Big Math for Little Kids

Big Math for Little Kids, a comprehensive program for 4- and 5-year-olds, develops and expands on the mathematics that children know and are capable of doing. The program uses activities and stories to develop ideas about number, shape, pattern, logical reasoning, measurement, operations on numbers, and space. The activities introduce the mathematical ideas in a coherent, carefully sequenced fashion, and are designed to promote curiosity and excitement about learning and doing mathematics. The program produces playful but purposeful learning of deep mathematical ideas, and encourages children to think about and express their mathematical thinking. Throughout the program, great emphasis is placed on the development of mathematical and mathematics-related language. Our observations suggest two broad questions for future research: What kinds of competence can children develop in the context of a rich mathematics environment? In what ways can mathematics learning promote language and literacy?     

The Interplay of Teacher and Student Actions in the Teaching and Learning of Geometric Proof

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.     

Scaffolding the development of effective collaboration and learning

Research in Mexican and British schools provides an empirical basis for arguing that, by the use of certain kinds of interactional strategies, teachers can enable children to become more able in managing individual and joint reasoning and learning activities in the classroom.The research described is based on a sociocultural conception and analysis of education, which focuses on the ways that children can be inducted into the communicative and intellectual activities of the classroom as a ‘community of enquiry’. The research has provided (a) an account of strategies teachers use, and for relating teacher’s scaffolding to the interactive process of knowledge construction; (b) an analysis of ways that children talk when working together on joint activities; and (c) a practical method for promoting children’s effective collaboration, communication, reasoning and learning, successfully tested with British and Mexican primary school children. The results of the research are discussed in relation to a Vygotskian conception of the relationship between ‘intermental’ (social) and ‘intramental’ (psychological) development.     

Early Numerical Competencies in 5- and 6-Year-Old Children With Autism Spectrum Disorder

Research Findings: To date, studies comparing the mathematical abilities of children with autism spectrum disorder (ASD) and typically developing children are scarce, and results remain inconclusive. In general, studies on this topic focus on mathematical abilities learned from elementary school onward, with little attention for possible precursors at younger ages. The current exploratory study focused on the important developmental period of preschool age, investigating 5 early numerical competencies in 30 high-functioning children with ASD and 30 age-matched control children: verbal subitizing, counting, magnitude comparison, estimation, and arithmetic operations. Children were examined at 5 or 6 years of age, attending the 3rd and final year of preschool. Overall, rather similar early number processing was found in children with and without ASD, although marginally significant results indicated a weaker performance of children with ASD on verbal subitizing and conceptual counting. Practice or Policy: Given the pervasiveness and impact of ASD on other domains of functioning, it is important to know that no general deficits in early numerical competencies were found in this study. However, some downward trends in mathematics performance were identified in children with ASD, which can serve as the basis for additional research in this field.     

Special education in Pakistan

The aim of this research was to examine the type of activities as well as the challenges faced by befrienders supporting persons with autism spectrum disorders (ASD). Twenty-nine volunteers befriending an equal number of families with children with ASD participated in this study. Content analysis was applied on visit logs (n = 465) completed by befrienders during a nine month period, supplemented by thematically analysed evidence derived from semi-structured interviews with befrienders at the end of their volunteering experience. In agreement with our expectations, findings indicate a variation in the type, number and location of activities as well as the challenges of befrienders, depending on the level of ability of persons with ASD. Befrienders in both groups were mostly involved in leisure activities, had greater social interaction with the high functioning autistic group and carried out more structured 1:1 activities with individuas with low functioning autism. Interestingly, befrienders in both groups did not differ significantly on their reported challenges. The implications of this befriending service as a means of social support for families with persons with ASD are discussed.     

Reasoning as a scientist: ways of helping children to use language to learn science

Sociocultural researchers have claimed that students' learning of science is a discursive process, with scientific concepts and ways of reasoning being learned through engagement in practical enquiry and social interaction as well as individualized activity. It is also often claimed that interacting with partners while carrying out scientific investigations is beneficial to students' learning and the development of their understanding. The research we describe investigated the validity of these claims and explored their educational implications. An experimental teaching programme was designed to enable children in British primary schools to talk and reason together and to apply these skills in their study of science. The results obtained indicate that (a) children can be enabled to use talk more effectively as a tool for reasoning and (b) talk‐based activities can have a useful function in scaffolding the development of reasoning and scientific understanding. The implications of the findings for educational policy and practice are discussed.     

