Is 2 + 2 = 4? Meta-analyses of brain areas needed for numbers and calculations
Research Highlights
►Normative adult neurofunctional model of mental arithmetic. ►Mathematical performance emerges from a large set of brain areas. ►Cingulate gyrus, insula and cerebellum play a role in mathematical cognition. ►Prefrontal cortices contribute differentially based on task demands. ►Laterality was modulated by arithmetic operations in parietal and prefrontal areas.
Introduction
We use numbers to tell time, report quantities and to estimate how much things cost. Numbers can be represented with words (e.g., three), objects (e.g., ♥ ♥ ♥) or Roman and Arabic numerals (e.g., III or 3). Many functional neuroimaging studies have investigated the brain regions that support numerical processes, e.g., comparing quantities or performing arithmetic operations such as subtraction and multiplication. Extant reviews of numerical processes in the neuroimaging literature are based on qualitative reports (Ansari, 2007, Ansari, 2008, Dowker, 2006, Neumarker, 2000, Nieder and Dehaene, 2009). As the field of functional neuroimaging has produced a substantial body of data, it is valuable and timely to compile this information using meta-analytical methods to provide a quantitative level of interpretation, which can help guide future studies.
Numbers are basic elements of mathematics which can be used for different operations such as counting, comparing quantities and ranking; number tasks do not involve calculations. In functional magnetic resonance imaging (fMRI) number task studies, stimuli were typically single digits which were later compared to other conditions such as single letters (Eger et al., 2003), arrays of dots that participants judged (e.g., based on size; Ansari et al., 2005, Ansari et al., 2007) or a visual stimulus that signalled participants to generate random numbers (Daniels et al., 2003). Campbell (1994) argued that the way numbers are presented (i.e., words, numbers or pictures) plays a key role in the processing or estimating numerical magnitude, whereas other researchers proposed that stimulus format is not a major factor for estimating numerical quantities (Dehaene and Cohen, 1995, McCloskey, 1992). The hypothesis that numerical magnitude estimates are largely unaffected by stimulus format is also supported by neuropsychological models which posit that numerical quantity is expressed in an abstract format in the intraparietal sulcus (Ansari, 2007). The left intraparietal sulcus was shown to activate for quantity estimations independent of stimulus format, whereas the right intraparietal sulcus responded to quantity only when Arabic numerals were used (Ansari, 2007). Thus, in this meta-analysis the intraparietal sulcus, which lies between the superior and inferior parietal lobules, was expected to be a key area among studies that used numbers as stimuli.
Calculation tasks that utilize arithmetic operations, such as subtraction and multiplication, require the subject to identify number quantities and then modify them based on the operational function. Arithmetic decisions pose different cognitive demands based on the number of steps they require (Agostino et al., 2010). Most neuroimaging studies on arithmetic processing used single step arithmetic problems (e.g., 3 + 4, 4 − 3, 4 × 3) with one-digit or a combination of one and two-digit numbers (e.g., Fehr et al., 2007). Other arithmetic operations also include manipulating numbers in successive operations (e.g., 4 − 3 + 5; Menon et al., 2000) or even solving integration problems (Krueger et al., 2008). In order to generate an answer, arithmetic operations generally require numbers to be monitored and manipulated. Activity in the prefrontal cortex has been linked to general-purpose cognitive functions such as working memory (Christoff and Gabrieli, 2000, Owen et al., 2005), with considerable emphasis on its role in monitoring or manipulating information, as required in calculation tasks. Researchers who study numerical processing and computations recognize that complex arithmetic tasks require more working memory resources than simple tasks (Fehr et al., 2007, Kong et al., 2005) and also report that training reduces the working memory load on the prefrontal lobes (Ischebeck et al., 2006).
Theories of numerical cognition differ in their assumptions about the components and mechanisms that underlie mathematical abilities. The ‘abstract-code’ model represents functionally independent mechanisms for numeral comprehension and numeral production (McCloskey, 1992). In contrast the ‘encoding-complex’ model predicts that arithmetic operations are not mediated by abstract codes, rather they are influenced primarily by modality-specific processes (e.g., visual and phonological codes; Campbell, 1994). Unlike the ‘abstract code’ and ‘encoding-complex’ model, the ‘triple-code’ model makes specific predictions of the neuroanatomical correlates of functions and mechanisms that underlie mental arithmetic (Dehaene, 1992, Dehaene and Cohen, 1995, Dehaene and Cohen, 1997). This is likely a major factor why this model is more frequently cited in functional neuroimaging studies, and it was claimed to be more predictive of data (Neumarker, 2000). Thus, we chose the ‘triple-code’ model as the comparison basis for the findings from the meta-analyses.
