Numerical Comparison Task Research Articles

Rapid Communication: The effect of mathematics anxiety on the processing of numerical magnitude

In an effort to understand the origins of mathematics anxiety, we investigated the processing of symbolic magnitude by high mathematics-anxious (HMA) and low mathematics-anxious (LMA) individuals by examining their performance on two variants of the symbolic numerical comparison task. In two experiments, a numerical distance by mathematics anxiety (MA) interaction was obtained, demonstrating that the effect of numerical distance on response times was larger for HMA than for LMA individuals. These data support the claim that HMA individuals have less precise representations of numerical magnitude than their LMA peers, suggesting that MA is associated with low-level numerical deficits that compromise the development of higher level mathematical skills.

The size congruity effect: Is bigger always more?

When comparing digits of different physical sizes, numerical and physical size interact. For example, in a numerical comparison task, people are faster to compare two digits when their numerical size (the relevant dimension) and physical size (the irrelevant dimension) are congruent than when they are incongruent. Two main accounts have been put forward to explain this size congruity effect. According to the shared representation account, both numerical and physical size are mapped onto a shared analog magnitude representation. In contrast, the shared decisions account assumes that numerical size and physical size are initially processed separately, but interact at the decision level. We implement the shared decisions account in a computational model with a dual route framework and show that this model can simulate the modulation of the size congruity effect by numerical and physical distance. Using other tasks than comparison, we show that the model can simulate novel findings that cannot be explained by the shared representation account.

Developmental Specialization in the Right Intraparietal Sulcus for the Abstract Representation of Numerical Magnitude

Because number is an abstract quality of a set, the way in which a number is externally represented does not change its quantitative meaning. In this study, we examined the development of the brain regions that support format-independent representation of numerical magnitude. We asked children and adults to perform both symbolic (Hindu-Arabic numerals) and nonsymbolic (arrays of squares) numerical comparison tasks as well as two control tasks while their brains were scanned using fMRI. In a preliminary analysis, we calculated the conjunction between symbolic and nonsymbolic numerical comparison. We then examined in which brain regions this conjunction differed between children and adults. This analysis revealed a large network of visual and parietal regions that showed greater activation in adults relative to children. In our primary analysis, we examined age-related differences in the conjunction of symbolic and nonsymbolic comparison after subtracting the control tasks. This analysis revealed a much more limited set of regions including the right inferior parietal lobe near the intraparietal sulcus. In addition to showing increased activation to both symbolic and nonsymbolic magnitudes over and above activation related to response selection, this region showed age-related differences in the distance effect. Our findings demonstrate that the format-independent representation of numerical magnitude in the right inferior parietal lobe is the product of developmental processes of cortical specialization and highlight the importance of using appropriate control tasks when conducting developmental neuroimaging studies.

Defective number module or impaired access? Numerical magnitude processing in first graders with mathematical difficulties

This study examined numerical magnitude processing in first graders with severe and mild forms of mathematical difficulties, children with mathematics learning disabilities (MLD) and children with low achievement (LA) in mathematics, respectively. In total, 20 children with MLD, 21 children with LA, and 41 regular achievers completed a numerical magnitude comparison task and an approximate addition task, which were presented in a symbolic and a nonsymbolic (dot arrays) format. Children with MLD and LA were impaired on tasks that involved the access of numerical magnitude information from symbolic representations, with the LA children showing a less severe performance pattern than children with MLD. They showed no deficits in accessing magnitude from underlying nonsymbolic magnitude representations. Our findings indicate that this performance pattern occurs in children from first grade onward and generalizes beyond numerical magnitude comparison tasks. These findings shed light on the types of intervention that may help children who struggle with learning mathematics.

Holistic Representation of Negative Numbers is Formed When Needed for the Task

Past research suggested that negative numbers are represented in terms of their components--the polarity marker and the number (e.g., Fischer & Rottmann, 2005; Ganor-Stern & Tzelgov, 2008). The present study shows that a holistic representation is formed when needed for the task requirement. Specifically, performing the numerical comparison task on positive and negative numbers presented sequentially required participants to hold both the polarity and the number magnitude in memory. Such a condition resulted in a holistic representation of negative numbers, as indicated by the distance and semantic congruity effects. This holistic representation was added to the initial components representation, thus producing a hybrid holistic-components representation.

