Proposal Background For Authors

Visuospatial Reasoning in Early Geometry Education Research by cognitive psychologists on visuospatial thinking has focused on how people represent and process visual and spatial information (e.g., Liben, 2006). Visuospatial thinking relies on spatial abilities but also on physical and/or mental manipulation of images. An understanding of visuospatial thinking is relevant not only to the field of cognitive psychology but also to education, geography, physiotherapy and occupational therapy, medicine, architecture, design, computer science, semiotics and animal cognition (Shah & Miyake, 2005). In addition, every school mathematics curriculum has a strand on space and geometry that begins on school entry. However, in order to achieve quality geometry education, teachers need to understand students’ visuospatial reasoning and its impact on learning. Visuospatial reasoning involves visuospatial thinking but also reasoning with spatial concepts. Visuospatial reasoning is important in other areas such as developing concepts about fractions and working with diagrams in problem solving both within mathematics and in other disciplines. Visuospatial reasoning begins with spatial prior-toschool experiences. For example, block play provides opportunities for visuospatial thinking through rotating and joining objects. Geometric and architectural concepts are also used in block play in making enclosed and connected spaces and balancing blocks (Clements & Samara, 2004; Ness & Farenga, 2004). The extent of visuospatial reasoning in prior-to-school contexts still requires substantial synthesis. A comprehensive review of spatial abilities was written in the late 1980s by Eliot (1987) and research on visual imagery was developed from a psychological perspective (Owens, 1993). Research on visuospatial reasoning began with studies of spatial abilities and Piaget’s concerns for how students think at different age levels. These theories were extended by Van Hiele who showed that intuition (incidental experience) and planned experience could impact on learning and the development of concepts. Other studies have extended the structuralist approaches and provided some alternatives (Lehrer & Chazan, 1998; Owens & Outhred, 2006) but there are many unanswered questions on early visuospatial thinking (Battista, 2007). The ICMI study group on geometry (Mammana & Villani, 1998) recognised that applications in computer graphics and images provided a return to the visual approaches to geometry. Geometry was seen as having a visualisation process, a construction process and a reasoning process. The study group noted the limitations of some of the

theories that have most influenced mathematics education, namely constructivism and some forms of van Hiele’s theoretical approach since multimodal cognitive activity cannot be reduced to a hierarchical development of spatial reasoning and geometry. This work will be extended by synthesising both previous and more recent research in the proposed book. In addition, there are an increasing number of sociocultural studies that could be considered to have similarities to prior-to-school research in that the geometric constructions are not arbitrary but have many practical advantages (Gerdes, 1999; Ness & Farenga, 2007). Sociocultural studies draw our attention to the diversity of mathematical ideas that are used by different sociocultural groups. The importance of recognising these visuospatial and geometric ways of reasoning expands not only our understanding of geometries but also how people actually learn to think spatially and geometrically. Language and other representations embed some of this thinking as indicated by sociocultural studies (Adler, 2001; Barton, 2007; Gerdes, 1999; Owens & Kaleva, 2008a & b). Equity in education will be enhanced by drawing on these sociocultural studies in this book (D’Ambrosio, 2006; Barton et al., 2008). The book will present the multiple facets of visuospatial reasoning and argue how spatial experiences in prior-to-school and school contexts within a broader cultural context are integral to the learning of geometry. The book will provide a broader purpose for teaching visuospatial reasoning beyond Euclidean geometry in recognition of its multiple sources and applications. It will assist teacher educators to extend teachers’ understanding of space and geometry curricula. In different settings and cultures, problem solving during spatial experiences provides a means of extending visuospatial reasoning and geometric concepts. Intuition, intention and attention are socially and psychologically formed and impact on visuospatial reasoning and problem solving. The book will present an argument for an expanded understanding of space and geometry from recent visuospatial thinking studies and sociocultural studies. Studies in early childhood indicate the importance of spatial experiences, recognition of pattern and structure, and transitions from home to out-of-home education. Problem solving is seen as a key to learning visuospatial reasoning and geometric concepts. However, this learning in school is within a broader cultural context that impacts on reasoning, language and other representations. Visuospatial reasoning also impacts on other mathematical learning and other school learning areas. The challenge is to develop adequate teacher education

and curricula to provide a richer framework for teaching visuospatial reasoning and geometry. The book will provide an international and cultural perspective with contributions from eight different countries.

