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Using neuroimaging as a lens through which to understand numerical and mathematical cognition has provided both a different and complementary level of analysis to the broader behavioural literature. In particular, functional magnetic resonance imaging (fMRI) has contributed to our understanding of numerical and mathematical processing by testing and expanding existing psychological theories, creating novel hypotheses, and providing converging evidence with behavioural findings. There now exist several examples where fMRI has provided unique insights into the cognitive underpinnings of basic number processing, arithmetic, and higher-level mathematics. In this review, we discuss how fMRI has contributed to five critical questions in the field including: 1) the relationship between symbolic and nonsymbolic processing; 2) whether arithmetic skills are rooted in an understanding of basic numerical concepts; 3) the role of arithmetic strategies in the development of arithmetic skills; 4) whether basic numerical concepts scaffold higher-level mathematical skills; and 5) the neurobiological origins of developmental dyscalculia. In each of these areas, we review how the fMRI literature has both complemented and pushed the boundaries of our knowledge on these central theoretical issues. Finally, we discuss limitations of current approaches and future directions that will hopefully lead to even greater contributions of neuroimaging to our understanding of numerical cognition.

The use of neuroscientific methods has enriched many fields including that of numerical and mathematical cognition. Early work with neuropsychological patients first informed theories of numerical and mathematical processing (e.g. Triple Code Model,

Much fMRI research is now reaching beyond simple localization and is furthering our understanding of the mechanisms underlying numerical and mathematical processes and their developmental trajectories. Functional brain imaging has also provided insights that were not possible, or more difficult, with behavioural measures (e.g.

In the present paper, we review several examples where fMRI has made a significant contribution to the field of numerical and mathematical cognition. In particular, we outline several major questions within numerical cognition and how functional brain imaging uniquely contributed to these questions. We focus our discussion on neuroimaging literature examining typically developing individuals (children and adults), as well as children with learning disabilities in mathematics. Finally, we provide an outlook towards potential future fMRI research in numerical and mathematical cognition.

It is thought that humans possess an evolutionarily ancient system to process numerical magnitudes (i.e. the number of items in a set) (for reviews see:

Symbolic numbers (Arabic digits), however, are a culturally acquired skill and need to be learned. How children learn the meaning of symbolic numbers (Arabic digits), and whether they are associated with nonsymbolic representations of number, has been a subject of significant debate in the field (

To date, a large body of research has accumulated to suggest that the intraparietal sulcus (IPS) processes numerical magnitudes. This stems from multiple lines of evidence including cases of neuropsychological patients with brain injuries (for a review see:

These findings may initially point to a shared neural representation of symbolic and nonsymbolic quantities, particularly within the IPS. However, these studies are limited in a significant way that curtails the inferences that can be made about format-dependent versus independent processing. In particular, simply demonstrating that two tasks co-activate the same brain region does not imply equivalent processing within that brain region and does not reveal anything about the nature of common activation. One instrumental neuroimaging tool that can begin to disambiguate the representations of symbolic and nonsymbolic processing within the same brain region is Multi-Voxel Pattern Analysis (MVPA). Standard GLM analyses often require that the data be spatially smoothed to increase the sensitivity to a particular task, resulting in brain activity that is averaged across data points (i.e. voxels – three dimensional pixels that are the unit of analysis in fMRI studies) within particular regions of interest (

Several studies have now employed MVPA techniques to determine whether symbolic and nonsymbolic formats have the same or different underlying representations. In the first study to examine whether machine-learning algorithms can successfully predict quantities within and across-formats,

Other kinds of MVPA analyses, such as representational similarity analysis (RSA), can also be used to examine the underlying neural representations. RSA determines the degree of similarity (or dissimilarity) between patterns of brain activity; greater representational overlap should be indicated by stronger correlations between spatial patterns of brain activity. To determine the degree to which symbolic and nonsymbolic formats have similar underlying representations,

The studies discussed above illustrate how fMRI has furthered our understanding of how symbolic and nonsymbolic numbers are represented. By examining how similar or dissimilar symbolic and nonsymbolic processing is at the neuronal level, inferences can be made about whether these two formats are rooted in the same representation. MVPA analyses have also revealed information beyond what we could glean from behavioural evidence alone; even though symbolic and nonsymbolic formats demonstrate similar behavioural ratio effects, analyses of spatial patterns of brain activation suggest that they may have different neural origins. This evidence indicates that the two number formats may have fundamentally different underlying neural representations, even though they activate some common brain regions. Therefore, neuroimaging has expanded our knowledge of how numbers are represented and has challenged existing theories that symbolic and nonsymbolic formats have the same underlying representation. This has led to new hypotheses about how symbols acquire their meaning (

Several key pieces of evidence from the behavioural and neuroimaging literature point to arithmetic skills being scaffolded on (i.e. built on) earlier basic numerical competencies, such as symbolic number knowledge. A large body of literature has shown that fluency with symbolic numbers and understanding quantity relations are correlated with adults’ and children’s math achievement (