Using eye tracking to study numerical cognition: the case of the ratio effect

In both behavioural and brain-imaging studies, numerical magnitude comparison tasks have been used to glean insights into the processing and representation of numerical magnitude. The present study examined the extent to which eye movement data can be used to investigate the neurocognitive processes underlying numerical magnitude processing. Twenty-two participants performed a numerical comparison task (deciding which of two Arabic numerals represents the larger numerical magnitude) while eye tracking data was recorded. The ratio between numbers (smaller/larger) was manipulated and ranged from 0.11 to 0.89. Consistent with previous reaction time and accuracy studies, the present results demonstrated significant main effects of ratio on the number of fixations, as well as a significant main effect of correct (numerically larger) versus incorrect (numerically smaller) number on the duration of fixations. Furthermore, data from the present investigation also revealed that participants made significantly more saccades between the two numbers for large relative to small ratio trials. Moreover, the ratio effects on eye movements were uncorrelated with the effect of numerical ratio on reaction times, suggesting that eye tracking measures of number comparison may tap into a different level of numerical magnitude processing than reaction time measures do.

Exploring the mental number line via the size congruity effect.

To address the ongoing debate about the origins of the size effect (faster comparison time for smaller than larger numbers, given a fixed intrapair distance), an indication of automatic number processing was searched for. Participants performed a quantity comparison task in which they had to decide which of two sketched cups contained more liquid, while ignoring the number superimposed on each cup. In the congruent condition, the larger number appeared on the cup containing more liquid, while in the incongruent condition the larger number appeared on the cup containing less liquid. The size effect was found in a numerical comparison task, while in the quantity comparison task the size congruity effect decreased as the magnitude of the irrelevant numbers increased. Together, these results suggest that the size effect reflects a basic feature of the mental number line.

Language effects in addition: How you say it counts

This study aimed to investigate the role of language in calculation. Two populations were compared, one with a base-10 language, and another (Basque) in which number words are constructed by combining multiples of 20 and units or teens (e.g., "35" is said "twenty and fifteen"). Experiment 1 asked participants to verbally solve additions presented as Arabic digits. Basque participants solved the additions that consisted of a multiple of 20 and a teen (e.g., 20 + 15) faster than controls with identical answers (e.g., 25 + 10). No differences were found in the base-10 language group. Experiment 2 replicated this result even if participants had to type the answer on a numerical keypad, instead of saying it. Hence, the structure of number words in each of the languages influenced the way additions were solved, even if language was not necessary for conducting the task. Finally, in Experiment 3, both language groups performed a numerical comparison task in which no effects of the structure of number words were obtained. Results of the three experiments are discussed in light of current models of numerical cognition.

Challenging the reliability and validity of cognitive measures: The case of the numerical distance effect

The numerical distance effect (NDE) is one of the most robust effects in the study of numerical cognition. However, the validity and reliability of distance effects across different formats and paradigms has not been assessed. Establishing whether the distance effect is both reliable and valid has important implications for the use of this paradigm to index the processing and representation of numerical magnitude in both behavioral and neuroimaging studies. In light of this, we examine the reliability and validity of frequently employed variants (and one new variant) of the numerical comparison task: two symbolic comparison variants and two nonsymbolic comparison variants. The results of two experiments demonstrate that measures of the NDE that use nonsymbolic stimuli are far more reliable than measures of the NDE that use symbolic stimuli. With respect to correlations between measures, we find evidence that the NDE that arises using symbolic stimuli is uncorrelated with the NDE that is elicited by using nonsymbolic stimuli. Results are discussed with respect to their implications for the use of the NDE as a metric of numerical processing and representation in research with both children and adults.

Developmental changes in the comparison of decimal fractions

Learning about decimal fractions is difficult because it requires an extension of the number concept built on natural numbers. The aim of the present study was to investigate developmental changes in children's misconceptions in decimal fraction processing. A large sample of children from Grades 3 to 6 performed a numerical comparison task on different categories of pairs of decimal fractions. The success rate and the type of error they made varied with age and categories. We distinguished the impact of the value of the digits from the impact of the length of the fractional part on children's pattern of responses. Although both kinds of impact affected the success rate, the digit values had a stronger impact and were mastered later than the length. Our results also showed that a zero just after the decimal point was understood better and earlier than a zero at the end of the fractional part of a number. Cluster analysis was conducted to determine groups of children who answered similarly regarding the response type across the various categories of decimal fractions. To interpret the data the conceptual change framework was used.

