Abstract

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.

INTRODUCTION

Because numerical magnitude is an abstract quality of a set, “10,” “ten,” and “••••••••••” each have the same central meaning. This characteristic of numbers has led some to propose a common neural representation of numerical magnitudes regardless of the format in which the magnitudes are represented. Such a format-independent activation has been found across numerical tasks in the intraparietal sulcus (IPS) (Dehaene, Piazza, Pinel, & Cohen, 2003).

Because numerical symbols and their meanings are culturally transmitted, the common activation for symbolic and nonsymbolic representation found in adults must be the outcome of a developmental trajectory. Involvement of the IPS in numerical representation can be found in children as young as 4 years old (Cantlon, Brannon, Carter, & Pelphrey, 2006), and its recruitment during numerical tasks increases across development for symbolic (Ansari, Garcia, Lucas, Hamon, & Dhital, 2005) and nonsymbolic (Ansari & Dhital, 2006) numerical magnitudes. These studies investigating age-related changes in brain responses during numerical magnitude processing have each focused on single numerical surface formats and therefore could not directly address the question of how an abstract representation of numerical magnitude emerges over developmental time.

Recently, the parietal lobe was implicated in the development of an abstract representation of numerical magnitude (Cantlon et al., 2008). In that study, adults and 6- to 7-year-old children performed symbolic (Hindu-Arabic numerals) and nonsymbolic (arrays of dots) numerical comparison. A conjunction analysis for each group identified regions that responded to both symbolic and nonsymbolic numerical magnitudes. A between-group comparison of the conjunction revealed that the left superior parietal lobe showed greater activity in adults relative to children for both symbolic and nonsymbolic numerical processing.

Although this study provided important and novel data characterizing the development of numerical magnitude processing, the study did not include control tasks, which are crucial for disentangling age-related differences in task-general processes from ontogenetic differences in one's process of interest. The authors described the age-related differences they found in left superior parietal lobe as reflective of age-related differences in the processing of an abstract representation of numerical magnitude. However, the absence of nonnumerical control tasks leaves open the possibility that the age-related changes in activation can be explained by nonnumerical variables, such as response selection, which has been associated with IPS activation (Göbel, Johansen-Berg, Behrens, & Rushworth, 2004). In other words, it is possible that the parietal cortex becomes increasingly activated during tasks requiring response selection and other cognitive functions over the course of development, as has been demonstrated in other studies (Morton, Bosma, & Ansari, 2009; Klingberg, Forssberg, & Westerberg, 2002). Thus, greater parietal activation in a number comparison paradigm in adults compared with children could reflect either greater activation of inferior parietal regions related to response selection or numerical magnitude processing. More generally, it is important to note that a conjunction of two conditions without the subtraction of a control cannot be used to investigate task-specific effects. This is because such an analysis is the conjunction of two or more task versus rest contrasts and will therefore necessarily elicit neural activation related to general input factors such as visual stimulation and general output factors such as motoric responses (e.g., button presses) that are common to both tasks. Therefore, the ability to make any claims about shared neural activation related to a task-specific process (such as numerical magnitude processing) requires the use of control tasks that account for the variance of common task-related processes that are unrelated to the processes of interest.

In the present study, we sought to reveal the brain regions that develop a format-independent representation of numerical magnitude over and above age-related changes in parietal mechanisms for response selection. In an fMRI experiment, children and adults performed both symbolic and nonsymbolic numerical comparison tasks as well as two tasks to control for age-related differences in response selection. To address the questions outlined earlier, we planned two analyses. Specifically, in a preliminary analysis, we contrasted the results of a conjunction analysis (symbolic ∩ nonsymbolic) between the two age groups. This analysis identified regions that become increasingly recruited over developmental time for the performance of both symbolic and nonsymbolic comparison tasks. The regions revealed by this analysis could be involved in any aspect of the comparison tasks including numerical processing, response selection, and so forth because this analysis conjoins two task versus rest contrasts. In our primary analysis, we again performed an age-related contrast of a conjunction analysis but also accounted for age-related changes in response selection by subtracting the variance related to the control task from our two experimental conditions ([symbolic − symbolic control] ∩ [nonsymbolic − nonsymbolic control]). Using this latter contrast, we were able to identify regions that undergo age-related changes in the abstract representation of numerical magnitude independently of domain-general changes in response selection. Note that although the control task for the symbolic stimuli is referred to as “symbolic control” throughout the manuscript, the stimuli themselves are not symbols.

We predicted that the preliminary analysis would yield widespread activation of areas of the parietal and occipital regions, related to the processing of the visual and motor components common to both symbolic and nonsymbolic comparison tasks as well as the processing of numerical magnitude. For our primary analysis, we predicted activation to be significantly more confined. In light of a large body of evidence implicating the IPS in the representation of numerical magnitude in adult participants (Piazza, Pinel, Le Bihan, & Dehaene, 2007; Ansari, Dhital, & Siong, 2006; Cohen Kadosh et al., 2005; Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004; Eger, Sterzer, Russ, Giraud, & Kleinschmidt, 2003; Fias, Lammertyn, Reynvoet, Dupont, & Orban, 2003; Pinel, Dehaene, Riviere, & Le Bihan, 2001; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999; Pinel et al., 1999), we predicted that this region would exhibit greater conjunction-related activation in adults relative to children. Moreover, we hypothesized that if this region were to be implicated in the age-related specialization for the representation of format-independent numerical magnitude, it would exhibit a significant numerical distance effect (greater activation for pairs of numerical magnitudes separated by a relatively small versus those separated by a relatively large numerical distance), reflecting its modulation by semantic aspects of numerical processing.

METHODS

Participants

Thirty-three children and 20 adults participated in this study. Thirteen children were excluded for excessive motion. No adults were excluded from the analysis. After exclusion, 19 children (12 girls; mean age = 8.25 years, range = 6.8–9.3 years) and 19 adults (10 women; mean age = 23.5 years, range = 18.4–28.25 years) were included in the analysis. The group of children consisted of 7- and 8-year-olds, with the exception of one 6-year-old who was only 4 months away from being 7 years of age and one 9-year-old who had only been 9 years of age for 3 months at the time of testing. All participants were right-handed and had normal or corrected-to-normal vision. Adults were recruited from graduate and undergraduate faculties at Dartmouth College, Hanover, New Hampshire. Children were recruited from elementary schools around Hanover, New Hampshire. Informed consent/assent procedures were approved by the Committee for the Protection of Human Subjects at Dartmouth College.

Only functional runs with less than 3 mm motion across the entire run and less than 2 mm motion between sequential functional volumes were included in the analysis. To be included, participants were required to have at least one run for each condition that met these motion parameters.

