Mathematics constitutes a core part of competencies that are important both in professional development and of the daily life of individuals and societies at large (Butterworth, 1999). Because of practical and theoretical reasons, one should not wonder why today’s efforts of many psychologists and cognitive scientists aim to discover and describe mechanisms of mathematical cognition shared by all human beings and these shaping individual differences in mathematical skills.
For several decades most research efforts focused on numerical processing. The cognitive foundation of the latter is traditionally called “the number sense.” Although its original meaning referred to the understanding of the change in a small collection of items after adding or removing an object, nowadays it extends to various number-related phenomena (Dehaene, 2011), especially approximation of non-symbolic numerosities, handled by the so-called approximate number system (ANS), and subitizing (i.e., the capacity for quick and effortless, but precise, assessment of the number of elements of small sets), handled by the object tracking system (OTS).
However, numerical cognition is much more inclusive since it also comprises operating with symbolic numbers (single and multi-digit numbers, fractions, etc.), namely, comparing them, counting, and conducting operations. These capacities are thought to be built upon domain-specific pre-linguistic mechanisms, such as ANS and OTS, but they also involve linguistic processing and domain-general cognitive resources (Hohol, Cipora, Willmes, & Nuerk, 2017). Spatial-numerical associations (SNAs; Cipora, Schroeder et al., 2018) constitute one of the most-explored aspects of numerical cognition, at least since observation that in a speeded bimanual decision task (e.g., deciding whether the number is even or odd), left-to-right readers respond faster to small magnitude numbers with their left hand and to large magnitude numbers with their right hand. This behavioral pattern, called the SNARC effect (see Dehaene, 2011), reflects a direction of the hypothetical “mental number line.” Importantly, SNAs are observed not only in elementary numerical processing since handling arithmetic operations is also associated with “mental movements” along the number line.
Even taking the diversity outlined above into account, numerical cognition should be distinguished from more advanced mathematical skills, e.g., solving algebraic equations or proving mathematical theorems, that require extensive knowledge acquired through long-term training. Many researchers claim that elementary numerical cognition constitutes scaffolding for arithmetic and the latter, in turn, for more advanced mathematical skills. The core idea is that full-fledged mathematical cognition cannot develop properly if elementary mental representations of numbers, both non-symbolic and symbolic, are not well-constructed (Dehaene, 2011; Lakoff & Núñez, 2000). On the other hand, while the performance in some tasks tackling elementary numerical representations correlates positively with mathematical achievements, other tasks reveal no correlation or mixed evidence (see Cipora, Schroeder et al., 2018). What is more, different tasks, and even their different modes, could correlate with full-fledged mathematical achievements in different ways. Schneider et al.’s (2017) meta-analysis shows a stronger association between mathematical achievements and performance in symbolic numerical tasks. However, note that in most previous research, the above association has been studied only in individuals characterized by typical––for a given age group––mathematical skills or in populations with mathematical learning deficits (e.g., developmental dyscalculia).
Very little is known about the presumed scaffolding of full-fledged mathematical cognition on the more basic numerical processing in mathematical experts. This is important since experts pass along prolonged training and purposeful practice that transform their cognitive mechanisms and build new ones. An in-depth study of expertise can contribute to understanding better performance in some tasks and control the influence of other variables (Sella & Cohen Kadosh, 2018). Unfortunately, professional mathematicians, with rather a few exceptions of behavioral (e.g., Cipora, Hohol et al., 2016; Hohol, Willmes et al., 2020; Sella et al., 2016; Meier et al., 2021) and neuroimaging (Amalric & Dehaene, 2016, 2019; Amalric et al., 2018; Popescu, Sader et al., 2019) studies still constitute an understudied group. These studies established a good starting point, but their conclusions are limited since they tested relatively small samples of professional mathematicians, usually only in single tasks or small task batteries, tapping into narrow aspects of numerical processing. Furthermore, many aspects of elementary numerical processing and SNAs, theoretically considered as the scaffolding of mathematics, have never been investigated in experts.
Despite several decades of strenuous research efforts, exploring how elementary numerical processing and SNAs relate to mathematical expertise still remains one of the essential purposes of the differential psychology of mathematics. The proposed project aims to investigate these aspects of numerical cognition by comparing two qualitatively different groups: professional mathematicians (doctoral students of mathematics and post-docs) and controls (carefully matched not only in terms of age, gender, and academic level but also in domain-general cognitive capacities) in a large set of tasks. This will be done mostly in behavioral terms, but neuroscientific measures will also be applied. The specific goals of the project include:
(1) testing hallmark numerical processing phenomena considered as scaffolding for the development of more advanced mathematical skills in the extensive battery of behavioral tasks, supported by using other methodologies (grip force sensors, oculography); these phenomena include non-symbolic and symbolic instances of analog numerical magnitude processing and various SNAs, which have never been studied in mathematicians or have been tested only on small samples and single tasks;
(2) investigating the individual prevalence of behavioral effects considered as hallmarks of elementary numerical processing and SNAs by using bootstrapping methods;
(3) studying the structure and connectivity of the mathematician’s brain on a larger scale than before, and investigating its functional characteristics in multi-digit number processing for the first time.
