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This article reviews the topics of affect, motivation, working memory, and their relationships to mathematics learning and performance. The underlying factors of interest, motivation, self-efficacy, and maths anxiety, as well as an approach concerning people’s beliefs about fixed versus malleable intelligence, can be grouped into an approach and an avoidance constellation of attitudes and beliefs, with opposite relationships to outcome measures of learning and mastery in maths. This article then considers the research on working memory, showing it to be central to arithmetic and maths processing, and also the principle mental component being disrupted by affective and emotional reactions during problem solving. After discussing the disruptive effects of maths anxiety, choking under pressure, and stereotype threat, the article closes with a brief consideration of how these affective disruptions might be minimized or eliminated.

This article reviews recent research exploring children’s abilities to perform approximate arithmetic with non-symbolic and symbolic quantities, and considers what role this ability might play in mathematics achievement. It has been suggested that children can use their approximate number system (ANS) to solve approximate arithmetic problems before they have been taught exact arithmetic in school. Recent studies provide evidence that preschool children can add, subtract, multiply, and divide non-symbolic quantities represented as dot arrays. Children can also use their ANS to perform simple approximate arithmetic with non-symbolic quantities presented in different modalities (e.g. sequences of tones) or even with symbolic representations of number. This article reviews these studies, and consider whether children’s performance can be explained through the use of alternative non-arithmetical strategies. Finally, it discusses the potential role of this ability in the learning of formal symbolic mathematics.

Children in the bottom quartile of mathematics achievement are at high risk for underemployment in adulthood. These children include the roughly 7% of students with a mathematical learning disability (MLD) and another 10% of students with persistent low achievement (LA) that is not attributable to intelligence. The poor mathematics achievement of children who compose groups of MLD and LA students appears to be related to one or several deficits; specifically, (1) a delay in the development and poor fidelity of the system for representing approximate magnitudes; (2) difficulty mapping Arabic numerals, number words, and rational numbers onto associated quantities; (3) poor conceptual understanding of some arithmetic concepts; (4) developmental delay in the learning of mathematical procedures; and (5) difficulty committing basic arithmetic facts to or retrieving them from long-term memory. Children with MLD also have concurrent working memory deficits that exacerbate their mathematics-specific deficits and delays.

Mathematical competence rests on developing knowledge of concepts and of procedures (i.e. conceptual and procedural knowledge). Although there is some variability in how these constructs are defined and measured, there is general consensus that the relations between conceptual and procedural knowledge are often bi-directional and iterative. The chapter reviews recent studies on the relations between conceptual and procedural knowledge in mathematics and highlights examples of instructional methods for supporting both types of knowledge. It concludes with important issues to address in future research, including gathering evidence for the validity of measures of conceptual and procedural knowledge and specifying more comprehensive models for how conceptual and procedural knowledge develop over time.

Early number competencies predict later mathematical learning. Weaknesses in number, number relations, and number operations can be reliably identified before school entry in first grade. Income status, associated early home and preschool opportunities, and general cognitive capacity all influence children’s level of numerical knowledge. Interventions based on a developmental progression and targeted to specific areas of number, such as the ability to count and sequence numbers, compare numerical quantities, and add and subtract small quantities, have shown positive, meaningful, and lasting effects on children’s achievement. Guided practice is effective when configured to support efficient counting strategies, frequent correct responding, and meta-cognitive behaviour and when contextualized with a strong focus on number knowledge tutoring.

Fictional worlds (pretending, reading, watching television) can teach and change children. This chapter discusses the influence of fictional worlds on children's prosocial and aggressive behavior, theory of mind, and acquisition of factual knowledge about the world. Neurological changes that accompany fictional presentation are also presented. In addition it discusses how Montessori education and play are similar (choice, interest, peers, embodied cognition), how Discovery Learning can be viewed as play, and the limitations of play as a vehicle for learning.

