Generative processing underlies the mutual enhancement of arithmetic fluency and math-grounding number sense

Number skills are popularly bound to arithmetic knowledge in its symbolic form, such as “five + nine = fourteen,” but mounting evidence suggests that these symbolic relations are actually grounded, i.e., computed (see Harnad, 1990) on noisy internal magnitude representations that bear our general understanding of numbers and further improve with math experience (Figure 1). Multiple lines of evidence support the idea of semantics-based arithmetic, including behavioral research on humans (Gallistel and Gelman, 1992), animals (Gallistel and Gelman, 2000; Rugani et al., 2009), development (Halberda et al., 2008), mathematical disability, i.e., dyscalculia (Butterworth, 1999; review, Butterworth et al., 2011), and computational modeling (Stoianov et al., 2004; review, Zorzi et al., 2005). Even more intimate relation between the number skills and the internal noisy magnitudes was recently demonstrated in several studies showing finer magnitude representations in subjects with greater arithmetic fluency (e.g., Nys et al., 2013; Piazza et al., 2013), also caused by extensive math studying during higher education (Lindskog et al., 2014). Here we discuss how these findings could be explained within a generative framework of cognition, according to which top-down predictive connections play a key role in the computing of lowto highlevel representations (e.g., Friston, 2010; Clark, 2013). The noisy internal magnitude representations also known as Approximate Number System (ANS), or Number Sense are systematically found in the intraparietal silcus and prefrontal cortices (Dehaene, 1997; Viswanathan and Nieder, 2013) and one principle method to investigate them is to characterize the ability to quickly and approximately estimate the number of objects seen (Jevons, 1871). This phylogenetic ability is qualified as a visual sense (Burr and Ross, 2008), the mechanism of which emerged in generative neural networks that learn to efficiently encode visual numerosities (Stoianov and Zorzi, 2012). One crucial property of the internal magnitude representations is their systematically increasing imprecision (Figure 1; Gallistel and Gelman, 2000; Dehaene, 2003) characterized by a subject-specific constant known as ANS acuity (Halberda et al., 2008), which at the behavioral level is associated with log-linear performance decrement as the magnitude increases. In numerosity comparison, the probability to select the greater numerosity is a sigmoid function of the log-ratio of the compared magnitudes that is characterized by a discriminability (Weber) fraction w describing the slope of the sigmoid, whereby the better the discriminability, the closer is the sigmoid to a step-function, for which w = 0 (Piazza et al., 2013; Cappelletti et al., 2014). The behavioral discriminability coefficient w is closely related to the internal ANS acuity (Piazza et al., 2013). The ANS acuity progressively improves along with development, with corresponding w = 1 in the first few months of life to about w = 0.24 in healthy adults (Piazza et al., 2010), to worsen then with ageing to more than w = 0.30 (Cappelletti et al., 2014). The intriguing question we explore here is whether ANS improves along with refinement of the mathematical knowledge it supports, that is, whether math-studying improves general quantity understanding. A hint about this was provided by a study on a curious Amazonian Mundurucù population with two levels of math education (Piazza et al., 2013). The effect of math-studying on ANS acuity, controlling for age, was impressive: w = 0.31 for adults that had never studied math and w = 0.19 for math-educated adults. Sure, this is a study on a particular population, with an educational system that permitted to find subjects allowing the dissociation between age and education (see also Nys et al., 2013), and it remained unclear whether prolonged math schooling in cultures with broad educational system is associated with further improvement of the ANS. Lindskog et al. (2014) investigated this issue with firstand third-year university students majoring in disciplines with various levels of initial math-expertise and amount of math-studying, ranked in the order: humanity-disciplines, with expected basic math-background, and no math-studying, business-disciplines with average math-background and applied math-studying, and math-disciplines with high math proficiency and mostly theoretical math-studying. The ANS-acuity of the students was evaluated using visual numerosity comparison tasks and supplementary assessment measured their arithmetic fluency. Overall, the arithmetic fluency increased along with the rank of the math-expertise and similarly, the