Task Design and Stimuli

Stimuli

All stimuli were white, presented on a black background measuring 800 × 600 pixels. Stimuli were presented equidistant from a fixation dot that appeared between individual trials (see Figure 1).

Figure 1. 

Experimental timing information and examples of stimuli.

Figure 1. 

Experimental timing information and examples of stimuli.

Symbolic task

In the symbolic task, two Arabic numerals 1–9, measuring 200 pixels in height, were presented side by side. Participants chose which side of the screen contained the larger number. A total of 36 stimulus pairs were laterally counterbalanced across the three runs of the experiment for a total of 72 trials per condition. Half of the 36 stimulus pairs consisted of small distance pairs (distances 1–3) and the other half consisted of large distance pairs (distances 5–7), presented in different blocks.

Nonsymbolic task

In the nonsymbolic comparison condition, participants determined which of two arrays of squares contained the larger quantity. The quantities presented in each nonsymbolic trial corresponded to each symbolic trial. To control for the possible confound of continuous variables, we systematically varied the density, the individual square size, and the total area of each array across trials to ensure that numerical quantity could not be reliably predicted from variables continuous with it. These stimuli were previously used in other recent research (Holloway, Price, & Ansari, 2010; Holloway & Ansari, 2008, 2009; Price, Holloway, Rasanen, Vesterinen, & Ansari, 2007). As with the symbolic stimuli, half of the 36 stimulus pairs consisted of small distance pairs (distances 1–3) and the other half consisted of large distance pairs (distances 5–7), administered in different blocks.

Control tasks

For each control task, participants judged which of two stimuli resembled a diagonal line. In this way, the control task, like the experimental task, involved making a selection between the two sides of the display. For the symbolic control task, the stimuli were created by dividing the Hindu-Arabic numerals into segments that were then rotated and reconnected in arbitrary shapes that either approximated a diagonal line or did not (see Figure 1). Using a similar procedure, the nonsymbolic control task was created by combining the separate squares into a shape that either resembled a diagonal line or did not. The linelike stimuli were all constructed from the larger numerosity.

Task Timing Parameters

A total of 12 functional runs were collected for most participants, three runs for each of the four conditions. In each run, 30 sec of initial fixation was followed by four 15-sec blocks of six trials each. Each trial was 2.5 sec in length, including 1200 msec of stimulus presentation followed by 1300 msec of fixation. Participants chose whether the right or the left side of the screen contained the correct answer using a right or left button press. Trial blocks were punctuated with 21 sec of rest fixation. A final block of fixation was presented for 27 sec before the run terminated. The total duration of each run was 3 min.

Data Acquisition

Images were acquired in a 3-T Phillips Intera Allegra whole-body MRI scanner using an eight-channel Phillips Sense head-coil. Functional images were acquired using a gradient-echo-planar T2* sequence sensitive to BOLD contrast. Functional images consisted of 30 noncontiguous slices acquired in an interleaved order (thickness = 4 mm, gap = 0.5 mm, 80 × 80 matrix, repetition time = 3000 msec, echo time = 35 msec, flip angle = 90°, field of view = 240 × 240 mm) covering the whole brain. For each run, 58 volumes were acquired. A high-resolution T1-weighted structural scan, consisting of 160 three-dimensional whole-brain high-resolution T1-weighted images collected in the sagittal plane and measuring 1 × 0.94 × 0.94 mm, was collected using a standard Phillips MPRage 3-D sequence.

Data Analysis

Imaging data were analyzed using Brain Voyager QX 1.10.4 (Brain Innovation, Maastricht, Netherlands). The functional images were corrected for differences in slice time acquisition, head motion, and linear trends and spatially smoothed with a 6-mm FWHM Gaussian smoothing kernel. The functional data were aligned to the structural images and then transformed into Talairach space (Talairach & Tournoux, 1988). A two-gamma hemodynamic response function modeled the expected BOLD signal (Friston et al., 1998).

Individual statistical maps were calculated for each participant estimating the two conjunctions. The preliminary analysis investigated age-related increases in the activation related to the conjunction of the fMRI data acquired during the comparison of symbolic and nonsymbolic numerical stimuli (symbolic ∩ nonsymbolic), whereas the primary analysis aimed at uncovering brain regions that showed developmental differences in their response to both symbolic and nonsymbolic comparison after subtracting out activation related to the control tasks [(symbolic − symbolic control) ∩ (nonsymbolic − nonsymbolic control)]. The individual conjunction maps were then averaged into two group-average maps. In this way, we could compare activation common to symbolic and nonsymbolic numerical magnitude processing between groups with and without controlling for developmental differences in response selection. For both analyses, a random-effects t statistic identified differences between the groups in the activation underlying the conjunction analysis. Because of the large difference in statistical power for each analysis, we corrected for multiple comparisons using different methods. The statistical map from the preliminary analysis was corrected for multiple comparisons using the Bonferroni method (p < .01). For the primary analysis, a random-effects t statistic (t = 3.02, p < .005) identified differences between the groups in the activation underlying the conjunction analysis. Note that for a conjunction analysis, the effective p value is the square of the p values for each component (in our case .0052). The resulting statistical map was then corrected for multiple comparisons (k = 5, p < .05) using cluster-size thresholding (Goebel, Esposito, & Formisano, 2006).

Finally, to inspect relevant regions for the distance effect, we submitted the parameter estimates (beta weights) of the primary analysis to a mixed design ANOVA examining the effect of distance (small vs. large) and format (symbolic vs. nonsymbolic). Note that the conjunction of stimulus format that was used to define the ROI analysis is distinct from the distance effect contrast. In other words, the contrast used to define the ROI is different from the analysis of the beta weights in the ROI analysis.

RESULTS

Behavioral Results

RT and accuracy of the comparison tasks were analyzed in four separate mixed design ANOVAs. The first pair examined the effects of format (symbolic vs. nonsymbolic), task (numerical vs. control), and group (children vs. adults). The second pair examined the effects of format (symbolic vs. nonsymbolic), distance (small vs. large), and group (children vs. adults). The second pair of analyses did not analyze the control tasks as these tasks did not include a distance component. All post hoc tests of means were corrected using the Bonferroni method with a significance threshold of .05.

Effects of Group, Format, and Task on RT

Adults were faster than children, F(1, 36) = 38.8, p < .001, η2 = .52; symbolic comparison was faster than nonsymbolic comparison, F(1, 36) = 7.7, p < .01, η2 = .18; and responses in control tasks were significantly faster than those in numerical comparison tasks, F(1, 36) = 174.1, p < .001, η2 = .83.