Rational numbers: Componential versus holistic representation of fractions in a magnitude comparison task

This study investigated whether the mental representation of the fraction magnitude was componential and/or holistic in a numerical comparison task performed by adults. In Experiment 1, the comparison of fractions with common numerators (x/a_x/b) and of fractions with common denominators (a/x_b/x) primed the comparison of natural numbers. In Experiment 2, fillers (i.e., fractions without common components) were added to reduce the regularity of the stimuli. In both experiments, distance effects indicated that participants compared the numerators for a/x_b/x fractions, but that the magnitudes of the whole fractions were accessed and compared for x/a_x/b fractions. The priming effect of x/a_x/b fractions on natural numbers suggested that the interference of the denominator magnitude was controlled during the comparison of these fractions. These results suggested a hybrid representation of their magnitude (i.e., componential and holistic). In conclusion, the magnitude of the whole fraction can be accessed, probably by estimating the ratio between the magnitude of the denominator and the magnitude of the numerator. However, adults might prefer to rely on the magnitudes of the components and compare the magnitudes of the whole fractions only when the use of a componential strategy is made difficult.

Numbers in the Blind's “Eye”

BackgroundAlthough lacking visual experience with numerosities, recent evidence shows that the blind perform similarly to sighted persons on numerical comparison or parity judgement tasks. In particular, on tasks presented in the auditory modality, the blind surprisingly show the same effect that appears in sighted persons, demonstrating that numbers are represented through a spatial code, i.e. the Spatial-Numerical Association of Response Codes (SNARC) effect. But, if this is the case, how is this numerical spatial representation processed in the brain of the blind?Principal FindingsHere we report that, although blind and sighted people have similarly organized numerical representations, the attentional shifts generated by numbers have different electrophysiological correlates (sensorial N100 in the sighted and cognitive P300 in the blind).ConclusionsThese results highlight possible differences in the use of spatial representations acquired through modalities other than vision in the blind population.

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.

A generalized fraction: An entity smaller than one on the mental number line.

The representation of fractions in long-term memory (LTM) was investigated by examining the automatic processing of such numbers in a physical comparison task, and their intentional processing in a numerical comparison task. The size congruity effect (SiCE) served as a marker of automatic processing and consequently as an indicator of the access to the primitives of numerical representation in LTM. Mixed pairs composed of a natural number and a fraction showed both a SiCE and a distance effect. The SiCE for mixed pairs was stable across relative sizes of natural numbers compared to the fraction digits (Experiment 4). However, comparing pairs of fractions revealed a strong influence of fractional components: An inverse SiCE was found for pairs of unit fractions (Experiment 1), while no SiCE was found for pairs of non-unit fractions (Experiments 2-3). This leads to the conclusions that: (1) there are no unique representations of distinct fraction values in LTM, and (2) there is a representation of a "generalized fraction" as an "entity smaller than one" that emerges from the notational structure common to all fractions.

Domain‐specific and domain‐general changes in children's development of number comparison

The numerical distance effect (inverse relationship between numerical distance and reaction time in relative number comparison tasks) has frequently been used to characterize the mental representation of number. The size of the distance effect decreases over developmental time. However, it is unclear whether this reduction simply reflects developmental changes in domain-general speed of processing and whether it is specific to numerical compared with non-numerical magnitude. To examine these open questions, we conducted a cross-sectional study with 6-, 7-, and 8-year-old children as well as adult college students. Participants performed comparisons on Arabic numerals, arrays of squares, squares of varying luminance and bars of varying height. To control for general age-related changes in reaction time, a measure of speed of processing was used as a covariate in the analysis. A significant developmental decrease in the distance effect was found across numerical and non-numerical comparison tasks over and above general changes in processing speed. However, this change was not found to differ as a function of format. These data suggest that developmental changes in the distance effect are reflective of changes in a domain-general comparison process, rather than domain-specific developmental changes in number representations. However, analysis of overall reaction times revealed significantly greater developmental changes for numerical relative to non-numerical comparison tasks. These findings highlight the importance of taking multiple measures into account when characterizing developmental changes in numerical magnitude processing. Implications for theories of numerical cognition and its development are discussed.

Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children’s mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6- to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.

Negative Numbers Are Generated in the Mind

The goal of the present study was to disentangle two possible representations of negative numbers--the holistic representation, where absolute magnitude is integrated with polarity; and the components representation, where absolute magnitude is stored separately from polarity. Participants' performance was examined in two tasks involving numbers from--100 to 100. In the numerical comparison task, participants had to decide which number of a pair was numerically larger/smaller. In the number line task, participants were presented with a spatial number line on which they had to place a number. The results of both tasks support the components representation of negative numbers. The findings suggest that processing of negative numbers does not involve retrieval of their meaning from memory, but rather the integration of the polarity sign with the digits' magnitudes.

The Neuronal Correlate of Bidirectional Synesthesia: A Combined Event-related Potential and Functional Magnetic Resonance Imaging Study

The neuronal correlate of a rare explicit bidirectional synesthesia was investigated with numerical and physical size comparison tasks using both functional magnetic resonance imaging and event-related potentials. Interestingly, although participant I.S. exhibited similar congruity effects for both tasks at the behavioral level, subsequent analyses of the imaging data revealed that different brain areas were recruited for each task, and in different time windows. The results support: (1) the genuineness of bidirectional synesthesia at the neuronal level, (2) the possibility that discrepancy in the neuronal correlates of synesthesia between previous studies might be task-related, and (3) the possibility that synesthesia might not be a unitary phenomenon.

Is 27 a Big Number? Correlational and Causal Connections Among Numerical Categorization, Number Line Estimation, and Numerical Magnitude Comparison

This study examined the generality of the logarithmic to linear transition in children's representations of numerical magnitudes and the role of subjective categorization of numbers in the acquisition of more advanced understanding. Experiment 1 (49 girls and 41 boys, ages 5-8 years) suggested parallel transitions from kindergarten to second grade in the representations used to perform number line estimation, numerical categorization, and numerical magnitude comparison tasks. Individual differences within each grade in proficiency for the three tasks were strongly related. Experiment 2 (27 girls and 13 boys, ages 5-6 years) replicated results from Experiment 1 and demonstrated a causal role of changes in categorization in eliciting changes in number line estimation. Reasons were proposed for the parallel developmental changes across tasks, the consistent individual differences, and the relation between improved categorization of numbers and increasingly linear representations.

Electrophysiological evidence for notation independence in numerical processing

BackgroundA dominant view in numerical cognition is that numerical comparisons operate on a notation independent representation (Dehaene, 1992). Although previous human neurophysiological studies using scalp-recorded event-related potentials (ERPs) on the numerical distance effect have been interpreted as supporting this idea, differences in the electrophysiological correlates of the numerical distance effect in symbolic notations (e.g. Arabic numerals) and non-symbolic notations (e.g. a set of visually presented dots of a certain number) are not entirely consistent with this view.Methods and resultsTwo experiments were conducted to resolve these discrepancies. In Experiment 1, participants performed a symbolic and a non-symbolic numerical comparison task ("smaller or larger than 5?") with numerical values 1–4 and 6–9 while ERPs were recorded. Consistent with a previous report (Temple & Posner, 1998), in the symbolic condition the amplitude of the P2p ERP component (210–250 ms post-stimulus) was larger for values near to the standard than for values far from the standard whereas this pattern was reversed in the non-symbolic condition. However, closer analysis indicated that the reversal in polarity was likely due to the presence of a confounding stimulus effect on the early sensory ERP components for small versus larger numerical values in the non-symbolic condition. In Experiment 2 exclusively large numerosities (8–30) were used, thereby rendering sensory differences negligible, and with this control in place the numerical distance effect in the non-symbolic condition mirrored the symbolic condition of Experiment 1.ConclusionCollectively, the results support the claim of an abstract semantic processing stage for numerical comparisons that is independent of input notation.