Solo and solid geometry : interpreting upper primary students' representations of the three-dimensional geometrical shapes and their development of three-dimensional visualization
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Within the study of the three-dimensional (3D) geometry in the primary school curriculum is the development of students’ spatial visualization, the ability to visualize the 3D geometrical shapes. Both the National Council of Teachers of Mathematics in the States and the Curriculum Development Council in Hong Kong emphasis that all students should have opportunities to visualize and work with 3D shapes in order to develop spatial skills fundamental to everyday life and to many careers. This research is a study about how a sample of 297 upper primary students in Hong Kong describe in (Chinese) words about, and draw on papers, the 3D geometrical shapes. Theses shapes include cube, triangular prism, pyramid and cylinder. The focus is on the interpretation of the diversity of the responses used in students’ descriptions and on the exploration of students’ development of visualization by their drawings of the four 3D shapes. It raises issues about upper primary students’ spatial ability and the need to conceptualize 3D spatial ability within board theoretical frameworks in the literature. The theoretical models of different aspects are explored: Bishop’s IFI and VP spatial abilities be adopted to define the scopes of spatial abilities; Van Hiele’s levels of geometry learning be the basis of the macro stages of cognitive development; and the Prestructural-Unistructural-Multistructural-Relational (P-U-M-R) sequence in the SOLO model be the micro levels of cognitive development growth. Epistemologically, the view of knowledge in this study is based on Vygotsky’s socialcultural perspective. The grounded theory approach and the interpretive approach are adopted for the research methodology. A fundamental finding is that the response patterns about the descriptions and the drawings of the 3D geometrical shapes are rich and diverse. Also, the findings support the P-U-M-R levels of the SOLO model and provide a set of descriptors of students’ developmental representation patterns for the four levels both in descriptions and drawings underpinned by the theoretical framework achieved in this study. This study concludes with recommendations for explicit instructions on the 3D geometrical shapes such as the 3D geometrical vocabulary, 3D drawing conventions, the static nature of representational drawings on papers, building up the coordination views, the partial occlusion technique, hands-on activities with real objects and relating 3D objects in the real world.
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