We found an interaction between format and group, indicating that the above effect of format was greater in adults than that in children, F(1, 36) = 26.9, p < .001, η2 = .43. We also observed a Format × Task × Group interaction, F(1, 36) = 44.8, p < .001, η2 = .55. For the adults, symbolic numerical comparison was significantly faster than nonsymbolic numerical comparison. The opposite was true of children for whom nonsymbolic numerical comparison was significantly faster than symbolic comparison. There was no effect of format on the control tasks for either group. No other interactions were found in the RT data. Most notably, we found no interaction between task and group. Although adults were faster than children in all tasks, the RT differences between experimental and control conditions were comparable for both groups. Therefore, any differences between children and adults for our primary conjunction analysis of the fMRI data cannot be attributed to differences in RTs between experimental and control tasks.

Effect of Group, Format, and Task on Accuracy

Accuracy was above 90% for both groups on all tasks. Adults were slightly more accurate than children, F(1, 36) = 8.1, p < .01, η2 = .18; accuracy was lower in nonsymbolic comparison than symbolic comparison, F(1, 36) = 14.6, p < .01, η2 = .29; and accuracy was lower in the numerical comparison tasks relative to the control tasks, F(1, 36) = 45.0, p < .001, η2 = .56.

Children made more errors than adults, but only during the numerical tasks, F(1, 36) = 11.8, p < .01, η2 = .25. In addition, across all participants, the effect of format differed by task, F(1, 36) = 10.2, p < .01, η2 = .22. The difference in accuracy between numerical and control tasks was smaller for symbolic comparison than it was for nonsymbolic comparison. No other interactions were found in the accuracy data.

Effect of Group, Format, and Distance on RT

In the numerical comparison tasks, adults were faster than children, F(1, 36) = 37.5, p < .001, η2 = .51; symbolic comparison took less time than nonsymbolic comparison, F(1, 36) = 6.0, p < .05, η2 = .14; and responses to small distances were slower than responses to large distances, F(1, 36) = 222.5, p < .001, η2 = .86.

The main effect of format differed by group, F(1, 36) = 43.5, p < .001, η2 = .55, with adults being faster at symbolic relative to nonsymbolic comparison, whereas children showed the opposite pattern. The main effect of distance also differed by format, F(1, 36) = 48.2, p < .001, η2 = .57. Specifically, the distance effect was smaller in symbolic comparison than that in nonsymbolic comparison. Finally, we found a significant Format × Distance × Group interaction, F(1, 36) = 12.2, p < .01, η2 = .25. In adults, the distance effect was larger in nonsymbolic relative to symbolic comparison. In children, the distance effect was equivalent in the symbolic and nonsymbolic tasks. No other significant interactions were found.

Effect of Group, Format, and Distance on Accuracy

In numerical comparison tasks, children made more errors than adults, F(1, 36) = 10.5, p < .01, η2 = .23, more errors were made in the nonsymbolic task than the symbolic task, F(1, 36) = 15.3, p < .001, η2 = .30, and more errors were made during comparison of small distances than large distances, F(1, 36) = 120.8, p < .001, η2 = .77. The effect of distance differed between groups, F(1, 36) = 22.3, p < .001, η2 = .38, as the distance effect was larger for children than for adults. Furthermore, the effect of distance differed between formats, F(1, 36) = 25.0, p < .001, η2 = .41. The distance effect was smaller for symbolic comparison than nonsymbolic comparison. No other significant interactions were found.

Imaging Results

The preliminary analysis (group difference of [symbolic ∩ nonsymbolic]) revealed greater conjunction-related activation in adults relative to children in bilateral regions of the IPS (see Figure 2) as well as the left precentral gyrus, the bilateral posterior inferior temporal gyrus bordering on the fusiform gyrus, the bilateral caudate and putamen, the bilateral anterior insula, the bilateral medial frontal gyrus, the right middle frontal gyri, and the medial regions of the cerebellum.

Figure 2. 

Statistical map illustrating greater activation in the bilateral parietal cortex for adults relative to children in the conjunction of symbolic and nonsymbolic comparison without subtracting activation related to the control tasks. z-coordinate for the axial brain slice shown = 47.

Figure 2. 

Statistical map illustrating greater activation in the bilateral parietal cortex for adults relative to children in the conjunction of symbolic and nonsymbolic comparison without subtracting activation related to the control tasks. z-coordinate for the axial brain slice shown = 47.

Results from the primary analysis (group difference between [symbolic − control] ∩ [nonsymbolic − control]) revealed several regions that showed greater conjunction-related activation in adults relative to children including the right inferior parietal lobe near the IPS. The analysis also revealed two further parietal regions, one bordering the supramarginal gyrus and a region in the superior parietal lobule (see Figure 3 and Table 1). Additional regions were found in the right anterior insula and left precentral gyrus. Stronger conjunction-related activation in children versus adults was found in the left inferior frontal gyrus (IFG), the right insula, the posterior cingulate gyrus, the subcallosal gyrus, and two regions in the left temporal pole near the superior temporal gyrus.

Figure 3. 

Statistical map illustrating greater activation for adults (white bars) than for children (gray bars) in the conjunction of symbolic and nonsymbolic comparison in the right inferior parietal lobe (IPL) after controlling for activation associated with response selection. Upper bar chart represents mean parameter estimates of activation in the IPL for symbolic comparison (S), symbolic control (SC), nonsymbolic comparison (N), and nonsymbolic control (NC). Lower bar chart represents mean parameter estimates in the IPL for symbolic small distance (SS), symbolic large distance (SL), nonsymbolic small distance (NS), and nonsymbolic large distance (NL). The y-axis depicts BOLD signal represented in Z scores. z-coordinate for the axial brain slice shown = 49.

Figure 3. 

Statistical map illustrating greater activation for adults (white bars) than for children (gray bars) in the conjunction of symbolic and nonsymbolic comparison in the right inferior parietal lobe (IPL) after controlling for activation associated with response selection. Upper bar chart represents mean parameter estimates of activation in the IPL for symbolic comparison (S), symbolic control (SC), nonsymbolic comparison (N), and nonsymbolic control (NC). Lower bar chart represents mean parameter estimates in the IPL for symbolic small distance (SS), symbolic large distance (SL), nonsymbolic small distance (NS), and nonsymbolic large distance (NL). The y-axis depicts BOLD signal represented in Z scores. z-coordinate for the axial brain slice shown = 49.

Table 1. 

