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Neuroimaging of Numerical and Mathematical Development
Written by:
Daniel Ansari, Ph.D., Department of Psychology and Graduate Program in Neuroscience, University of Western Ontario
Published online:
2009-10-14 11:23:08
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In recent years, the advent of non-invasive neuroimaging methods, such as functional Magnetic Resonance Imaging (fMRI) has provided an important complement to behavioral investigations of cognitive processes. Together with traditional neuropsychological studies of brain-damaged patients, these investigations have lead to the creation of a new field of research: Cognitive Neuroscience. The combination of behavioral and neuroscientific empirical studies is starting to reveal how the brain enables the mind. More recently, this approach has been applied to the study of the typical and atypical developmental trajectories of cognitive processes, such as numerical cognition.

This entry reviews the current state of the insights, which have been gleaned from the application of modern functional and structural neuroimaging methods into the development of children's numerical and mathematical skills.

Both the study of brain-damaged patients and, more recently, non-invasive neuroimaging studies of healthy participants, have revealed a set of brain areas that are associated with the processing of numeracy and arithmetic. Specifically, a wide range of functional neuroimaging studies point to areas of the parietal cortex (see Figure 1 below) as playing a critical role in the processing of both numerical and mathematical problems. In a recent meta-analysis (Dehaene, Piazza, Pinel, & Cohen, 2003) of neuroimaging studies in which adult participants were asked to solve numerical (e.g., number comparison and estimation) or mathematical (e.g., mental arithmetic) tasks, the authors reported the presence of three key areas in the parietal cortex that are thought to subserve numerical and mathematical cognition. Specifically, the meta-analysis suggests that when participants process numerical quantity (e.g., while deciding which of two numbers is numerically larger), they activate bilateral (left and right side) areas of the so-called intraparietal sulcus (IPS). In contrast, when adults are asked to calculate, activation is most commonly found in a left-lateralized parietal region called the angular gyrus (AG). In addition to these two regions, Dehaene et al. identified a third region in both the left and right superior parietal cortex that, according to these authors, is associated with the attentional and visuo-spatial resources required by numerical and mathematical processing tasks.

Figure 1. Key areas in the parietal cortex.

This meta-analysis demonstrates the significant progress that has been made in our understanding of the brain regions involved in both numerical and mathematical processing. However, to date, most of the neuroimaging studies investigating the neural mechanisms underlying numeracy and mathematical processing have been conducted with healthy adult participants. Therefore, much less is known about whether, and if so how, the brain areas associated with numerical and mathematical processing change over developmental time and how such developmental processes differ among children who present with mathematical difficulties (i.e., Developmental Dyscalculia; DD).

Key Research Questions

Based on the background of the information discussed above, two key research questions can be delineated:
1. Do the brain mechanisms underlying numeracy and mathematics change over developmental time? If so, how?
2. Do children with DD activate the same brain regions as their typically developing peers?

Recent Research Results

While neuroimaging studies into the developmental trajectories underlying numeracy and mathematics lag significantly behind empirical research that has investigated the neural correlates of these functions in the adult brain, there now exists a growing body of both functional and structural neuroimaging studies with developmental subject populations.