幼儿园学习环境质量与幼儿发展结果的相关分析及其阈值效应

幼儿园学习环境与幼儿发展结果之间存在密切的关系。本研究以来自我国5个省区89所幼儿园267个班级的1592名幼儿为研究对象,采用广义相加混合模型,分析幼儿园学习环境质量各要素与幼儿语言、数学认知、情感社会性发展之间的关系。结果发现,学习环境与幼儿发展结果之间的关系具有一定的复杂性,其中园舍建筑和幼儿语言、数学认知存在非线性关联;户外空间与设施和幼儿语言、数学认知存在线性关联;区域规划与材料和幼儿语言存在线性关联,与幼儿情感社会性存在非线性关联;心理氛围与幼儿情感社会性存在线性关联,与幼儿语言、数学认知存在非线性关联;班级空间与设施和幼儿发展不相关。与此同时,园舍建筑、心理氛围、区域规划与材料对幼儿发展结果分别具有一定的阈值效应。高质量的幼儿园学习环境阈值范围为4.3~6.0分,表明教育经费投入需要保证幼儿园学习环境质量在此阈值之上,学习环境对幼儿发展的促进效果才能显现。     

Comparison of the effects of mainstream and special school on National Curriculum outcomes in children with autism spectrum disorder: an archive‐based analysis

The literature dealing with the inclusion of children with autism spectrum disorder (ASD) in mainstream schools has increased over recent years, propelled by the argument that it will improve the quality of life, educational performance and social development of ‘included’ children. This area of research is currently an important one for the development of policy and practice. The literature on inclusion dealing with the inclusion of children with ASD is limited, so the implementation of inclusion has preceded research. The current study investigated whether children in mainstream placements show enhanced performance, relative to those in specialist provisions. The study used a combination of primary and secondary data analysis to explore the impact of inclusion on children with ASD in four authorities in the south east of England. The results suggest that mainstream children have no greater academic success than children in specialist provision. The study suggests that a number of specific provisions are involved in promoting success, such as Speech and Language Therapy, and the impact of Learning Support Assistants, and these are also reviewed and discussed.     

Developmental change in the ERP responses to familiar faces in toddlers with autism spectrum disorders versus typical development

Individuals with autism spectrum disorder (ASD) show differences in face processing abilities from early in development. To examine whether these differences reflect an atypical versus delayed developmental trajectory, neural responses to familiar and unfamiliar faces in twenty-four 18- to 47-month-old children with ASD were compared with responses of thirty-two 12- to 30-month-old typically developing children. Results of 2 experiments revealed that neural responses to faces in children with ASD resembled those observed in younger typically developing children, suggesting delayed development. Electrophysiological responses to faces were also related to parent report of adaptive social behaviors for both children with ASD and typical development. Slower development of the face processing system in ASD may be related to reduced self-directed "expected" experience with faces in early development.     

A comparison of parent and professional perceptions of children with autism spectrum disorder in Portugal

Improving early intervention in Portugal for children with autism spectrum disorder (ASD) requires parents and professionals to collaborate in assessing and planning intervention programmes. This article analyses parental and professional assessments of children with ASD, in the dimensions of social communication, repetitive behaviours and restricted interests, and sensory processing. The study presents contrasting perspectives from professionals and parents from a range of socio‐academic backgrounds. The sample consists of 136 Portuguese children (aged three to six years old) with ASD and involves children, parents and professionals from across the country. All children, at the time, were in preschool and receiving early special education intervention. Parents and professionals used the Assessment Scale for Children with ASD to complete the assessments. The results show that parents rate children's development and learning more positively compared to professionals; and that this difference is more significant in the field of social communication. Further, parents with higher academic qualifications showed statistically significant differences in comparison with professionals in the social communication dimension. These findings are discussed in relation to the literature and some suggestions are made regarding certain practices related to intervention in early childhood.     

From social interaction to individual reasoning: an empirical investigation of a possible socio-cultural model of cognitive development

This study explores the theory that individual reasoning ability, as measured using standard reasoning tests, has part of its origin in dialogue with others. In the study, 64 eight- and nine-year-old children were taught the use of ‘exploratory talk’, a type of talk in which joint reasoning is made explicit. The relationship between the talk of the children and the solving of Raven's test problems was studied using discourse analysis of groups working together. The findings of the study support four claims: that use of exploratory talk can improve group reasoning, that exploratory talk can be taught, that the teaching of exploratory talk can successfully transfer between educational contexts and that individual results on a standard non-verbal reasoning test significantly improved as a result of the intervention teaching exploratory talk. Our results offer support for the hypothesis that experience of social reasoning can improve scores on measures of individual reasoning. The stronger hypothesis that general cognitive development is a product of induction into social reasoning remains in doubt.     

