The development of numerical cognition in children and artificial systems: a review of the current knowledge and proposals for multi-disciplinary research
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paper_title: The Connection Between Spatial and Mathematical Ability Across Development

paper_content:

In this article, we review approaches to modeling a connection between spatial and mathematical thinking across development. We critically evaluate the strengths and weaknesses of factor analyses, meta-analyses, and experimental literatures. We examine those studies that set out to describe the nature and number of spatial and mathematical skills and specific connections between these abilities, especially those that included children as participants. We also find evidence of strong spatial-mathematical connections and transfer from spatial interventions to mathematical understanding. Finally, we map out the kinds of studies that could enhance our understanding of the mechanisms by which spatial and mathematical processing are connected and the principles by which mathematical outcomes could be enhanced through spatial training in educational settings.
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paper_title: Embodiment as a unifying perspective for psychology

paper_content:

Adaptive action is the function of cognition. It is constrained by the properties of evolved brains and bodies. An embodied perspective on social psychology examines how biological constrains give expression to human function in socially situated contexts. Key contributions in social psychology have highlighted the interface between the body and cognition, but theoretical development in social psychology and embodiment research remain largely disconnected. The current special issue reflects on recent developments in embodiment research. Commentaries from complementary perspectives connect them to social psychological theorizing. The contributions focus on the situatedness of social cognition in concrete interactions, and the implementation of cognitive processes in modal instead of amodal representations. The proposed perspectives are highly compatible, suggesting that embodiment can serve as a unifying perspective for psychology. Copyright © 2009 John Wiley & Sons, Ltd.
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paper_title: Finger Counting and Numerical Cognition

paper_content:

Following a recent surge of interest in finger-based number knowledge, we invited empirical and conceptual contributions to assess the feasibility of a Research Topic on this issue. We received a considerable number of submissions, many of which were further improved by constructive and interactive peer-review and ultimately appeared as part of the Research Topic “Handy numbers: Finger counting and numerical cognition.” We wish to thank all authors and reviewers, as well as the publisher’s support team around Meghan Hodge, for their excellent work. ::: ::: This enthusiastic response from the research community confirmed our expectation that the time is ripe to consider the domain of number knowledge from the theoretical perspective of embodied cognition. This domain is particularly challenging for an embodied perspective on human cognition because mental arithmetic was thought to consist of abstract and amodal symbol manipulation. By disregarding the acquisition, implementation, or retrieval context of such knowledge, numerical cognition provided an ideal example of abstract information processing (e.g., Groen and Parkman, 1972). Yet in recent years a flurry of reports documented just such sensory and motor contributions to numerical cognition, and the contributions gathered for the present research topic on “handy numbers” provide an up-to-date survey of this development. ::: ::: The published contributions make clear that there is no agreement about the relevance of finger counting for numerical cognition. For example, finger associations might not be a necessary component of number knowledge acquisition (Crollen et al., 2011), they might merely reflect immature retrieval strategies (Kaufmann et al., 2011), and some aspects of finger-based number representation might actually hinder the initial learning process (Beller and Bender, 2011). Nevertheless, all empirical contributions to this research topic support a role of fingers in numerical cognition: Spatial–numerical associations, previously attributed to reading habits, may at least partly have their origin in finger counting routines (Fischer and Brugger, 2011; Riello and Rusconi, 2011). They are prevalent in finger counting systems of many cultures (Previtali et al., 2011; Domahs et al., 2012), affect a wide range of behaviors (Fischer and Brugger, 2011), depend on hand orientation (Previtali et al., 2011), and possibly on finger gnosia (Costa et al., 2011; Reeve and Humberstone, 2011). Furthermore, finger usage and finger-based number representations may vary considerably according to cultural influences (Bender and Beller, 2011; Domahs et al., 2012). Mental addition is selectively impaired by passive hand movements (Imbo et al., 2011) and shows sub-base five effects that can be attributed to hand-based representations (Klein et al., 2011). Addition also activates finger-related cortical structures (Krinzinger et al., 2011). ::: ::: Findings such as these highlight the special status of finger representations in numerical cognition (Di Luca and Pesenti, 2011) and require a conceptual rethinking. This can begin by aligning educational and neuroscientific perspectives (Moeller et al., 2011) or by contextualizing them within the embodied cognition framework (Fischer and Brugger, 2011).
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paper_title: The neuronal code for number

paper_content:

Primates have a functional network in frontal and parietal cortices that allows them to quantify the number of elements in a stimulus; that is, its numerosity or cardinality. In this Review, Andreas Nieder examines how neurons in this network process cardinal numbers.
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paper_title: How the Body Shapes the Way We Think: A New View of Intelligence

paper_content:

How could the body influence our thinking when it seems obvious that the brain controls the body? In How the Body Shapes the Way We Think, Rolf Pfeifer and Josh Bongard demonstrate that thought is not independent of the body but is tightly constrained, and at the same time enabled, by it. They argue that the kinds of thoughts we are capable of have their foundation in our embodiment -- in our morphology and the material properties of our bodies. This crucial notion of embodiment underlies fundamental changes in the field of artificial intelligence over the past two decades, and Pfeifer and Bongard use the basic methodology of artificial intelligence -- "understanding by building" -- to describe their insights. If we understand how to design and build intelligent systems, they reason, we will better understand intelligence in general. In accessible, nontechnical language, and using many examples, they introduce the basic concepts by building on recent developments in robotics, biology, neuroscience, and psychology to outline a possible theory of intelligence. They illustrate applications of such a theory in ubiquitous computing, business and management, and the psychology of human memory. Embodied intelligence, as described by Pfeifer and Bongard, has important implications for our understanding of both natural and artificial intelligence.
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paper_title: Spatial Ability for STEM Domains : Aligning Over 50 Years of Cumulative Psychological Knowledge Solidifies Its Importance

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The importance of spatial ability in educational pursuits and the world of work was examined, with particular attention devoted to STEM (science, technology, engineering, and mathematics) domains. Participants were drawn from a stratified random sample of U.S. high schools (Grades 9–12, N 400,000) and were tracked for 11 years; their longitudinal findings were aligned with pre-1957 findings and with contemporary data from the Graduate Record Examination and the Study of Mathematically Precocious Youth. For decades, spatial ability assessed during adolescence has surfaced as a salient psychological attribute among those adolescents who subsequently go on to achieve advanced educational credentials and occupations in STEM. Results solidify the generalization that spatial ability plays a critical role in developing expertise in STEM and suggest, among other things, that including spatial ability in modern talent searches would identify many adolescents with potential for STEM who are currently being missed.
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paper_title: Applying embodied cognition : from useful interventions and their theoretical underpinnings to practical applications

paper_content:

Embodied trainings allowing children to move their whole body in space have recently been shown to foster the acquisition of basic numerical competencies (e.g. magnitude understanding, addition performance). Following a brief summary of recent embodied training studies, we integrate the different results into a unified model framework to elucidate the working mechanisms of embodied trainings: Mapping processes, interaction between different regions of personal space, and the integration of different spatial frames of reference are addressed as potential factors underlying the effectiveness of embodied numerical trainings. In the concluding section, we elaborate on the practical applications of embodied numerical trainings in educational setting. We discuss under which circumstances embodied trainings work best, that is, for which age group and/or which numerical content embodied trainings should be most beneficial and which aspects need to be considered when aiming at applying embodied numerical trainings in formal educational settings like kindergartens or schools.
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paper_title: Six views of embodied cognition

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The emerging viewpoint of embodied cognition holds that cognitive processes are deeply rooted in the body’s interactions with the world. This position actually houses a number of distinct claims, some of which are more controversial than others. This paper distinguishes and evaluates the following six claims: (1) cognition is situated; (2) cognition is time-pressured; (3) we off-load cognitive work onto the environment; (4) the environment is part of the cognitive system; (5) cognition is for action; (6) offline cognition is body based. Of these, the first three and the fifth appear to be at least partially true, and their usefulness is best evaluated in terms of the range of their applicability. The fourth claim, I argue, is deeply problematic. The sixth claim has received the least attention in the literature on embodied cognition, but it may in fact be the best documented and most powerful of the six claims.
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paper_title: Cognition in Practice: Mind, Mathematics and Culture in Everyday Life

paper_content:

List of figures List of tables Preface 1. Introduction: psychology and anthropology I Part I. Theory in Practice: 2. Missionaries and cannibals (indoors) 3. Life after school 4. Psychology and anthropology II Part II. Practice in Theory: 5. Inside the supermarket (outdoors) and from the veranda 6. Out of trees of knowledge into fields for activity 7. Through the supermarket 8. Outdoors: a social anthropology of cognition in practice Notes References.
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paper_title: Core numerical skills for learning mathematics in children aged five to eight years – a working model for educators

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The aim of this study was to model the most crucial numerical factors to the development of mathematical skills among children aged five to eight years (i.e. kindergarten, preschool, first and second graders). We categorised numerical skills into four main groups based on the results of longitudinal studies. A series of analyses of test batteries designed to measure the development of mathematical skills in children yielded results in support of this construct. Based on our findings we propose a working model for teachers of core numerical skills that focuses on four major factors: (1) symbolic and non-symbolic number sense; (2) understanding mathematical relations (early mathematical-logical principles, arithmetic principles, mathematical operational symbols, place-value and base-ten system); (3) counting skills (knowledge of number-symbols, number word-sequence, enumeration with concrete objects); and (4) basic skills in arithmetic (arithmetic combinations, addition and subtraction skills with number sy...
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paper_title: You can count on your fingers: The role of fingers in early mathematical development

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Even though mathematics is considered one of the most abstract domains of human cognition, recent work on embodiment of mathematics has shown that we make sense of mathematical concepts by using insights and skills acquired through bodily activity. Fingers play a significant role in many of these bodily interactions. Finger-based interactions provide the preliminary access to foundational mathematical constructs, such as one-to-one correspondence and whole-part relations in early development. In addition, children across cultures use their fingers to count and do simple arithmetic. There is also some evidence for an association between children’s ability to individuate fingers (finger gnosis) and mathematics ability. Paralleling these behavioral findings, there is accumulating evidence for overlapping neural correlates and functional associations between fingers and number processing. In this paper, we synthesize mathematics education and neurocognitive research on the relevance of fingers for early mathematics development. We delve into issues such as how the early multimodal (tactile, motor, visuospatial) experiences with fingers might be the gateway for later numerical skills, how finger gnosis, finger counting habits, and numerical abilities are associated at the behavioral and neural levels, and implications for mathematics education. We argue that, taken together, the two bodies of research can better inform how different finger skills support the development of numerical competencies, and we provide a road map for future interdisciplinary research that can yield to development of diagnostic tools and interventions for preschool and primary grade classrooms.
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paper_title: The role of gesture in communication and thinking

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Abstract People move their hands as they talk – they gesture. Gesturing is a robust phenomenon, found across cultures, ages, and tasks. Gesture is even found in individuals blind from birth. But what purpose, if any, does gesture serve? In this review, I begin by examining gesture when it stands on its own, substituting for speech and clearly serving a communicative function. When called upon to carry the full burden of communication, gesture assumes a language-like form, with structure at word and sentence levels. However, when produced along with speech, gesture assumes a different form – it becomes imagistic and analog. Despite its form, the gesture that accompanies speech also communicates. Trained coders can glean substantive information from gesture – information that is not always identical to that gleaned from speech. Gesture can thus serve as a research tool, shedding light on speakers' unspoken thoughts. The controversial question is whether gesture conveys information to listeners not trained to read them. Do spontaneous gestures communicate to ordinary listeners? Or might they be produced only for speakers themselves? I suggest these are not mutually exclusive functions – gesture serves as both a tool for communication for listeners, and a tool for thinking for speakers.
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paper_title: Embodied Mental Imagery in Cognitive Robots

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This chapter is focused on discussing the concept of mental imagery as a fundamental cognitive capability to enhance the performance of cognitive robots . Indeed, the emphasis will be on the embodied imagery mechanisms applied to build artificial cognitive models of motor imagery and mental simulation to control complex behaviors of humanoid platforms, which represent the artificial body.
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paper_title: Developmental robotics: a survey

paper_content:

Developmental robotics is an emerging field located at the intersection of robotics, cognitive science and developmental sciences. This paper elucidates the main reasons and key motivations behind the convergence of fields with seemingly disparate interests, and shows why developmental robotics might prove to be beneficial for all fields involved. The methodology advocated is synthetic and two-pronged: on the one hand, it employs robots to instantiate models originating from developmental sciences; on the other hand, it aims to develop better robotic systems by exploiting insights gained from studies on ontogenetic development. This paper gives a survey of the relevant research issues and points to some future research directions.
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paper_title: Autonomous learning in humanoid robotics through mental imagery.

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In this paper we focus on modeling autonomous learning to improve performance of a humanoid robot through a modular artificial neural networks architecture. A model of a neural controller is presented, which allows a humanoid robot iCub to autonomously improve its sensorimotor skills. This is achieved by endowing the neural controller with a secondary neural system that, by exploiting the sensorimotor skills already acquired by the robot, is able to generate additional imaginary examples that can be used by the controller itself to improve the performance through a simulated mental training. Results and analysis presented in the paper provide evidence of the viability of the approach proposed and help to clarify the rational behind the chosen model and its implementation.
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paper_title: The iCub humanoid robot : An open-systems platform for research in cognitive development

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We describe a humanoid robot platform - the iCub - which was designed to support collaborative research in cognitive development through autonomous exploration and social interaction. The motivation for this effort is the conviction that significantly greater impact can be leveraged by adopting an open systems policy for software and hardware development. This creates the need for a robust humanoid robot that offers rich perceptuo-motor capabilities with many degrees of freedom, a cognitive capacity for learning and development, a software architecture that encourages reuse & easy integration, and a support infrastructure that fosters collaboration and sharing of resources. The iCub satisfies all of these needs in the guise of an open-system platform which is freely available and which has attracted a growing community of users and developers. To date, twenty iCubs each comprising approximately 5000 mechanical and electrical parts have been delivered to several research labs in Europe and to one in the USA.
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paper_title: A Neural Network model for spatial mental imagery investigation: A study with the humanoid robot platform iCub

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Understanding the process behind the human ability of creating mental images of events and experiences is a still crucial issue for psychologists. Mental imagery may be considered a multimodal biological simulation that activates the same, or very similar, sensorial and motor modalities that are activated when we interact with the environment in real time. Neuro-psychological studies show that neural mechanisms underlying real-time visual perception and mental visualization are the same when a task is mentally recalled. Nevertheless, the neural mechanisms involved in the active elaboration of mental images might be different from those involved in passive elaborations. The enhancement of this active and creative imagery is the aim of most psychological and educational processes, although, more empirical effort is needed in order to understand the mechanisms and the role of active mental imagery in human cognition. In this work we present some results of on ongoing investigation about mental imagery using cognitive robotics. Here we focus on the capability to estimate, from proprioceptive and visual information, the position into a soccer field when the robot acquires the goal. Results of simulation with the iCub platform are given to show that the computational model is able to efficiently estimate the robot's position. The final objective of our work is to replicate with a cognitive robotics model the mental imagery when it is used during the training phase of athletes that are allowed to imaginary practice to score a goal.
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paper_title: Lateral specialization in unilateral spatial neglect: a cognitive robotics model

