Mathematics and Virtual Culture: an Evolutionary Perspective on Technology and Mathematics Education

This paper suggests that from a cognitive-evolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical. This progression was driven by three cognitive advances: the ability to ‘represent’ events, the development of symbolic reference, and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture: a ‘virtual’ culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills. This revised version was published online in July 2006 with corrections to the Cover Date.     

In quest of the functional architecture of the developing mind: The Aristotelian project

This paper summarizes a theory of cognitive development that argues that the mind develops across three fronts. The first refers to a general processing system that defines the general potentials of mind to develop cognitive strategies and skills. The second refers to a hypercognitive system that governs self-understanding and self-regulation. The third involves a set of specialized structural systems which are responsible for the representation and processing of different reality domains. There are specific forces which are responsible for this organization of mind. These are expressed in the paper in terms of a set of five organizational principles. The developmental course of the major systems is outlined. Developmental change is ascribed by the theory to the interaction between the various systems. Different types of development require different change mechanisms. Several studies are outlined in the paper which provide empirical support to each of these propositions. The theoretical and educational implications of this project are discussed in the concluding section of the paper.     

Metacognition, Theory of Mind, and Self-Control: The Relevance of High-Level Cognitive Processes in Development, Neuroscience, and Education

ABSTRACT— The cognitive control of behavior is critical for success in school. The emergence of self-control in development has been linked to the ability to represent one's own and others' mental states (theory of mind and metacognition). Despite rapid progress in exploring the neural correlates of both mind reading and executive function in recent years, to date, the implications of these high-level cognitive processes for issues relevant to education have hardly been addressed. The present special issue brings together developmental perspectives on the relation of self-control, theory of mind, and metacognition; theoretical and empirical contributions on the implications of theory of mind and self-control for teaching and learning; and brief reviews of the state of the art in cognitive neuroscience on these high-level cognitive processes in adolescents and adults.     

Developing theories of the mind: children's and adults' concepts of mental activities

8-year-olds', 10-year-olds', and adults' concepts of mental activities involved in acts of knowing were examined in an attempt to gain insight into developmental changes in underlying theories of mind. Subjects rated the similarity of how the mind is used in a variety of common activities, each of which primarily involved either memory, comprehension, attention, or inference. Analyses of conceptual structure revealed 2 developmental changes. Between 8 and 10 years of age, children came to see the most important relation among mental activities to be the degree to which they involved memory. Between 10 years of age and adulthood, Comprehension and Attention first appeared as distinct, coherent concepts. At all ages, Memory and Inference were seen as distinct types of mental activities. These findings suggested that a sophisticated understanding of the representational nature of the mind might not be acquired before the latter part of middle childhood and might be mediated by an understanding of the central role played by memory.     

Beyond Conceptual Change: Using Representations to Integrate Domain-Specific Structural Models in Learning Mathematics

ABSTRACT— Effective teaching should focus on representational change, which is fundamental to learning and education, rather than conceptual change, which involves transformation of theories in science rather than the gradual building of knowledge that occurs in students. This article addresses the question about how to develop more efficient strategies for promoting representational change across cognitive development. I provide an example of an integrated structural model that highlights the underlying cognitive structures that connect numbers, mathematical operations, and functions. The model emphasizes dynamic multiple representations that students can internalize within the number line and which lead to developing a dynamic mental structure. In teaching practice, the model focuses on a counting task format, which integrates a variety of activities, specifically addressing motor, visual, and verbal skills, as well as various types of learning transfer.     

A Framework for Re‐thinking Learning in Science from Recent Cognitive Science Perspectives

Recent accounts by cognitive scientists of factors affecting cognition imply the need to reconsider current dominant conceptual theories about science learning. These new accounts emphasize the role of context, embodied practices, and narrative‐based representation rather than learners’ cognitive constructs. In this paper we analyse data from a longitudinal study of primary school children’s learning to outline a framework based on these contemporary accounts and to delineate key points of difference from conceptual change perspectives. The findings suggest this framework provides strong theoretical and practical insights into how children learn and the key role of representational negotiation in this learning. We argue that the nature and process of conceptual change can be re‐interpreted in terms of the development of students’ representational resources.     

Going beyond. In tribute to Annette Karmiloff-Smith / Ir más allá. En homenaje a Annette Karmiloff-Smith

This article provides an overview of some of Annette Karmiloff-Smith’s many contributions and the impact of her research on the study of cognitive and linguistic development. Her research work exceeded the dichotomous explanations of development, broadened the limits of developmental periods that are worth examining, emphasized the importance of notational knowledge as a developmental domain and demonstrated the need for research in cognitive science to go beyond describing successful performance on certain tasks in order to more deeply account for representational change.     