Left Column Contains Talairach Coordinates, Cluster Size, and Average t Statistics across Cluster for Each Activation from the Primary Analysis

Location of Peak Voxel
x
y
z
k
t
Adults
Children
S
SC
N
NC
S
SC
N
NC
Right inferior parietal lobe (near IPS) 40 −44 50 155 3.19 0.81 0.60 0.84 0.54 0.12 0.26 0.19 0.27 
Right inferior parietal lobe (near supramarginal gyrus) 32 −48 36 414 3.37 0.68 0.42 0.58 0.44 0.00 0.09 −0.10 0.11 
Right superior parietal lobe 30 −64 50 350 3.40 0.64 0.67 0.83 0.58 0.11 0.26 0.20 0.41 
Right anterior insula 36 17 371 3.32 0.53 0.32 0.62 0.35 0.07 −0.02 0.11 0.04 
Left precentral gyrus −34 −17 61 145 3.18 0.13 0.10 0.07 0.09 −0.19 0.13 −0.14 0.09 
Left IFG −24 16 −13 163 3.22 0.06 0.13 −0.01 0.14 0.01 −0.29 0.01 −0.10 
Right insula 47 −17 16 168 3.27 −0.01 0.07 −0.08 0.00 −0.07 −0.17 −0.07 −0.38 
Left superior temporal gyrus −44 −18 522 3.25 0.02 0.13 −0.15 0.10 −0.12 −0.29 −0.10 −0.24 
Left superior temporal gyrus −43 13 −28 283 3.38 −0.16 0.09 −0.11 0.08 −0.06 −0.14 −0.02 −0.21 
Posterior cingulate gyrus −54 16 304 3.24 −0.02 0.19 0.02 0.23 −0.04 −0.04 0.14 −0.07 
Subcallosal gyrus 13 −10 193 3.42 −0.14 0.02 −0.14 −0.02 −0.06 −0.19 −0.02 −0.11 
Location of Peak Voxel
x
y
z
k
t
Adults
Children
S
SC
N
NC
S
SC
N
NC
Right inferior parietal lobe (near IPS) 40 −44 50 155 3.19 0.81 0.60 0.84 0.54 0.12 0.26 0.19 0.27 
Right inferior parietal lobe (near supramarginal gyrus) 32 −48 36 414 3.37 0.68 0.42 0.58 0.44 0.00 0.09 −0.10 0.11 
Right superior parietal lobe 30 −64 50 350 3.40 0.64 0.67 0.83 0.58 0.11 0.26 0.20 0.41 
Right anterior insula 36 17 371 3.32 0.53 0.32 0.62 0.35 0.07 −0.02 0.11 0.04 
Left precentral gyrus −34 −17 61 145 3.18 0.13 0.10 0.07 0.09 −0.19 0.13 −0.14 0.09 
Left IFG −24 16 −13 163 3.22 0.06 0.13 −0.01 0.14 0.01 −0.29 0.01 −0.10 
Right insula 47 −17 16 168 3.27 −0.01 0.07 −0.08 0.00 −0.07 −0.17 −0.07 −0.38 
Left superior temporal gyrus −44 −18 522 3.25 0.02 0.13 −0.15 0.10 −0.12 −0.29 −0.10 −0.24 
Left superior temporal gyrus −43 13 −28 283 3.38 −0.16 0.09 −0.11 0.08 −0.06 −0.14 −0.02 −0.21 
Posterior cingulate gyrus −54 16 304 3.24 −0.02 0.19 0.02 0.23 −0.04 −0.04 0.14 −0.07 
Subcallosal gyrus 13 −10 193 3.42 −0.14 0.02 −0.14 −0.02 −0.06 −0.19 −0.02 −0.11 

Middle (adults) and right (children) columns contain parameter estimates for each region for each of the four conditions of the conjunction analysis.

We then tested whether activation in the right IPS, as illustrated in Figure 3, was significantly greater than zero for each condition in each group. Bonferroni-corrected single-sample t tests were performed on the beta values for each group. These tests revealed that all conditions significantly activated the right IPS in adults (all p values <.05), and all conditions, with the exception of the symbolic condition, were significantly active in the right IPS in children (all p values <.05).

To test our hypothesis that the IPS elicited by the between-groups conjunction analysis would be involved in the semantic processing of numerical information, we examined whether this region showed a numerical distance effect in adults and in children by contrasting activation elicited by task blocks consisting of number pairs with a relatively small numerical distance and blocks of trials in which the Hindu-Arabic numerals or arrays of squares were separated by a relatively large numerical distance. A distance effect is reflected by greater neural activation in response to small numerical distances relative to large numerical distances. To ensure that the results reflected numerical processing (distance effect) rather than age-related differences in performance (accuracy and response times), we used RTs and accuracy scores for both symbolic and nonsymbolic comparison as covariates in our analysis. Thus, we performed a mixed design ANCOVA examining the effects of format, distance, and group while covarying RT and accuracy. We found a significant effect of distance across both groups, F(1, 36) = 6.55, p < .05. We also found a Distance × Group interaction that approached significance, F(1, 36) = 4.11, p < .051. No other main effects or interactions were found to be significant.

To clarify the marginally significant Distance × Group interaction, we examined the effects of format and distance in each group independently. To control for the Type I error inflation caused by conducting multiple ANOVAs, we corrected the two ANOVAs using the Bonferroni method. The ANOVAs revealed a significant effect of distance for adults, F(1, 18) = 13.8, p < .01, and no interaction between format and distance (see bar chart in Figure 3). In contrast, the children showed no overall effect of distance in this region but did show a significant Format × Distance interaction, F(1, 18) = 6.6, p < .05. Bonferroni-corrected t tests (p < .05) revealed that this interaction was characterized by a significant distance effect in the nonsymbolic condition but no significant distance effect in the symbolic condition. This lack of significant modulation of the right IPS in the symbolic condition mirrors the findings presented earlier, which showed that all conditions with the exception of the symbolic condition were significantly activated in the children. Although the IPS shows age-related increases in activation, the BOLD responses in each of the four Distance × Format conditions was not correlated with chronological age in the group of children, presumably because of the narrow age range tested (all r values <.14, all p values >.5).

Previous studies (Cantlon et al., 2008; Diester & Nieder, 2007) have suggested that the IFG could be involved in the development of numerical representation. Specifically, this region is thought to play a role in shaping parietally mediated numerical representations as children learn numerical symbols such as the Hindu-Arabic numerals. We therefore examined whether the IFG activation for the conjunction analysis that was found to be greater for children compared with adults exhibits a significant distance effect. Using an identical mixed design ANCOVA to the one used for the IPS earlier, we examined the effect of format, distance, and group on the activity in the IFG. No significant main effects or interactions were found (all p values >.2).