Specifically, the ‘triple-code’ model predicts that numbers are processed in three numerical surface formats: (1) a visual Arabic code represented by strings of digits, (2) an analogic quantity and magnitude code and (3) verbal code represented by words (Dehaene and Cohen, 1997), by distinct brain areas: (1) bilateral activity in inferior ventral occipito-temporal areas underlying visual Arabic code, (2) activity in inferior parietal areas underlying quantity and magnitude judgments and (3) the left perisylvian areas underlying verbal code. Within this framework, simple single-digit calculations can be solved either through a direct route using operands (e.g., 2 × 5) transcoded into verbal code (two times five), which would elicit the rote memory of this operation (e.g., two times five equals ten), or through an indirect semantic route in which the operands represent quantities on which semantically meaningful manipulations can be performed. The indirect route is typically taken when rote memory for a problem is unavailable, such as in subtraction problems (Dehaene and Cohen, 1997). The direct route is reported to elicit activity in the left cortico-subcortical loop through basal ganglia and thalamus, and the indirect route recruits areas in the inferior parietal cortex and the left perisylvian language network (Dehaene and Cohen, 1997). Thus, within this model, key regions for calculation tasks include bilateral inferior parietal areas responsible for semantic knowledge about numerical quantities, and the cortico-thalamic loop responsible for storing rote sequences of simple-arithmetic facts. Elementary operations applied on numbers rely on rote verbal memory and semantic manipulations associated with magnitudes. Dehaene and Cohen (1997) argued that addition and multiplication rely mostly on rote verbal memory (direct route), whereas subtraction relies mostly on quantitative manipulations (indirect route), and that these two processes are reflected in the brain as two main cortical networks for calculation. The role of the prefrontal cortex in this model was that of strategy choice and planning; hierarchical involvement of prefrontal regions and possible hemispheric asymmetries were not clearly specified.
Using activation likelihood estimation (ALE; Laird et al., 2005, Turkeltaub et al., 2002) we explored the brain areas involved in both number and calculation tasks and provide normative fMRI atlases for these processes in a standard stereotaxic space. In doing so, we first identified what brain structures participated in numerical and computational processes. Secondly we clarified brain structures that participated in processing different types of arithmetic operations (i.e., addition, subtraction and multiplication).
Section snippets
Literature search and article selection
The literature was searched using the standard search engine of Web of Science (http://www.isiknowledge.com). We looked for keywords (fMRI, number and math) and (fMRI and arithmetic) to identify articles published between January 1st 1990 and January 31st 2009. These articles were also restricted to include human participants and be written in English. This search, which yielded a total of 268 studies, was subjected to two successive criteria to identify articles that used fMRI and number
Results
Methodological information was extracted from each study. Table 1 shows demographic information of the datasets and selected contrasts of number and calculation tasks. A total of 698 participants took part in these studies. Nine studies did not report gender; of the remaining studies, 47.7% were female participants. The vast majority of the studies that reported handedness (79.25%) tested subjects who were right-handed (99.68%). Six studies did not report the age of the participants. When an
Discussion
Neurofunctional activity associated with number and calculation tasks was examined using quantitative ALE meta-analyses. There were three main findings from these meta-analyses:
- (a)
Although a large overlap existed among areas with significant ALE values during number and calculation tasks, the regions in which they differed were most notable, such as distinct areas of prefrontal cortices.
- (b)
Solving calculation tasks elicited ALE values in more prefrontal areas than solving number tasks. This
Conclusions
The ability to process numbers and perform computations relies on a large number of brain regions. For many years the triple-code model (Dehaene and Cohen, 1995, Dehaene and Cohen, 1997) has provided a framework for research in mental arithmetic. We have demonstrated that mathematical performance emerges from areas extensively discussed and studied under this model; however, we also show that another set of areas, not part of this framework, demonstrate significant probabilities of being
Acknowledgments
We wish to thank Drew Morris for advice on data analyses and Peng You for help with creating the figures. This project was funded in part by NSERC to MJT (138502-09).
References (143)
- et al.
Executive functions underlying multiplicative reasoning: problem type matters
J. Exp. Child Psychol.
(2010) Does the parietal cortex distinguish between “10”, “ten”, and ten dots?
Neuron
(2007)- et al.
Dissociating response conflict from numerical magnitude processing in the brain: an event-related fMRI study
NeuroImage
(2006) - et al.
Tuning of the human left fusiform gyrus to sublexical orthographic structure
NeuroImage
(2006) - et al.
How green is the grass on the other side? Frontopolar cortex and the evidence in favor of alternative courses of action
Neuron
(2009) - et al.
Neural correlates of social and nonsocial emotions: an fMRI study
NeuroImage
(2006) - et al.
Cognitive and emotional influences in anterior cingulate cortex
Trends Cogn. Sci.
(2000) Architectures for numerical cognition
Cognition
(1994)Hemispheric specialization in prefrontal cortex: effects of verbalizability, imageability and meaning
J. Neurolinguist.
(2003)- et al.
A common network of functional areas for attention and eye movements
Neuron
(1998)