Primary audience Mathematics education researchers and postgraduate prior-to-school and primary/elementary teachers. It will provide a comprehensive background in the area of space and geometry education for researchers in this area but also for lecturers whose main research interests are in other areas. Secondary audience: It provides a sound background of early year development for postgraduate secondary teachers and should be a recommended text for reading lists in all mathematics education courses (undergraduate and postgraduate).

Possible Sections and Chapters Section Social and research contexts 1

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Chapters

Visuospatial reasoning in context This introductory chapter will introduce visuospatial reasoning and the contexts in which it is used and developed. It will provide • a preliminary definition of visuospatial reasoning; • an overview of prior-to-school contexts, primary and middle school contexts, and sociocultural contexts; • the importance of visuospatial reasoning for education and for future areas of learning such as physical activity, geographic information systems, vocations, and diagrammatic and model representations; • parameters for the book intended for mathematics educators rather than geographers or other vocations; the focus on prior-to-school, primary and middle school education within a cultural context; • outline of the book. Visuospatial thinking This chapter will provide our initial understanding of visuospatial thinking. It will synthesise the psychological literature on spatial abilities and visual imagery building on earlier research from the 1980s and 1990s with an expanded framework on generative learning. The topic will be developed by incorporating more recent research on • perception, • spatial abilities, • visual imagery, • learning in context, • visuospatial thinking. The role of intuition, attention and intention in visuospatial reasoning Critical to our understanding of the link between cognition and context are the following topics:

intuition or incidental learning, noticing and attending, intention and awareness, visuospatial reasoning in emerging and perceptual strategies. Visuospatial reasoning in diverse cultures School geometry education has developed from a Euclidean understanding of space. Diverse cultures present alternative approaches: • alternative perspectives, • practical purposes and visuospatial reasoning, • decision making, • valuing place, • representing space as place, • language and locating, • holistic views - relationships and mathematics, • patterning, • designing and making. • • • •

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Prior-to-school contexts 5

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School Contexts 8

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Spatial patterning Visuospatial reasoning is enhanced by recognition of spatial patterns and structure: • spatial patterning in prior-to-school experiences, • recognising structure as a perceptual strategy. Early spatial experiences Visuospatial reasoning develops through spatial experiences in play and investigation such as • spatial, geometric and architectural reasoning in block play, • shape analysis in environmental study, • place-based experiences in space and place: mapping, engagement, and valuing, Home to prior-to-school settings Transitions from one context to another create problems to be solved through • transitions in spatial and geometric thinking, • designing and imagining in creative play, • language, • measures, • frames of reference, • cultural transitions, • transitions from family to prior-to-school setting . Learning through problem solving Classroom context within a sociocultural context has an impact on problem solving and subsequently on visuospatial reasoning and spatial concepts. Important are • significant aspects of classroom context, • sociocultural contexts and the classroom, • the impact of classroom context on cognitive processing, • developing early visual imagery and effective imagery involving patterning, dynamic change and abstraction. Dynamic visuospatial reasoning Early dynamic visuospatial reasoning revolves around • shape changes and creating shapes, • orientation changes of shapes, • embedding and disembedding shapes,