The behavioral data we report earlier showed a significant Task × Group interaction in accuracy, suggesting that accuracy differed between the groups in the numerical comparison tasks. Although we used performance measures as covariates in the analysis, it is still possible that group differences in accuracy performance accounts for some of the age-related differences in neural activation. We therefore selected 14 children and 14 adults who were equated on both accuracy (Task × Group, F = 2.92, p = .09, ns) and RT (Task × Group, F = 2.22, p = .14, ns). We then conducted, on the whole-brain level, a group contrast of the conjunction of symbolic and nonsymbolic comparison after subtracting out their controls. The same network of regions we found in the primary analysis earlier was revealed to be significant, albeit at a slightly lower threshold, including greater activation in the right IPS for adults relative to children. This provides evidence that the age-related differences in the activation of the right IPS in response to abstract numerical processing cannot be explained by performance differences between children and adults.

DISCUSSION

Neuroimaging data from adults (Libertus, Woldorff, & Brannon, 2007; Piazza et al., 2007; Venkatraman, Ansari, & Chee, 2005; Piazza et al., 2004) have revealed that the IPS processes numerical magnitude across multiple stimulus formats such as number words, Arabic numerals, or collections of dots (for another viewpoint, see Cohen Kadosh & Walsh, 2009).

When characterizing the neural underpinnings of numerical processing, it is important to acknowledge that children learn symbolic representations of number over developmental time. Hence, format-independent processing of symbolic and nonsymbolic numerical magnitude must be subject to a process of ontogenetic specialization. An outstanding question in the literature, therefore, is whether the format-independent representation of numerical magnitude in the parietal cortex undergoes age-related changes over and above response selection.

Using two complementary conjunction analyses, we revealed that areas of the right parietal cortex exhibit a greater response for the conjunction of symbolic and nonsymbolic numerical magnitude comparison in adults relative to children. Our preliminary analysis revealed that bilateral inferior parietal regions show greater activation in adults relative to children during the symbolic and nonsymbolic numerical comparison tasks. However, after controlling for brain activation related to response selection, we found that the only the right parietal lobe showed significant age-related increases in the activation related to the conjunction of symbolic and nonsymbolic numerical magnitude comparison. In view of these data, we contend that although both the left and the right parietal lobe undergo ontogenetic changes in numerical task performance, only the right parietal cortex undergoes a process of developmental specialization for a format-independent representation of numerical magnitude over and above response selection.

The conjunction analysis revealed brain regions previously shown to be involved in numerical processing including the right IPS (Piazza et al., 2004) and the right superior parietal lobe (Eger et al., 2003). Moreover, we showed that the neural activity in the IPS was modulated by numerical distance, which demonstrates that this region not only responds to numerical tasks in multiple formats but also is meaningfully modulated by the numerical semantics of the tasks. Thus, our results are consistent with an extensive body of previous research, which has highlighted the role of the adult inferior parietal cortex in the abstract representation of numerical magnitude (Dehaene et al., 2003). However, our study extends these previous findings in two ways.

First, we demonstrate that the right IPS undergoes developmental specialization for the abstract representation of numerical magnitude. Although previous developmental studies have made similar claims about the development of numerical representation (Ansari & Dhital, 2006; Cantlon et al., 2006; Ansari et al., 2005), each of them reported brain activation during the processing of a single numerical stimulus format. A significant departure from this “single format” approach was taken by Cantlon et al. (2008), who reported data from an experiment similar to the one presented here. Our study converges with that study by showing age-related changes in the processing of both symbolic and nonsymbolic numerical stimuli in the parietal cortex. However, the study reported by Cantlon et al. was limited in that it could not disentangle numerical processing from domain-general processes such as response selection (Göbel et al., 2004).

In contrast, our study required participants to perform two control tasks to account for brain activity reflecting response selection components of the comparison tasks. Therefore, the age-related changes in the right parietal cortex for the conjunction of symbolic and nonsymbolic number comparison cannot be reduced to age-related changes in response selection. The importance of using control tasks is highlighted by a comparison of the two conjunction maps. When the analysis did not use control tasks, bilateral parietal regions were found to be involved in the development of abstract numerical processing. However, when the control tasks were subtracted from the comparison tasks, only the right parietal cortex showed significant age-related modulation.

Although the control tasks used in this study were useful in clarifying the role of the right parietal cortex in the development of an abstract representation of number, it should be noted that the adult participants were faster than the children in all tasks. However, the magnitude of the difference in RTs between experimental and control conditions did not differ between the two age groups. Because the differences in activations between experimental and control conditions for symbolic and nonsymbolic comparison form the basis of the conjunction analysis, the observed age-related effects cannot be explained by group differences in RT.

In contrast to the RT data, accuracy performance differed between children and adults, but only for the numerical comparison tasks. We therefore matched a subset of children and adults on both accuracy and RT and performed the whole-brain primary analysis again on these 28 participants. The analysis, albeit at a lower threshold presumably because of the smaller number of participants in each group, revealed the same pattern of findings as those found for the whole sample of participants. This strongly suggests that the pattern of age-related differences in neural activation we have reported cannot be explained simply by differences in numerical task performance.

A comparison of our results with those of Cantlon et al. (2008) reveals two major differences between our central finding (developmental increases in activation in right IPS) and their key finding (developmental increases in activation in the left superior parietal lobe). The first difference between the results of the two studies is the locus of the activations. Earlier, we presented evidence that suggests the IPS undergoes developmental increases in activation, over and above differences in performance and response selection that is related to the processing of both symbolic and nonsymbolic numerical magnitudes. The IPS has repeatedly been implicated in the processing of numerical information and is thought to house a neural representation of numerical magnitude that is common across all numerical stimulus formats (Dehaene et al., 2003). Our data converge with and extend this viewpoint to suggest that this format-independent representation of numerical magnitude emerges gradually over developmental time. The data presented by Cantlon et al. implicate the superior parietal lobe in the development of a common representation for symbolic and nonsymbolic numerical magnitude in the brain. Although the superior parietal lobe is anatomically distinct from the IPS, it should be noted that activity in this region has been correlated with numerical processing tasks in other studies (Venkatraman et al., 2005; Fias et al., 2003; Pinel et al., 2001). However, the specific role played by the superior parietal lobe in numerical magnitude processing is not clear. Some researchers have suggested that it reflects attention processes required during the manipulation of numerical magnitudes (Dehaene et al., 2003).

