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Home , Numerical cognition

Emotions underlying early numerical in

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 sense in children and show that natural curiosity is constantly used to study foundational in infancy. We conclude this article by outlining recommendations for improving the sense at an early age and underline future directions of study.

Keywords: mathematical , emotions, knowledge, foundational mathematics, surprise, curiosity. Introduction

Foundational mathematics has received considerable 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 by coordinating various brain functions such as , 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 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 , 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 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 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 (Bremmer et al. 2001).

Numerical abilities seem to be present in humans across cultures and regardless of formal 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 (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 to understanding and meaning making (Wynn, 1990). However, the evolution of numerical abilities has been observed across the phylogeny (Hauser, 2000). In humans, selective neurons have been observed to respond solely to certain numerosities (Dehaene and Neider, 2009) and this has led some authors have proposed various possible evolutionary functions for the number sense and the selection pressures that have led to its psychological importance in humans. For example, Boyer (2001) considers foundational mathematics (counting and adding) as adaptive traits from which higher-level mathematics are derived as byproducts of rudimentary quantitative skills. Other authors have proposed to view quantitative abilities as a communication tool (Dehaene, 2011).

The ability to discriminate between quantities is generally considered to be inborn as a hardwired genetically endowed ability (Tosto et al., 2018). One of the most fundamental questions concerning the number sense is to determine whether it can also be acquired, learned or improved. Humans display evidence of preverbal number knowledge at a very early age (Wynn, 1992) and other animal species as well (Hauser, 2000). It is generally admitted that humans have a fundamental system for understanding and mentally representing numbers. This internal system for representing quantities is sometimes referred to as the object file system (Feigenson & Carey, 2003). The object file system is a cognitive system dealing with precise representation of a small number of individual objects. Research suggests that infants use a mental model allowing them to keep track of individual arrays of objects in their immediate surroundings (Pylyshyn, 2001). This visual index, or pointer, is believed to allow short-term operations on the object. The main function of the system is to allow the tracking of multiple objects in space and time. Here, abstraction and quantitative abilities are thought to serve the perceptual process.

Recent experiments indicate a fundamental limit on the object file system. The fundamental bound is often referred to as the system’s signature (Spelke, 2007; Feigenson et al., 2004; Feigenson & Carey, 2003; Feigenson et al., 2002). Infants, as young as fourteen-month-old, have been observed to be able to distinguish the exact numerosity of arrays containing up to three objects (Feigenson & Carey, 2003). At six months old, this ability for exact calculation does not seem to extend beyond the number four, where the child starts making errors in counting (Dehaene 1997). The emotion of surprise is a crucial indicator in the study of the threshold on the object file system. For example, in a generic experiment, to test the bound on numerical cognition, human infants are presented with a box where they should search and retrieve objects, after being specified the amount of objects expected (Starkey, 1980). The search time is measured and when they fail to find the correct number of objects, it is found to increase. For example, when two balls were hidden and after the retrieval of one object, the infants searched for the second ball. Similarly, when three balls were hidden after retrieving two objects, infants searched for the third ball. However, experimental data suggest that they fail to recognize the numerosity four (Starkey, 1980). After being presented with a box containing four balls, once these balls were hidden and after retrieving two balls, infants stopped searching. Interestingly, a similar search time is observed when only two balls are present in the box. This suggests a fundamental bound on the object file system.

Some experiments involve children younger than a day. Newborn infants ranging from 21 to 144 hours old tested for their ability to discriminate between differing numerosities were found to possess the ability to discriminate quantities (Antell & Keating, 1983). Researchers use the length of fixation points to determine such ability and so in these experiments human surprise and expectations play a crucial role in identifying the presence of numerical skills. Among others, these experiments demonstrate that infants can discriminate between numerosities 2 and 3 but not 4 versus 5 or 6 (Antell & Keating, 1983).

