Teaching and Learning Spatial Thinking with Young Students: the Use and Influence of External Representations

Previous research suggests spatial thinking is fundamental to mathematics learning (Bronowski, 1947; Clements & Sarama, 2007, 2011), and acts as a predictor for future mathematical achievement levels (Battista, 1990; Gunderson et al., 2012). However, research with regard to spatial thinking is almost non-existent in early years mathematics classrooms (Bruce, Moss, & Ross, 2012; Clements & Sarama, 2011; Newcombe & Frick, 2010; Sarama & Clements, 2009, 2011; Stipek, 2013), and how to teach it in these contexts has received little attention. Fewer studies again have focused on the use of virtual manipulatives in influencing young students’ spatial thinking (Highfield & Mulligan, 2007; Ng & Sinclair, 2015). Despite a recent surge in studies exploring the influence of virtual manipulatives in mathematics classrooms, little is known about how these manipulatives compare to physical manipulatives, especially in regard to the changes that occur in the social interactions between teacher and students during the learning process. To date, there has been no comparative study conducted that explores the influence of different external representations (e.g., physical manipulatives and virtual manipulatives) on both the teaching and the learning aspects within mathematics classrooms. The purpose of this research is to explore the use of external representations (i.e., physical manipulatives as compared to virtual manipulatives) in the mathematics classroom and how these representations support young, disadvantaged students’ spatial thinking. The use of manipulatives is a common starting point for the teaching and learning of spatial thinking. Previous research on manipulative use (both physical and virtual) in mathematics education has yielded positive results with regard to student learning (Clements, 1999; Heddens, 1997; Highfield & Mulligan, 2007; Riconscente, 2013; Siemon et al., 2011; Warren, 2006; Warren & Miller, 2013). Recent studies indicate that these newer digital technologies promote interactions between visual and kinaesthetic learning, which have been shown to support the teaching and learning of spatial thinking (Battista, 2008; Bruce, McPherson, Sabeti, & Flynn, 2011; Clements & Sarama, 2011; Highfield & Mulligan, 2007; Jorgensen & Lowrie, 2012; Sinclair, de Freitas, & Ferrara, 2013; Sinclair & Moss, 2012). However, results from comparative studies between physical manipulatives and virtual manipulatives have been varied (e.g., Brown, 2007; Olkum, 2003; Suh, 2005). It is proposed that different types of manipulatives influence the teaching and learning of spatial thinking in different ways. By viewing the learning of spatial thinking through a sociocultural perspective, aspects of the teaching and learning of spatial learning in mathematics classrooms can be scrutinised. A review of the literature generated two research questions that informed the research design of this study. These were: 1. What influence do different external representations (e.g., physical manipulatives and virtual manipulatives) have on young students’ learning of spatial thinking? 2. What changes occur in the teaching and learning of spatial thinking when using different external representations (e.g., physical manipulatives and virtual manipulatives)? Given that the study focused on exploring students’ spatial thinking as they construct their knowledge from the interactions they experience with external representations, an interpretive paradigm was an appropriate epistemological, ontological and methodological stance adopted for the research. Vygotsky’s (1978) sociocultural theory provided a lens to interpret the interaction between teacher and students. Practical application of this theory permitted a narrowing lens to pinpoint particular aspects of the teaching of spatial thinking and students’ learning of spatial thinking. Within this study, these practical applications included the use of Anghileri’s “hierarchy of scaffolding practices” (2006) and Sfard’s “commognitive approach” (2008). The methodology for the study included teaching experiments. Data collection methods incorporated the use of pre-test, post-test and post post-testing using spatial testing material and observations of lessons from a teaching experiment (n = 68) comprising six lessons (three based on spatial orientation concepts and three based on spatial visualisation concepts). Findings from this study provide further insights into the teaching and learning of spatial thinking. First, the use of manipulatives (either physical or virtual) appears to be important to students’ learning of spatial thinking. Furthermore, the use of virtual manipulatives increases the communicative functions used by students, thus benefiting their spatial thinking. Second, teachers need to be able to instantaneously access deep content and pedagogical knowledge in order to maintain their role as “more knowledgeable other” and continually contribute to the teaching and learning of spatial thinking. Finally, teaching and learning appears to be positively influenced when both the teacher and students are major contributors to the classroom discourse. This study contributes to the understanding of how different external representations influence the teaching and learning of spatial thinking. Theoretical contributions to new knowledge include a hypothesised theory on the interaction between teacher, student and manipulatives type. Implications for future classroom practice include placing importance on the use of manipulatives and communication in mathematics classrooms. Furthermore, teachers need to be aware that their ability to instantaneously access deep levels of content and pedagogical knowledge to further develop students’ spatial thinking is essential and that for optimum learning to occur, both the teacher and students need to be major contributors to the teaching and learning process

Relational trauma and its impact on late-adopted children

This paper describes work with two children, placed for late adoption who have suffered relational trauma. The paper explores the long-term consequences of such trauma, which includes problems with affect regulation, difficulties in generalising from one experience to another and shifts between phantasies of omnipotent control and sudden helplessness. Using drawings from one boy's therapy, it is argued that many children adopted at a later age live in two worlds, both internal and external, and internal objects and memories from the past vie with new experiences and representations for ascendancy within the child's mind. Which is more real: the world of the past or the present? The paper describes how these children experienced sudden and troubling shifts in focus as they were catapulted from feeling states belonging to one world to the other. The paper ends with a consideration of how findings from neuroscience may help us to understand these sudden shifts and overall argues for a pulling together of psychoanalytic thinking and child development research findings to support the child in psychotherapy