The second difference between the results of our analysis and those of Cantlon et al. (2008) is the laterality of the activation. Our results converge with several other studies highlighting the crucial role of the right IPS in the representation of numerical magnitude in the adult brain (Cappelletti, Lee, Freeman, & Price, 2009; Dormal & Pesenti, 2009; Piazza et al., 2007; Chochon, Cohen, van de Moortele, & Dehaene, 1999). Our data converge with these studies and diverge from Cantlon et al., who found left-lateralized parietal activation. However, other studies have demonstrated left-lateralized activation in numerical processing (Cohen Kadosh et al., 2005; Pesenti, Thioux, Seron, & De Volder, 2000). Pesenti et al. (2000), for example, showed left parietal activation in a sample of adult participants in response to symbolic numerical processing (Hindu-Arabic numerals) even after using appropriate control tasks. Thus, symbolic numerical magnitude representation can be found (in adults) in the left IPS, even when response selection is systematically controlled for.

The majority of studies in numerical neurocognition have implicated bilateral activation in the IPS during magnitude processing (Ansari et al., 2006; Castelli, Glaser, & Butterworth, 2006; Cohen Kadosh et al., 2005; Colvin, Funnell, & Gazzaniga, 2005; Piazza et al., 2004; Pinel, Piazza, Le Bihan, & Dehaene, 2004; Eger et al., 2003; Fias et al., 2003; Pinel et al., 2001; Dehaene et al., 1999). The extent to which numerical representation is lateralized may depend on individual differences and sampling-related effects and is therefore currently unresolved. Future research is required to clarify the specific roles of the right and left parietal lobes in numerical processing. However, it is important to note that the studies highlighted earlier, which demonstrate bilateral parietal involvement in numerical processing, almost exclusively investigated numerical magnitude processing in the adult brain. Because the data reported in this article focuses on the comparison of brain activation in children and adults, the current results do not speak directly to the nature of numerical representation in the adult brain but instead suggest that the right IPS becomes more activated by numerical magnitude processing, which is common to both symbolic and nonsymbolic representations, over developmental time. Moreover, none of the previously cited adult studies looked at the activation related to the conjunction of different numerical stimulus formats. There may be different laterality effects for format-dependent and format-independent processing of numerical magnitude in the parietal cortex (Cohen Kadosh, Cohen Kadosh, Kaas, Henik, & Goebel, 2007; Piazza et al., 2007).

Although considering the earlier caveats concerning laterality, our data demonstrate that before removing variance accounted for by the control tasks, age-related increases in activity are found across the bilateral parietal lobe. After using the control tasks in our analysis, only the right parietal lobe showed significant developmental differences. Thus, the present findings may contribute to a better understanding of lateralization of the abstract processing of numerical magnitude and its developmental trajectory.

Several reasons could underlie the differences between our study and that of Cantlon et al. (2008), but the most salient methodological difference between the studies is the use of nonnumerical tasks to control for response selection. It is therefore highly likely that the majority of the differences in the data between these two studies resulted from this methodological difference. However, other possible explanations exist. For example, Cantlon et al. examined 6- to 7-year old children, whereas our study focused on children aged 7–8 years. The difference in age range could account for some of the differences between the results of the two studies, and future studies should compare these two age groups. In addition, Cantlon et al. presented quantities as large as 50, whereas our study used only numbers 1–9.

Despite their differences, our study and that of Cantlon et al. (2008) stand apart as the only two experiments investigating the development of an abstract representation of numerical magnitude in the brain. Additional research that will clarify the commonalities and reconcile the differences between the two studies are required to further elucidate the development of format-independent numerical processing.

In addition to revealing an age-related increase in format-independent activation in the IPS, our analysis of the distance effect in the IPS activation reveals further clues as to the role of this region in the development of numerical processing. IPS activation in adults showed a distance effect in response to both symbolic and nonsymbolic numerical comparison. In contrast to the adults, our analysis indicated that activation elicited in children reflected a distance effect in the nonsymbolic condition that was absent in the symbolic condition. This suggests that in addition to becoming increasingly active in response to numerical tasks over developmental time, the IPS becomes increasingly sensitive to the semantics of numerical symbols.

It is important to note that although the children in our study show no distance effect in the right IPS during the symbolic condition, evidence from the present and other studies suggests that children show a behavioral distance effect (De Smedt, Verschaffel, & Ghesquiere, 2009; Holloway & Ansari, 2008, 2009; Rousselle & Noel, 2007; Ansari et al., 2005; Sekuler & Mierkiewicz, 1977). Several neuroimaging studies have suggested that prefrontal regions, and specifically the IFG, are recruited more heavily by children relative to adults during numerical comparison tasks (Cantlon et al., 2008; Ansari & Dhital, 2006; Ansari et al., 2005). These findings have led researchers to hypothesize that the IPS, over the course of developmental time, becomes tuned to the specific numerical properties of numerical symbols through communication with the pFC (Cantlon et al., 2008; Diester & Nieder, 2007).

Importantly, the parietal distance effects revealed by our study were located in regions that responded more to numerical comparison than the control tasks, thus excluding the possibility that the parietal distance effects are merely related to differences in the response selection demands of small and large distances. Moreover, because performance differences (RT and accuracy) were used as covariates, the age-related differences in the distance effect in this region cannot be explained solely by developmental improvement in task performance. In addition, a subset of the children and adults who were equated on both RT and accuracy showed the same pattern of activation in the parietal cortex. This provides further evidence that the age-related differences in IPS activation are not due simply to age-related differences in performance.

The second way in which our study extends earlier research is by demonstrating that the right inferior parietal lobe is involved in processing a representation of numerical magnitude across numerical stimulus formats. In general, previous studies have been limited by their focus on numerical tasks using either symbolic or nonsymbolic stimuli, but not both. Using a stringent conjunction analysis, our study demonstrates the role of the right IPS during both symbolic and nonsymbolic numerical comparison. Two other studies of adults have used both symbolic and nonsymbolic stimuli to address slightly different questions. One of these studies revealed the role of the IPS in symbolic and nonsymbolic calculation tasks (Venkatraman et al., 2005). That study used a conjunction analysis and found bilateral IPS activation in response to symbolic and nonsymbolic addition in the IPS and superior parietal lobe. The fact that the results of the present study diverge from the results of Venkatraman et al. (2005) is likely due to the latter's examination of arithmetic processing. Arithmetic tasks can be expected to invoke several processes in addition to numerical representation, and it is likely that the differences in processes investigated resulted in the differences in neural correlates reported by the two studies. Furthermore, the study by Venkatraman et al. examined the format-independent neural correlates in adults, whereas the current study elucidated the developmental processes underlying the neural correlates of format-independent processing of numerical magnitude. A second, more recent study, examined similarities in the brain response to passive viewing of symbolic and nonsymbolic quantities (Piazza et al., 2007). Convergent with our results, that study showed that the adult right IPS responded to numerical magnitudes regardless of whether the magnitudes were presented as numerals or nonsymbolic arrays. The present results are commensurate with the data from Piazza et al. (2007) and suggest that the right IPS is involved in the representation of both symbolic and nonsymbolic numerical magnitude and furthermore demonstrate that a format-independent parietal representation of numerical magnitude is the outcome of a developmental process. However, for results suggesting format-independent representation for words and Arabic numerals in the left but not right IPS, see a different fMRI adaptation study (Cohen Kadosh et al., 2007).