Another set of experiments on fundamental numerical bounds consists in exposing children to images presenting different quantities of similar objects. In this context, human infants were found able to discriminate between numerosities two and three, just a few days after birth (Antell & Keating, 1983). Six to eight months old infants were shown slides of two different objects repeatedly. Immediately, the slides were changed to contain three objects. Infants looked longer at these new images indicating their renewed interest in the new image (Starkey, 1980). In these experiments, infants were able to distinguish from 2 and 3 objects but not for larger sets involving 4 and 6 objects (Antell & Keating, 1983).

Such numerical abilities seem to operate across sensory modalities. This knowledge derives from data where six and eight months old infants are presented with images of n objects along with corresponding amount of sequences of sounds. It was found that infants spent more time looking at the images that match the appropriate number of sounds perceived. Conversely, infants showed no interest in the pictures when it did not match with the sounds (Starkey et al.,1990). In terms of neurophysiology, a great body of work seems to indicate that the (IPS) is closely associated with abstract, amodal representation of numbers (for review see Dehaene and Neider, 2009). The prefrontal and posterior parietal cortices are ideal brain structures for an abstract encoding of quantity. They receive highly processed multimodal input (Duhamel et al. 1998, Lewis & Van Essen 2000, Bremmer et al. 2001) and code for number that can apply to all sensory modalities. This research has also identified specific neurons that respond to numerosity (Dehaene & Neider 2009). These numerosity selective neurons were tuned to a certain number of items through visual number display. During the experiment, numerosity selective neurons show maximum activity when presented with the preferred number and decreased activity when presented with a numerosity distant from the preferred number.

The evolution of the number sense has also been studied across human phylogeny. As we have seen, human infants were found to discriminate quantity 2 from quantity 3 (Dehaene, 1997). Occasionally, they can distinguish 3 from 4, but not 4 from 5, or 4 from 6. At this level of counting, differences have been observed among primates (Rumbaugh et al., 1987; Hauser, 2000). The abilities of adult chimpanzees for example seem superior to those of human infants as they seem to be able to perceive up to the numerosity 7 (Rumbaugh et al., 1987). When presented with trays containing various pieces of chocolate, chimpanzees are not only able to discriminate between these quantities but also to compare them. This suggests an innate ability for computation (Rumbaugh et al., 1987). Neurons responding to three sounds or three light flashes have been observed across species (Dehaene & Changeux, 1993). According to Hauser, rhesus monkeys can be taught to understand ordinal relations from 1 to 9, but only under very intensive training (Hauser, 2000).

3. The importance of surprise in the study of numerical cognition

In the context of the experiments detailed in the previous section, the emotion of surprise is constantly used as a window into numerical abilities. In fact, virtually all data acquired about numerical cognition in human children is deduced from experiments concerned with the avoidance of surprising states. The emotion of surprise has received considerable attention in the context of active inference where surprise is described in informational terms (Schwartenbeck et al., 2013). Surprise is a higher bound on natural curiosity that results from prior expectations about the state of the world (Schoeller, 2016). The facial muscle associated to this emotion makes it an ideal candidate to learn about early human competency and human core knowledge (Spelke, 2007). Due to constraint, researchers studying early numerical skills often rely on the babies’ interest in novelty and surprising states (Antell & Keating, 1983). This makes natural curiosity and human surprise core dimensions in the experimental study of early cognitive and numerical abilities.

One of the most basic models of the mind-brain is provided by the theory of active inference (Friston et al., 2006). In the framework of active inference, agents do not try to maximize reward but rather minimize surprise (about future states of the world). Active inference describes the human brain as an organ of prediction constantly minimizing uncertainty and prediction error. In the experiments described in the previous section, the longer the gaze the more surprised the child is expected to be. In this context, the brain-mind is constantly avoiding violation of expectation (see also Perlovsky, 2016). The basic emotion at stake is surprise, which measures the level of satisfaction of natural curiosity. Behavioral surprise thus reveals the subject’s prior expectations and the children’s core knowledge and number abilities. The experimental study of numerical abilities in young human children is fundamentally related to natural curiosity and the emotion of surprise.