Do the brain mechanisms underlying numeracy and mathematics change over developmental time? If so, how?
In one of the first neuroimaging studies of arithmetic development, Rivera, Reiss, Eckert, and Menon (2005) used fMRI to study the neural correlates of addition and subtraction in healthy participants between the ages of 8 and 19 years. Participants viewed addition and subtraction problems (e.g., 5+3=8 or 7-4=2) and had to judge whether the result displayed was correct or incorrect. Moreover, participants completed a control task in which they had to judge whether a sequence of 5 digits contained a 0 or not (e.g., 61059 or 93263). When brain activation associated with the arithmetic task was subtracted from that associated with the control task, the brain areas involved in calculation could be revealed. The authors then correlated the results of the difference between the calculation and the control task with the chronological ages of the participants. Using this analysis, Rivera et al. could reveal in which brain areas the activation either positively or negatively correlated with chronological age. This correlational analysis uncovered both brain circuits whose activation during mental arithmetic increases with chronological age as well as brain regions in which the younger participants exhibited greater activation than their older peers. More specifically, Rivera et al. found that activation in the left inferior parietal cortex (including the left supramarginal gyrus and the anterior aspect of the left angular gyrus) increased with the chronological age of the participants. In contrast, areas of the prefrontal and anterior cingulate cortex as well as subcortical regions such as the hippocampus and basal ganglia exhibited decreasing activation related to the arithmetic task with increasing age. These findings therefore suggest that there are indeed dramatic changes in the neural correlates of addition and subtraction as a function of chronological age. The increasing activation of the left inferior parietal cortex suggests that the involvement of this region in adults is the outcome of a process of developmental specialization. Furthermore, the decreasing activation of the prefrontal cortex may suggest that functions such as attention and working memory as well as the use of immature problem-solving strategies, which have frequently been associated with these frontal brain regions, are more engaged during calculation by young children, compared to adolescents and young adults. Finally, the decreasing activation in the hippocampus may be evidence that over developmental time calculation requires less engagement of systems involved in the formation of long-term memory (such as arithmetic facts) in the brain.

The data reported by Rivera et al. (2005) suggests that the development of calculation skills is associated with a dynamic pattern of changes in the brain circuitry engaged during calculation. These findings raise the question of whether similar developmental changes can also be found when more basic tasks are tested. When participants are asked to compare which of two numbers is numerically larger, the time it takes them to perform this relative magnitude judgment is related to the difference or numerical distance between the two numbers (Moyer & Landauer, 1967). More specifically, it takes longer to judge which of two numbers is numerically larger when the numerical distance is comparatively small. In other words, there exists a highly reliable inverse relationship between reaction times and numerical distance. This effect has become a standard measure in studies of numerical magnitude processing, as it is thought to reveal features of the underlying representations of numerical magnitude. Consequently, in neuroimaging studies with adults, the effect of numerical distance has been investigated and a growing body of research has revealed that numerical distance modulated activation in areas of the IPS in both hemispheres (Pinel, Dehaene, Riviere, & LeBihan, 2001). Since behavioral studies have shown that the distance effect decreases over developmental time (Holloway & Ansari, 2008; Sekuler & Mierkiewicz, 1977), it is possible that there exist age-related changes in the functional neuroanatomy associated with numerical distance. In order to address this question, Ansari, Garcia, Lucas, Hamon, and Dhital (2005) used fMRI to measure brain activation in both children (mean age 10 years) and adults while they compared the relative magnitudes of single-digit Arabic numerals. The results of this study revealed that, consistent with prior evidence, numerical distance modulated activation in bilateral regions of the parietal cortex. In the group of children, however, the strongest effect of numerical distance on brain activation was found in the prefrontal cortex. Thus, consistent with the above-reviewed developmental study of calculation (Rivera et al., 2005) there appears to be a shift away from the engagement of the prefrontal cortex towards increasing engagement of the parietal cortex over developmental time.

In a subsequent study, Ansari and Dhital (2006) examined whether similar age-related changes in the neural correlates of the numerical distance effect could be found when participants were asked to compare non-symbolic numerical magnitudes. In an fMRI study, both adults and children were asked to judge which of two arrays of squares was numerically larger and the numerical distance between the groups of squares was systematically varied. The results, consistent with those reported by Ansari et al. (2005), revealed a greater effect of numerical distance on the parietal cortex in adults compared to children. That is the activation of the parietal cortex was greater when participants were comparing pairs of numbers separated by a relatively small (e.g. 1 vs. 2 squares) compared to a large (2 vs. 9 squares) numerical distance.

Taken together, the studies discussed above demonstrate age-related specialization of brain circuits for both calculation and numerical magnitude processing. The available evidence suggests that the left inferior parietal cortex (comprising the supramarginal and angular gyri) increases in calculation-related activation as a function of chronological age. Conversely, bilateral regions of the IPS are more engaged during numerical magnitude processing in adults compared to children. Other data also support this fronto-parietal shift in the activation underlying both calculation and numerical magnitude processing (Cantlon et al., 2009; Kaufmann et al., 2006; Kucian, von Aster, Loenneker, Dietrich, & Martin, 2008).