The link between basic number processing and arithmetic has also been demonstrated with brain-behaviour correlations. For example,

Even though basic number processing skills and arithmetic have been related to one another using brain-behaviour correlations, it is necessary to examine these processes within the same participants in order to determine whether arithmetic and basic number processing skills have the same underlying neural basis. To date, only a handful of studies have carried out within-subjects investigations into the shared activation for basic numerical tasks and arithmetic. Some evidence has pointed to the idea that basic number processing skills and arithmetic may converge or diverge in the brain depending on participants’ problem solving strategies. For example,

In concert with behavioural evidence, these neuroimaging findings suggest that arithmetic is closely associated with basic number processing skills (e.g. symbol-quantity and symbol-symbol relationships), and the relationship between these skills may be dependent on the cognitive operation being performed. Future research will need to examine the degree to which the four arithmetic operations relate to basic number processing tasks at the neural level and whether the degree of overlap is greater for operations that are more demanding of magnitude processing skills. It is also important to acknowledge that most of the conclusions on the relationship between arithmetic and basic number processing skills are based on brain-behaviour correlations or co-activation of two tasks (with a few exceptions, e.g.

Different cognitive strategies are implemented depending on the type of arithmetic problem presented (ie. addition vs subtraction) and the difficulty of the problem. In particular, some problems will be solved by retrieving the solution from memory, whereas other problems will be solved by using a more procedural and time intensive strategy such as counting or decomposing the problem into smaller parts. In adults, problems such as simple multiplication and addition tend to be retrieved from memory whereas subtraction and division problems are more likely to be solved through more effortful procedural strategies (

Research from neuropsychological patients provided some of the first evidence that there may be cortical distinctions for arithmetic problems that are typically retrieved (e.g. multiplication) versus those that tend to be calculated (e.g. subtraction). Patients with lesions to the left temporo-parietal cortex showed impaired performance on multiplication but intact performance on subtraction (

Verbal strategy reports and manipulations of problem size have also been used to investigate the functional and neurophysiological correlates of arithmetic strategies. Small problems, or problems that are typically solved using retrieval, have been shown to activate perisylvian language regions in the left hemisphere, particularly the left angular and supramarginal gyri (

Training studies have also been instrumental in examining the causal mechanisms underlying experience-dependent changes in strategy use, and the corresponding changes to the neural correlates of arithmetic (for a review see

Accounting for strategy use has also been also been instrumental in determining whether operation effects (e.g. faster reaction times and higher accuracy on multiplication compared to subtraction) are due to fundamental differences in how different operations are solved, or whether they are a consequence of different frequencies of retrieval and procedural strategies. This distinction is particularly important because some studies have interpreted activation differences between arithmetic operations as fundamental differences in the representations and processes of these operations. To address this question, two studies have compared different arithmetic operations after accounting for the types of strategies being performed in adults (

Examining the neural distinctions between calculation and retrieval has also provided an important validation of behavioural methods that have uncovered distinct cognitive processes for different strategies. For example, it is possible that distinct verbal reports could have had similar underlying processes and neural substrates. Therefore, neuroimaging can strengthen existing hypotheses on the cognitive processes of arithmetic strategy use and can also provide insights into the kinds of cognitive operations being performed. For example, retrieval has been found to rely on brain regions that have traditionally been associated with language and phonological decoding (

Are sophisticated mathematical skills (e.g. calculus, algebra, complex geometry) grounded in more basic mathematical concepts such as an understanding of numerical magnitudes and single-digit arithmetic? Neuroimaging is an attractive method for investigating this question because it can be used to determine whether higher-level mathematical skills rely on brain regions that subserve more basic numerical skills, or whether these more sophisticated mathematical processes rely on an entirely different network of regions due to differing cognitive demands.

Some evidence has accumulated to support the hypothesis that higher-level mathematics are grounded in more basic numerical skills. For instance,

Another study involving mathematicians has found parallel conclusions (

Together, these findings suggest that regions within the parietal cortex may serve as a neuroanatomical scaffold for the acquisition of higher-level mathematical skills, and that higher-level mathematical concepts are rooted in more basic arithmetical skills and numerical concepts. Presently, the evidence linking basic numerical concepts and higher-level mathematical skills is largely correlational in nature. Future research will need to examine how these processes unfold longitudinally to determine whether more basic numerical concepts precede and provide a scaffold for more complex mathematical problem solving, or whether the acquisition of such higher-level skills changes the neural correlates of basic number processing, or both. It will also be important to determine whether other cognitive skills that commonly rely on parietal circuits (e.g. attention, working memory, processing speed) mediate the relationship between basic mathematical concepts and higher-level mathematical skills (as discussed in previous sections). Yet, the fMRI literature to date offers a unique insight into the relationship between basic and higher-level mathematical processes by elucidating the possible neurobiological mechanisms for the development and relationship between these skills.

