In Chapter 2, I focus on the acquisition of number concepts related to natural numbers. I review nativist views, as well as Dehaene’s early view that number concepts arise from estimations due to the approximate numbers system. I end up focusing in most detail on the bootstrapping account of Carey and Beck, according to which the object tracking system is the key cognitive resource used in number concept acquisition. However, I endorse a hybrid account that also includes an important role for the approximate numerosity system. I then review some of the criticism against the bootstrapping account, concluding that, while more empirical data is needed to establish its correctness and details, currently it provides the most plausible account of early number concept acquisition.

In Chapter 1, I present empirical research on proto-arithmetical abilities, that is, subitising and estimating, and emphasise the importance of distinguishing them from arithmetical abilities. I review the empirical literature on the cognitive basis of proto-arithmetical abilities, focusing mostly on the core cognitive theory of the object tracking system (OTS) and approximate numerosity system (ANS). Although the topic requires further research, I proceed with the view that the OTS and ANS are different cognitive systems and are responsible for the subitising and estimating abilities, respectively.

Numbers are an integral part of our daily lives, essential for making sound decisions. Surprisingly, numerical abilities, often termed number sense, begin developing early in life, shaping our foundational understanding of mathematics. This chapter explores the concept of number sense, demonstrating that even young children exhibit sensitivity to numerical magnitudes in everyday problem-solving scenarios. We delve into the transition from non-symbolic to symbolic numbers and its impact on brain development. As children acquire symbolic numerical skills, brain regions supporting number sense are influenced, and experiences refine these representations. We also explore individual differences in mathematical competence and their neural correlates. Furthermore, we discuss the implications of math interventions on brain development, emphasizing the importance of nurturing numeracy skills from an early age. This understanding has far-reaching implications for education policies, ensuring that every child has the opportunity to unlock their numerical wisdom. This chapter illuminates the journey from number sense to mathematical mastery in the developing mind.

How is number represented without language? Research on numerical cognition in humans and nonhuman primates suggests two possible shared systems of processing. The most widely demonstrated system comes from research with humans, nonhuman primates, and a wide variety of other species. This system, known as the approximate number system, is characterized by ratio-dependent representation of a wide range of numerical values. A smaller body of research points to a second system, the object file system, which represents small numerical values with precision. In addition to reviewing this literature, we will also attend to the role of training in numerical discrimination, the use of number when other continuous variables are present, and the arithmetic abilities of nonhuman primates.

Although fish represent approximately half of vertebrates, the quantitative abilities of fish have been investigated only recently. Two methodological approaches commonly used with mammals and birds have been used: the observation of spontaneous behaviour and training procedures. In the former, fish are observed in their preference for reaching the larger or smaller quantities of biologically relevant stimuli (in most cases, whether they join a larger shoal when placed in an unfamiliar environment). In the latter, fish are trained to select the larger or the smaller of two sets of abstract objects (e.g., two-dimensional figures that differ in numerosity). These studies showed that different fish species process numerical information in a similar way to that of mammals and birds. In this chapter, we review the relevant literature, giving particular regard to the strength and potential weaknesses of the two methodological approaches.