It should be noted, however, that Cantlon, Brannon, Carter, and Pelphrey (2006) found similar activation of the parietal cortex in both 4-year-old children and adults during non-symbolic numerical(arrays of dots) magnitude processing. In their study participants were presented with repeated images displaying the same number of dots (e.g., 16) in different spatial configurations. Following several repetitions, participants were presented with a novel numerosity (e.g., 32). By mapping the brain regions activated by the presentation of the novel numerosity it is possible to uncover brain regions that are sensitive to changes in the number of items displayed. The similarity in parietal activation between children and adults revealed by Cantlon et al. in this passive number 'perception' task raises the possibility that developmental differences revealed in other studies have to do with the neurocognitive mechanisms engaged when translating representations of number and arithmetic into responses. Understanding the difference in brain activation in passive compared to active task is an important avenue for future research.

Do children with DD activate the same brain regions as their typically developing peers?
The above reviewed studies demonstrate that, among typically developing children, the functional neuroanatomy underlying numerical and mathematical processing undergoes age-related changes. This raises the question of whether the neural correlates of calculation and numerical magnitude processing differ in children with mathematical difficulties (or DD).

To date, there are only very few empirical studies that can address this question. However some progress has been made. In the first functional neuroimaging study of children with Developmental Dyscalculia (DD; see this Encyclopedia entry by Cohen Kadosh, 2009 ) more diffuse and weaker activation was found in the IPS. In a more recent study, Price, Holloway, Rasanen, Vesterinen, and Ansari (2007) investigated the effect of non-symbolic (e.g., arrays of squares) numerical distance on brain activation in children with DD as well as a group of typically developing, age-matched children. These authors found that while the right IPS of typically developing children was modulated by numerical distance, there was lower activation and no distance effect on IPS activation in the group of children with DD. These data suggest that there is an impairment of numerical magnitude representation in DD. A similar pattern of data was recently reported by Mussolin et al. (2010) for symbolic (Arabic numerals) numerical magnitude comparison.

In addition to measuring brain activation patterns and how these differ between individuals with and without DD, it is also possible to compare the physical structure of the brain between participants. In other words, using structural images of the brain acquired using Magnetic Resonance Imaging (MRI), the volume of brain tissue (such as gray and white matter) can be calculated and compared between participants with and without DD. A recent comparison of gray matter volume (amount of gray matter) between children with DD and their typically developing peers revealed reduced gray matter volumes in the right parietal cortex as well as the frontal brain regions (Rotzer et al., 2007). In other words, the individuals with DD had less brain tissue compared to typically developing children in regions that are well known to be involved in the processing of numerical magnitude. Thus DD is associated with both atypical structure and function of the IPS.

Future Directions

The above overview illustrates that progress towards a greater understanding of the neural mechanisms underlying the typical and atypical development of numeracy and mathematical skills is being made. However, there are many outstanding questions that should be addressed in the future. From a theoretical perspective, it is currently unclear which factors mediate and explain the age-related changes found in brain activation. It is possible that these are the consequence of brain maturation, education, experience or an interaction between these factors. Systematic research is needed to understand what causes the developmental changes in the neural correlates of numeracy and mathematics. Understanding, for example, how education influences the representation of numerical symbols in the brain (such as Arabic numerals) will help to better understand the effects of education on the brain.

A related future direction is the need for more longitudinal studies to provide a more detailed understanding of how the brain activation underlying numeracy and arithmetic changes over developmental time. Such longitudinal studies will also be crucial for a greater understanding of the stability of atypical activation profiles found among children with DD. From an educational perspective, neuroimaging could become an important tool for tracking the outcomes associated with structured intervention/remediation programs. In literature on reading remediation, neuroimaging methods such as fMRI have been used to show brain activation changes mediated by intervention programs (Shaywitz et al., 2004; Shaywitz & Shaywitz, 2009; Temple et al., 2003) Similar studies should be employed to establish whether research-based remediation programs (for more details, see this Encyclopedia entry by Wilson & Räsänen, 2008) lead to normalization of atypical brain activation profiles in children with DD.

