Magnitude Comparison Task Research Articles

The conceptual/procedural distinction belongs to strategies, not tasks: A comment on Gabriel et al. (2013)

The conceptual/procedural distinction belongs to strategies, not tasks: A comment on Gabriel et al. (2013)

Is there a generalized magnitude system in the brain? Behavioral, neuroimaging, and computational evidence

Some 10 years ago, it was proposed that different quantities, including time, space or number, are processed by a common mechanism (Walsh, 2003; Pinel et al., 2004; Cohen Kadosh et al., 2005), henceforth called generalized magnitude system (GMS). This presented an important attempt at theoretically integrating data across various literatures, and became very influential (e.g., Burr et al., 2010; Lourenco and Longo, 2011; Bonato et al., 2012). However, in this paper we will evaluate the concept in a critical way. We first argue that such a mechanism faces conceptual difficulties. Second, we look at empirical findings that were proposed to support a GMS and will offer a different explanation of these findings based on computational modeling and recent empirical observations. Third, we review positive evidence in favor of distinct magnitude mechanisms. We discuss alternatives for a GMS in a final paragraph.

Count on dopamine: influences of COMT polymorphisms on numerical cognition

Catechol-O-methyltransferase (COMT) is an enzyme that is particularly important for the metabolism of dopamine. Functional polymorphisms of COMT have been implicated in working memory and numerical cognition. This is an exploratory study that aims at investigating associations between COMT polymorphisms, working memory, and numerical cognition. Elementary school children from 2th to 6th grades were divided into two groups according to their COMT val158met polymorphism [homozygous for valine allele (n = 61) vs. heterozygous plus methionine homozygous children or met+ group (n = 94)]. Both groups were matched for age and intelligence. Working memory was assessed through digit span and Corsi blocks. Symbolic numerical processing was assessed through transcoding and single-digit word problem tasks. Non-symbolic magnitude comparison and estimation tasks were used to assess number sense. Between-group differences were found in symbolic and non-symbolic numerical tasks, but not in working memory tasks. Children in the met+ group showed better performance in all numerical tasks while val homozygous children presented slower development of non-symbolic magnitude representations. These results suggest COMT-related dopaminergic modulation may be related not only to working memory, as found in previous studies, but also to the development of magnitude processing and magnitude representations.

Visual stimulus parameters seriously compromise the measurement of approximate number system acuity and comparative effects between adults and children.

It has been suggested that a simple non-symbolic magnitude comparison task is sufficient to measure the acuity of a putative Approximate Number System (ANS). A proposed measure of the ANS, the so-called “internal Weber fraction” (w), would provide a clear measure of ANS acuity. However, ANS studies have never presented adequate evidence that visual stimulus parameters did not compromise measurements of w to such extent that w is actually driven by visual instead of numerical processes. We therefore investigated this question by testing non-symbolic magnitude discrimination in seven-year-old children and adults. We manipulated/controlled visual parameters in a more stringent manner than usual. As a consequence of these controls, in some trials numerical cues correlated positively with number while in others they correlated negatively with number. This congruency effect strongly correlated with w, which means that congruency effects were probably driving effects in w. Consequently, in both adults and children congruency had a major impact on the fit of the model underlying the computation of w. Furthermore, children showed larger congruency effects than adults. This suggests that ANS tasks are seriously compromised by the visual stimulus parameters, which cannot be controlled. Hence, they are not pure measures of the ANS and some putative w or ratio effect differences between children and adults in previous ANS studies may be due to the differential influence of the visual stimulus parameters in children and adults. In addition, because the resolution of congruency effects relies on inhibitory (interference suppression) function, some previous ANS findings were probably influenced by the developmental state of inhibitory processes especially when comparing children with developmental dyscalculia and typically developing children.

Young adult male carriers of the fragile X premutation exhibit genetically modulated impairments in visuospatial tasks controlled for psychomotor speed

BackgroundA previous study reported enhanced psychomotor speed, and subtle but significant cognitive impairments, modulated by age and by mutations in the fragile X mental retardation 1 (FMR1) gene in adult female fragile X premutation carriers (fXPCs). Because male carriers, unlike females, do not have a second, unaffected FMR1 allele, male fXPCs should exhibit similar, if not worse, impairments. Understanding male fXPCs is of particular significance because of their increased risk of developing fragile X-associated tremor/ataxia syndrome (FXTAS).MethodsMale fXPCs (n = 18) and healthy control (HC) adults (n = 26) aged less than 45 years performed two psychomotor speed tasks (manual and oral) and two visuospatial tasks (magnitude comparison and enumeration). In the magnitude comparison task, participants were asked to compare and judge which of two bars was larger. In the enumeration task, participants were shown between one and eight green bars in the center of the screen, and asked to state the total number displayed. Enumeration typically proceeds in one of two modes: subitizing, a fast and accurate process that works only with a small set of items, and counting, which requires accurate serial-object detection and individuation during visual search. We examined the associations between the performance on all tasks and the age, full-scale intelligent quotient, and CGG repeat length of participants.ResultsWe found that in the magnitude comparison and enumeration tasks, male fXPCs exhibited slower reaction times relative to HCs, even after controlling for simple reaction time.ConclusionsOur results indicate that male fXPCs as a group show impairments (slower reaction times) in numerical visuospatial tasks, which are consistent with previous findings. This adds to a growing body of literature characterizing the phenotype in fXPCs who are asymptomatic for FXTAS. Future longitudinal studies are needed to determine how these impairments relate to risk of developing FXTAS.

