A case for neuroscience in mathematics education

Mathematics lies at the heart of science and technology impacting on the economic performance of societies since ancient times (OECD, 2010). At the level of individuals too, the development of mathematical proficiency appears correlated with individual development and career prospects across a wide range of professions (RAND Mathematics Study Panel and Loewenberg Ball, 2003). It does not come as a surprise to realize that mathematics education traces back several thousand years. However, still very little is known about the fundamental principles of how individuals learn mathematics and at which age education should start. The issue is far from trivial as it is commonly assumed that mathematics is a special subject area perhaps requiring specific motivation, interest and teaching methods in order to be learned efficiently (National Council of Teachers of Mathematics, 2000). Here, we are attempting to make a case for neuroscience methodology as a modern tool capable of contributing to the debate, where a special but not exclusive emphasis is on brain development. Note that for the purpose of this opinion paper, neuroscience is essentially equated with magnetic resonance imaging (MRI), as MRI based approaches currently constitute mainstream research in this field of study according to our understanding. Developmental studies are increasing our understanding of maturational changes in the human brain (Blakemore, 2012). In particular, structural MRI studies reveal an increase in white matter volume during childhood and adolescence suggesting an increase of connectivity in the developing brain (Giedd and Rapoport, 2010). Interestingly, gray matter volume is characterized by an invertedU shaped curve peaking at different age in different brain regions (Giedd et al., 1999), which suggests a non-linear, heterogeneous trajectory where proficiencies mature at different times and speeds dependent on which brain regions are most important for a given skill. For example, it is commonly agreed that the intuitive sense of number or quantity is an early ability that can be observed already in infants and that can predict mathematical proficiency later in life (Starr et al., 2013). In addition to structural studies, functional neuroimaging provides further insight relevant to mathematics education. For example, a developmental functional MRI study of mental arithmetic has shown that the pattern of brain activation changes with student age (Rivera et al., 2005). Importantly, these age-related changes were associated with functional maturation rather than alterations in gray matter density. Moreover, functional studies can help to elucidate the role of specific brain regions in mathematical processing. For example, it has been suggested that the intuitive understanding of quantities is associated with activity in the intra-parietal sulcus (Dehaene, 1997) and, more generally, parietal cortices that are involved in various mathematical tasks from number comparison to complex processing such as proportional and deductive reasoning (e.g., Kroger et al., 2008; Vecchiato et al., 2013). However, additional studies are needed to establish links between development of brain structures and their functional maturation. Many neuroimaging studies have focused on development of arithmetic skills in children and adults (for a review see Zamarian et al., 2009). Again, different parts of the parietal cortex, such as bilateral intra-parietal sulcus and left angular gyrus, are shown to have a crucial role in mental calculations (e.g., De Smedt et al., 2011; Grabner et al., 2013). In contrast, other brain areas appear to mature relatively late, such as prefrontal association areas thought to be involved in mathematical cognition and other higher-order processes developing throughout childhood and adolescence (Blakemore, 2012). Such insight might shed some light on the transition from concrete arithmetic to the symbolic language of algebra, where students have to develop abstract reasoning skills that enable them to generalize, model, and analyze mathematical equations and theorems (e.g., Qin et al., 2004; Lee et al., 2007; Anderson et al., 2012). Ultimately, mathematical proficiency will require the coordinated action of many brain regions as exemplified by an influential model of algebraic equation solving (Anderson et al., 2008). Based largely on functional MRI studies of brain activation, the model stipulates distinguishable functional modules that map onto anatomically separate brain regions. For example, a visual module that extracts information about the equation is associated with the fusiform gyrus. An imagery module holding a representation of the equation and performing transformations on the equation is located in posterior parietal cortices. A module responsible for retrieval of previously learned algebraic rules is associated with the left prefrontal cortex. Such models are