting top-down projections from PPA. By using arbitrary shapes and numbers we minimized visual, categorical, or functional influences on fMRI measurement, providing evidence for a possible neural mechanism underlying the representation of abstract conceptual size within the ventral visual stream.

Adaptive processing of fractions--evidence from eye-tracking

Recent evidence indicated that fraction pair type determined whether a particular fraction is processed holistically, componentially or in a hybrid manner. Going beyond previous studies, we investigated how participants adapt their processing of fractions not only to fraction type, but also to experimental context. To examine adaptation in fraction processing, we recorded participants' eye-fixation behaviour in a fraction magnitude comparison task. Participants' eye fixation behaviour indicated componential processing of fraction pairs with common components for which the decision-relevant components are easy to identify. Importantly, we observed that fraction processing was adapted to experimental context: Evidence for componential processing was stronger, when experimental context allowed valid expectations about which components are decision-relevant. Taken together, we conclude that fraction processing is adaptive beyond the comparison of different fraction types, because participants continuously adjust to the experimental context in which fractions are processed.

Does the learning of two symbolic sets of numbers affect the automaticity of number processing in children?

We explored the effects of learning two different symbolic sets of numerals (Arabic and Indian) on the development of automatic number processing. Children in the school we examined learn Indian numerals between first and third grades. In third grade, they switch to a new set of numerals (i.e., Arabic numbers). Participants in this study performed a numerical Stroop-like task in which they assessed the numerical value or physical size of stimuli varying along these two dimensions. Each participant saw either Arabic or Indian numerals. The results of the size congruity effect in the physical task, for both Indian and Arabic numerals, suggest that studying two sets of numerals interferes with the acquisition of an automatic association of a numerical symbol and magnitude. This is true both for the first learned set of numerals (i.e., Indian numerals) and for the second one (i.e., Arabic numerals). Furthermore, we found an absence of the distance effect, which further supports this conclusion. This learning program gave us the unique opportunity to examine the connection between symbolic sets and the mental representation of numbers in a novel fashion.

Quantifier processing can be dissociated from numerical processing: evidence from semantic dementia patients

Quantifiers such as frequency adverbs (e.g., "always", "never") and quantity pronouns (e.g., "many", "none") convey quantity information. Whether quantifiers are processed as numbers or as general semantics has been a matter of much debate. Some neuropsychological and fMRI studies have found that the processing of quantifiers depends on the numerical magnitude comprehension system, but others have found that quantifier processing is associated with semantic representation. The selective impairment of language in semantic dementia patients provides a way to examine the above controversy. We administered a series of neuropsychological tests (i.e., language processing, numerical processing and semantic distance judgment) to two patients with different levels of severity in semantic dementia (mild vs. severe). The results showed that the two patients had intact numerical knowledge, but impairments in semantic processing. Moreover, the patient with severe/late semantic dementia showed more impairment in quantifier and semantic processing than the patient with mild/early semantic dementia. We concluded that quantifier processing is associated with general semantic processing, not with numerical processing.