Symbolic and Non-Symbolic Number Magnitude Processing in Children with Developmental Dyscalculia

The aim of this study was to evaluate if children with Developmental Dyscalculia (DD) exhibit a general deficit in magnitude representations or a specific deficit in the connection of symbolic representations with the corresponding analogous magnitudes. DD was diagnosed using a timed arithmetic task. The experimental magnitude comparison tasks were presented in non-symbolic and symbolic formats. DD and typically developing (TD) children showed similar numerical distance and size congruity effects. However, DD children performed significantly slower in the symbolic task. These results are consistent with the access deficit hypothesis, according to which DD children's deficits are caused by difficulties accessing magnitude information from numerical symbols rather than in processing numerosities per se.

On the development of Arabic three-digit number processing in primary school children

The development of two-digit number processing in children, and in particular the influence of place-value understanding, has recently received increasing research interest. However, place-value influences leading to decomposed processing have not yet been investigated for multi-digit numbers beyond the two-digit number range in children. Therefore, we evaluated the separate influences of hundreds, tens, and units on three-digit number processing by means of the hundred distance effect, the decade-hundred compatibility effect, and the unit-hundred compatibility effect in a longitudinal design from Grade 2 to Grade 4. In a number magnitude comparison task, a strong hundred distance effect indicated that the magnitudes of the hundreds digits were predominantly processed. We also observed indexes of decomposed parallel processing of hundreds and units digits but not of hundreds and tens digits. Regarding the developmental trajectories, the hundred distance effect and the unit-hundred compatibility effect showed a reliable trend to increase with grade level. However, both the significance and the increase with grade level of decomposed parallel processing were observed to be less consistent than expected. The latter is discussed in terms of different processing strategies as well as specificities differentiating between two- and three-digit numbers. Taken together, these are the first data showing decomposed processing of three-digit numbers in children. Yet, it must be noted that the results also indicate that findings from two-digit number processing cannot simply be generalized to the three-digit number range.

Multi-digit number processing beyond the two-digit number range: A combination of sequential and parallel processes

Investigations of multi-digit number processing typically focus on two-digit numbers. Here, we aim to investigate the generality of results from two-digit numbers for four- and six-digit numbers. Previous studies on two-digit numbers mostly suggested a parallel processing of tens and units. In contrast, the few studies examining the processing of larger numbers suggest sequential processing of the individual constituting digits. In this study, we combined the methodological approaches of studies implying either parallel or sequential processing. Participants completed a number magnitude comparison task on two-, four-, and six-digit numbers including unit-decade compatible and incompatible differing digit pairs (e.g., 32_47, 3<4 and 2<7 vs. 37_52, 3<5 but 7>2, respectively) at all possible digit positions. Response latencies and fixation behavior indicated that sequential and parallel decomposition is not exclusive in multi-digit number processing. Instead, our results clearly suggested that sequential and parallel processing strategies seem to be combined when processing multi-digit numbers beyond the two-digit number range. To account for the results, we propose a chunking hypothesis claiming that multi-digit numbers are separated into chunks of shorter digit strings. While the different chunks are processed sequentially digits within these chunks are processed in parallel.

Malleability of the approximate number system: effects of feedback and training

Prior research demonstrates that animals and humans share an approximate number system (ANS), characterized by ratio dependence and that the precision of this system increases substantially over human development. The goal of the present research was to investigate the malleability of the ANS (as measured by Weber fraction) in adult subjects in response to feedback and to explore the relationship between ANS acuity and acuity on another magnitude comparison task. We tested each of 20 subjects over six 1-h sessions. The main findings were that (a) Weber fractions rapidly decreased when trial-by-trial feedback was introduced in the second session and remained stable over continued training, (b) Weber fractions remained steady when trial-by-trial feedback was removed in session 6, (c)Weber fractions from the number comparison task were positively correlated with Weber fractions from a line length comparison task, (d) improvement in Weber fractions in response to feedback for the number task did not transfer to the line length task, (e) finally, the precision of the ANS was positively correlated with math, but not verbal, standardized aptitude scores. Potential neural correlates of the perceptual information and decision processes are considered, and predictions regarding the neural correlates of ANS malleability are discussed.