Social interaction,relationships, and the development of causal discourse and conflict management

Links between the development of social understanding and social processes are considered, in the context of data on the development of causal reasoning and argument in conflict, drawn from a longitudinal study of 50 children observed in conversation with their mothers, siblings, and friends at 33 and 47 months of age. Children showed powers of causal reasoning before they were 3 years, capabilities that showed rapid developmental change; their interest in cause was particularly focussed on psychological causality and social action. The contexts of disputes were especially important. Differences were found in children’s arguments with their mothers, siblings, and friends, with more ‘other-oriented’ reasoning shown with friends. It is argued that the data support Vygotskyan ideas on the significance of social interaction in developmental advance, that the pragmatics of what children and their partners are trying to do must be taken into account in considering what children may be learning in interactions, and that ‘social competence’ should not be treated as an individual within-child trait, but that we should move towards a ‘relationships’ approach to the development of social understanding.     

Language as resource: a key notion for understanding the complexity of mathematics learning

Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits and limitations. In this paper, I examine prototypes in a context in which they seem to play an important role: graphical representations of the calculus concept of the definite integral. I use student data to empirically describe the makeup of the definite integral prototype image, and I report on the frequency of its appearance among student, instructor, and textbook image data. I end by discussing the possible benefits and drawbacks of this particular prototype, as well as what the results of this study may say about prototypes more generally.     

Generation Conservation: Children’s Developing Folkbiological and Moral Conceptions of Protecting Endangered Species

Research Findings: This study investigated folkbiological concepts that structure children’s moral reasoning regarding conservation. Participants (N = 52; 7- and 10-year-olds, gender balanced) were interviewed regarding their values, moral obligations, and rights concerns for endangered and extinct animals. Across the 2 ages, children drew on the animal’s living status and teleology, as well as ecological relations, to ground their moral reasoning. Developmentally speaking, the younger children employed both nature-centered and human-centered moral reasoning, whereas the older children employed a complex form of moral reasoning that integrated human models of moral thinking with distinct biological understanding. Practice or Policy: The findings can improve education and policy that supports the next generation of conservationists. Educators can draw on these findings to develop class material and activities that teach children about ecological relations—knowledge children can employ to construct their moral arguments for conservation and protecting the environment. The findings revealed that children think about and value endangered animals in deep and beautiful ways. Policymakers can draw on the findings to frame their decision making—drawing more deeply on children’s values of the natural world to inform policy decisions. Drawing on children’s values in the development of environmental policy engages them in the policy process.     

Teaching social skills to children with autism spectrum disorders using tiers of support: A guide for school‐based professionals

The quintessential characteristic of autism spectrum disorders (ASD) is having difficulty with basic social interactions. Due to the pervasiveness of their social skills difficulties, it is necessary that supports for increasing social skills of children with ASD are included as part of the educational plan within schools. The purpose of this article is to provide educators with a framework for using multiple tiers of instruction to teach social skills to children with ASD. Specifically, a number of evidence‐based strategies for teaching social skills are summarized within a systemic, multitiered approach. It is anticipated that such a model proves helpful as educators address issues related to service delivery provisions for students with ASD. © 2010 Wiley Periodicals, Inc.     

The Inclusion of Children with ASD: Using the Theory of Planned Behaviour as a Theoretical Framework to Explore Peer Attitudes

This study used the Theory of Planned Behaviour to explore the attitudes, behavioural intentions and behaviour of 318 mainstream primary school children in an urban East London borough towards peers with Autism Spectrum Disorders (ASD). Pupils were presented with a vignette about a hypothetical peer with ASD then completed self-report questionnaires and peer socio-metric measures. The results showed that children’s attitudes, the social pressure they felt from others and the amount of control they felt they had over their own behaviour, significantly predicted their behavioural intentions to befriend a peer with ASD. The strongest association was between perceived behavioural control and behavioural intentions. A significant association was also found between children’s behavioural intentions and actual behaviour towards an included peer. Support was therefore found for the use of the Theory of Planned Behaviour as a valuable tool by which to explore the social inclusion of children with ASD.     

Knowledge Construction and Collective Practice: At the Intersection of Learning,Talk, and Social Configurations in a Computer-Mediated Mathematics Classroom

In this study I investigated how students' mathematical activities, and thereby their mathematical understandings, change as a function of their participation in different social configurations. I examined how the interplay between 2 social configurations-local investigations at a computer simulation and whole-class discussions-contributes to how 7th-grade students learn probabilistic reasoning. I used 2 case studies to investigate (a) how different forms of participation are linked to different social configurations, and (b) how specific discourse practices and ways of reasoning propagate across the classroom and are adopted by individual students. The analyses suggest that classroom mathematical practices are developed, in part, for the social or communicative purpose of settling disputes and not purely for their rational or cognitive value to individuals. Results also provide insight into how to design and orchestrate classroom practice, particularly computer-mediated inquiry, to foster individual learning that is situated within a classroom community oriented toward the construction of a shared understanding of probability.