The rule-assessment approach and education

This paper describes the rule-assessment approach to cognitive development and summarizes work using it that is relevant to education. The basic question that motivated the rule-assessment approach is how people's existing knowledge influences their ability to learn. Addressing this issue requires a means of assessing what people already know, a means for studying how they learn, and a means for examining basic processes that may underlie developmental differences in knowledge and learning. Research using the rule-assessment approach is summarized in terms of eight conclusions about these issues. Each of the conclusions is illustrated with empirical examples.     

Learning to redesign teacher education: a conceptual framework to support program change

University-based teacher education faces intensifying pressure to prove its effectiveness. This has prompted renewed interest in program redesign. In this article, we argue that enacting meaningful redesign requires university-based teacher educators to learn new ways of thinking and acting not only with teacher candidates but also with their university and K-12 colleagues. Drawing on Engeström’s expansive learning theory and our own efforts to redesign our teacher education program along a practice-based teacher education model, we delineate a conceptual framework and illustrate how it supports the organizational learning necessary to enact change in university-based teacher education. The expansive learning redesign framework specifies conceptual tools and social practices to guide, enact, and sustain university-based teacher education redesign. It acknowledges the multi-organizational nature of university-based teacher education and provides tools for engaging K-12 educators in the redesign of teacher education.     

心理理论研究述评

心理理论的研究是近几年来认知发展研究的热点．总结近二十年来对心理理论的大量研究和该领域各方面的成果,对儿童心理理论发展的历史进行了初步整理,概述了心理理论的概念幕定、理论模型和主要研究方向．在对儿童心理理论的研究现状进行综述的基础上,探讨了其中存在的问题及未来的发展趋势．     

What's in a Name? A Comparison of Instructional Systems Design,Organization Development,and Human Performance Technology/Improvement and Their Contributions to Performance Improvement

What are the similarities and differences between instructional systems design, organization development, and human performance technology/improvement (HPT/HPI)? This article defines these fields and highlights how they contribute to the performance improvement domain. With Rossett's view of HPT as a “perspective, or habit of mind, that inclines us towards data, analysis, systems, alignment and partnerships” (A. Rossett, personal communication, October 28, 2012), HPT/HPI will be presented as the overarching framework for selecting and measuring instructional systems design or organization development interventions.     

学生道德素质培养的策略及思考

发展性教育已经成为当今教育的主题．让学生学会做人是发展性教育观念下的重要内容，而培养学生良好的道德素质则是让学生学会做人的关键。本文从高中学生的认知特点出发，就培养高中学生道德素质的重要性、实践策略和途径等方面进行了初步的探讨。     

Cognitive Development in Gifted Children: Toward a More Precise Understanding of Emerging Differences in Intelligence

To truly understand gifted performance, it is necessary to merge research on giftedness with current thinking in cognitive development and intelligence. This article presents traditional research on gifted children's cognitive development then considers how the application of newer models and theories from the field of cognitive development can be combined with research on giftedness to change the way people think about gifted performance. First four factors that have often been associated with giftedness are discussed from the perspectives of cognitive developmental psychology and gifted education. Next, emphasis is placed on investigating the strategic development of gifted children. Specifically, R. S. Siegler's (Emerging Minds: The Process of Change in Children's Thinking, Oxford University Press, New York, 1996) model of strategy development is addressed in terms of what it may contribute to understanding gifted cognition. Finally, future lines of research using models from cognitive development and complex systems models of development are recommended.     

Changing the Metacognitive Orientation of a Classroom Environment to Stimulate Metacognitive Reflection Regarding the Nature of Physics Learning

Problems persist with physics learning in relation to students' understanding and use of representations for making sense of physics concepts. Further, students' views of physics learning and their physics learning processes have been predominantly found to reflect a ‘surface’ approach to learning that focuses on mathematical aspects of physics learning that are often passed on via textbooks and lecture-style teaching. This paper reports on a teacher's effort to stimulate students' metacognitive reflection regarding their views of physics learning and their physics learning processes via a pedagogical change that incorporated the use of a representational framework and metaphors. As a consequence of the teacher's pedagogical change, students metacognitively reflected on their views of physics and their learning processes and some reported changes in their views of what it meant to understand physics and how they might learn and understand physics concepts. The findings provide a basis for further explicit teaching of representational frameworks to students in physics education as a potential means of addressing issues with their physics learning.     