Representation of numerosity in posterior parietal cortex

Humans and animals appear to share a similar representation of number as an analog magnitude on an internal, subjective scale. Neurological and neurophysiological data suggest that posterior parietal cortex (PPC) is a critical component of the circuits that form the basis of numerical abilities in humans. Patients with parietal lesions are impaired in their ability to access the deep meaning of numbers. Acalculiac patients with inferior parietal damage often have difficulty performing arithmetic (2 + 4?) or number bisection (what is between 3 and 5?) tasks, but are able to recite multiplication tables and read or write numerals. Functional imaging studies of neurologically intact humans performing subtraction, number comparison, and non-verbal magnitude comparison tasks show activity in areas within the intraparietal sulcus (IPS). Taken together, clinical cases and imaging studies support a critical role for parietal cortex in the mental manipulation of numerical quantities. Further, responses of single PPC neurons in non-human primates are sensitive to the numerosity of visual stimuli independent of low-level stimulus qualities. When monkeys are trained to make explicit judgments about the numerical value of such stimuli, PPC neurons encode their cardinal numerical value; without such training PPC neurons appear to encode numerical magnitude in an analog fashion. Here we suggest that the spatial and integrative properties of PPC neurons contribute to their critical role in numerical cognition.

A unitary or multiple representations of numerical magnitude? - the case of structure in symbolic and non-symbolic quantities.

Currently, there is a controversial debate on whether there is an abstract representation of number magnitude, multiple different ones, or multiple different ones that project onto a unitary representation. The current study aimed at evaluating this issue by means of a magnitude comparison task involving Arabic numbers and structured as well as unstructured non-symbolic patterns of squares. In particular, we were interested whether a specific numerical effect, the unit-decade compatibility effect reflecting decomposed processing of tens and units complying with the place-value structure of the Arabic number system, is affected by input notation. Indeed, a reliable unit-decade compatibility effect was observed in the symbolic-digital notation condition but was absent for unstructured non-symbolic notation. However, for structured non-symbolic notation a – albeit negative – compatibility effect was observed as well. Theses results are hard to reconcile with the notion of an abstract representation of number magnitude. Instead, our data support the existence of multiple representations of numerical magnitude. In addition, the current data indicate that it may not be a question of symbolic vs. non-symbolic notation only but also an issue of the structuring of the input notation. While unstructured non-symbolic quantities seemed to be processed holistically we found evidence suggesting at least partially decomposed processing not only for symbolic Arabic numbers but also for structured non-symbolic quantities.

The development of the SNARC effect: Evidence for early verbal coding

In a recent study, Gevers and colleagues (2010, Journal of Experimental Psychology: General, Vol. 139, pp. 180–190) showed that the SNARC (spatial numerical association of response codes) effect in adults results not only from spatial coding of magnitude (e.g., mental number line hypothesis) but also from verbal coding. Because children are surrounded by rulers, number lines, and the like in the classroom, it is intuitively appealing to assume that they first use their mental number line to represent numbers and that only later in development a verbal recoding of magnitude information takes place. However, this hypothesis has never been tested. The goal of the current study was to define the developmental pattern of both accounts (spatial and verbal) in explaining the SNARC effect. To this end, 9- and 11-year-olds were tested in a magnitude comparison task. Surprisingly, clear and robust evidence for verbal coding of magnitude information was observed in both age groups. Our results imply that the ability to use verbal coding of magnitude information is robustly present early in formal schooling.

Multi-Representation of Symbolic and Nonsymbolic Numerical Magnitude in Chinese Number Processing

Numerical information can be conveyed by either symbolic or nonsymbolic representation. Some symbolic numerals can also be identified as nonsymbolic quantities defined by the number of lines (e.g., I, II, III in Roman and , , in Japanese Kanji and Chinese). Here we report that such multi-representation of magnitude can facilitate the processing of these numerals under certain circumstances. In a magnitude comparison task judging 1 to 9 (except 5) Chinese and Arabic numerals presented at the foveal (at the center) or parafoveal (3° left or right of the center) location, multi-representational small-value Chinese numerals showed a processing advantage over single-representational Arabic numerals and large-value Chinese numerals only in the parafoveal condition, demonstrated by lower error rates and faster reaction times. Further event-related potential (ERP) analysis showed that such a processing advantage was not reflected by traditional ERP components identified in previous studies of number processing, such as N1 or P2p. Instead, the difference was found much later in a N400 component between 300–550 msec over parietal regions, suggesting that those behavioral differences may not be due to early processing of visual identification, but later processing of subitizing or accessing mental number line when lacking attentional resources. These results suggest that there could be three stages of number processing represented separately by the N1, P2p and N400 ERP components. In addition, numerical information can be represented simultaneously by both symbolic and nonsymbolic systems, which will facilitate number processing in certain situations.