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AbstractIn this paper, we present the experimental results of an embodied cognitive robotic approach for modelling the human cognitive deficit known as unilateral spatial neglect (USN). To this end, we introduce an artificial neural network architecture designed and trained to control the spatial attentional focus of the iCub robotic platform. Like the human brain, the architecture is divided into two hemispheres and it incorporates bio-inspired plasticity mechanisms, which allow the development of the phenomenon of the specialization of the right hemisphere for spatial attention. In this study, we validate the model by replicating a previous experiment with human patients affected by the USN and numerical results show that the robot mimics the behaviours previously exhibited by humans. We also simulated recovery after the damage to compare the performance of each of the two hemispheres as additional validation of the model. Finally, we highlight some possible advantages of modelling cognitive dysfunctions of the human brain by means of robotic platforms, which can supplement traditional approaches for studying spatial impairments in humans.
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paper_title: The intelligent ASIMO: system overview and integration

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We present the system overview and integration of the ASIMO autonomous robot that can function successfully in indoor environments. The first model of ASIMO is already being leased to companies for receptionist work. In this paper, we describe the new capabilities that we have added to ASIMO. We explain the structure of the robot system for intelligence, integrated subsystems on its body, and their new functions. We describe the behavior-based planning architecture on ASIMO and its vision and auditory system. We describe its gesture recognition system, human interaction and task performance. We also discuss the external online database system that can be accessed using internet to retrieve desired information, the management system for receptionist work, and various function demonstrations.
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paper_title: Developmental Robotics: From Babies to Robots

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Developmental robotics is a collaborative and interdisciplinary approach to robotics that is directly inspired by the developmental principles and mechanisms observed in children's cognitive development. It builds on the idea that the robot, using a set of intrinsic developmental principles regulating the real-time interaction of its body, brain, and environment, can autonomously acquire an increasingly complex set of sensorimotor and mental capabilities. This volume, drawing on insights from psychology, computer science, linguistics, neuroscience, and robotics, offers the first comprehensive overview of a rapidly growing field. After providing some essential background information on robotics and developmental psychology, the book looks in detail at how developmental robotics models and experiments have attempted to realize a range of behavioral and cognitive capabilities. The examples in these chapters were chosen because of their direct correspondence with specific issues in child psychology research; each chapter begins with a concise and accessible overview of relevant empirical and theoretical findings in developmental psychology. The chapters cover intrinsic motivation and curiosity; motor development, examining both manipulation and locomotion; perceptual development, including face recognition and perception of space; social learning, emphasizing such phenomena as joint attention and cooperation; language, from phonetic babbling to syntactic processing; and abstract knowledge, including models of number learning and reasoning strategies. Boxed text offers technical and methodological details for both psychology and robotics experiments.
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paper_title: A Mass-Produced Sociable Humanoid Robot: Pepper: The First Machine of Its Kind

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As robotics technology evolves, we believe that personal social robots will be one of the next big expansions in the robotics sector. Based on the accelerated advances in this multidisciplinary domain and the growing number of use cases, we can posit that robots will play key roles in everyday life and will soon coexist with us, leading all people to a smarter, safer, healthier, and happier existence.
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paper_title: How Can the Human Mind Occur in the Physical Universe

paper_content:

1. Cognitive Architecture 2. The Modular Organization of the Mind 3. Human Associative Memory 4. The Adaptive Control of Thought 5. What Does It Take to Be Human? Lessons From High School Algebra 6. How Can the Human Mind Occur?
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paper_title: Cognitive developmental robotics as a new paradigm for the design of humanoid robots

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This paper proposes cognitive developmental robotics(CDR) as a new principle for the design of humanoid robots. This principle may provide ways of understanding human beings that go beyond the current level of explanation found in the natural and social sciences. Furthermore, a methodological emphasis on humanoid robots in the design of artificial creatures holds promise because they have many degrees of freedom and sense modalities and, thus, must face the challenges of scalability that are often side-stepped in simpler domains. We examine the potential of this new principle as well as issues that are likely to be important to CDR in the future. © 2001 Elsevier Science B.V. All rights reserved.
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paper_title: The iCub cognitive humanoid robot: an open-system research platform for enactive cognition

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This paper describes a multi-disciplinary initiative to promote collaborative research in enactive artificial cognitive systems by developing the iCub : a open-systems 53 degree-of-freedom cognitive humanoid robot. At 94 cm tall, the iCub is the same size as a three year-old child. It will be able to crawl on all fours and sit up, its hands will allow dexterous manipulation, and its head and eyes are fully articulated. It has visual, vestibular, auditory, and haptic sensory capabilities. As an open system, the design and documentation of all hardware and software is licensed under the Free Software Foundation GNU licences so that the system can be freely replicated and customized. We begin this paper by outlining the enactive approach to cognition, drawing out the implications for phylogenetic configuration, the necessity for ontogenetic development, and the importance of humanoid embodiment. This is followed by a short discussion of our motivation for adopting an open-systems approach. We proceed to describe the iCub's mechanical and electronic specifications, its software architecture, its cognitive architecture. We conclude by discussing the iCub phylogeny, i.e. the robot's intended innate abilities, and an scenario for ontogenesis based on human neonatal development.
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paper_title: Cognitive Developmental Robotics: A Survey

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Cognitive developmental robotics (CDR) aims to provide new understanding of how human's higher cognitive functions develop by means of a synthetic approach that developmentally constructs cognitive functions. The core idea of CDR is ldquophysical embodimentrdquo that enables information structuring through interactions with the environment, including other agents. The idea is shaped based on the hypothesized development model of human cognitive functions from body representation to social behavior. Along with the model, studies of CDR and related works are introduced, and discussion on the model and future issues are argued.
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paper_title: Robotic platform for teaching maths in junior high school

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This paper describes a structured methodology to improve the attention in math classes based on a NAO robot. Although there are several papers that use NAO as a social robot, this paper goes further integrating the robot inside the classes as well as an assistive tool to support the teachers. A multidisciplinary work was implemented with the junior high school in which the NAO robot was used. Two focus groups were established for the study. One of them use the robot during the classes and the other one was running the classes in conventional manner. A comparison about the students’ attention was conducted to validate the proposed methodology. The results show that the level of attention and motivation were improved.
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paper_title: A Comparison of Kindergarten Storytelling by Human and Humanoid Robot with Different Social Behavior

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In this paper, we present a study on the influence of different social behavior on preschool children's perception of stories narrated either by a humanoid robot or by a human teacher. Four conditions were considered: static human, static robot, expressive human and expressive robot. Two stories, with knowledge and emotional content, were narrated in two different encounters. After each story, children draw what they remember of the story. We examined drawings of 81 children to study whether the sociability of the teacher (robot or human) could influence elements and details recorded. Results suggest a positive effect of the expressive behavior in robot storytelling, whose efficacy is comparable to the human with the same behavior or better if the expressive robot is compared with a static inexpressive human.
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paper_title: Building successful long child-robot interactions in a learning context

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The CoWriter activity involves a child in a rich and complex interaction where he has to teach handwriting to a robot. The robot must convince the child it needs his help and it actually learns from his lessons. To keep the child engaged, the robot must learn at the right rate, not too fast otherwise the kid will have no opportunity for improving his skills and not too slow otherwise he may loose trust in his ability to improve the robot' skills. We tested this approach in real pedagogic/therapeutic contexts with children in difficulty over repeated long sessions (40-60 min). Through 3 different case studies, we explored and refined experimental designs and algorithms in order for the robot to adapt to the troubles of each child and to promote their motivation and self confidence. We report positive observations, suggesting commitment of children to help the robot, and their comprehension that they were good enough to be teachers, overcoming their initial low confidence with handwriting.
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paper_title: Children teach a care-receiving robot to promote their learning: field experiments in a classroom for vocabulary learning

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In contrast to conventional teaching agents (including robots) that were designed to play the role of human teachers or caregivers, we propose the opposite scenario in which robots receive instruction or care from children. We hypothesize that by using this care-receiving robot, we may construct a new educational framework whose goal is to promote children's spontaneous learning by teaching through their teaching the robot. In this paper, we describe the introduction of a care-receiving robot into a classroom at an English language school for Japanese children (3--6 years of age) and then conduct an experiment to evaluate if the care-receiving robot can promote their learning using English verbs. The results suggest that the idea of a care-receiving robot is feasible and that the robot can help children learn new English verbs efficiently. In addition, we report on investigations into several forms of teaching performed by children, which were revealed through observations of the children, parent interviews, and other useful knowledge. These can be used to improve the design of care-receiving robots for educational purposes.
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paper_title: A Review on the Use of Robots in Education and Young Children

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Introduction With the rapid development of technology in the 21st century, the use of multi-media tool in education has become increasingly popular. Notwithstanding their usual engineering applications, robots are being used more in schools. According to Beran et al. (2011), children are also playing more with technologically advanced devices during their playtime. Subsequently, studies were conducted to investigate robot use's influence on children's cognition, language, interaction, social and moral development (Wei et al., 2011; Kozima & Nakagawa, 2007; Shimada, Kanda & Koizumi, 2012; Kahn et al., 2012). Recent studies (Wei, Hung, Lee & Chen, 2011; Highfield, 2010; Chen, Quadir & Teng, 2011) reported that robot use encourages interactive learning, making children more engaged in their learning activities. This increase research on robot application to education needs systematic look at the direction taken this past decade in order to elucidate a roadmap for future studies. Recent reviews on the use of robots in education show the challenges faced by researchers in this field. Benitti (2012) points out that more than 70 papers could have qualified in his review work but only 10 provided quantitative measurement on the use of robots in education. From these ten papers, only those that discuss the potential of using robots in all level of education and highlight the non-engineering benefits were selected. Mubin et al. (2013) analysed research works from through the actual robots used. The major factors identified were robot's role, type (physical form), behaviour (capabilities and interaction capacity), learning activity type, and venue (inside or outside of classroom) where learning takes place. Mubin et al. (2013) and Benitti (2012) find similarity on the topics where robots were being used in education--learning language, science, and technology. Although Mubin et al. (2013) differs by pointing out the various roles played by the robot in education--as tutor, tool, or peer. The reviews provide good starting points for researchers, the criteria (Benitti, 2012) and perspective (Mubin et al., 2013) taken by these two papers could potentially miss those that could be relevant to researchers in the field. Moreover, other factors critical in the use of robot in education may have been overlooked, like the effect of design on interaction or the importance of parent's perception in the success of implementing a robot-in-education project. The aim of this paper is to assess the effectiveness of using robots in studies published within the last decade. We look at effectiveness as having four sub-factors--the study type done by the researcher, the influence of the robots on the behaviour and development of students, the perception of stakeholders (parents, educators and children) about the robots, and the importance of design or robot appearance. To achieve this aim, we would focus on articles on the application of robots in early childhood and lower level education and evidence for the factors would be analysed. The rest of the paper is organized as follows. The review approach, especially the search and selection strategies, is discussed in details in the next section. The discussions on the four factors above are described in the succeeding sections. The conclusion provides a summary and presents the remaining challenges in this research field. Review approach To limit the papers to be reviewed, we implemented a search and selection strategy using specific keywords in electronic databases. We started with 369 articles and narrowed it down to 27. Search strategy Articles reviewed were limited to those published in English from 2003-2013. To gather as many papers as possible, five major databases were searched: IEEE Xplore, Academic Search Premier, ERIC (Educational Resources Information Center, ScienceDirect, and SpringerLink. …
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paper_title: Scenarios of robot-assisted play for children with cognitive and physical disabilities

paper_content:

This article presents a novel set of ten play scenarios for robot-assisted play for children with special needs. This set of scenarios is one of the key outcomes of the IROMEC project that investigated how robotic toys can become social mediators, encouraging children with special needs to discover a range of play styles, from solitary to collaborative play (with peers, carers/teachers, parents etc.). The target user groups in the project were children with Mild Mental Retardation,1 children with Severe Motor Impairment and children with Autism. The play scenarios were developed against specific educational and therapeutic objectives that were discussed with panels of experts (teachers therapists parents) in various countries, during several user panel meetings for each of the above mentioned target user groups. These objectives were classified with reference to the ICF-CY, the International Classification of Functioning – version for Children and Youth. The article presents a detailed description of the play scenarios, each with its relevant educational and therapeutic objectives in five key developmental areas (i.e. sensory development, communication and interaction, Cognitive development, motor development and social and emotional development). While the play scenarios described in this paper originally were developed for and tested with the above user groups and with the IROMEC robot, the play scenarios can potentially be applied to other user groups and to a wide range of other applications involving human-robot interaction using different robotic toys. Keywords: human-robot interaction; assistive technology; play scenarios; autism; mild mental retardation; sever motor impairment; assisted play
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paper_title: Pre-schoolers’ Interest and Caring Behaviour Around a Humanoid Robot

paper_content:

This exploratory case study involved a humanoid robot, NAO, and four pre-schoolers. NAO was placed in an indoor playground together with other toys and appeared as a peer who played, talked, danced and said stories. Analysis of video recordings focused on children’s behaviour around NAO and how the robot gained children’s attention and interest. First, the study demonstrated that pre-schoolers easily interact with this humanoid robot and showed particular interest to NAO when he danced and when he was in need of help (e.g., fell down) demonstrating caring behaviour such as kisses, hugs and cuddling. Further, the study contributes a systematically developed coding scheme capturing the verbal and non-verbal behaviour of young children around NAO.
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paper_title: Towards interactive robots in autism therapy: Background, motivation and challenges

paper_content:

This article discusses the potential of using interactive environments in autism therapy. We specifically address issues relevant to the Aurora project, which studies the possible role of autonomous, mobile robots as therapeutic tools for children with autism. Theories of mindreading, social cognition and imitation that informed the Aurora project are discussed and their relevance to the project is outlined. Our approach is put in the broader context of socially intelligent agents and interactive environments. We summarise results from trials with a particular mobile robot. Finally, we draw some comparisons to research on interactive virtual environments in the context of autism therapy and education. We conclude by discussing future directions and open issues.
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paper_title: Robots in Education and Care of Children with Developmental Disabilities: A Study on Acceptance by Experienced and Future Professionals

paper_content:

Research in the area of robotics has made available numerous possibilities for further innovation in the education of children, especially in the rehabilitation of those with learning difficulties and/or intellectual disabilities. Despite the scientific evidence, there is still a strong scepticism against the use of robots in the fields of education and care of people. Here we present a study on the acceptance of robots by experienced practitioners (specialized in the treatment of intellectual disabilities) and university students in psychology and education sciences (as future professionals). The aim is to examine the factors, through the Unified Theory of Acceptance and Use of Technology (UTAUT) model, that may influence the decision to use a robot as an instrument in the practice. The overall results confirm the applicability of the model in the context of education and care of children, and suggest a positive attitude towards the use of the robot. The comparison highlights some scepticism among the practitioners, who perceive the robot as an expensive and limited tool, while students show a positive perception and a significantly higher willingness to use the robot. From this experience, we formulate the hypothesis that robots may be accepted if more integrated with standard rehabilitation protocols in a way that benefits can outweigh the costs.
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paper_title: Does appearance matter in the interaction of children with autism with a humanoid robot

paper_content:

This article studies the impact of a robot’s appearance on interactions involving four children with autism. This work is part of the Aurora project with the overall aim to support interaction skills in children with autism, using robots as ‘interactive toys’ that can encourage and mediate interactions. We follow an approach commonly adopted in assistive robotics and work with a small group of children with autism. This article investigates which robot appearances are suitable to encourage interactions between a robot and children with autism. The children’s levels of interaction with and response to different appearances of two types of robots are compared: a small humanoid doll, and a life-sized ‘Theatrical Robot’ (a mime artist behaving like a robot). The small humanoid robot appeared either as a human-like ‘pretty doll’ or as a ‘robot’ with plain features. The Theatrical Robot was presented either as an ordinary human, or with plain clothing and a featureless, masked face. The results of these trials clearly indicate the children’s preference in their initial response for interaction with a plain, featureless robot over the interaction with a human-like robot. In the case of the life-size Theatrical Robot, the response of children towards the plain/robotic robot was notably more social and pro-active. Implications of these results for our work on using robots as assistive technology for children with autism and their possible use in autism research are discussed.
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paper_title: Social robot tutoring for child second language learning

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An increasing amount of research is being conducted to determine how a robot tutor should behave socially in educational interactions with children. Both human-human and human-robot interaction literature predicts an increase in learning with increased social availability of a tutor, where social availability has verbal and nonverbal components. Prior work has shown that greater availability in the nonverbal behaviour of a robot tutor has a positive impact on child learning. This paper presents a study with 67 children to explore how social aspects of a tutor robot's speech influences their perception of the robot and their language learning in an interaction. Children perceive the difference in social behaviour between `low' and `high' verbal availability conditions, and improve significantly between a pre- and a post-test in both conditions. A longer-term retention test taken the following week showed that the children had retained almost all of the information they had learnt. However, learning was not affected by which of the robot behaviours they had been exposed to. It is suggested that in this short-term interaction context, additional effort in developing social aspects of a robot's verbal behaviour may not return the desired positive impact on learning gains.
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paper_title: Modelling Learning to Count in Humanoid Robots

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paper_title: Robots in Education and Care of Children with Developmental Disabilities: A Study on Acceptance by Experienced and Future Professionals

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Research in the area of robotics has made available numerous possibilities for further innovation in the education of children, especially in the rehabilitation of those with learning difficulties and/or intellectual disabilities. Despite the scientific evidence, there is still a strong scepticism against the use of robots in the fields of education and care of people. Here we present a study on the acceptance of robots by experienced practitioners (specialized in the treatment of intellectual disabilities) and university students in psychology and education sciences (as future professionals). The aim is to examine the factors, through the Unified Theory of Acceptance and Use of Technology (UTAUT) model, that may influence the decision to use a robot as an instrument in the practice. The overall results confirm the applicability of the model in the context of education and care of children, and suggest a positive attitude towards the use of the robot. The comparison highlights some scepticism among the practitioners, who perceive the robot as an expensive and limited tool, while students show a positive perception and a significantly higher willingness to use the robot. From this experience, we formulate the hypothesis that robots may be accepted if more integrated with standard rehabilitation protocols in a way that benefits can outweigh the costs.
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paper_title: The Mathematical Brain

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The concept of numbers and the ability to recognize and process them is innate, part of everyone's intellectual apparatus whether they've had formal education or not. This "number instinct" is not dependent on basic intelligence or general knowledge, a fact which has implications for neuroscience and poses the question: why did man evolve with such specialized neural apparatus. It has been that the social development of humans has been crucially affected by language, yet numbers have also been critical in the advancement of human culture. Every child goes through a stage of learning to count using their ten fingers, much as early Homo Sapiens must have done. If number learning is a natural and universal function of the brain, why do so many suffer from dyscalculia? This text, containing theories and anecdote, is an investigation into the bizarre world of numbers. It examines the role of education, good or bad, in the development of mathematical disorders.
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paper_title: Learning and development of embodied numerosity

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Recent empirical evidence indicates that seemingly abstract numerical cognitions are rooted in sensory and bodily experiences. In particular in finger counting finger-based representations reflect a specific case of embodied cognition, we termed embodied numerosity. Furthermore, we suggest that finger-based representations should be considered a distinct representation of number (magnitude) and argue that this representation is activated automatically whenever we encounter a number. We discuss in what way such a theoretical framework can account for the associations of fingers and numbers observed so far. In the final part, we evaluate whether the concept of embodied numerosity should be generalized beyond finger-based representations with particular focus on whether bodily-sensory experiences (such as moving the whole body along the mental number line) may corroborate numerical capabilities. In a series of intervention studies, we consistently observed more pronounced training effects for our embodied numerosity trainings for different age groups, different digital media, different number ranges, and different control conditions. Taken together, we conclude that embodied representations of number (magnitude) exist, are not limited to finger-based representations, and influence number processing in a systematic and functional way that can be used to foster the efficiency of numerical trainings.
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paper_title: Finger counting habits modulate spatial-numerical associations

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Abstract This study explored the contribution of finger counting habits to the association of numbers with space (the SNARC effect). First, a questionnaire study indicated that two-thirds of 445 adults started counting on their left hand, regardless of their handedness. Secondly, a group of 53 “left-starters” but not a group of 47 “right-starters” showed a SNARC effect in a parity task. A significant difference in the strength of the effect between groups suggests that finger counting habits indeed contribute to the association between numbers and space in adults.
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paper_title: Finger Numeral Representations: More than Just Another Symbolic Code

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Representing numerosities with finger configurations offers children the opportunity to learn and internalize fundamental properties of natural numbers through sensory–motor interactions with the world. Recent findings show that even educated adults use their fingers as a visuo-motor support to process, represent, and communicate numerosities. Indeed, using fingers to represent numerosities prototypically has been shown to give the corresponding finger configurations a special status in long-term memory: these configurations are recognized and processed faster than other finger configurations, providing a direct access to number magnitude, what other finger configurations do less efficiently. This occurs for configurations stemming from finger counting (i.e., the way the fingers are raised to count for oneself; for example, thumb, index, and middle fingers for numerosity {3}) and from finger montring (i.e., the way fingers are raised to show numerosities to someone else; for example, index, middle, and ring fingers for numerosity {3}). However, finger numeral representation is not just another way of representing numerical magnitudes mentally. We argue that it also contributes to acquiring, building, and then accessing number semantics, and that, compared to other numerical representations, it provides an extra value by rooting number meaning in a culturally shared yet non-arbitrary and self-experienced sensory–motor representation.
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paper_title: Arithmetic in the developing brain: A review of brain imaging studies

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Abstract Brain imaging studies on academic achievement offer an exciting window on experience-dependent cortical plasticity, as they allow us to understand how developing brains change when children acquire culturally transmitted skills. This contribution focuses on the learning of arithmetic, which is quintessential to mathematical development. The nascent body of brain imaging studies reveals that arithmetic recruits a large set of interconnected areas, including prefrontal, posterior parietal, occipito-temporal and hippocampal areas. This network undergoes developmental changes in its function, connectivity and structure, which are not yet fully understood. This network only partially overlaps with what has been found in adults, and clear differences are observed in the recruitment of the hippocampus, which are related to the development of arithmetic fact retrieval. Despite these emerging trends, the literature remains scattered, particularly in the context of atypical development. Acknowledging the distributed nature of the arithmetic network, future studies should focus on connectivity and analytic approaches that investigate patterns of brain activity, coupled with a careful design of the arithmetic tasks and assessments of arithmetic strategies. Such studies will produce a more comprehensive understanding of how the arithmetical brain unfolds, how it changes over time, and how it is impaired in atypical development.
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paper_title: Embodied numerosity: Implicit hand-based representations influence symbolic number processing across cultures

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Abstract In recent years, a strong functional relationship between finger counting and number processing has been suggested. Developmental studies have shown specific effects of the structure of the individual finger counting system on arithmetic abilities. Moreover, the orientation of the mental quantity representation (“number line”) seems to be influenced by finger counting habits. However, it is unclear whether the structure of finger counting systems still influences symbolic number processing in educated adults. In the present transcultural study, we pursued this question by examining finger-based sub-base-five effects in an Arabic number comparison task with three different groups of participants (German deaf signers, German and Chinese hearing adults). We observed sub-base-five effects in all groups, but particularly so for both German groups who use an explicit sub-base-five system in their finger counting habits. It is concluded that bodily experiences – namely finger counting – influence the structure of the abstract mental number representations even in adults. Thus, the present findings support the general idea that even seemingly abstract cognition may at least partially be rooted in our bodily experiences.
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paper_title: The function of gesture in learning to count: more than keeping track *

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Preschoolers count objects most accurately when they gesture as they count. This study tests two possible explanations for this effect. One is that gesture helps children keep track of counted items. Another is that gesture helps children coordinate saying the number words and tagging the items. Twenty preschoolers counted chips under three types of conditions: with gesture prohibited, with active gesture, and with a puppet gesturing as children counted aloud. The puppet conditions were intended to distinguish the benefits of keeping track with gesture from the benefits of active gesture. Children counted more accurately when they or the puppet gestured than when gesture was prohibited. However, children's errors differed when they and the puppet gestured. When children gestured themselves, they made errors keeping track, but when the puppet gestured, they made errors coordinating number words and items. Thus, active gesture helps children both to keep track and to coordinate tagging the items and saying the number words. In these ways, active gesture helps children implement their knowledge of one-to-one correspondence.
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paper_title: The role of fingers in number processing in young children

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The aim of the present study was to investigate the relationship between finger counting and numerical processing in 4- to 7-year-old children. Children were assessed on a variety of numerical tasks and we examined the correlations between their rates of success and their frequency of finger use in a counting task. We showed that children’s performance on finger pattern comparison and identification tasks did not correlate with the frequency of finger use. However, this last variable correlated with the percentages of correct responses in an enumeration task (i.e., Give-N task), even when the age of children was entered as a covariate in the analysis. Despite this correlation, we showed that some children who never used their fingers in the counting task were able to perform optimally in the enumeration task. Overall, our results support the conclusion that finger counting is useful but not necessary to develop accurate symbolic numerical skills. Moreover, our results suggest that the use of fingers in a counting task is related to the ability of children in a dynamic enumeration task but not to static tasks involving recognition or comparison of finger patterns. Therefore, it could be that the link between fingers and numbers remain circumscribed to counting tasks and do not extent to static finger montring situations.
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paper_title: Gesture as a window onto children’s number knowledge

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Before learning the cardinal principle (knowing that the last word reached when counting a set represents the size of the whole set), children do not use number words accurately to label most set sizes. However, it remains unclear whether this difficulty reflects a general inability to conceptualize and communicate about number, or a specific problem with number words. We hypothesized that children’s gestures might reflect knowledge of number concepts that they cannot yet express in speech, particularly for numbers they do not use accurately in speech (numbers above their knower-level). Number gestures are iconic in the sense that they are item-based (i.e., each finger maps onto one item in a set) and therefore may be easier to map onto sets of objects than number words, whose forms do not map transparently onto the number of items in a set and, in this sense, are arbitrary. In addition, learners in transition with respect to a concept often produce gestures that convey different information than the accompanying speech. We examined the number words and gestures 3- to 5-year-olds used to label small set sizes exactly (1–4) and larger set sizes approximately (5–10). Children who had not yet learned the cardinal principle were more than twice as accurate when labeling sets of 2 and 3 items with gestures than with words, particularly if the values were above their knower-level. They were also better at approximating set sizes 5–10 with gestures than with words. Further, gesture was more accurate when it differed from the accompanying speech (i.e., a gesture–speech mismatch). These results show that children convey numerical information in gesture that they cannot yet convey in speech, and raise the possibility that number gestures play a functional role in children’s development of number concepts.
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paper_title: The mental representation of parity and number magnitude

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Nine experiments of timed odd-even judgments examined how parity and number magnitude are accessed from Arabic and verbal numerals. With Arabic numerals, Ss used the rightmost digit to access a store of semantic number knowledge. Verbal numerals went through an additional stage of transcoding to base 10. Magnitude information was automatically accessed from Arabic numerals. Large numbers preferentially elicited a rightward response, and small numbers a leftward response. The Spatial-Numerical Association of Response Codes (SNARC) effect depended only on relative number magnitude and was weaker or absent with letters or verbal numerals. Direction did not vary with handedness or hemispheric dominance but was linked to the direction of writing, as it faded or even reversed in right-to-left writing Iranian Ss
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paper_title: Does finger sense predict addition performance?

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The impact of fingers on numerical and mathematical cognition has received a great deal of attention recently. However, the precise role that fingers play in numerical cognition is unknown. The current study explores the relationship between finger sense, arithmetic and general cognitive ability. Seventy-six children between the ages of 5 and 12 participated in the study. The results of stepwise multiple regression analyses demonstrated that while general cognitive ability including language processing was a predictor of addition performance, finger sense was not. The impact of age on the relationship between finger sense, and addition was further examined. The participants were separated into two groups based on age. The results showed that finger gnosia score impacted addition performance in the older group but not the younger group. These results appear to support the hypothesis that fingers provide a scaffold for calculation and that if that scaffold is not properly built, it has continued differential consequences to mathematical cognition.
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paper_title: Number and language: how are they related?