4. Experimental study of surprise related to numerical abilities

Human elementary computing abilities have been tested in a wealth of experimental studies by examining natural curiosity and interest. These generally involve hiding object behind a curtain, and measuring behavioral data such as sucking rate. For example, in one experimental procedure, a toy is hidden behind a screen, when another identical toy is added next to it. When the screen is removed, exposing the number of toys hidden, human children as young as 4.5 months were found to look longer at the impossible event “1+1=1” than at the logical event “1+1=2”. These data is used to infer that these children are able to understand that “1+1=2” (Wynn, 1992). Infants stare longer at unexpected events, which hint towards their ability to compute and expect for possible events (Wynn, 1992). In these data, surprise plays an essential role in our ability to infer numerical cognition and understanding.

Another type of procedure consists in measuring the sucking rate of children to infer their interest in the task. Four days old human infants were tested for their ability to discriminate between two and three syllable words. Due to the age constraint, the investigators choose to use the sucking rhythm of the babies to measure their interest in the task. Babies suck on a nipple connected to a pressure transducer and a computer. Whenever the child hears a word with two syllables, it begins to suck vigorously. After few trials the child’s sucking rhythm goes back to a normal rate. When a three syllables word is introduced, the sucking rate increases.

It has been suggested that the mechanism allowing humans to measure does not have equally spaced progression between numbers (Dehaene, 2000). For instance, the distance between 1 and 2 is not same as that from 8 to 9. The mental ruler tends to compress larger numbers into smaller spaces. Mental representations of quantities in humans appear to depict a logarithmic order of numbers, which leads to increased response time and errors while distinguishing between closer numbers when presented in larger quantities (Dehaene 1997). In a number discrimination task when presented with Arabic numerals between 31 and 99 as being smaller or larger than 65, subjects showed a steady rise in response time as the target number got closer to 65, supporting the theory of compressed representations of quantities (Dehaene et al. 1990). This distance effect predicts that the perceived difference between one million and one million and one is less rapidly detected than the difference between one and two. Generally, the distance effect states that the processing rate of numbers is a function of numerical distance and the greater the distance, the shorter the response. In other words, very similar quantities are difficult to discriminate. Studies on the distance effect seem to indicate that quantities are largely processed unconsciously and that distance plays a major role in this processing. Humans can perceive distant numbers, faster and more clearly. They are able to discriminate numbers easily and with fewer errors when distanced consequently. This is important when attempting to design tools to facilitate mathematical understanding in young children. Perhaps the distance effect is due to neurophysiological properties of the neural network of quantity estimation that makes it hard for the system to perform operations of this kind. This might be tide to an evolutionary pressure of the environment in which human cognition developed over the largest part of our history, the fast estimation of large quantities in a natural environment may be more important than the costly estimation of small quantities (Kahneman, 2011).

Another line of data relates to core knowledge about physical regularities (Spelke, 2007) and highlights the importance of movement in numerical inferences. Infants’ numerical expectations are based on object trajectory, not on object identity (Spelke, 2007). This is coherent with the view of natural curiosity as adaptation to change (Schoeller, 2017). In fact, it is known that absence of change leads to absence of cognition (Yarbus, 1967). Indeed, this is also coherent with the framework of active inference as physical change implies causes to be inferred (Friston, 2006). Spelke has found numerical inferences in babies to be determined by spatiotemporal trajectory of objects. If a single object cannot cause the motion without violating the laws of physics, babies infer that multiple objects must be present. Even if that implies that the object is constantly changing in shape, size, and color. This seems to suggest that the ability to estimate quantity develops prior to that of recognizing objects. Humans can distinguish amounts before identifying the things counted.

5. Conclusion

It has been recognized that the study of emotions underlying human numerical cognition requires much more attention from scientists (D'Mello & Graesser, 2012). Here, we have shown that the emotion of surprise plays an important role in the experimental study of early numerical cognition as a way to read data about the internal state of the child. This yields important consequences in regards to both the study of numerical cognition but also human curiosity and exploration at large (Berlyne, 1960; Schoeller et al., 2018). Based on the data reviewed, we conclude this short article by proposing ways in which early numerical cognition can be impaired or facilitated as well as proposing future directions for the study of the emotions underlying early numerical cognition.