Enhanced Manual and Oral Motor Reaction Time in Young Adult Female Fragile X Premutation Carriers

A previous study reported preliminary results of enhanced processing of simple visual information in the form of faster reaction times, in female fragile X premutation carriers (fXPCs). In this study, we assessed manual and oral motor reaction times in 30 female fXPCs and 20 neurotypical (NT) controls. Participants completed two versions of the reaction time task; one version required a manual motor response and the other version required an oral motor response. Results revealed that the female fXPCs displayed faster reaction times for both manual and oral motor responses relative to NT controls. Molecular measures including CGG repeat length, FMR1 mRNA levels, and age were not associated with performance in either group. Given previously reported age and CGG repeat modulated performance on a magnitude comparison task in this same group of premutation carriers, results from the current study seem to suggest that female fXPCs may have spared basic psychomotor functionality.

Young adult female fragile X premutation carriers show age- and genetically-modulated cognitive impairments

The high frequency of the fragile X premutation in the general population and its emerging neurocognitive implications highlight the need to investigate the effects of the premutation on lifespan cognitive development. Until recently, cognitive function in fragile X premutation carriers (fXPCs) was presumed to be unaffected by the mutation. Here we show that young adult female fXPCs show subtle, yet significant, age- and FMR1 gene mutation-modulated cognitive impairments as tested by a quantitative magnitude comparison task. Our results begin to define the neurocognitive endophenotype associated with the premutation in adults, who are at risk for developing a neurodegenerative disorder associated with the fragile X premutation. Results from the present study may potentially be applied toward the design of early interventions wherein we might be able to target premutation carriers most at risk for degeneration for preventive treatment.

Overlapping representations of numerical magnitude and motion direction in the posterior parietal cortex: A TMS-adaptation study

The human posterior parietal cortex (PPC) is involved in the encoding of both visual motion and numerical magnitude. In non human primates, neurons have been found in PPC that are selective for both motion direction and magnitude. Whether such neurons also exist in human PPC is not known. Here we investigated this hypothesis using state-dependent transcranial magnetic stimulation (TMS). Participants were adapted to a specific motion direction (either leftward or rightward), after which they performed a magnitude comparison task, with TMS applied at the onset of each trial. Our hypothesis was that neurons tuned to leftward motion may also be sensitive to small magnitudes and neurons tuned to rightward motion may also be sensitive to large magnitudes, a mapping that may have developed via spatial attentional mechanisms. Our results supported this view by showing that the effect of PPC TMS on small and large numbers depended on the motion direction being adapted, thus suggesting that there may be a functional overlap in neuronal representations of motion direction and numerical magnitude in human PPC.

Sensori-motor spatial training of number magnitude representation

An adequately developed spatial representation of number magnitude is associated with children's general arithmetic achievement. Therefore, a new spatial-numerical training program for kindergarten children was developed in which presentation and response were associated with a congruent spatial numerical representation. In particular, children responded by a full-body spatial movement on a digital dance mat in a magnitude comparison task. This spatial-numerical training was more effective than a non-spatial control training in enhancing children's performance on a number line estimation task and a subtest of a standardized mathematical achievement battery (TEDI-MATH). A mediation analysis suggested that these improvements were driven by an improvement of children's mental number line representation and not only by unspecific factors such as attention or motivation. These results suggest a benefit of spatial numerical associations. Rather than being a merely associated covariate, they work as an independently manipulated variable which is functional for numerical development.

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.

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.

Comparing 5/7 and 2/9: Adults can do it by accessing the magnitude of the whole fractions

This study investigated adults' ability to compare the magnitude of fractions without common components (e.g., 5/7 and 3/8), and the representation accessed in that process. We hypothesized that the absence of common components would enhance access to the magnitude of the fractions (i.e., a holistic representation) rather than a direct comparison of the numerators or the denominators. This hypothesis was tested in four between-subject conditions. Two types of experimental pairs were used that differed in the congruity of the magnitude of the denominator and the magnitude of the fraction. Each type of experimental pair was presented either alone or with filler pairs that introduced variability into the congruity of the components. In all four conditions, accuracy was above chance and the effect of the distance between the fractions on response times was significant, indicating an access to the magnitude of the fractions. Nevertheless, the variability of the congruity of the components had also a significant effect on performance, suggesting that the relative magnitude of the components was also processed. In conclusion, the representation of the fraction magnitude is hybrid, rather than purely holistic, in a magnitude-comparison task on fractions without common components.