Conclusions

The present entry provides a broad overview of the existing body of neuroimaging studies of numerical and mathematical development. Taken together the available evidence demonstrates that the neural correlates of numerical and mathematical development change as a function of chronological age. More specifically, for both calculation and basic numerical magnitude processing, areas of the parietal cortex increase in their engagement over developmental time, while prefrontal regions of the brain decrease in their involvement. Furthermore, a growing, but still small, body of research suggests that this developmental specialization of the parietal cortex for numerical and mathematical processing is disrupted in children with DD. Functional activation studies have revealed reduced activation in the parietal cortex during numerical and mathematical processing in individuals with DD and structural evidence suggests that these individuals have less gray matter in the parietal cortex than their typically developing peers.

In conclusion, neuroimaging of numerical and mathematical development is very much in its infancy and much more research is needed to answer questions concerning the specific nature of developmental changes, or absence thereof, in children who have mathematical difficulties. Furthermore, much more work is needed to connect the neuroimaging literature with mathematics education and remediation programs.
References
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Ansari, D., Garcia, N., Lucas, E., Hamon, K., & Dhital, B. (2005). Neural correlates of symbolic number processing in children and adults. Neuroreport, 16(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(5), e125.

Cantlon, J. F., Libertus, M. E., Pinel, P., Dehaene, S., Brannon, E. M., & Pelphrey, K. A. (2009). The neural development of an abstract concept of number. Journal of Cognitive Neuroscience, 21(11), 2217-2229.

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Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Three parietal circuits for number processing. Cognitive Neuropsychology, 20(3-6), 487-506.

Holloway, I. D., & Ansari, D. (2008). Domain-specific and domain-general changes in children's development of number comparison. Developmental Science, 11(5), 644-649.

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Kucian, K., von Aster, M., Loenneker, T., Dietrich, T., & Martin, E. (2008). Development of neural networks for exact and approximate calculation: A FMRI study. Developmental Neuropsychology, 33(4), 447-473.

Moyer, R. S., & Landauer, T. K. (1967). Time required for judgements of numerical inequality. Nature, 215(109), 1519-1520.

Mussolin, C., De Volder, A., Grandin, C., Schlogel, X., Nassogne, M. C., & Noel, M. P. (2010). Neural correlates of symbolic number comparison in developmental dyscalculia. Journal of Cognitive Neuroscience, 22, 860-874.

Pinel, P., Dehaene, S., Riviere, D., & LeBihan, D. (2001). Modulation of parietal activation by semantic distance in a number comparison task. Neuroimage, 14(5), 1013-1026.

Price, G. R., Holloway, I., Rasanen, P., Vesterinen, M., & Ansari, D. (2007). Impaired parietal magnitude processing in developmental dyscalculia. Current Biology, 17(24), R1042-1043.

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Rotzer, S., Kucian, K., Martin, E., Aster, M. V., Klaver, P., & Loenneker, T. (2007). Optimized voxel-based morphometry in children with developmental dyscalculia. Neuroimage, 39(1), 417-422.

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Shaywitz, B. A., & Shaywitz, S. E. (2009). Brain imaging in studies of reading and dyslexia. Encyclopedia of Language and Literacy Development (pp. 1-6). London, ON: Canadian Language and Literacy Research Network. Retrieved October 7, 2009, from http://www.literacyencyclopedia.ca/pdfs/topic.php?topId=281

Temple, E., Deutsch, G. K., Poldrack, R. A., Miller, S. L., Tallal, P., Merzenich, M. M., et al. (2003). Neural deficits in children with dyslexia ameliorated by behavioral remediation: Evidence from functional MRI. Proceedings of the National Academy of Sciences of the United States of America, 100(5), 2860-2865.

Wilson, A. J., & Räsänen, P. (2008). Effective interventions for numeracy difficulties/disorders. Encyclopedia of Language and Literacy Development (pp. 1-11). London, ON: Canadian Language and Literacy Research Network. Retrieved September 28, 2009, from http://www.literacyencyclopedia.ca/pdfs/topic.php?topId=259
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