Emotions underlying early numerical cognition in humans Kishore Sivakumar 1, 2, Felix Schoeller 1,2, * 1 Centre de recherches interdisciplinaires, université Paris Descartes, 75005 Paris, France 2 U1001, Institut national de la santé et de la recherche médicale, 75014 Paris, France * Correspondence: [email protected] Research Highlights: Recent research on early numerical cognition is reviewed. Natural curiosity and the emotion of surprise play a key role in the study on early numerical knowledge Some recommendations to facilitate numerical cognition are devised. Abstract. This article investigates emotions related to early numerical cognition. We review the available data on the numbers sense in human children and show that natural curiosity is constantly used to study foundational mathematics in infancy. We conclude this article by outlining recommendations for improving the number sense at an early age and underline future directions of study. Keywords: mathematical problem solving, emotions, knowledge, foundational mathematics, surprise, curiosity. Introduction Foundational mathematics has received considerable attention in the science of how mathematical practices emerge in human children. However, little is known about the basic emotions underlying this process. Foundational mathematics is the basic knowledge required for any mathematical practice (Geary, 1994). Acquired at the youngest age in human children, this knowledge ranges from early numerical competencies to basic arithmetic skills (Starkey, 1992). At a general level, foundational mathematics is concerned with quantities, symbols and calculations. Quantities are understood through the number sense (Dehaene, 2011), symbols through symbolic knowledge (Geary & vanMarle, 2018), and calculations through elementary arithmetical skills (Spelke, 2007). At each level of understanding, emotions guide and regulate learning by coordinating various brain functions such as perception, attention, and cognition. These mathematical emotions can be observed and studied through behavioral, physiological and metacognitive studies. In this article, we review the experimental data concerning how such emotional dynamics influence mathematical processes through basic behavioral response such as surprise and curiosity. We conclude by proposing applications to facilitate basic mathematical learning and outline new directions of research. At a general level, learning emotions can be ordered into two categories depending on their valence –i.e., positive or negative. Positive emotions facilitate mathematical problem solving and are related to the feeling of confidence and to the exploratory drive. As we detail in the first section, both confidence and curiosity are prerequisite for mathematical cognition. It is generally recognized that the child must be interested in the problem and that he should be positive about his own ability to solve the problem (Geary, 1994). In other words, a child cannot be tested on his ability to solve any given mathematical problem, which fails to understand. To a certain extent, this explains why negative emotions such as anxiety, fear, and frustration have received considerable attention from scientists in the context of mathematical problem solving (Dowker et al., 2016). Negative emotions have been recognized as major obstacles to mathematical learning (Maloney, et al., 2011) but depending on the situation, self- confidence can also be an obstacle to mathematical accuracy (Eirickson & Heit, 2015). Through specific observational and experimental methods, a large body of data has been accumulated about the essential features of numerical emotions through age and experience. These data concern subjects ranging all the way from early infants understanding quantities (Spelke, 2007) to the level of expert mathematicians (Zeki, 2014). Indeed, a large portion of the data concerns students learning basic arithmetic skills. To a certain extent, the evolution of numerical cognition along phylogeny and its genetic underpinnings have been studied as well (Hauser, 2000; Tosto et al., 2014). One of the major challenges concerning this large body of data is the gap between first-person subjective data and third-person objective data. Given the importance of metacognition for problem solving in humans, it is necessary to overcome this difficulty in order to design optimal mathematical experiences to facilitate mathematical understanding. Here we propose that the study of human curiosity may help bridge this gap between science and disciplines concerned with the evolution of the number sense. We conclude this article with simple propositions to improve mathematical skills in children. 1. Foundations of the cognitive system From the point of view of developmental psychology, humans are genetically equipped with a small number of systems of core knowledge serving as a foundation to learn new skills, including numerical abilities (Leslie & Keeble, 1987; Spelke, 1990; Spelke, 2007). Four fundamental core knowledge systems are thought to govern human cognition at birth and regulate the mental representation of objects, actions, numbers and space (Spelke, 2004). The existence of a fifth core knowledge system for representing social interactions has been proposed in (Spelke, 2007). Each system functions according to a set of principles, which differentiates objects and support inferences about their behavior. Numerical cognition arises out of at least three of these core knowledge systems present at birth in the human child: object representation, number representation, and geometry representation. As we detail throughout this article, signature limits can be identified for each system across the species, throughout ages and across cultures (Everett, 2005). How does numerical cognition arises from the building blocks of cognition? One of the crucial core knowledge systems for the emergence of numerical cognition is the system of object representation. This system acts on the spatiotemporal principles of cohesion, continuity and contact (i.e., objects do not interact at distance) (Aguiar & Baillargeon, 1999). If an object exists at a time and exists later at a different time, then it must have existed during the interval – i.e., an object cannot temporarily disappear over time. The ability to perceive solid objects through space and time aids human infants to identify boundaries, shapes and motion of object and some of these abilities are observed in visually inexperienced human infants as well as newly hatched chicks (Valenza et al, 2006; Regolin & Vallortigara, 1995, Lea, Slater and Ryan 1995). Even infants with significant visual experience do not have systems for reasoning about inanimate non-object entities such as foods and liquid (objects that are not solid). Adults also fail to track objects that do not obey spatio- temporal constraints. These constraints tend to be universal in humans as suggested by studies in Amazonian tribes (Everett, 2005). This system for tracking and perceiving the behavior of individual objects is a prerequisite for numerical cognition. Even though they are subject to multiple debates, it is generally admitted that the core system of numerical cognition is regulated by three fundamental principles (Spelke, 2007). First, number representations are imprecise, and their imprecision increases linearly with the cardinal value (Izard 2006). Second, number representations are abstract: they apply to diverse entities experienced through multiple sensory modalities (Spelke & Kinzler, 2007). Finally, numerical representations can be compared and combined through operations such as addition and subtraction. These three principles are coherent with the data available concerning the number sense and early numerical abilities in humans. For example, the imprecision of number representation mirrors what is known empirically as the distance effect (Dehaene 1990). As we detail subsequently, the abstractness of number representation is coherent with the available data about their neurophysiology (Bremmer et al. 2001). Numerical abilities seem to be present in humans across cultures and regardless of formal education and training (Spelke & Kinzler, 2007). The numerical system interacts closely with the core knowledge system for representing geometry. This core system is thought to deal with knowledge about distance, angle and the sense of extended surfaces in the surroundings. Surprisingly, this system does represent non- geometric properties as well such as odor, color and texture and it fails in conditions where moveable objects are involved. Non-human and human infants orient themselves in accord with the layout of the geometry (Spelke, 2007). Human adults seem to take more advantage of landmarks, but they too rely primarily on surface geometric characteristics. Evidence from studies done on Amazonian tribes suggests that sensitivity to the geometry of the surfaces is universal (Everett, 2005). It is in this context that the number sense and numerical abilities develop in human infants. 2. Experimental studies of early numerical abilities Before solving any algebraic problem (Taylor, 1918) or geometrical problem (Metzler, 1912), students must first be able to understand numbers (Gelman & Galistel, 1978). This is referred to in the literature as the number sense or early numeracy (e.g., Jordan, 2007; Wagner & Davis, 2010). Early numerical cognition is the building block for foundational mathematics. In humans, the number sense is fundamentally related