Rejoinder in Defense of the Standard model of the Mind: From Whimsical to Systematic Science

This rejoinder discusses the issues raised by the three commentators of the target article. In agreement with Hunt, it is accepted that, as any overarching theory in any science, it does involve postulates about the mind that must be scrutinized by further research. However, its main postulates are well supported by the findings of specifically designed experiments. In response to Olson, it is maintained that this theory, as an integration of the standard models of the human mind in cognitive, psychometric, and differential psychology, deals better than its competitors with educationally important questions about what, how, and when to teach important knowledge. In fact, the mechanisms of specified understanding are refined enough to explicate what meaning making is possible at different ages. In agreement with Anderson, it is noted that the theory may direct curriculum developers how to develop the curriculum in different subjects and teachers colleges how to educate teachers so that learning in the classroom may become deeper, more stable, and transferable, thereby increasing the understanding and learning autonomy of students.     

Defending Qualitative Change: The View From Dynamical Systems Theory

A central controversy in developmental science, enflamed by nativist accounts, is whether development is best viewed as a series of qualitative or continuous changes. This article defends the notion of qualitative change from the perspective of dynamical systems theory (DST). Qualitative change within DST refers to the shift that occurs when a system goes from one attractor state through an instability into a different attractor state. Such changes occur on the second‐to‐second timescale of behavior. Thus, developmental analysis must always stay local, grounded in the real‐time attractor states around which behavior is organized. This article also demonstrates that qualitative and continuous change should not be cast in opposition. They are aligned concepts that work together across multiple timescales.     

Mental retardation general intelligence and modularity

This article presents a case for distinguishing between mental retardation as a general deficit of thinking and mental retardation that might result from the global effects of a specific deficit in a cognitive module. Using Anderson's (1992a) theory of the minimal cognitive architecture of intelligence and developmental, I show how this distinction can explain the pattern of intellectual strengths and weaknesses in Savant syndrome, Williams syndrome, Down syndrome, and autism. In addition, I discuss the developmental versus difference view and the distinction between organic and cultural familial mental retardation in the light of this theory. I conclude that not only is there no inherent incompatibility between the constructs of general intelligence and modularity of mind but that both are essential to understanding the different patterns of abilities and developmental profiles found in individuals with low IQ.     

FIRST COMES THE THEORY, THEN THE PRACTICE? ON THE ACQUISITION OF GENERAL PEDAGOGICAL KNOWLEDGE DURING INITIAL TEACHER EDUCATION

Teacher education systems worldwide are confronted with the essential question of how to foster both future teachers’ theoretical and practical knowledge and to adequately enable future teachers to connect their theoretical and practical knowledge for teaching. This article investigates how future teachers acquire general pedagogical knowledge (GPK) as a central component of teacher knowledge during initial teacher education, exemplified by pre-service teachers in Germany, where initial teacher education is divided into a first phase with a heavy focus on theoretical, academic study, and a second phase where future teachers learn how to apply their theoretical knowledge in the classroom. Data from teacher knowledge studies Teacher Education and Development Study in Mathematics and Längsschnittliche Erhebung pädagogischer Kompetenzen von Lehramtsstudierenden/Longitudinal Survey of Student Teachers’ Pedagogical Competencies are used to compare future teachers’ GPK at different teacher education stages (the beginning, after 2 years, and end of training). Findings show the more advanced future teachers are in the course of their initial teacher education, the better they perform in the test measuring GPK. When analyzing subscales of the test measuring cognitive dimensions of GPK, as would be expected declarative-conceptual knowledge (measured by cognitive dimensions “recall” and “understand/analyze”) was gained predominantly during the theoretical study (first phase), whereas future teachers who had additionally passed through the practical second phase performed much better on the practical knowledge test subscale (measured by the cognitive dimension “generate”). Research findings are discussed with regard to the development of teacher expertise during initial teacher education, and recommendations for future policy directions with respect to teacher education are given.     

Learning How to Learn,Art Education and the ‘Background’

This paper describes the conditions that make learning how to learn in art possible. Rather than privileging risk-taking or the playfulness of primary process thinking, I will use a Wittgensteinian understanding of mind to argue against this approach in favour of one that raises students’ awareness of the cultural and cognitive ‘backgrounds’ (forms of life and language games) against which our individual actions and risks are framed and understood. Instead of building art education on a metaphysical view of the self, this paper will advocate a social constructivist approach to curriculum, pedagogy and assessment to create a cognitive map for ‘the background’ of art. When these three are practised in a mutual way they connect individual minds to the resources or mind of culture which, in turn, makes lifelong learning possible, reclaiming the idea of art-for-life’s sake.