Further support for the specific role of the right IPS in numerical representation comes from a recent article demonstrating hemispheric differences in numerical processing (Cappelletti et al., 2009). In that study, participants performed either conceptual or perceptual comparisons on either numbers or object names. A contrast of the conceptual comparison of numbers with the conceptual comparison of objects revealed greater bilateral parietal activation for number comparisons. However, after equating the RTs of number comparison with the RTs of object naming, the left parietal activations were no longer significant. In contrast, the right parietal regions showed significantly more activation for conceptual numerical judgments even after response times were equated. The authors concluded that the activation in the left parietal lobe is indicative of several different processes such as response selection, which are involved in the processing of both numbers and object names, whereas activation in the right parietal lobe reflects number-specific processing. The findings from the two analyses of the present study are convergent with the suggestion of Cappelletti et al. (2009) that the left IPS is related to more general processes engaged during number processing, whereas activation of the right IPS is found in numerical tasks, even after controlling for response selection and/or response times.

The data of Piazza et al. (2007), Cappelletti et al. (2009), and our own data also converge with two recent studies of developmental dyscalculia, where atypical function and structure of the right IPS have been reported (Rotzer et al., 2008; Price et al., 2007). Within the context of these empirical findings, our data suggest that the age-related differences in the right IPS reflect the development of a brain region involved in the abstract representation of numerical magnitude and thereby highlight the importance of taking a developmental perspective on the functional organization of the brain.

Acknowledgments

This work was supported by grants from the NSF Science of Learning Center Program (SBE-0354400), the Natural Sciences and Engineering Council of Canada, the Canada Foundation for Innovation, the Ontario Ministry for Research and Innovation, and the Canadian Institutes of Health Research. Gwyn Taylor, Ian Lyons, Bibek Dhital, Lucia van Eimeren, and Nicholas Garcia assisted in data collection and analysis.

Reprint requests should be sent to Daniel Ansari, Department of Psychology, University of Western Ontario, London, Ontario, Canada N6A 3K7, or via e-mail: daniel.ansari@uwo.ca.