Developmental dyscalculia (DD) is a learning disorder that refers to children who have an impairment in learning arithmetic facts, have poor calculation and math reasoning abilities, and have problems processing numerical information (

Determining how symbolic and nonsymbolic magnitudes are processed at the neural level can provide further evidence towards the core number or access deficit hypotheses. For instance, if children with DD show atypical activation during both nonsymbolic and symbolic magnitude processing, this would support a core number deficit account of DD. In contrast, if differences were only observed for symbolic magnitude processing then the most parsimonious explanation would be an access deficit account. Few studies have systematically contrasted these theories by comparing symbolic and nonsymbolic activation in a sample of children. To date, most studies have separately investigated the neural correlates of symbolic or nonsymbolic processing in children with DD. This research has hinted at some candidate regions that differ between typically developing children and children with DD, and has also provided some insights into the underlying deficits in DD. For example,

Beyond symbolic and non-symbolic number processing, it has also been shown that children with DD exhibit atypical activation of the right IPS during a visuo-spatial working memory task (

The small number of studies and conflicting findings currently preclude our ability to determine whether the available literature points to a core defective number module or access deficit account of DD. However, these studies highlight how neuroimaging can help determine the etiology of DD and revise existing theories. Future research will need to examine both symbolic and nonsymbolic processing within the same sample of children with DD to disentangle which theory is best supported. Other more sensitive analytic techniques, such as MVPA, will also be fundamental in investigating how symbolic and nonsymbolic processing is qualitatively different in children with DD (

Functional brain imaging provides an additional level of analysis that complements behavioural methods and also provides additional tools with which to address several outstanding questions in the field of numerical cognition. In this review, we have outlined several instances in which fMRI has supported and furthered our understanding of the cognitive underpinning of numerical and mathematical skills including: 1) whether symbolic and nonsymbolic magnitudes have the same underlying representation; 2) whether basic number processing skills provide the basis on which arithmetic is learned; 3) the role of strategies in the development of arithmetic skills; 4) how arithmetic and basic number processing skills scaffold higher-level mathematics; and 5) the neurobiological correlates of developmental dyscalculia. In some cases, fMRI research has supported and converged with behavioural investigations of numerical and mathematical processing, but there are some instances in which it has significantly extended our knowledge. For instance, fMRI has begun to disentangle whether symbolic and nonsymbolic magnitudes have the same underlying representation (

There have also been instances in which neural indices have had greater predictive power than behavioural measures alone (e.g.

Examining the neurocognitive underpinnings of numerical and mathematical thinking can also have broader implications for how multiple cognitive systems interact within the brain. For instance, learning mathematics may not only change brain circuitry underlying math, but may also shape the circuitry underlying other functions (

Even though fMRI has significantly added to our understanding of how numbers are represented, it is important to acknowledge some of its limitations. For example, it is often difficult to control for performance differences between groups (e.g. adults versus children, or dyscalculic children versus controls), and it can be difficult to determine whether neural differences between groups are a result of performance differences or fundamental differences on the dimension of interest (e.g. development or math impairment). One way to overcome this limitation is to use passive paradigms that do not rely on behavioural performance (e.g. fMR-adaptation paradigms). Such techniques can be particularly useful when there are performance differences because no overt response is necessary (for a greater discussion of this issue see

Using functional brain imaging as a lens through which to understand numerical and mathematical cognition has proved to be fruitful in complementing and expanding our existing knowledge on core theoretical issues such as how numbers are represented and how individuals learn to calculate. It has pushed the boundaries of our knowledge on the development of numerical and mathematical skills in both typically and atypically developing individuals. Importantly, fMRI can help examine how relatively recent cultural inventions, such as number symbols, become represented in the brain, and how evolutionarily ancient systems in the brain may or may not provide restrictions and shape the acquisition of mathematical thinking. Going forward, the use of fMRI to study questions within the field of numerical and mathematical cognition must follow recent recommendations for improved practices, including adequately powered studies and best practices for data analysis (

It is important to clearly state that the use of any neuroimaging methodology (e.g. near infrared spectroscopy (NIRS), electroencephalography (EEG), etc.) to constrain our understanding of numerical cognition should not be viewed as a superior level of analysis, but rather a complementary one. Neuroimaging findings that confirm what has already been established using behavioural methods are of equal utility as are those that challenge and expand our understanding of a given phenomenon. Applying neuroimaging methods to better understand numerical and mathematical cognition is one of many tools available to researchers in the field and should be viewed as being on the same level playing field as other approaches.

This work was supported by grants from the Canada Institute of Health Research (CIHR), National Sciences and Engineering Research Council of Canada (NSERC), and the Steacie Memorial Fellowship (NSERC) to DA, and the Vanier Canada Graduate Scholarship and Ontario Graduate Scholarship to AAM.

The authors have declared that they have no competing interests.

The authors have no support to report.