technology for developing dynamic visuospatial reasoning, • recognising change, • positioning self in space and place. Learning visuospatial reasoning within a cultural context This chapter will outline appropriate classroom experiences and provide results from schools using an emphasis on visuospatial reasoning and cultural context through • globalised experiences, • Indigenous experiences, • multicultural classes. Learning spatial and geometric concepts This chapter will indicate how teaching strategies will encourage students to develop visuospatial reasoning and spatial concepts with overarching ideas like • variance and invariance in forming concepts, • different contexts for geometric concepts e.g. angle, • abstracting geometric concepts, • the impact of geometric concepts on visuospatial reasoning. Learning measurement and fraction concepts This chapter will emphasise visuospatial reasoning within a school and cultural context • in recognizing attributes of objects and spaces, • in deciding how to measure, • in developing fraction concepts, • in developing pattern and relationship concepts The role of language and technology Cultural contexts have an impact on the development of relationships and intentions in visuospatial reasoning through • language and multiple languages • frames of reference for space and place, • developing positional concepts • learning to map, • technology in developing visuospatial reasoning about space and place. Visuospatial reasoning in Non-Euclidean school geometry and other school learning areas Visuospatial reasoning is used in • transformations, kinematics, and other geometries, • visual representations for mathematical problem solving, • visual representations in problem solving for society and its environment, • artistic creations. •

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Educational Challenge 15 The Challenge This chapter will set out the challenge for educating teachers on visuospatial reasoning in geometry. It will report on studies of mathematics teachers illustrating • the challenge of teachers’ knowledge, • sociocultural limitations on teachers’ knowledge, • curricula and policy. 16 The Way Forward This chapter presents studies and curricula that move towards solutions through • programs that develop teachers’ knowledge and skills in visuospatial reasoning, • programs that develop teachers’ cultural knowledge,

curriculum initiatives that present recognition of cultural perspectives, Theoretical framework for visuospatial reasoning in context • a new theoretical framework for developing visuospatial reasoning in geometry within a cultural context, • implications for prior-to-school settings, • implications for school settings, • concluding remarks. •

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Length of each chapter

Each chapter should be provided on the Springer template. Each chapter should be 20 pages in length plus up to 5 pages of references.

About the Editors

Dr Kay Owens is Australian and has been a teacher at all levels of education specialising in mathematics, health education and social sciences. She lived in Papua New Guinea for 15 years where she worked at the PNG University of Technology in the Mathematics & Computing Department and at Balob Teachers College. She has worked at the University of Western Sydney and is currently teaching in western NSW at Charles Sturt University, Dubbo campus. Her doctoral study and several subsequent studies have been in the field of problem solving in the space and geometry strand of mathematics. Her current research is on measurement activities among the diverse Indigenous communities of PNG in order to record, analyse and encourage these cultural mathematics to be maintained. Her other current research focuses on quality teaching of space, geometry, measurement and early literacy (specifically mathematical literacy) in schools with a significant number of Indigenous students. Keith Jones, Associate Professor of Pedagogy and Curriculum, is from the United Kingdom. His research interests are in mathematics education especially in the acquisition and use of knowledge, cognitive aspects of mathematics learning, geometrical reasoning, proof, computers in mathematics education, mathematical problem-solving, and mathematics teacher education and professional development. He is director of the Collaborative Group for Research in Mathematics Education at the University of Southampton. He instigated and now co-ordinates the Geometry Working Group for the British Society for Research into Learning Mathematics (BSRLM) and he is a member of the editorial board of the journal Research in Mathematics Education, the official journal of the society. During 2006 he co-edited four special issues of the International Journal of Technology in Mathematics Education, and, in

2000, a special volume of Educational Studies in Mathematics on proof and proving. He has provided reports including advice to the Royal Society to report on the teaching and learning of geometry pre-19. Philip Clarkson, Professor of Education and Associate Director of the Mathematics Teaching and Learning Research Centre, has taught at the Australian Catholic University since 1985. This followed nearly five years as Director of a Research Centre at the Papua New Guinea University of Technology, and prior to that as a lecturer at Monash University and tertiary colleges in Melbourne. He began his professional life as a secondary school teacher. He has served as President of the Mathematics Education Research Group of Australasia and was foundation editor of Mathematics Education Research Journal. Research interests are wide ranging from evaluation of schools, education systems and research programs, through to various areas of mathematics education. He has had to an enduring interest in communication, with language and visually, in the context of learning mathematics.