REFERENCES

REFERENCES
Ansari
,
D.
, &
Dhital
,
B.
(
2006
).
Age-related changes in the activation of the intraparietal sulcus during nonsymbolic magnitude processing: An event-related functional magnetic resonance imaging study.
Journal of Cognitive Neuroscience
,
18
,
1820
1828
.
Ansari
,
D.
,
Dhital
,
B.
, &
Siong
,
S. C.
(
2006
).
Parametric effects of numerical distance on the intraparietal sulcus during passive viewing of rapid numerosity changes.
Brain Research
,
1067
,
181
188
.
Ansari
,
D.
,
Garcia
,
N.
,
Lucas
,
E.
,
Hamon
,
K.
, &
Dhital
,
B.
(
2005
).
Neural correlates of symbolic number processing in children and adults.
NeuroReport
,
16
,
1769
1773
.
Cantlon
,
J. F.
,
Brannon
,
E. M.
,
Carter
,
E. J.
, &
Pelphrey
,
K. A.
(
2006
).
Functional imaging of numerical processing in adults and 4-y-old children.
PLoS Biology
,
4
,
e125
.
Cantlon
,
J. F.
,
Libertus
,
M. E.
,
Pinel
,
P.
,
Dehaene
,
S.
,
Brannon
,
E. M.
, &
Pelphrey
,
K. A.
(
2008
).
The neural development of an abstract concept of number.
Journal of Cognitive Neuroscience
,
21
,
2217
2229
.
Cappelletti
,
M.
,
Lee
,
H. L.
,
Freeman
,
E. D.
, &
Price
,
C. J.
(
2009
).
The role of right and left parietal lobes in the conceptual processing of numbers.
Journal of Cognitive Neuroscience
,
22
,
331
346
.
Castelli
,
F.
,
Glaser
,
D. E.
, &
Butterworth
,
B.
(
2006
).
Discrete and analogue quantity processing in the parietal lobe: A functional MRI study.
Proceedings of the National Academy of Sciences, U.S.A.
,
103
,
4693
4698
.
Chochon
,
F.
,
Cohen
,
L.
,
van de Moortele
,
P. F.
, &
Dehaene
,
S.
(
1999
).
Differential contributions of the left and right inferior parietal lobules to number processing.
Journal of Cognitive Neuroscience
,
11
,
617
630
.
Cohen Kadosh
,
R.
,
Cohen Kadosh
,
K.
,
Kaas
,
A.
,
Henik
,
A.
, &
Goebel
,
R.
(
2007
).
Notation-dependent and -independent representations of numbers in the parietal lobes.
Neuron
,
53
,
307
314
.
Cohen Kadosh
,
R.
,
Henik
,
A.
,
Rubinsten
,
O.
,
Mohr
,
H.
,
Dori
,
H.
,
van de Ven
,
V.
,
et al
(
2005
).
Are numbers special? The comparison systems of the human brain investigated by fMRI.
Neuropsychologia
,
43
,
1238
1248
.
Cohen Kadosh
,
R.
, &
Walsh
,
V.
(
2009
).
Numerical representation in the parietal lobes: Abstract or not abstract?
Behavioral and Brain Sciences
,
32
,
313
328
.
Colvin
,
M. K.
,
Funnell
,
M. G.
, &
Gazzaniga
,
M. S.
(
2005
).
Numerical processing in the two hemispheres: Studies of a split-brain patient.
Brain and Cognition
,
57
,
43
52
.
De Smedt
,
B.
,
Verschaffel
,
L.
, &
Ghesquiere
,
P.
(
2009
).
The predictive value of numerical magnitude comparison for individual differences in mathematics achievement.
Journal of Experimental Child Psychology
,
103
,
469
479
.
Dehaene
,
S.
,
Piazza
,
M.
,
Pinel
,
P.
, &
Cohen
,
L.
(
2003
).
Three parietal circuits for number processing.
Cognitive Neuropsychology
,
20
,
487
506
.
Dehaene
,
S.
,
Spelke
,
E.
,
Pinel
,
P.
,
Stanescu
,
R.
, &
Tsivkin
,
S.
(
1999
).
Sources of mathematical thinking: Behavioral and brain-imaging evidence.
Science
,
284
,
970
974
.
Diester
,
I.
, &
Nieder
,
A.
(
2007
).
Semantic associations between signs and numerical categories in the prefrontal cortex.
PLoS Biology
,
5
,
e294
.
Dormal
,
V.
, &
Pesenti
,
M.
(
2009
).
Common and specific contributions of the intraparietal sulci to numerosity and length processing.
Human Brain Mapping
,
30
,
2466
2476
.
Eger
,
E.
,
Sterzer
,
P.
,
Russ
,
M. O.
,
Giraud
,
A. L.
, &
Kleinschmidt
,
A.
(
2003
).
A supramodal number representation in human intraparietal cortex.
Neuron
,
37
,
719
725
.
Fias
,
W.
,
Lammertyn
,
J.
,
Reynvoet
,
B.
,
Dupont
,
P.
, &
Orban
,
G. A.
(
2003
).
Parietal representation of symbolic and nonsymbolic magnitude.
Journal of Cognitive Neuroscience
,
15
,
47
56
.
Friston
,
K. J.
,
Fletcher
,
P.
,
Josephs
,
O.
,
Holmes
,
A.
,
Rugg
,
M. D.
, &
Turner
,
R.
(
1998
).
Event-related fMRI: Characterizing differential responses.
Neuroimage
,
7
,
30
40
.
Göbel
,
S. M.
,
Johansen-Berg
,
H.
,
Behrens
,
T.
, &
Rushworth
,
M. F.
(
2004
).
Response-selection-related parietal activation during number comparison.
Journal of Cognitive Neuroscience
,
16
,
1536
1551
.
Goebel
,
R.
,
Esposito
,
F.
, &
Formisano
,
E.
(
2006
).
Analysis of functional image analysis contest (FIAC) data with brainvoyager QX: From single-subject to cortically aligned group general linear model analysis and self-organizing group independent component analysis.
Human Brain Mapping
,
27
,
392
401
.
Holloway
,
I.
, &
Ansari
,
D.
(
2008
).
Domain-specific and domain-general changes in children's development of number comparison.
Developmental Science
,
11
,
644
649
.
Holloway
,
I. D.
, &
Ansari
,
D.
(
2009
).
Mapping numerical magnitudes onto symbols: The distance effect and children's mathematical competence.
Journal of Experimental Child Psychology
,
103
,
17
29
.
Holloway
,
I. D.
,
Price
,
G. R.
, &
Ansari
,
D.
(
2010
).
Common and segregated neural pathways for the processing of symbolic and nonsymbolic numerical magnitude: An fMRI study.
Neuroimage
,
49
,
1006
1017
.
Klingberg
,
T.
,
Forssberg
,
H.
, &
Westerberg
,
H.
(
2002
).
Increased brain activity in frontal and parietal cortex underlies the development of visuospatial working memory capacity during childhood.
Journal of Cognitive Neuroscience
,
14
,
1
10
.
Libertus
,
M. E.
,
Woldorff
,
M. G.
, &
Brannon
,
E. M.
(
2007
).
Electrophysiological evidence for notation independence in numerical processing.
Behavioral and Brain Functions
,
3
,
1
.
Morton
,
J. B.
,
Bosma
,
R.
, &
Ansari
,
D.
(
2009
).
Age-related changes in brain activation associated with dimensional shifts of attention: An fMRI study.
Neuroimage
,
46
,
249
256
.
Pesenti
,
M.
,
Thioux
,
M.
,
Seron
,
X.
, &
De Volder
,
A.
(
2000
).
Neuroanatomical substrates of arabic number processing, numerical comparison, and simple addition: A PET study.
Journal of Cognitive Neuroscience
,
12
,
461
479
.
Piazza
,
M.
,
Izard
,
V.
,
Pinel
,
P.
,
Le Bihan
,
D.
, &
Dehaene
,
S.
(
2004
).
Tuning curves for approximate numerosity in the human intraparietal sulcus.
Neuron
,
44
,
547
555
.
Piazza
,
M.
,
Pinel
,
P.
,
Le Bihan
,
D.
, &
Dehaene
,
S.
(
2007
).
A magnitude code common to numerosities and number symbols in human intraparietal cortex.
Neuron
,
53
,
293
305
.
Pinel
,
P.
,
Dehaene
,
S.
,
Riviere
,
D.
, &
Le Bihan
,
D.
(
2001
).
Modulation of parietal activation by semantic distance in a number comparison task.
Neuroimage
,
14
,
1013
1026
.
Pinel
,
P.
,
Le Clec
,
H. G.
,
van de Moortele
,
P. F.
,
Naccache
,
L.
,
Le Bihan
,
D.
, &
Dehaene
,
S.
(
1999
).
Event-related fMRI analysis of the cerebral circuit for number comparison.
NeuroReport
,
10
,
1473
1479
.
Pinel
,
P.
,
Piazza
,
M.
,
Le Bihan
,
D.
, &
Dehaene
,
S.
(
2004
).
Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments.
Neuron
,
41
,
983
993
.
Price
,
G. R.
,
Holloway
,
I.
,
Rasanen
,
P.
,
Vesterinen
,
M.
, &
Ansari
,
D.
(
2007
).
Impaired parietal magnitude processing in developmental dyscalculia.
Current Biology
,
17
,
R1042
R1043
.
Rotzer
,
S.
,
Kucian
,
K.
,
Martin
,
E.
,
von Aster
,
M.
,
Klaver
,
P.
, &
Loenneker
,
T.
(
2008
).
Optimized voxel-based morphometry in children with developmental dyscalculia.
Neuroimage
,
39
,
417
422
.
Rousselle
,
L.
, &
Noel
,
M. P.
(
2007
).
Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing.
Cognition
,
102
,
361
395
.
Sekuler
,
R.
, &
Mierkiewicz
,
D.
(
1977
).
Children's judgments of numerical inequality.
Child Development
,
48
,
630
633
.
Talairach
,
J.
, &
Tournoux
,
P.
(
1988
).
Co-planar steotaxic atlas of the human brain
(M. Rayport, Trans.).
Stuttgart
:
Thieme Medical Publishers
.
Venkatraman
,
V.
,
Ansari
,
D.
, &
Chee
,
M. W.
(
2005
).
Neural correlates of symbolic and nonsymbolic arithmetic.
Neuropsychologia
,
43
,
744
753
.