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Does the ability to develop numerical concepts depend on our ability to use language? We consider the role of the vocabulary of counting words in developing numerical concepts. We challenge the ‘bootstrapping’ theory which claims that children move from using something like an object-file – an attentional process for responding to small numerosities – to a truly arithmetic one as a result of their learning the counting words. We also question the interpretation of recent findings from Amazonian cultures that have very restricted number vocabularies. Our review of data and theory, along with neuroscientific evidence, imply that numerical concepts have an ontogenetic origin and a neural basis that are independent of language.
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paper_title: Mind the gap between both hands: Evidence for internal finger-based number representations in children's mental calculation

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Abstract At a certain stage of development, virtually all children use some kind of external finger-based number representation. However, only little is known about how internal traces of this early external representation may still influence calculation even when finger calculation ceases to be an efficient tool in mental calculation. In the present study, we provide evidence for a disproportionate number of split-five errors (i.e., errors with a difference of ±5 from the correct result) in mental addition and subtraction (e.g., 18 − 7 = 6). We will argue that such errors may have different origins. For complex problems and initially also for simple problems they are due to failure to keep track of ‘full hands’ in counting or calculation procedures. However, for simple addition problems split-five errors may later also be caused by mistakes in directly retrieving the result from declarative memory. In general, the present results are interpreted in terms of a transient use of mental finger patterns – in particular the whole hand pattern – in children's mental calculation.
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paper_title: Numbers are represented in egocentric space: Effects of numerical cues and spatial reference frames on hand laterality judgements

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Abstract Convergent findings demonstrate that numbers can be represented according to a spatially oriented mental number line. However, it is not established whether a default organization of the mental number line exists (i.e., a left-to-right orientation) or whether its spatial arrangement is only the epiphenomenon of specific task requirements. To address this issue we performed two experiments in which subjects were required to judge laterality of hand stimuli preceded by small, medium or large numerical cues; hand stimuli were compatible with egocentric or allocentric perspectives. We found evidence of a left-to-right number–hand association in processing stimuli compatible with an egocentric perspective, whereas the reverse mapping was found with hands compatible with an allocentric perspective. These findings demonstrate that the basic left-to-right arrangement of the mental number line is defined with respect to the body-centred egocentric reference frame.
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paper_title: The Influence of Implicit Hand-Based Representations on Mental Arithmetic

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Recently, a strong functional relationship between finger counting and number processing has been suggested. It has been argued that bodily experiences such as finger counting may influence the structure of the basic mental representations of numbers even in adults. However, to date it remains unclear whether the structure of finger counting systems also influences educated adults’ performance in mental arithmetic. In the present study, we pursued this question by examining finger-based sub-base-five effects in an addition production task. With the standard effect of a carry operation (i.e., base 10 crossing) being replicated, we observed an additional sub-base-five effect such that crossing a sub-base 5 boundary led to a relative response time increase. For the case of mental arithmetic sub-base-five effects have previously been reported only in children. However, it remains unclear whether finger-based numerical effects in mental arithmetic reflect an important but transitory step in the development of arithmetical skills. The current findings suggest that even in adults embodied representations such as finger counting patterns modulate arithmetic performance. Thus, they support the general idea that even seemingly abstract cognition in adults may at least partly be rooted in our bodily experiences.
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paper_title: Embodiment in Mathematics Teaching and Learning: Evidence From Learners' and Teachers' Gestures

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Gestures are often taken as evidence that the body is involved in thinking and speaking about the ideas expressed in those gestures. In this article, we present evidence drawn from teachers' and learners' gestures to make the case that mathematical knowledge is embodied. We argue that mathematical cognition is embodied in 2 key senses: It is based in perception and action, and it is grounded in the physical environment. We present evidence for each of these claims drawn from the gestures that teachers and learners produce when they explain mathematical concepts and ideas. We argue that (a) pointing gestures reflect the grounding of cognition in the physical environment, (b) representational gestures manifest mental simulations of action and perception, and (c) some metaphoric gestures reflect body-based conceptual metaphors. Thus, gestures reveal that some aspects of mathematical thinking are embodied.
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paper_title: Development of number combination skill in the early school years: when do fingers help?

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Children's change over time in frequency of finger use on number combinations was examined in relation to their change in accuracy. Performance was tracked longitudinally over 11 time points, from the beginning of kindergarten (mean age = 5.7 years) to the end of second grade (n= 217). Accuracy in number combinations increased steadily during the time period while frequency of finger use declined. Correlations between finger use and accuracy decreased gradually, ranging from 0.60 in kindergarten to −0.15 at the end of second grade. Low-income children showed linear growth in frequency of finger use while middle-income children slowed down by second grade and even started to decline. Although girls and boys showed similar growth patterns in frequency and accuracy, boys used their fingers less often than girls and were more accurate. The findings indicate that finger use is most adaptive when children are first learning number combinations, but this benefit lessens over time.
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paper_title: Effects of Finger Counting on Numerical Development – The Opposing Views of Neurocognition and Mathematics Education

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Children typically learn basic numerical and arithmetic principles using finger-based representations. However, whether or not reliance on finger-based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education. From the neurocognitive perspective, finger counting provides multisensory input, which conveys both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis, or “finger sense,” enhances mathematical skills. Therefore neurocognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting. More precisely, mathematics education recommends first using finger counting, then concrete structured representations and, finally, mental representations of numbers to perform numerical operations. Taken together, these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be discussed.
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paper_title: The child's understanding of number

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1. Focus on the Preschooler 2. Training Studies Reconsidered 3. More Capacity Than Meets the Eye: Direct Evidence 4. Number Concepts in the Preschooler? 5. What Numerosities Can the Young Child Represent? 6. How Do Young Children Obtain Their Representations of Numerosity? 7. The Counting Model 8. The Development of the How-To-Count Principles 9. The Abstraction and Order-Irrelevance Counting Principles 10. Reasoning about Number 11. Formal Arithmetic and the Young Child's Understanding of Number 12. What Develops and How Conclusions References Index
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paper_title: You can count on the motor cortex: Finger counting habits modulate motor cortex activation evoked by numbers

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The embodied cognition framework suggests that neural systems for perception and action are engaged during higher cognitive processes. In an event-related fMRI study, we tested this claim for the abstract domain of numerical symbol processing: is the human cortical motor system part of the representation of numbers, and is organization of numerical knowledge influenced by individual finger counting habits? Developmental studies suggest a link between numerals and finger counting habits due to the acquisition of numerical skills through finger counting in childhood. In the present study, digits 1 to 9 and the corresponding number words were presented visually to adults with different finger counting habits, i.e. left- and right-starters who reported that they usually start counting small numbers with their left and right hand, respectively. Despite the absence of overt hand movements, the hemisphere contralateral to the hand used for counting small numbers was activated when small numbers were presented. The correspondence between finger counting habits and hemispheric motor activation is consistent with an intrinsic functional link between finger counting and number processing.
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paper_title: Numbers within Our Hands: Modulation of Corticospinal Excitability of Hand Muscles during Numerical Judgment

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Developmental and cross-cultural studies show that finger counting represents one of the basic number learning strategies. However, despite the ubiquity of such an embodied strategy, the issue of whether there is a neural link between numbers and fingers in adult, literate individuals remains debated. Here, we used transcranial magnetic stimulation to study changes of excitability of hand muscles of individuals performing a visual parity judgment task, a task not requiring counting, on Arabic numerals from 1 to 9. Although no modulation was observed for the left hand muscles, an increase in amplitude of motor-evoked potentials was found for the right hand muscles. This increase was specific for smaller numbers (1 to 4) as compared to larger numbers (6 to 9). These findings indicate a close relationship between hand/finger and numerical representations.
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paper_title: A developmental fMRI study of nonsymbolic numerical and spatial processing

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Abstract This functional magnetic resonance imaging (fMRI) study systematically investigates whether there is a neurofunctional overlap of nonsymbolic numerical and spatial cognition in (intra)parietal regions in children and adults. The study also explores the association between finger use and (nonsymbolic) number processing across development. Twenty-four healthy individuals (12 children, 12 adults) were asked to make nonsymbolic numerical and spatial (experimental tasks) as well as color discriminations (control task). Using identical stimulus material across the three tasks disentangled nonsymbolic number representations from general attentional mechanisms, visual–spatial processing and response selection requirements. In both age groups, behavioral distance effects were obtained upon processing numerical (but not spatial and/or color) stimuli. Baseline imaging effects revealed age-dependent, partly overlapping activations of nonsymbolic numerical and spatial processing in the right posterior superior parietal lobe (PSPL) in adults only. Interestingly, regions more activated in children relative to adults were centred on bilateral supramarginal gyrus (SMG) and lateral portions of the anterior intraparietal sulcus (IPS), further extending to adjacent right post- and precentral gyrus, the latter of which has been reported to support grasping previously (Simon et al., 2002). Overall, our results are first evidence for an age-dependent neurofunctional link between areas supporting finger use and nonsymbolic number processing and furthermore, might be suggestive of a special role of fingers for the development of number magnitude representations and early arithmetic.
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paper_title: Common substrate for mental arithmetic and finger representation in the parietal cortex

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The history of mathematics provides several examples of the use of fingers to count or calculate. These observations converge with developmental data showing that fingers play a critical role in the acquisition of arithmetic knowledge. Further studies evidenced specific interference of finger movements with arithmetic problem solving in adults, raising the question of whether or not finger and number manipulations rely on common brain areas. In the present study, functional magnetic resonance imaging (fMRI) was used to investigate the possible overlap between the brain areas involved in mental arithmetic and those involved in finger discrimination. Solving subtraction and multiplication problems was found to increase cerebral activation bilaterally in the horizontal part of the intraparietal sulcus (hIPS) and in the posterior part of the superior parietal lobule (PSPL). Finger discrimination was associated with increased activity in a bilateral occipito-parieto-precentral network extending from the extrastriate body area to the primary somatosensory and motor cortices. A conjunction analysis showed common areas for mental arithmetic and finger representation in the hIPS and PSPL bilaterally. Voxelwise correlations further showed that finger discrimination and mental arithmetic induced a similar pattern of activity within the parietal areas only. Pattern similarity was more important for the left than for the right hIPS and for subtraction than for multiplication. These findings provide the first evidence that the brain circuits involved in finger representation also underlie arithmetic operations in adults. © 2012 Elsevier Inc.
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paper_title: Masked priming effect with canonical finger numeral configurations

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Discrete numerosities can be represented by various finger configurations. The impact of counting strategies on these configurations and their possible semantic status were investigated in young adults. Experiment 1 showed that young adults named numerical finger configurations faster when they conformed to their own canonical finger-counting habits than when they did not. Experiment 2 showed that numeral finger configurations used as unconsciously presented primes speeded up numerical comparative judgements of Arabic numeral targets. Participants responded faster and made fewer errors with numerical than with non-numerical primes, and when primes and targets were congruent (i.e., leading to the same response). Moreover, this priming effect generalised to novel never consciously seen numerosities for canonical configurations but not for non-canonical ones. These results support the idea that canonical finger configurations automatically activate number semantics whereas non-canonical ones do not.
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paper_title: Mathematics at Your Fingertips: Testing a Finger Training Intervention to Improve Quantitative Skills

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Previous research indicates that the use of fingers as representations of ordinal and cardinal number is an important part of young children's mathematics learning. Further to this, some studies have shown that a finger training intervention can improve young children's quantitative skills. In this article, we argue that fingers represent a means for children to bridge between different external representations of number (including verbal, symbolic and non-symbolic representations). Therefore we predicted that an intervention that combined finger training with experience playing games involving multiple representations would lead to greater increases in quantitative skills than either aspect of the intervention alone. One hundred and thirty-seven children aged between six and seven years old took part in an intervention study over the course of four weeks. The study tested the impact of five different conditions on participants' quantitative skills, their finger gnosis, and their ability to compare magnitudes of two non-symbolic representations of number. Relative to a control group, those children receiving a finger training intervention saw gains in finger gnosis skills (the ability to differentiate fingers when touched, without visual cues). Those children who played number games saw an increase in their non-symbolic magnitude comparison skills. However, only those children who experienced both aspects of the intervention saw increases in quantitative skills, which were assessed using an instrument informed by Gelman and Gallistel's (1978) five principles of counting. The findings show that a finger training intervention, when combined with intensive exposure to multiple representations of number, can support young children's development of quantitative skills. This adds to evidence in the literature regarding the role of fingers in children's mathematics learning, and may have implications for pedagogical approaches.
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paper_title: Finger gnosia: a predictor of numerical abilities in children?

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This paper aimed to test the specificity of predicting power of finger gnosia on later numerical abilities in school-age children and to contribute to the understanding of this effect. Forty-one children were tested in the beginning of Grade 1 on finger gnosia, left-right orientation (another sign of the Gerstmann "syndrome"), and global development. Fifteen months later, numerical and reading abilities were assessed. Analyses of the results indicated that, contrary to the general measures of cognitive development, performance in the finger gnosia test was a good predictor of numerical skills 1 year later but not of reading skills, which proves the specificity of that predictor. The same conclusion was also true for the left-right orientation. However, finger gnosia could equally predict performance in numerical tasks that do or do not rely heavily on finger representation or on magnitude representation. Results are discussed in terms of the localizationist and the functional hypotheses.
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paper_title: Perception of numbers by human infants.

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Infants are capable of discriminating, representing, and remembering particular small numbers of items. A perceptual enumeration process called subitizing, present in 2-year-olds, probably underlies this capacity. This finding indicates that some number capacity is present before the onset of verbal counting, and it suggests that verbal counting may have precursors present during infancy.
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paper_title: Five- to 7-Year-Olds’ Finger Gnosia and Calculation Abilities

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The research examined the relationship between 65 5- to 7-year-olds’ finger gnosia, visuo-spatial working memory and finger-use solving single digit addition problems. Their non-verbal IQ and basic RT were also assessed. Previous research has found significant changes in children’s representational abilities between five and seven years. One aim of the research was to determine whether changes in finger representational abilities (finger gnosia) occur across these ages and whether they are associated with finger-use in computation. A second aim was to determine whether visuo-spatial working memory is associated with finger gnosia and computation abilities. We used latent class profile analysis to identify patterns of similarities and differences in finger gnosia and computation/finger-use abilities. The analysis yielded four finger gnosia subgroups that differed in finger representation ability. It also yielded four finger/computation subgroups that differed in the relationship between finger-use and computation success. Analysis revealed associations between computation finger-use/success subgroups, finger gnosia subgroups, and visuo-spatial working memory. A multinomial logistic regression analysis showed that finger gnosia subgroup membership and visuo-spatial working memory uniquely contribute to a model predicting finger-use in computation group membership. The results show that finger gnosia abilities change in the early school years, and that these changes are associated with the ability to use fingers to aid computation.
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paper_title: Does finger training increase young children's numerical performance?

paper_content:

Butterworth (1999) suggested that fingers are important in representing numerosities. Furthermore, scores on a finger gnosis test are a better predictor of numerical performance up to 3 years later than intellectual measures (Marinthe et al., 2001; Noel, 2005). We hypothesised that training in finger differentiation would increase finger gnosis and might also improve numerical performance. Accordingly, 47 first-grade children were selected and divided into 3 groups: children with poor finger gnosis who followed the finger-differentiation training programme (G1), a control-intervention who were trained in story comprehension (G2), and a group with high finger gnosis scores who just continued with normal school lessons (G3). The finger training consisted of 2 weekly sessions of half an hour each, for 8 weeks. Before the training period, children in G3 performed better in finger gnosis and enumeration than children in the two other groups. After the training period this pattern remained for the children in G2 and G3, but the children in G1 were significantly better than those in G2 at finger gnosis, representation of numerosities with fingers, and quantification tasks; they also tended to be better at the processing of Arabic digits. These results indicate that improving finger gnosis in young children is possible and that it can provide a useful support to learning mathematics. Such an approach could be particularly appropriate for children with a developmental Gerstmann syndrome. Theoretically, these results are important because they suggest a functional link between finger gnosis and number skills.
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paper_title: The Universal History of Numbers: From Prehistory to the Invention of the Computer

paper_content:

PART ONE. Chapter 1: Historical Summary of Arithmetic, Numerical Notation, and Writing Systems. chapter 2: From the Particular to the General: Arithmetic Leads to Algebra. Chapter 3: From Calculation to Calculus. Chapter 4: Binary Arithmetic and Other Non-Decimal Systems. PART TWO. Chapter 5: From Clockwork Calculator to Computer: The History of Automatic Calculation. Chapter 6: What is a Computer? PART TWO. Chapter 7: Information, the New Universal Dimension. Conclusion: Intelligence, Science, and the Future of Mankind. List of Abbreviations. Bibliography. Index.
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paper_title: Finding Structure in Time

paper_content:

Time underlies many interesting human behaviors. Thus, the question of how to represent time in connectionist models is very important. One approach is to represent time implicitly by its effects on processing rather than explicitly (as in a spatial representation). The current report develops a proposal along these lines first described by Jordan (1986) which involves the use of recurrent links in order to provide networks with a dynamic memory. In this approach, hidden unit patterns are fed back to themselves; the internal representations which develop thus reflect task demands in the context of prior internal states. A set of simulations is reported which range from relatively simple problems (temporal version of XOR) to discovering syntactic/semantic features for words. The networks are able to learn interesting internal representations which incorporate task demands with memory demands; indeed, in this approach the notion of memory is inextricably bound up with task processing. These representations reveal a rich structure, which allows them to be highly context-dependent while also expressing generalizations across classes of items. These representations suggest a method for representing lexical categories and the type/token distinction.
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paper_title: Computational Evidence for the Subitizing Phenomenon as an Emergent Property of the Human Cognitive Architecture

paper_content:

A computational modeling approach was used to test one possible explanation for the limited capacity of the subitizing phenomenon. Most existing models of this phenomenon associate the subitizing span with an assumed structural limitation of the human information processing system. In contrast, we show how this limit might emerge as the combinatorics of the space of enumeration problems interacts with the human cognitive architecture in the context of an enumeration task. Subitizing-like behavior was generated in two different models of enumeration, one based on the ACT-R cognitive architecture and the other based on the principles of parallel distributed processing (PDP). Our results provide good qualitative fits to results obtained in a variety of empirical studies.
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paper_title: LESIONING MCCLOSKEY AND LINDEMANN’S (1992) MATHNET: THE EFFECT OF DAMAGE LOCATION AND AMOUNT

paper_content:

McCloskey and Lindemann [32] provide a simulation of brain damage on a neural network architecture and offer evidence that different lesions to a same network can lead to different error distributions. We briefly review the various kinds of networks that have been proposed to simulate various arithmetical fact retrieval phenomena and we present a simple network designed to make some computational constraints apparent. Additionally, we replicate McCloskey and Lindemann’s [32] simulation by training 5 different artificial “subjects” and inflicting various types of damage upon each. Examination of the behaviour of our version of the network after different amounts of damage to its various connection blocks confirms that the error pattern may vary. These variations in the error patterns can be analyzed. The data may help to clarify the functioning of the network and give insight into the reasons why it produces several effects observed in human behaviour.
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paper_title: Development of Elementary Numerical Abilities: A Neuronal Model

paper_content:

Despite their lack of language, human infants and several animal species possess some elementary abilities for numerical processing. These include the ability to recognize that a given numerosity is being presented visually or auditorily, and, at a later stage of development, the ability to compare two nume-rosities and to decide which is larger. We propose a model for the development of these abilities in a formal neuronal network. Initially, the model is equipped only with unordered numerosity detectors. It can therefore detect the numerosity of an input set and can be conditioned to react accordingly. In a later stage, the addition of a short-term memory network is shown to be sufficient for number comparison abilities to develop. Our computer simulations account for several phenomena in the numerical domain, including the distance effect and Fechner's law for numbers. They also demonstrate that infants' numerosity detection abilities may be explained without assuming that infants can count. The neurobiological bases of the critical components of the model are discussed.
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paper_title: Perceiving numerosity from birth

paper_content:

Leibovich et al. opened up an important discussion on the nature and origins of numerosity perception. The authors rightly point out that non-numerical features of stimuli influence this ability. Despite these biases, there is evidence that from birth, humans perceive and represent numerosities, and not just non-numerical quantitative features such as item size, density, and convex hull.
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paper_title: A recurrent neural network that learns to count

paper_content:

Parallel distributed processing (PDP) architectures demonstrate a potentially radical alternative to the traditional theories of language processing that are based on serial computational models. However, learning complex structural relationships in temporal data presents a serious challenge to PDP systems. For example, automata theory dictates that processing strings from a context-free language (CFL) requires a stack or counter memory device. While some PDP models have been hand-crafted to emulate such a device, it is not clear how a neural network might develop such a device when learning a CFL. This research employs standard backpropagation training techniques for a recurrent neural network (RNN) in the task of learning to predict the next character in a simple deterministic CFL (DCFL). We show that an RNN can learn to recognize the structure of a simple DCFL. We use dynamical systems theory to identify how network states reflect that structure by building counters in phase space. The work is an empirical investigation which is complementary to theoretical analyses of network capabilities, yet original in its specific configuration of dynamics involved. The application of dynamical systems theory helps us relate the simulation results to theoretical results, and the learning task enables us to highlight some issues for understanding dynamical systems that process language with counters.
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paper_title: ACT-R: a theory of higher level cognition and its relation to visual attention

paper_content:

The ACT-R system is a general system for modeling a wide range of higher level cognitive processes. Recently, it has been embellished with a theory of how its higher level processes interact with a visual interface. This includes a theory of how visual attention can move across the screen, encoding information into a form that can be processed by ACT-R. This system is applied to modeling several classic phenomena in the literature that depend on the speed and selectivity with which visual attention can move across a visual display. ACT-R is capable of interacting with the same computer screens that subjects do and, as such, is well suited to provide a model for tasks involving human-computer interaction. In this article, we discuss a demonstration of ACT-R's application to menu selection and show that the ACT-R theory makes unique predictions, without estimating any parameters, about the time to search a menu. These predictions are confirmed.
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paper_title: Backpropagation through time: what it does and how to do it

paper_content:

Basic backpropagation, which is a simple method now being widely used in areas like pattern recognition and fault diagnosis, is reviewed. The basic equations for backpropagation through time, and applications to areas like pattern recognition involving dynamic systems, systems identification, and control are discussed. Further extensions of this method, to deal with systems other than neural networks, systems involving simultaneous equations, or true recurrent networks, and other practical issues arising with the method are described. Pseudocode is provided to clarify the algorithms. The chain rule for ordered derivatives-the theorem which underlies backpropagation-is briefly discussed. The focus is on designing a simpler version of backpropagation which can be translated into computer code and applied directly by neutral network users. >
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paper_title: A learning algorithm for Boltzmann Machines

paper_content:

The computational power of massively parallel networks of simple processing elements resides in the communication bandwidth provided by the hardware connections between elements. These connections can allow a significant fraction of the knowledge of the system to be applied to an instance of a problem in a very short time. One kind of computation for which massively parallel networks appear to be well suited is large constraint satisfaction searches, but to use the connections efficiently two conditions must be met: First, a search technique that is suitable for parallel networks must be found. Second, there must be some way of choosing internal representations which allow the preexisting hardware connections to be used efficiently for encoding the constraints in the domain being searched. We describe a general parallel search method, based on statistical mechanics, and we show how it leads to a general learning rule for modifying the connection strengths so as to incorporate knowledge about a task domain in an efficient way. We describe some simple examples in which the learning algorithm creates internal representations that are demonstrably the most efficient way of using the preexisting connectivity structure.
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paper_title: Children's Logical and Mathematical Cognition: Progress in Cognitive Development Research.

paper_content:

1 Conservation-Nonconservation: Alternative Explanations.- Conservation and the Appreciation of an Identity Rule.- Operational and Nonoperational Conservation.- Nonconservation and the Overreliance on Perceptual Cues.- Pseudononconservation.- Nonoperational Conservation.- Conclusions.- Future Research: The Development of the Identity Rule.- Reference Notes.- References.- 2 The Acquisition and Elaboration of the Number Word Sequence.- Acquisition of the Sequence.- Elaboration of the Sequence.- Conclusion.- Reference Notes.- References.- 3 Children's Concepts of Chance and Probability.- Piagetian Theory.- Subsequent Studies.- Theoretical Implications.- References.- 4 The Development of Quantity Concepts: Perceptual and Linguistic Factors.- Linguistic Factors and the Development of Quantity Concepts.- A Taxonomy of Quantity Concepts.- The Relationship between Language and Thought in the Child.- Study 1: Concept versus Language.- Study 2: Does Language Training Facilitate Concept Acquisition?.- Study 3: Visual versus Verbal Functions.- Study 4: Training of Cognitive and Language Abilities.- Study 5: Cognitive Development of Children with Impaired Language Development.- Study 6: The Abstraction of the Concept of Number.- Conclusion.- Reference Notes.- References.- 5 Culture and the Development of Numerical Cognition: Studies among the Oksapmin of Papua New Guinea.- Methodology and Cross-Cultural Number Research.- The Oksapmin Community.- Studies on Numerical Cognition among the Oksapmin.- Concluding Remarks.- Reference Notes.- References.- 6 Children's Concept Learning as Rule-Sampling Systems with Markovian Properties.- Concept Learning as Rule Sampling.- Some Questions about Concept Learning.- Some Experimental Evidence.- Remark.- References.
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paper_title: Representation of Number in Animals and Humans: A Neural Model

paper_content:

This article addresses the representation of numerical information conveyed by nonsymbolic and symbolic stimuli. In a first simulation study, we show how number-selective neurons develop when an initially uncommitted neural network is given nonsymbolic stimuli as input (e.g., collections of dots) under unsupervised learning. The resultant network is able to account for the distance and size effects, two ubiquitous effects in numerical cognition. Furthermore, the properties of the network units conform in detail to the characteristics of recently discovered number-selective neurons. In a second study, we simulate symbol learning by presenting symbolic and nonsymbolic input simultaneously. The same number-selective neurons learn to represent the numerical meaning of symbols. In doing so, they show properties reminiscent of the originally available number-selective neurons, but at the same time, the representational efficiency of the neurons is increased when presented with symbolic input. This finding presents a concrete proposal on the linkage between higher order numerical cognition and more primitive numerical abilities and generates specific predictions on the neural substrate of number processing.
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paper_title: Semantic priming in number naming

paper_content:

The issue of semantic and non-semantic conversion routes for numerals is still debated in numerical cognition. We report two number-naming experiments in which the target numerals were preceded by another numeral (prime). The primes and targets could be presented either in arabic (digit) notation or in verbal (alphabetical) notation. The results reveal a semantically related distance effect: Latencies are fastest when the prime has the same value as the target and increase when the distance between prime and target increases. We argue that the present results are congruent with the idea that the numerals make access to an ordered semantic number line common to all notations, as the results are the same for within-notation priming (arabic-arabic or verbal- verbal) and between-notations priming (arabic-verbal or verbal-arabic). The present results also point to a rapid involvement of semantics in the naming of numerals, also when the numerals are words. As such, they are in line with recent claims of rapid ...
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paper_title: Reinforcement Learning: An Introduction

paper_content:

Reinforcement learning, one of the most active research areas in artificial intelligence, is a computational approach to learning whereby an agent tries to maximize the total amount of reward it receives when interacting with a complex, uncertain environment. In Reinforcement Learning, Richard Sutton and Andrew Barto provide a clear and simple account of the key ideas and algorithms of reinforcement learning. Their discussion ranges from the history of the field's intellectual foundations to the most recent developments and applications. The only necessary mathematical background is familiarity with elementary concepts of probability. The book is divided into three parts. Part I defines the reinforcement learning problem in terms of Markov decision processes. Part II provides basic solution methods: dynamic programming, Monte Carlo methods, and temporal-difference learning. Part III presents a unified view of the solution methods and incorporates artificial neural networks, eligibility traces, and planning; the two final chapters present case studies and consider the future of reinforcement learning.
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paper_title: Model-based reinforcement learning for humanoids: A study on forming rewards with the iCub platform

paper_content:

Technological advancements in robotics and cognitive science are contributing to the development of the field of cognitive robotics. Modern robotic platforms are able to exhibit the ability to learn and reason about complex tasks and to follow behavioural goals in complex environments. Nevertheless, many challenges still exist. One of these great challenges is to equip these robots with cognitive systems that allow them to deal with less constrained situations, beyond constrained scenarios as in industrial robotics. In this work we explore the application of the Reinforcement Learning (RL) paradigm to study the autonomous development of robot controllers without a priori supervised learning. Such a model-based RL architecture is discussed for the cognitive implications of applying RL in humanoid robots. To this end we show a developmental framework for RL in robotics and its implementation and testing for the iCub robotic platform in two novel experimental scenarios. In particular we focus on iCub simulation experiments with comparisons between internal perception-based reward signals and external ones, in order to compare learning performance of the robot guided by its own perception of action's outcomes with the one when the robot has its actions externally evaluated.
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paper_title: Number and language: how are they related?

paper_content:

Does the ability to develop numerical concepts depend on our ability to use language? We consider the role of the vocabulary of counting words in developing numerical concepts. We challenge the ‘bootstrapping’ theory which claims that children move from using something like an object-file – an attentional process for responding to small numerosities – to a truly arithmetic one as a result of their learning the counting words. We also question the interpretation of recent findings from Amazonian cultures that have very restricted number vocabularies. Our review of data and theory, along with neuroscientific evidence, imply that numerical concepts have an ontogenetic origin and a neural basis that are independent of language.
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paper_title: Perceiving numbers causes spatial shifts of attention

paper_content:

Number symbols are part of our everyday visual world. Here we show that merely looking at numbers causes a shift in covert attention to the left or right side, depending upon the number's magnitude. This observation implies obligatory activation of number meaning and signals a tight coupling of internal and external representations of space.
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paper_title: Representation of the Quantity of Visual Items in the Primate Prefrontal Cortex

paper_content:

Deriving the quantity of items is an abstract form of categorization. To explore it, monkeys were trained to judge whether successive visual displays contained the same quantity of items. Many neurons in the lateral prefrontal cortex were tuned for quantity irrespective of the exact physical appearance of the displays. Their tuning curves formed overlapping filters, which may explain why behavioral discrimination improves with increasing numerical distance and why discrimination of two quantities with equal numerical distance worsens as their numerical size increases. A mechanism that extracts the quantity of visual field items could contribute to general numerical ability.
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paper_title: Modelling Learning to Count in Humanoid Robots

paper_content:

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paper_title: Two Plus Three Is Five: Discovering Efficient Addition Strategies without Metacognition

paper_content:

Two Plus Three Is Five: Discovering Efficient Addition Strategies without Metacognition Steven S. Hansen (sshansen@stanford.edu) Cameron McKenzie (crlmck@stanford.edu) James L. McClelland (mcclelland@stanford.edu) Department of Psychology Stanford University, Stanford, CA 94305 USA Abstract several early models did have this character (Neches, 1987). However, the paucity of solution paths that involved faulty strategies appear to rule out the „take a random step‟ style exploration used by most reinforcement learning models (Crawley, Shrager, & Siegler, 1997). Trial and error accounts were thus rejected, and replaced by a theory that posited a metacognitive mechanism with explicit, domain- specific content knowledge to filter out flawed strategy proposals. This mechanism allows the discovery of new strategies while producing only reasonable errors. However, it remains unclear how children could acquire the complex knowledge required to judge the appropriateness of their own strategy proposals. The acquisition of the metacognitive filter is neither explained nor explicitly modeled. The approach taken in this paper is to circumvent this difficulty by proposing that a metacognitive filter may not be necessary in the first place. We accomplish this by modifying a standard trial and error, reinforcement-learning- based paradigm to be biased towards previously learnt actions. We note that children learning the addition task have already learnt to count out numbers of objects, count on their fingers, and perform addition using a finger- counting strategy (Siegler & Jenkins, 1989). As we shall demonstrate, instantiating a model with biases towards these actions obviates the need for a metacognitive filter. We also expand the notion of retrieval – a „strategy‟ that circumvents the need to engage in a structured sequence of behaviors by simply recalling the correct final answer to a problem – by suggesting that retrieval might also occur for appropriate subparts of a larger problem. Our model makes several novel predictions about the discovery process and questions the notion that selection and discovery processes necessarily take place at the level of complete strategies. When learning addition, children appear to perform a remarkable feat: as they practice counting out sums on their fingers, they discover more efficient strategies while avoiding conceptually flawed procedures. Existing models that seek to explain how children discover good strategies while avoiding bad ones postulate metacognitive filters that reject faulty strategies. However, this leaves unexplained how the domain- specific knowledge required to evaluate a strategy might be acquired prior to addition being mastered. We introduce a biased exploration model, which demonstrates that new addition strategies can be discovered without invoking metacognitive filtering. This model provides a fit to data comparable to previous models, with the considerable advantage of avoiding an appeal to knowledge whose source is not itself explained. Specifically, we fit the pattern of changes in strategy use over time as children learn addition, as well as the overall error rate and error types reported empirically. The model suggests that the critical element allowing strategy discovery may be prior learning, rather than metacognitive strategy evaluation. We close by offering several empirical predictions and propose that what others have called strategies might often be decomposable into elements that can be assembled on the fly as problem solving unfolds in real time. Keywords: M athematical Cognition; Strategy Discovery; Reinforcement Learning; M etacognition. Introduction Single-digit addition is one of the first hurdles children master on their way to mathematical competence. Given the importance of mathematics to educational attainment, it is unsurprising that the process by which children learn addition has received considerable attention (eg, Siegler & Jenkins, 1989). A remarkable observation from these studies is that, once they are equipped with the ability to count out sums on their fingers, children spontaneously (without instruction) exhibit faster strategies. Despite this willingness to innovate, children rarely arrive at a strategy that, when executed correctly, leads to the wrong answer. This poses a real problem for trial and error theories of learning. As they acquire new, faster strategies, how do children know which strategies to avoid? Several attempts have been made to model the evolution of children‟s approaches to solving simple addition problems. The apparent absence of explicit instruction in the use of particular observed s trategies would normally make reinforcement learning a candidate mechanism, and indeed Background The data our model attempts to account for comes from a study examining 4 and 5 year olds‟ discoveries of new finger addition strategies (Siegler & Jenkins, 1989). Children are assumed to come to the task knowing a correct but inefficient strategy, and are observed in a series of sessions spread over approximately three months as they solve simple addition problems . Over this time, children gradually acquire strategies that lead to faster completion of the task.
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paper_title: A competitive neural model of small number detection

paper_content:

The ability to represent numbers is a key attribute for both humans and animals. Recent developments in the understanding of numerical processing has led to the proposal that humans utilise two independent representations of number, one for real numbers and another for integers. We describe a computational model of small number detection to explore the relationship between these core systems of number. We use a combination of unsupervised and supervised neural networks to simulate the interaction between the real and integer representations. For real values we use a self-organised spatial representation of number. For integer values we use a supervised network motivated by linguistic processing. During training and testing, the networks exhibit behavioural characteristics such as the number size and numerical distance effects. Each representation is combined using the mixture-of-experts architecture that allows us to model the subitization limit (the maximum number of visual stimuli that can be accurately quantified almost immediately) as the competitive allocation of representations for number detection, where the crossover point between deploying the real and integer representations of number is obtained through a process of learning. Our results suggest that the existence of two core systems of number is at least computationally plausible and further suggests that the subitization limit emerges through the interaction of spatial and linguistic numerical processing. This provides computational evidence for one way in which small and large numbers are related in humans.
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paper_title: Coding of Cognitive Magnitude Compressed Scaling of Numerical Information in the Primate Prefrontal Cortex

paper_content:

Whether cognitive representations are better conceived as language-based, symbolic representations or perceptually related, analog representations is a subject of debate. If cognitive processes parallel perceptual processes, then fundamental psychophysical laws should hold for each. To test this, we analyzed both behavioral and neuronal representations of numerosity in the prefrontal cortex of rhesus monkeys. The data were best described by a nonlinearly compressed scaling of numerical information, as postulated by the Weber-Fechner law or Stevens' law for psychophysical/sensory magnitudes. This nonlinear compression was observed on the neural level during the acquisition phase of the task and maintained through the memory phase with no further compression. These results suggest that certain cognitive and perceptual/sensory representations share the same fundamental mechanisms and neural coding schemes.
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paper_title: SCADS: A model of children’s strategy choices and strategy discoveries

paper_content:

Preschoolers show surprising competence in choosing adaptively among alternative strategies and in discovering new approaches. The SCADS computer simulation illustrates how simple processes can generate this impressive competence. The model's behavior parallels data on children's addition in at least eight ways: It uses diverse strategies over prolonged periods of time, makes adaptive choices among strategies, discovers the same strategies as children, discovers strategies in the same sequence as children, makes discoveries without trial and error, makes discoveries without having experienced failure, narrowly generalizes new approaches, and generalizes more broadly following challenging problems. SCADS thus indicates plausible sources of young children's surprising competence at strategy choice and strategy discovery.
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paper_title: Beyond the mental number line: A neural network model of number–space interactions

paper_content:

It is commonly assumed that there is an interaction between the representations of number and space (e.g., Dehaene, Bossini, & Giraux, 1993; Walsh, 2003), typically ascribed to a mental number line. The exact nature of this interaction has remained elusive, however. Here we propose that spatial aspects are not inherent to number representations, but that instead spatial and numerical representations are separate. However, cultural factors establish ties between them. By extending earlier models (Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006; Verguts, Fias, & Stevens, 2005) based on this hypothesis, the authors present computer simulations showing that a model incorporating this idea can account for data from a series of studies. These results suggest that number-space interactions are emergent properties resulting from the interaction between different brain areas.
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paper_title: Children's Counting and Concepts of Number

paper_content:

I Number Words.- 1 Introduction and Overview of Different Uses of Number Words.- 2 The Number-Word Sequence: An Overview of Its Acquisition and Elaboration.- II Correspondence Errors in Counting Objects.- 3 Correspondence Errors in Children's Counting.- 4 Effects of Object Arrangement on Counting Correspondence Errors and on the Indicating Act.- 5 Effects of Object Variables and Age of Counter on Correspondence Errors Made When Counting Objects in Rows.- 6 Correspondence Errors in Children's Counting: A Summary.- III Concepts of Cardinality.- 7 Children's Early Knowledge About Relationships Between Counting and Cardinality.- 8 Later Conceptual Relationships Between Counting and Cardinality: Addition and Subtraction of Cardinal Numbers.- 9 Uses of Counting and Matching in Cardinal Equivalence Situations: Equivalence and Order Relations on Cardinal Numbers.- IV Number Words, Counting, and Cardinality: The Increasing Integration of Sequence, Count, and Cardinal Meanings.- 10 Early Relationships Among Sequence Number Words, Counting Correspondence, and Cardinality.- 11 An Overview of Changes in Children's Number Word Concepts from Age 2 Through 8.- References.- Author Index.
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paper_title: Numbers and space: A computational model of the SNARC effect.

paper_content:

The SNARC (spatial numerical associations of response codes) effect reflects the tendency to respond faster with the left hand to relatively small numbers and with the right hand to relatively large numbers (S. Dehaene, S. Bossini, & P. Giraux, 1993). Using computational modeling, the present article aims to provide a framework for conceptualizing the SNARC effect. In line with models of spatial stimulus-response congruency, the authors modeled the SNARC effect as the result of parallel activation of preexisting links between magnitude and spatial representation and short-term links created on the basis of task instructions. This basic dual-route model simulated all characteristics associated with the SNARC effect. In addition, 2 experiments tested and confirmed new predictions derived from the model.
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paper_title: A model of exact small-number representation

paper_content:

To account for the size effect in numerical comparison, three assumptions about the internal structure of the mental number line (e.g., Dehaene, 1992) have been proposed. These are magnitude coding (e.g., Zorzi & Butterworth, 1999), compressed scaling (e.g., Dehaene, 1992), and increasing variability (e.g., Gallistel & Gelman, 1992). However, there are other tasks besides numerical comparison for which there is clear evidence that the mental number line is accessed, and no size effect has been observed in these tasks. This is contrary to the predictions of these three assumptions. Moreover, all three assumptions have difficulties explaining certain symmetries in priming studies of number naming and parity judgment. We propose a neural network model that avoids these three assumptions but, instead, uses place coding, linear scaling, and constant variability on the mental number line. We train the model on naming, parity judgment, and comparison and show that the size effect appears in comparison, but not in naming or parity judgment. Moreover, no asymmetries appear in primed naming or primed parity judgment with this model, in line with empirical data. Implications of our findings are discussed. This work was supported by Grant P5/04 from the Interuniversity Attraction Poles Program—Belgian Science Policy and by a GOA grant from the Ghent University Research Council to W.F.
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paper_title: Learning Deep Architectures for AI

paper_content:

Can machine learning deliver AI? Theoretical results, inspiration from the brain and cognition, as well as machine learning experiments suggest that in order to learn the kind of complicated functions that can represent high-level abstractions (e.g. in vision, language, and other AI-level tasks), one would need deep architectures. Deep architectures are composed of multiple levels of non-linear operations, such as in neural nets with many hidden layers, graphical models with many levels of latent variables, or in complicated propositional formulae re-using many sub-formulae. Each level of the architecture represents features at a different level of abstraction, defined as a composition of lower-level features. Searching the parameter space of deep architectures is a difficult task, but new algorithms have been discovered and a new sub-area has emerged in the machine learning community since 2006, following these discoveries. Learning algorithms such as those for Deep Belief Networks and other related unsupervised learning algorithms have recently been proposed to train deep architectures, yielding exciting results and beating the state-of-the-art in certain areas. Learning Deep Architectures for AI discusses the motivations for and principles of learning algorithms for deep architectures. By analyzing and comparing recent results with different learning algorithms for deep architectures, explanations for their success are proposed and discussed, highlighting challenges and suggesting avenues for future explorations in this area.
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paper_title: Modeling language and cognition with deep unsupervised learning: a tutorial overview

paper_content:

Deep unsupervised learning in stochastic recurrent neural networks with many layers of hidden units is a recent breakthrough in neural computation research. These networks build a hierarchy of progressively more complex distributed representations of the sensory data by fitting a hierarchical generative model. In this article we discuss the theoretical foundations of this approach and we review key issues related to training, testing and analysis of deep networks for modeling language and cognitive processing. The classic letter and word perception problem of McClelland and Rumelhart (1981) is used as a tutorial example to illustrate how structured and abstract representations may emerge from deep generative learning. We argue that the focus on deep architectures and generative (rather than discriminative) learning represents a crucial step forward for the connectionist modeling enterprise, because it offers a more plausible model of cortical learning as well as way to bridge the gap between emergentist connectionist models and structured Bayesian models of cognition.
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paper_title: Deep learning

paper_content:

Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.
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paper_title: Emergence of a 'visual number sense' in hierarchical generative models

paper_content:

Numerosity estimation is phylogenetically ancient and foundational to human mathematical learning, but its computational bases remain controversial. Here we show that visual numerosity emerges as a statistical property of images in 'deep networks' that learn a hierarchical generative model of the sensory input. Emergent numerosity detectors had response profiles resembling those of monkey parietal neurons and supported numerosity estimation with the same behavioral signature shown by humans and animals.
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paper_title: Long-Short Term Memory Networks for Modelling Embodied Mathematical Cognition in Robots

paper_content:

Mathematical competence can endow robots with the necessary capability for abstract and symbolic processing, which is required for higher cognitive functions such as natural language understanding. But, so far, only few attempts have been made to model mathematical cognition in robots. This paper presents an experimental evaluation of the Long- Short Term Memory networks for modeling the simple mathematical operation of single-digits addition in a cognitive robot. To this end, the robotic model creates an association between the proprioceptive information from finger counting and the handwritten digits of the MNIST dataset. In practice, the model executes two tasks concurrently: it recognizes the handwritten digits in a sequence and sums them. The results show that the association with fingers can improve the robot precision, as observed in children. Also, the robot makes a disproportionate number of split-five errors similarly to what observed in studies with children and adults, hence giving evidence to support the hypothesis that these errors are due the use of a five-fingers counting system.
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paper_title: The iCub learns numbers: An embodied cognition study

paper_content:

Thanks to recent technological advances and the increasing interest towards the Cognitive Developmental Robotics (CDR) paradigm, many popular platforms for scientific research have been designed in order to resemble the shape of the human body. The motivation behind this strongly humanoid design is the embodied cognition hypothesis, which affirms that all aspects of cognition are shaped by aspects of the body. Thus CDR is based on a synthetic approach that aims to provide new understanding on how human beings develop their higher cognitive functions. Following this paradigm we have developed an artificial model, based on artificial neural networks, to explore finger counting and the association of number words (or tags) to the fingers, as bootstrapping for the representation of numbers in the humanoid robot iCub. In this paper, we detail experiments of our model with the iCub robotic platform. Results of the number learning with propri-oceptive data from the real platform are reported and compared with the ones obtained instead, with the simulated platform. These results support the thesis that learning the number words in sequence, along with finger configurations helps the building of the initial representation of number in the robot. Moreover, the comparison between the real and simulated iCub gives insights on the use of these platforms as a tool for CDR.
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paper_title: An Embodied Developmental Robotic Model of Interactions between Numbers and Space

paper_content:

In this paper we describe an embodied developmental model of the interactions between the neural representations of numbers and space in the humanoid robot iCub. We show how a simple developmental process that mimics real-world cultural biases leads to the emergence of certain properties of the number and space representation system that enable the robot to reproduce well-known experimental phenomena. We demonstrate the validity of the proposed approach by showing that it leads to the reproduction of three psychological phenomena connected with number processing, namely size and distance effects, the SNARC effect and the Posner-SNARC effect.
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paper_title: Grounding fingers, words and numbers in a cognitive developmental robot

paper_content:

The young math learner must make the transition from a concrete number situation, such as that of counting objects (fingers often being the most readily available), to that of using a written symbolic form that stands for the quantities the sets of objects come to represent. This challenging process is often coupled to that of learning a verbal number system that is not always transparent to children. A number of theoretical approaches have been advanced to explain aspects of how this transition takes place in cognitive development. The results obtained with the model presented here, show that a symbol grounding approach can be used to implement aspects of this transition in a cognitive robot. In the current extended version, the model develops finger and word representations, through the use of finger counting and verbal counting strategies, together with the visual representations of learned number symbols, which it uses to perform basic arithmetic operations. In the final training phases, the model is able to do this using only the number symbols as addends. We consider this an example of symbolic grounding, in that through the direct sensory experience with the body (finger counting), a category of linguistic symbol is learned (number words), and both types of representations subsequently serve to ground higher level (numerical) symbols, which are later used exclusively to perform the arithmetic operations.
---
paper_title: Reducing the Dimensionality of Data with Neural Networks

paper_content:

High-dimensional data can be converted to low-dimensional codes by training a multilayer neural network with a small central layer to reconstruct high-dimensional input vectors. Gradient descent can be used for fine-tuning the weights in such “autoencoder” networks, but this works well only if the initial weights are close to a good solution. We describe an effective way of initializing the weights that allows deep autoencoder networks to learn low-dimensional codes that work much better than principal components analysis as a tool to reduce the dimensionality of data.
---
paper_title: The function of gesture in learning to count: more than keeping track *

paper_content:

Preschoolers count objects most accurately when they gesture as they count. This study tests two possible explanations for this effect. One is that gesture helps children keep track of counted items. Another is that gesture helps children coordinate saying the number words and tagging the items. Twenty preschoolers counted chips under three types of conditions: with gesture prohibited, with active gesture, and with a puppet gesturing as children counted aloud. The puppet conditions were intended to distinguish the benefits of keeping track with gesture from the benefits of active gesture. Children counted more accurately when they or the puppet gestured than when gesture was prohibited. However, children's errors differed when they and the puppet gestured. When children gestured themselves, they made errors keeping track, but when the puppet gestured, they made errors coordinating number words and items. Thus, active gesture helps children both to keep track and to coordinate tagging the items and saying the number words. In these ways, active gesture helps children implement their knowledge of one-to-one correspondence.
---
paper_title: TRoPICALS: A Computational Embodied Neuroscience Model of Compatibility Effects

paper_content:

Perceiving objects activates the representation of their affordances. For example, experiments on compatibility effects showed that categorizing objects by producing certain handgrips (power or precision) is faster if the requested responses are compatible with the affordance elicited by the size of objects (e.g., small or large). The article presents a neural-network architecture that provides a general framework to account for compatibility effects. The model was designed with a methodological approach (computational embodied neuroscience) that aims to provide increasingly general accounts of brain and behavior (4 sources of constraints are used: neuroscientific data, behavioral data, embodied systems, reproduction of learning processes). The model is based on 4 principles of brain organization that we claim underlie most compatibility effects. First, visual perception and action are organized in the brain along a dorsal neural pathway encoding affordances and a ventral pathway encoding goals. Second, the prefrontal cortex within the ventral pathway gives a top-down bias to action selection by integrating information on stimuli, context, and goals. Third, reaction times depend on dynamic neural competitions for action selection that integrate bottom-up and top-down information. The congruence or incongruence between affordances and goals explains the different reaction times found in the experiments. Fourth, as words trigger internal simulations of their referents, they can cause compatibility effects as objects do. We validated the model by reproducing and explaining 3 types of compatibility effects and showed its heuristic power by producing 2 testable predictions. We also assessed the explicative power of the model by comparing it with related models and showed how it can be extended to account for other compatibility effects.
---
paper_title: An embodied model for handwritten digits recognition in a cognitive robot

paper_content:

This paper presents an embodied model for recognition of handwritten digits in a cognitive developmental robot scenario. Inspired by neuro-psychological data, the model integrates three modules: a stacked auto-encoder network to process the visual information, a feedforward neural controller for the fingers, and a generalized regression network that associates number digits to finger configurations. Results from developmental learning experiments show an improvement in the digits' recognition rate thanks to the inclusion of the robot fingers in the training especially in its early stages (epochs) or with a low number of examples. This behaviour can be linked to that observed in psychological studies with children, who seem to benefit of finger counting only in the initial stage of mathematical learning. These results suggest the potential of the embodied approach to favour the creation of a psychologically plausible developmental model for mathematical cognition in robots and to support the creation of more complex models of human-like behaviours.
---
paper_title: Fast Optimal Leaf Ordering For Hierarchical Clustering

paper_content:

We present the first practical algorithm for the optimal linear leaf ordering of trees that are generated by hierarchical clustering. Hierarchical clustering has been extensively used to analyze gene expression data, and we show how optimal leaf ordering can reveal biological structure that is not observed with an existing heuristic ordering method. For a tree with n leaves, there are 2 n−1 linear orderings consistent with the structure of the tree. Our optimal leaf ordering algorithm runs in time O(n 4 ), and we present fur
---
paper_title: A Deep Learning Neural Network for Number Cognition: A bi-cultural study with the iCub

paper_content:

The novel deep learning paradigm offers a highly biologically plausible way to train neural network architectures with many layers, inspired by the hierarchical organization of the human brain. Indeed, deep learning gives a new dimension to research modeling human cognitive behaviors, and provides new opportunities for applications in cognitive robotics. In this paper, we present a novel deep neural network architecture for number cognition by means of finger counting and number words. The architecture is composed of 5 layers and is designed in a way that allows it to learn numbers from one to ten by associating the sensory inputs (motor and auditory) coming from the iCub humanoid robotic platform. The architecture performance is validated and tested in two developmental experiments. In the first experiment, standard backpropagation is compared with a deep learning approach, in which weights and biases are pre-trained by means of a greedy algorithm and then refined with backpropagation. In the second experiment, six bi-cultural number learning conditions are compared to explore the impact of different languages (for number words) and finger counting strategies. The developmental experiments confirm the validity of the model and the increase in efficiency given by the deep learning approach. Results of the bi-cultural study are presented and discussed with respect to the neuro-psychological literature and implications of the results for learning situations are briefly outlined.
---
paper_title: Mind the gap between both hands: Evidence for internal finger-based number representations in children's mental calculation

paper_content:

Abstract At a certain stage of development, virtually all children use some kind of external finger-based number representation. However, only little is known about how internal traces of this early external representation may still influence calculation even when finger calculation ceases to be an efficient tool in mental calculation. In the present study, we provide evidence for a disproportionate number of split-five errors (i.e., errors with a difference of ±5 from the correct result) in mental addition and subtraction (e.g., 18 − 7 = 6). We will argue that such errors may have different origins. For complex problems and initially also for simple problems they are due to failure to keep track of ‘full hands’ in counting or calculation procedures. However, for simple addition problems split-five errors may later also be caused by mistakes in directly retrieving the result from declarative memory. In general, the present results are interpreted in terms of a transient use of mental finger patterns – in particular the whole hand pattern – in children's mental calculation.
---
paper_title: Modelling Learning to Count in Humanoid Robots

paper_content:

In reference to IEEE copyrighted material which is used with permission in this thesis, the IEEE does not endorse any of Plymouth University's products or services. Internal or personal use of this material is permitted. If interested in reprinting/republishing IEEE copyrighted material for advertising or promotional purposes or for creating new collective works for resale or redistribution, please go to http://www.ieee.org/publications_standards/publications/rights/rights_link.html to learn how to obtain a License from RightsLink.
---
paper_title: Developmental Robotics: From Babies to Robots

paper_content:

Developmental robotics is a collaborative and interdisciplinary approach to robotics that is directly inspired by the developmental principles and mechanisms observed in children's cognitive development. It builds on the idea that the robot, using a set of intrinsic developmental principles regulating the real-time interaction of its body, brain, and environment, can autonomously acquire an increasingly complex set of sensorimotor and mental capabilities. This volume, drawing on insights from psychology, computer science, linguistics, neuroscience, and robotics, offers the first comprehensive overview of a rapidly growing field. After providing some essential background information on robotics and developmental psychology, the book looks in detail at how developmental robotics models and experiments have attempted to realize a range of behavioral and cognitive capabilities. The examples in these chapters were chosen because of their direct correspondence with specific issues in child psychology research; each chapter begins with a concise and accessible overview of relevant empirical and theoretical findings in developmental psychology. The chapters cover intrinsic motivation and curiosity; motor development, examining both manipulation and locomotion; perceptual development, including face recognition and perception of space; social learning, emphasizing such phenomena as joint attention and cooperation; language, from phonetic babbling to syntactic processing; and abstract knowledge, including models of number learning and reasoning strategies. Boxed text offers technical and methodological details for both psychology and robotics experiments.
---
paper_title: Robotic model of the contribution of gesture to learning to count

paper_content:

In this paper a robotic connectionist model of the contribution of gesture to learning to count is presented. By formulating a recurrent artificial neural network model of the phenomenon and assessing its performance without and with gesture it is demonstrated that the proprioceptive signal connected with gesture carries information which may be exploited when learning to count. The behaviour of the model is similar to that of human children in terms of the effect of gesture and the size of the counted set, although the detailed patterns of errors made by the model and human children are different.
---
paper_title: Beyond the mental number line: A neural network model of number–space interactions

paper_content:

It is commonly assumed that there is an interaction between the representations of number and space (e.g., Dehaene, Bossini, & Giraux, 1993; Walsh, 2003), typically ascribed to a mental number line. The exact nature of this interaction has remained elusive, however. Here we propose that spatial aspects are not inherent to number representations, but that instead spatial and numerical representations are separate. However, cultural factors establish ties between them. By extending earlier models (Gevers, Verguts, Reynvoet, Caessens, & Fias, 2006; Verguts, Fias, & Stevens, 2005) based on this hypothesis, the authors present computer simulations showing that a model incorporating this idea can account for data from a series of studies. These results suggest that number-space interactions are emergent properties resulting from the interaction between different brain areas.
---
paper_title: You can count on the motor cortex: Finger counting habits modulate motor cortex activation evoked by numbers

paper_content:

The embodied cognition framework suggests that neural systems for perception and action are engaged during higher cognitive processes. In an event-related fMRI study, we tested this claim for the abstract domain of numerical symbol processing: is the human cortical motor system part of the representation of numbers, and is organization of numerical knowledge influenced by individual finger counting habits? Developmental studies suggest a link between numerals and finger counting habits due to the acquisition of numerical skills through finger counting in childhood. In the present study, digits 1 to 9 and the corresponding number words were presented visually to adults with different finger counting habits, i.e. left- and right-starters who reported that they usually start counting small numbers with their left and right hand, respectively. Despite the absence of overt hand movements, the hemisphere contralateral to the hand used for counting small numbers was activated when small numbers were presented. The correspondence between finger counting habits and hemispheric motor activation is consistent with an intrinsic functional link between finger counting and number processing.
---
paper_title: Emergence of a 'visual number sense' in hierarchical generative models

paper_content:

Numerosity estimation is phylogenetically ancient and foundational to human mathematical learning, but its computational bases remain controversial. Here we show that visual numerosity emerges as a statistical property of images in 'deep networks' that learn a hierarchical generative model of the sensory input. Emergent numerosity detectors had response profiles resembling those of monkey parietal neurons and supported numerosity estimation with the same behavioral signature shown by humans and animals.
---
paper_title: Making fingers and words count in a cognitive robot

paper_content:

Evidence from developmental as well as neuroscientific studies suggest that finger counting activity plays an important role in the acquisition of numerical skills in children. It has been claimed that this skill helps in building motor-based representations of number that continue to influence number processing well into adulthood, facilitating the emergence of number concepts from sensorimotor experience through a bottom-up process. The act of counting also involves the acquisition and use of a verbal number system of which number words are the basic building blocks. Using a Cognitive Developmental Robotics paradigm we present results of a modeling experiment on whether finger counting and the association of number words (or tags) to fingers, could serve to bootstrap the representation of number in a cognitive robot, enabling it to perform basic numerical operations such as addition. The cognitive architecture of the robot is based on artificial neural networks, which enable the robot to learn both sensorimotor skills (finger counting) and linguistic skills (using number words). The results obtained in our experiments show that learning the number words in sequence along with finger configurations helps the fast building of the initial representation of number in the robot. Number knowledge, is instead, not as efficiently developed when number words are learned out of sequence without finger counting. Furthermore, the internal representations of the finger configurations themselves, developed by the robot as a result of the experiments, sustain the execution of basic arithmetic operations, something consistent with evidence coming from developmental research with children. The model and experiments demonstrate the importance of sensorimotor skill learning in robots for the acquisition of abstract knowledge such as numbers.
---
paper_title: Comparison of parametric representations for monosyllabic word recognition in continuously spoken sentences

paper_content:

Several parametric representations of the acoustic signal were compared with regard to word recognition performance in a syllable-oriented continuous speech recognition system. The vocabulary included many phonetically similar monosyllabic words, therefore the emphasis was on the ability to retain phonetically significant acoustic information in the face of syntactic and duration variations. For each parameter set (based on a mel-frequency cepstrum, a linear frequency cepstrum, a linear prediction cepstrum, a linear prediction spectrum, or a set of reflection coefficients), word templates were generated using an efficient dynamic warping method, and test data were time registered with the templates. A set of ten mel-frequency cepstrum coefficients computed every 6.4 ms resulted in the best performance, namely 96.5 percent and 95.0 percent recognition with each of two speakers. The superior performance of the mel-frequency cepstrum coefficients may be attributed to the fact that they better represent the perceptually relevant aspects of the short-term speech spectrum.
---
paper_title: Representation of Number in Animals and Humans: A Neural Model

paper_content:

This article addresses the representation of numerical information conveyed by nonsymbolic and symbolic stimuli. In a first simulation study, we show how number-selective neurons develop when an initially uncommitted neural network is given nonsymbolic stimuli as input (e.g., collections of dots) under unsupervised learning. The resultant network is able to account for the distance and size effects, two ubiquitous effects in numerical cognition. Furthermore, the properties of the network units conform in detail to the characteristics of recently discovered number-selective neurons. In a second study, we simulate symbol learning by presenting symbolic and nonsymbolic input simultaneously. The same number-selective neurons learn to represent the numerical meaning of symbols. In doing so, they show properties reminiscent of the originally available number-selective neurons, but at the same time, the representational efficiency of the neurons is increased when presented with symbolic input. This finding presents a concrete proposal on the linkage between higher order numerical cognition and more primitive numerical abilities and generates specific predictions on the neural substrate of number processing.
---
paper_title: An embodied model for handwritten digits recognition in a cognitive robot

paper_content:

This paper presents an embodied model for recognition of handwritten digits in a cognitive developmental robot scenario. Inspired by neuro-psychological data, the model integrates three modules: a stacked auto-encoder network to process the visual information, a feedforward neural controller for the fingers, and a generalized regression network that associates number digits to finger configurations. Results from developmental learning experiments show an improvement in the digits' recognition rate thanks to the inclusion of the robot fingers in the training especially in its early stages (epochs) or with a low number of examples. This behaviour can be linked to that observed in psychological studies with children, who seem to benefit of finger counting only in the initial stage of mathematical learning. These results suggest the potential of the embodied approach to favour the creation of a psychologically plausible developmental model for mathematical cognition in robots and to support the creation of more complex models of human-like behaviours.
---
paper_title: Mind the gap between both hands: Evidence for internal finger-based number representations in children's mental calculation

paper_content:

Abstract At a certain stage of development, virtually all children use some kind of external finger-based number representation. However, only little is known about how internal traces of this early external representation may still influence calculation even when finger calculation ceases to be an efficient tool in mental calculation. In the present study, we provide evidence for a disproportionate number of split-five errors (i.e., errors with a difference of ±5 from the correct result) in mental addition and subtraction (e.g., 18 − 7 = 6). We will argue that such errors may have different origins. For complex problems and initially also for simple problems they are due to failure to keep track of ‘full hands’ in counting or calculation procedures. However, for simple addition problems split-five errors may later also be caused by mistakes in directly retrieving the result from declarative memory. In general, the present results are interpreted in terms of a transient use of mental finger patterns – in particular the whole hand pattern – in children's mental calculation.
---
paper_title: The Influence of Implicit Hand-Based Representations on Mental Arithmetic

paper_content:

Recently, a strong functional relationship between finger counting and number processing has been suggested. It has been argued that bodily experiences such as finger counting may influence the structure of the basic mental representations of numbers even in adults. However, to date it remains unclear whether the structure of finger counting systems also influences educated adults’ performance in mental arithmetic. In the present study, we pursued this question by examining finger-based sub-base-five effects in an addition production task. With the standard effect of a carry operation (i.e., base 10 crossing) being replicated, we observed an additional sub-base-five effect such that crossing a sub-base 5 boundary led to a relative response time increase. For the case of mental arithmetic sub-base-five effects have previously been reported only in children. However, it remains unclear whether finger-based numerical effects in mental arithmetic reflect an important but transitory step in the development of arithmetical skills. The current findings suggest that even in adults embodied representations such as finger counting patterns modulate arithmetic performance. Thus, they support the general idea that even seemingly abstract cognition in adults may at least partly be rooted in our bodily experiences.
---
paper_title: Making fingers and words count in a cognitive robot

paper_content:

Evidence from developmental as well as neuroscientific studies suggest that finger counting activity plays an important role in the acquisition of numerical skills in children. It has been claimed that this skill helps in building motor-based representations of number that continue to influence number processing well into adulthood, facilitating the emergence of number concepts from sensorimotor experience through a bottom-up process. The act of counting also involves the acquisition and use of a verbal number system of which number words are the basic building blocks. Using a Cognitive Developmental Robotics paradigm we present results of a modeling experiment on whether finger counting and the association of number words (or tags) to fingers, could serve to bootstrap the representation of number in a cognitive robot, enabling it to perform basic numerical operations such as addition. The cognitive architecture of the robot is based on artificial neural networks, which enable the robot to learn both sensorimotor skills (finger counting) and linguistic skills (using number words). The results obtained in our experiments show that learning the number words in sequence along with finger configurations helps the fast building of the initial representation of number in the robot. Number knowledge, is instead, not as efficiently developed when number words are learned out of sequence without finger counting. Furthermore, the internal representations of the finger configurations themselves, developed by the robot as a result of the experiments, sustain the execution of basic arithmetic operations, something consistent with evidence coming from developmental research with children. The model and experiments demonstrate the importance of sensorimotor skill learning in robots for the acquisition of abstract knowledge such as numbers.
---
paper_title: A model of exact small-number representation

paper_content:

To account for the size effect in numerical comparison, three assumptions about the internal structure of the mental number line (e.g., Dehaene, 1992) have been proposed. These are magnitude coding (e.g., Zorzi & Butterworth, 1999), compressed scaling (e.g., Dehaene, 1992), and increasing variability (e.g., Gallistel & Gelman, 1992). However, there are other tasks besides numerical comparison for which there is clear evidence that the mental number line is accessed, and no size effect has been observed in these tasks. This is contrary to the predictions of these three assumptions. Moreover, all three assumptions have difficulties explaining certain symmetries in priming studies of number naming and parity judgment. We propose a neural network model that avoids these three assumptions but, instead, uses place coding, linear scaling, and constant variability on the mental number line. We train the model on naming, parity judgment, and comparison and show that the size effect appears in comparison, but not in naming or parity judgment. Moreover, no asymmetries appear in primed naming or primed parity judgment with this model, in line with empirical data. Implications of our findings are discussed. This work was supported by Grant P5/04 from the Interuniversity Attraction Poles Program—Belgian Science Policy and by a GOA grant from the Ghent University Research Council to W.F.
---
paper_title: Interactions between number and space in parietal cortex

paper_content:

Since the time of Pythagoras, numerical and spatial representations have been inextricably linked. We suggest that the relationship between the two is deeply rooted in the brain's organization for these capacities. Many behavioural and patient studies have shown that numerical-spatial interactions run far deeper than simply cultural constructions, and, instead, influence behaviour at several levels. By combining two previously independent lines of research, neuroimaging studies of numerical cognition in humans, and physiological studies of spatial cognition in monkeys, we propose that these numerical-spatial interactions arise from common parietal circuits for attention to external space and internal representations of numbers.
---
paper_title: Developmental Robotics: From Babies to Robots

paper_content:

Developmental robotics is a collaborative and interdisciplinary approach to robotics that is directly inspired by the developmental principles and mechanisms observed in children's cognitive development. It builds on the idea that the robot, using a set of intrinsic developmental principles regulating the real-time interaction of its body, brain, and environment, can autonomously acquire an increasingly complex set of sensorimotor and mental capabilities. This volume, drawing on insights from psychology, computer science, linguistics, neuroscience, and robotics, offers the first comprehensive overview of a rapidly growing field. After providing some essential background information on robotics and developmental psychology, the book looks in detail at how developmental robotics models and experiments have attempted to realize a range of behavioral and cognitive capabilities. The examples in these chapters were chosen because of their direct correspondence with specific issues in child psychology research; each chapter begins with a concise and accessible overview of relevant empirical and theoretical findings in developmental psychology. The chapters cover intrinsic motivation and curiosity; motor development, examining both manipulation and locomotion; perceptual development, including face recognition and perception of space; social learning, emphasizing such phenomena as joint attention and cooperation; language, from phonetic babbling to syntactic processing; and abstract knowledge, including models of number learning and reasoning strategies. Boxed text offers technical and methodological details for both psychology and robotics experiments.
---
paper_title: You can count on the motor cortex: Finger counting habits modulate motor cortex activation evoked by numbers

paper_content:

The embodied cognition framework suggests that neural systems for perception and action are engaged during higher cognitive processes. In an event-related fMRI study, we tested this claim for the abstract domain of numerical symbol processing: is the human cortical motor system part of the representation of numbers, and is organization of numerical knowledge influenced by individual finger counting habits? Developmental studies suggest a link between numerals and finger counting habits due to the acquisition of numerical skills through finger counting in childhood. In the present study, digits 1 to 9 and the corresponding number words were presented visually to adults with different finger counting habits, i.e. left- and right-starters who reported that they usually start counting small numbers with their left and right hand, respectively. Despite the absence of overt hand movements, the hemisphere contralateral to the hand used for counting small numbers was activated when small numbers were presented. The correspondence between finger counting habits and hemispheric motor activation is consistent with an intrinsic functional link between finger counting and number processing.
---
paper_title: Common substrate for mental arithmetic and finger representation in the parietal cortex

paper_content:

The history of mathematics provides several examples of the use of fingers to count or calculate. These observations converge with developmental data showing that fingers play a critical role in the acquisition of arithmetic knowledge. Further studies evidenced specific interference of finger movements with arithmetic problem solving in adults, raising the question of whether or not finger and number manipulations rely on common brain areas. In the present study, functional magnetic resonance imaging (fMRI) was used to investigate the possible overlap between the brain areas involved in mental arithmetic and those involved in finger discrimination. Solving subtraction and multiplication problems was found to increase cerebral activation bilaterally in the horizontal part of the intraparietal sulcus (hIPS) and in the posterior part of the superior parietal lobule (PSPL). Finger discrimination was associated with increased activity in a bilateral occipito-parieto-precentral network extending from the extrastriate body area to the primary somatosensory and motor cortices. A conjunction analysis showed common areas for mental arithmetic and finger representation in the hIPS and PSPL bilaterally. Voxelwise correlations further showed that finger discrimination and mental arithmetic induced a similar pattern of activity within the parietal areas only. Pattern similarity was more important for the left than for the right hIPS and for subtraction than for multiplication. These findings provide the first evidence that the brain circuits involved in finger representation also underlie arithmetic operations in adults. © 2012 Elsevier Inc.
---
paper_title: Cognitive Developmental Robotics: A Survey

paper_content:

Cognitive developmental robotics (CDR) aims to provide new understanding of how human's higher cognitive functions develop by means of a synthetic approach that developmentally constructs cognitive functions. The core idea of CDR is ldquophysical embodimentrdquo that enables information structuring through interactions with the environment, including other agents. The idea is shaped based on the hypothesized development model of human cognitive functions from body representation to social behavior. Along with the model, studies of CDR and related works are introduced, and discussion on the model and future issues are argued.
---
paper_title: Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles

paper_content:

Promoting self-explanation (i.e., generating explanations for oneself in an attempt to make sense of new information) is a recommended study strategy and instructional practice. A meta-analysis of the literature on prompting self-explanation to improve mathematics learning confirmed that prompted self-explanation leads to a small to moderate improvement in procedural knowledge, conceptual knowledge and procedural transfer when assessed immediately after the intervention. However, evidence that self-explanation reliably promotes learning within a classroom context or retention of knowledge over a delay is much more limited. Moderator analyses indicated that the effect on immediate outcomes was stronger if scaffolding of high-quality explanation was provided but did not vary based on whether time on task was controlled across conditions. Based on the research literature, we propose instructional recommendations for mathematics educators: (a) scaffold high-quality explanations via training on self-explanation or structuring the self-explanation responses, (b) design explanation prompts so they do not sacrifice attention to other important content, (c) prompt learners to explain correct information, and (d) prompt learners to explain why common misconceptions are incorrect. We conclude with issues for future research, such as the need for additional research on effective use of self-explanation in classroom contexts and the merits of self-explanation relative to alternative instructional techniques.
---
paper_title: Guided Play Principles and Practices

paper_content:

Competing trends in early childhood education emphasize the need for strong curricular approaches and for unfettered exploration. We propose an approach to early learning that avoids this false dichotomy: guided play. Guided play takes advantage of children’s natural abilities to learn through play by allowing them to express their autonomy within a prepared environment and with adult scaffolding. We provide examples of how guided-play situations have been implemented in past work, as well as evidence that guided play is successful for education across a range of content—perhaps even more successful than other pedagogical approaches.
---
paper_title: Preschoolers' counting: Principles before skill

paper_content:

Three- to 5-year-old children participated in one of 4 counting experiments. On the assumption that performance demands can mask the young child's implicit knowledge of the counting principles, 3 separate experiments assessed a child's ability to detect errors in a puppet's application of the one-one, stable-order and cardinal count principles. In a fourth experiment children counted in different conditions designed to vary performance demands. Since children in the errror-detection experiments did not have to do the counting, we predicted excellent performance even on set sizes beyond the range a young child counts accurately. That they did well on these experiments supports the view that errors in counting—at least for set sizes up to 20—reflect performance demands and not the absence of implicit knowledge of the counting principles. In the final experiment, where children did the counting themselves, set size did affect their success. So did some variations in conditions, the most difficult of which was the one where children had to count 3-dimensional objects which were under a plexiglass cover. We expected that this condition would interfere with the child's tendency to point and touch objects in order to keep separate items which have been counted from those which have not been counted.
---
paper_title: Beyond gamification: reconceptualizing game-based learning in early childhood environments

paper_content:

The recent promotion and adoption of digital game-based learning (DGBL) in K-12 education presents compelling opportunities as well as challenges for early childhood educators who seek to critically, equitably and holistically support the learning and play of today's so-called digital natives. However, with most DGBL initiatives focused on the increasingly standardized ‘accountability’ models found in K-12 educational institutions, the authors ask whose priorities, identities and notions of play this model reinforces or neglects. Drawing on the literatures of early childhood studies, game-based learning, and game studies, they seek to illuminate the informal contexts of play within the ‘hidden’ and ‘null’ curricula of DGBL that do not fit within the efficiency models of mainstream education in North America. In the absence of a common critical or theoretical foundation for DGBL, they propose a conceptual framework that challenges what they regard to be the institutionally nullified dimensions of autonomy, play, affinity and space that are essential to DGBL. They contend that these dimensions are ideally situated within the inclusive and play-based curriculum early childhood learning environments, and that the early years constitute a critically significant, yet overlooked, location for more holistic and inclusive thinking on DGBL.
---

Title: The development of numerical cognition in children and artificial systems: a review of the current knowledge and proposals for multi-disciplinary research

Section 1: Introduction
Description 1: This section introduces the significance of numerical cognition, its study in various disciplines, and the role of embodied experiences in the development of mathematical skills.

Section 2: Cognitive developmental robotics
Description 2: Discusses the exploration of abstract cognition through modeling numerical abilities in artificial intelligence, with a focus on cognitive developmental robotics (CDR) and its applications.

Section 3: Robots in the education of children
Description 3: Explores the use of robots in educational settings to teach various subjects, particularly to young children and those with learning difficulties, and the potential benefits and challenges.

Section 4: Objectives of this study and its structure
Description 4: Outlines the study's objectives to provide background knowledge and future interdisciplinary research directions, detailing the paper's structure.

Section 5: Finger gestures: an embodied tool for number processing
Description 5: Examines the key role of finger gestures in early numerical cognition, from subitising to counting and beyond, including the neural link between finger movements and numerical processing.

Section 6: Embodiment in teaching and learning mathematics: the role of fingers
Description 6: Discusses the influence of embodiment, particularly finger use, on children's learning of mathematical concepts and the implications for mathematics education.

Section 7: Simulation of number cognition in artificial cognitive systems and developmental robots
Description 7: Reviews major computational models for simulating numerical cognition in artificial systems and developmental robots, distinguishing classical models from CDR approaches.

Section 8: Discussion
Description 8: Highlights key themes from the reviewed work that could drive future CDR research and outlines the need for interdisciplinary collaboration, co-designed studies, and open data.

Section 9: Conclusion
Description 9: Summarizes the review's findings, emphasizing the importance of interdisciplinary work, computational models, and robotic assistance in education to further the understanding and application of numerical cognition.