This chapter explores how numbers are represented amongst children in different cultures, and shows how this can enrich our understanding of mathematical cognition. It focuses on two specific, related topics: the representation of multi-digit numbers and the scaling of a mental number line. The authors consider whether linguistic differences in number structures directly influence children’s understanding of place value. They also consider whether cross-cultural and developmental differences in the quality of children’s mental representations of number are direct influences on mathematical skill. Together, these two topics allow us to consider evidence for the existence of cross-cultural difference in mathematics and investigate factors that might underlie them. The authors propose that whilst the interpretation of data needs to proceed cautiously, valuable insights can be gained from relevant research.

Young children initially learn to ‘count’ without understanding either what counting means, or what numerical quantities the individual number words pick out. Over a period of many months, children assign progressively more sophisticated meanings to the number words, linking them to discrete objects, to quantification, to numerosity, and so on. Eventually, children come to understand the logic of counting. Along with this knowledge comes an implicit understanding of the successor function, as well as of the principle of equinumerosity, or exact equality between sets. Thus, when children arrive at a mature understanding of counting, they have (for the first time in their lives) a way of mentally representing exact, large numbers.

Before children begin school, there is a wide range of individual differences in children’s early numerical knowledge. Theoretical and empirical work from the sociocultural perspective suggests that children’s experiences in the early home environment and with informal number activities can contribute to these differences. This article draws from this work to hypothesize that differences in the home explain, in part, why the numerical knowledge of children from low-income backgrounds trails behind that of peers from middle-class backgrounds. By integrating sociocultural perspectives with a theoretical analysis of children’s mental number line, the authors created an informal learning activity to serve as an intervention to promote young children’s numerical knowledge. Our studies have shown that playing a simple number board game can promote the numerical knowledge of young children from low-income backgrounds. The authors discuss how informal learning activities can play a critical role in the development of children’s early maths skills.

This chapter describes the four main sources of individual differences in arithmetic that have been identified through research with children and adults. Numerical quantitative knowledge invokes basic cognitive processes that are either numerically specific or are recruited to be used in quantitative tasks (e.g. subitizing, discrimination acuity for approximate quantities). Attentional skills, including executive attention and various aspects of working memory are important, especially for more complex procedures. Linguistic knowledge is used within arithmetic to learn number system rules and structures, specific number words, and in developing and executing counting processes. Strategic abilities, which may reflect general planning and awareness skills, are involved in selecting procedures and solving problems adaptively. Other important sources of individual differences include automaticity of knowledge related to practice, experiences outside school, and the specific language spoken. Suggestions are made for further research that would be helpful in establishing a full picture of individual differences in arithmetic.

Cross-sectional and longitudinal data consistently indicate that mathematical difficulties are more prevalent in older than in younger children (e.g. Department of Education, 2011). Children’s trajectories can take a variety of shapes such as linear, flat, curvilinear, and uneven, and shape has been found to vary within children and across tasks (J Jordan, Mulhern, and Wylie, 2009). There has been an increase in the use of statistical methods which are specifically designed to study development, and this has greatly improved our understanding of children’s mathematical development. However, the effects of many cognitive and social variables (e.g. working memory and verbal ability) on mathematical development are unclear. It is likely that greater consistency between studies will be achieved by adopting a componential approach to study mathematics, rather than treating mathematics as a unitary concept.

This chapter reviews recent research investigating children’s Spontaneous Focusing On Numerosity (SFON) and considers the role it might play in the development of counting and arithmetical skills. SFON refers to a process of spontaneously (i.e. not prompted by others) focusing attention on the exact number of a set of items or incidents. This attentional process triggers exact number recognition and using the recognized exact number in action. The chapter describes how SFON tendency can be assessed, and suggests the measures of it to be indicators of the amount of a child’s self-initiated practice in using exact enumeration in his or her natural surroundings. The studies show that SFON tendency in early childhood is positively and domain-specifically related to the development of numerical skills up to the end of primary school. Promoting SFON tendency could be a potential way of preventing learning difficulties in mathematics.