From Parmenidean Identity to Beyond Classical Idealism and Epistemic Constructivism

Rockmore’s paper offers a nice discussion on how classical German idealism provides a plausible account of the Parmenidean insight that thought and being are identical and suggests that idealist epistemic constructivism is arguably the most promising approach to cognition. In this short commentary, I will explore the implications of adopting other interpretations of Parmenidean identity thesis, which arguably lead to different conclusions than the ones drawn by Rockmore. En route to disavow the distinction between ontology and epistemology, I argue that one may adopt an approach on cognition which would be immunized to worries that prompt Rockmore’s elaboration and also embrace (at least) some of its benefits

Outline of a new approach to the nature of mind

I propose a new approach to the constitutive problem of psychology ‘what is mind?’ The first section introduces modifications of the received scope, methodology, and evaluation criteria of unified theories of cognition in accordance with the requirements of evolutionary compatibility and of a mature science. The second section outlines the proposed theory. Its first part provides empirically verifiable conditions delineating the class of meaningful neural formations and modifies accordingly the traditional conceptions of meaning, concept and thinking. This analysis is part of a theory of communication in terms of inter-level systems of primitives that proposes the communication-understanding principle as a psychological invariance. It unifies a substantial amount of research by systematizing the notions of meaning, thinking, concept, belief, communication, and understanding and leads to a minimum vocabulary for this core system of mental phenomena. Its second part argues that written human language is the key characteristic of the artificially natural human mind. Overall, the theory both supports Darwin’s continuity hypothesis and proposes that the mental gap is within our own species

Patterns of Metacognitive Skills and External Representation of Students in Chemistry Problem Solving

This research aims to examine the pattern of external representation and metacognitive skills in chemistry problem solving for students of chemistry education at Tadulako University. This picture will enrich the treasures of thinking skills in the field of science, namely how students/prospective teachers think in the context of metacognitive skills and how these students display an external representation system on chemistry concepts. The subject of this qualitative study was obtained through a purposive random sampling of 97 students who have been programmed Basic Chemistry course 2017/2018 academic year. Subjects were selected based on the results of the screening using a metacognitive skills assessment questionnaire (MCAI). Two problems solved by the subject by setting one-on-one thinking aloud. Subjects who complete the issue by setting further interviews at different times. The problem-solving activity was recorded using a video camera. The subject uses a type of representation and aspects of metacognitive skills and how to use the sequence of activities, analyzed in detail. The results showed that students of chemical education used metacognitive skills and external representations. So students solve chemical problems. Students also can improve knowledge retention related to linking new knowledge with previous knowledge, creating ideas, organizing, and even synthesizing knowledge. Students become productive and find solutions for problem-solving wel

A Missing Step In Kant’s Refutation of Idealism

This paper contends that Kant’s argument in the Refutation of Idealism section of the Critique of Pure Reason misses a step which allows Kant to move illicitly from inner experience to outer objects. The argument for persistent outer objects does not comprehensively address the skeptic’s doubts as it leaves room for the question about the necessary connection between representations and outer objects. A second fundamental issue is the ability of transcendental idealism to deliver the account of outer objects, as required by the Refutation of Idealism itself

Teaching mathematics : self-knowledge, pupil knowledge and content knowledge

Mathematical learning is significantly influenced by the quality of mathematics teaching (Hiebert and Grouws 2007). In spite of the evidence for teachers seeking to do what they believe to be in the best interests of their learners (Schuck 2009; Gholami and Husu 2010), research and policy reports (within the UK and beyond) draw attention to insufficient mathematical attainment (Williams 2008; Eurydice 2011). Why is there this discrepancy? On the one hand, teachers are open to improving their professional practices (Escudero and S´anchez 2007), and on the other, the findings of mathematical education research make little or no impact on teachers’ practice (Wiliam 2003), even although teachers themselves think that they are enacting new or revised practices (Speer 2005)

Toward a Semiotic Framework for Using Technology in Mathematics Education: The Case of Learning 3D Geometry

This paper proposes and examines a semiotic framework to inform the use of technology in mathematics education. Semiotics asserts that all cognition is irreducibly triadic, of the nature of a sign, fallible, and thoroughly immersed in a continuing process of interpretation (Halton, 1992). Mathematical meaning-making or meaningful knowledge construction is a continuing process of interpretation within multiple semiotic resources including typological, topological, and social-actional resources. Based on this semiotic framework, an application named VRMath has been developed to facilitate the learning of 3D geometry. VRMath utilises innovative virtual reality (VR) technology and integrates many semiotic resources to form a virtual reality learning environment (VRLE) as well as a mathematical microworld (Edwards, 1995) for learning 3D geometry. The semiotic framework and VRMath are both now being evaluated and will be re-examined continuously

Mapping Big Data into Knowledge Space with Cognitive Cyber-Infrastructure

Big data research has attracted great attention in science, technology, industry and society. It is developing with the evolving scientific paradigm, the fourth industrial revolution, and the transformational innovation of technologies. However, its nature and fundamental challenge have not been recognized, and its own methodology has not been formed. This paper explores and answers the following questions: What is big data? What are the basic methods for representing, managing and analyzing big data? What is the relationship between big data and knowledge? Can we find a mapping from big data into knowledge space? What kind of infrastructure is required to support not only big data management and analysis but also knowledge discovery, sharing and management? What is the relationship between big data and science paradigm? What is the nature and fundamental challenge of big data computing? A multi-dimensional perspective is presented toward a methodology of big data computing.Comment: 59 page