Emphasis on prescribed procedures such as subtracting with two-digit numerals may inhibit this construction process. As mathematics activities are planned, it is important to provide opportunities for students to construct abstract composite units in both geometric and numeric settings. Unitizing seems to be a fundamental mental operation in coming to act mathematically. Evidence of unitizing and coordinating the units constructed was associated with advances in mathematical thinking. Students who constructed abstract composite units in tiling did so also in adding and subtracting whole numbers. Some students constructed rather sophisticated abstract composite units to facilitate their tiling with a particular shape while others had difficulty making a covering. The tiling activity of the students was analyzed for evidence of the construction and coordination of units. A consistent parallel was found in the sophistication of the types of units constructed in a geometric setting (tiling the plane) with their numeric activity. Data were gathered from students in grade three through six, with four students being observed over a three year period. The main thesis of this paper is that the construction and coordination of abstract units is central to mathematical activity in both numerical and geometric settings. Included is a bibliography of the writings of both authors, whose doctoral dissertations were presented to the University of Utrecht in 1957. Part 2 contains the last article written by Dina van Hiele-Geldof entitled "Didactics of Geometry as Learning Processes for Adults." Part 3 provides a summary of Pierre van Hiele's dissertation entitled "The Problems of Insight in Connection with School Children's Insight into the Subject Matter of Geometry" and an article by the same author about a child's thought and geometry. Part 1 of the document includes the dissertation of Dina van Hiele-Geldof entitled "The Didactics of Geometry in the Lowest Class of Secondary School" and a summary of the dissertation written by Dina van Hiele-Geldof. These translations were done as part of a research project entitled "An Investigation of the van Hiele Model of Thinking in Geometry among Adolescents" which was supported by a grant from the National Science Foundation. It is the purpose of this monograph to present English translations of some significant works of the van Hieles. #Translation tessellation software#Technological implications of the software are discussed as well as the potential for K-8 classroom use.Īfter observing secondary school students having great difficulty learning geometry in their classes, Dutch educators Pierre van Hiele and Dina van Hiele-Geldof developed a theoretical model involving five levels of thought development in geometry. Students' zones of proximal development were bridged, as students both collaborated and practiced independently, gaining a deep understanding of how transformations are used to create tessellations. Students were engaged in rich social interactions, exchanging predictions, and defending their reasoning. By combining a hands-on activity in which these and other transformations were demonstrated, followed by student exploration of the software, resulted in an unplanned, but welcomed, Vygotskian environment. Prior to discussing the topic of transformations, students were only slightly familiar with two types of transformations, namely, rotations and reflections. In the winter of 2000, a class of K-8 preservice teachers were introduced to TesselMania! Deluxe, a software product that visually and manually engages students in the study of transformations but, more specifically, tessellations. Various different textures will be gained to tessellate figures by mathematical transformations including translation, rotation, reflection and glide reflection on a surface. In the present study we will try to extend Escher's tessellation artworks into 3D for interior designing. The overall objective of the study is to emphasize how the use of interior design and different scientific disciplines could get a wealth to a place, how the situation activate the psychological perceptions and importance according to compliance with human-object-surrounding. For example, floor tiles could exactly be an Escher's tessellation artwork as in John August's Gecko Stone. An aplication of tessellation in interior designing is accepted as a decorative work. Tilings appear in all our surroundings, i.e., floor tiles, walls, ceilings, separators and even surfaces of equipment. Tiling is a very common way in general architecture. Escher's tessellation artworks are creative figures in the 2D plane. Tessellation is an arrangement of closed shapes that completely cover the plane without overlapping or leaving gaps.
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