In principles, it should be possible to make use of the distance effect to design tools to facilitate mathematical learning. The distance effect states that a large contrast between numbers accelerates understanding of the relation. When attempting to explain or teach foundational mathematics, one should therefore start with large disparities between numbers, rather than small differences that are more difficult to assess. This theoretical proposal based on the logarithmic structure of numerical cognition requires further experimental study but could provide an interesting application for the distance effect in facilitating early numerical understanding.

Given the multimodality of early numerical abilities and the fact that numerical cognition operates over various sensory domains, it seems that exposing children to multiple sources of sensory information (visual indeed but also tactile and acoustic) combined with various motor actions should facilitate the acquisition of numerical skills (see e.g., Novack et al., 2014). Multiple modalities activating the higher-level amodal neural network of quantity estimation could increase the rate of learning in human children. Specific visual techniques related to the structure of perception could also be developed to facilitate understanding of quantitative information (for many example of such techniques see Tufte, 1986). These visual techniques can increase the rate of information processing by being already consonant with the hierarchical structure of (Palmer, 1977). Indeed, such possible techniques must also rely on the numerical system’s signature (the bound of the object file system) and match physical expectations and numerical inference generally (e.g., in a train station, the third platform is expected to be found next to second one and before the fourth platform). An even more basic numerical inference concerns the trajectory of objects, where numerical expectations are based on object trajectory, rather than object identity (Spelke, 2007). Indeed, such techniques should conform to the principles of cohesion, continuity and contact detailed in the first section. In general, the problem or cases presented to the child must match her prior expectation about the world in order to maintain an optimal amount of surprise to maintain interest.

Given the relation between early numerical skills and exploratory behavior (Menon, 2017), it should be possible to apply knowledge about human curiosity to improve numerical skills in early infancy. Since the numerical system seems to be fundamentally related to natural curiosity, teachers should strive to avoid boring states (i.e., high redundancy), while limiting surprise (i.e., high amount of change). Studies should examine in greater detail the optimal amount of change that the system finds intrinsically rewarding (Oudeyer, in press; Perlovsky, 2014). In this regard, studies in adult humans and expert mathematicians (Schoeller, 2015; Zeki, 2014) can yield promising insights concerning mathematical abilities at a higher level, the biology of mathematical problem solving (Hauser, 2000) and perhaps the computational principles behind mathematical learning and discovery (Katz et al., 2016).

In terms of psychophysiology, natural curiosity can be studied at the behavioral level and used as an indicator for human activity of meaning making (Schoeller & Perlovsky, 2016). We can therefore learn about human numerical cognition by studying natural curiosity and its expression through eye gazing (Mirza et al., 2016), facial muscles (Reeve, 1993), body gestures (Wakefield et al., 2018), and specific physiological correlates (Jepma et al., 2012). In this regard, a very promising line of study concerns the relation between gestures and abstraction where the data seem to suggest that gestures facilitate generalization in human children (Wakefield et al., 2018; Golden-Meallow, 2014; Novack et al., 2014).

Finally, a core problem to be studied is indeed the role of social emotions in mathematical learning. Although mathematics is a solitary activity, it has been shown that the role of the teacher and family is crucial in learning mathematics (Dowker et al., 2012). This is still to be studied in early numerical cognition. Early numerical competencies are built in at a very young age. In terms of the social implications of these developmental skills, research shows that early numerical competencies are the most reliable indicators of later mathematical performance in students (Krajewski & Schneider, 2009). Evidences from various longitudinal studies from Europe suggest that the basic counting skills in preschools’ age as a reliable predictor of mathematical achievement in subsequent grades (Aunola, et al., 2004, Passolunghi, et al., 2007). Conversely, poor numerical competencies are a good predictor for anxiety and issues in the mathematical classroom. Recent studies seem to show that considerable progress can be made in teaching mathematics to young children using scientific knowledge of how numerical cognition develops (Griffin & Case, 1997). These studies yield promising results (Dilon et al, 2017; Banerji et al., 2017) and highlight the importance of improving our knowledge of mathematical emotions.

6. Conflict of interest

The authors do not have any conflict of interest. 7. References

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