24 SES 09, Studies in Geometry
Reforms in mathematics education recommend that geometry instruction should include the study of three-dimensional geometric figures and provide students with opportunities to use spatial skills in problem-solving tasks (NCTM, 2000). It is argued that geometry instruction should give increased attention on the analysis of three-dimensional figures as well as to the continued development of students' skills in visualization and pictorial representation of three-dimensional figures. In line with these recommendations, geometry curriculla all over the world are designed so that geometry instruction starts with two- and three-dimensional figures as early as in first grade. Prisms and pyramids are two of those three-dimensional figures that are most well known by, at least most familiar to, the students among the three-dimensional solids. More specifically, prisms and pyramids are subclasses of a polyhedron which is defined as a three-dimensional shape consisting of the union of polygonal faces resulting in an enclosed region without any holes (Leavy and Hourigan, 2015).
Although we live in a three-dimensional world, understanding and representing these figures is not an easy task for many students. Especially when instruction is limited only to a static paper-and-pencil representation of these figures, students’ three-dimensional visualization becomes more complex. Instruction in three-dimensional geometry usually presents students with planar two-dimensional representations of solids rather than with actual models (Battistia, 1999). It is well known that students have great difficulties conceptualizing solids that are presented on two-dimensional surfaces (Accascina & Rogora 2006; Parzysz, 1988). Similarly, even preservice and inservice mathematics teachers have difficulty in drawing or defining prisms (Gökkurt & Soylu, 2016; Bozkurt & Koc, 2012). Therefore, visualization and conceptualization of solids are complex cognitive processes, and both require the development of students’ abilities to decode and encode spatial information (Markopoulos et al., 2015)
There have been some attempts to characterize geometric thinking related to three-dimenisonal geometry. First of these is the van Hiele model of geometric thinking. It posits that students progress through five levels of geometric thinking. While van Hiele’s model deals with geometric thinking in general, Battista and his colleagues focused on more three-dimensional thinking (Battista & Clements, 1996; Battista, 2007). In addition, Battista (2007) refined the van Hiele levels and suggested the existence of some sub-levels. Battista claimed that children’s visual/holistic thinking and descriptive/analytic thinking continually grow throughout their development, with descriptive/analytic reasoning becoming more common as children gain experience. By drawing on research in cognitive psychology and mathematics education, Gutierrez (1996) developed a theoretical framework to characterize the field of visualization in mathematics education. In his framework, visualization which is integrated by four elements (mental images, external representations, processes of visualization, and abilities of visualization) is considered as a kind of reasoning activity based on the use of visual or spatial elements, either mental or physical. Another group of study regarding geometric thinking is done by Vinner and his colleagues (Vinner, 1991; Hershkowitz, Bruckheimer, & Vinner, 1987). They considered students’ and teachers’ geometric thinking in terms of concept images and concept definitions and suggested that a concept image is the collection of mental images that an individual has of a given concept and observed that these images can be complete, partial, or incorrect. Taken together these approaches suggest that as students learn about polyhedra, they progress from looking at shapes from a global perspective to looking at parts of shapes to seeing relationships between the parts (Ambrose & Kenehan, 2009). In this study, we want to explore preservice and inservice mathematics teachers’ conceptualization of prisms and pyramids while they were identifying, drawing and defining them.
Participants of this study were 38 preservice mathematics teachers and 5 mathematics teachers. More specifically, the preservice teachers consisted of students from different years (12 form the second year, 12 from the third year and 14 from the fourth year) of a mathematics teacher education program. All of the preservice teachers completed a three-credit geometry content course in the first year of the program. Mathematics teachers were enrolled in a master program when the study was conducted. Teaching experiences of mathematics teachers varied between 1 and 10 years. Data of the study was collected using a test and interviews. The test consisted of two parts. In the first part, the participants were asked to define, to write critical attributes of and to draw examples of a prism and a pyramid. In the second part, drawings of 14 solids consisting of examples and nonexamples of prisms and pyramids were given and they were asked to classify them as belonging to prisms or pyramids. The test was administered by the first researcher during partcipants’ regular lessons and there was no time limitation. After the participants completed the test, they were invited for an interview about their responses on the test. Interviews with volunteers were conducted by the first researcher and lasted at least 15 min and at most 35 min. During the interviews, participants were asked to explain why they gave those answers and how they were thinking. Audio recordings of interviews are transcribed and were analyzed by using content analysis method. In analysis, we focused on which properties the participants attend to when they identify, define and draw prisms and pyramids. We coded each participant’s written work and interview data to determine the types of reasoning employed by her/him. By scrutinizing the data line by line, we identified aspects of participants’ geometric thinking. For example, we focused on whether they base their reasoning on visual clues such as apex or properties while they explain their responses.
Although analysis of the data is in progress, preliminary analysis showed that preservice teachers have deficiencies in defining, drawing and identifying prism and pyramids. A considerable number of preservice teachers stated that a prism is a solid having a volume, edges and an altitude. Even some claimed that pyramids are special cases of prisms. It is also important to note that preservice teachers’ knowledge of pyramids is limited to the case of a square pyramid which is the shape of the Egyptian Pyramids. Participants had an insufficient set of properties to define pyramids and prisms. They mostly relied on noncritical attributes while identifying and defining the given solids. It is also observed that a static paper and pencil representation of three-dimensional figures creates obstacles for the visualization of these solids. In other words, static paper and pencil representations of three dimensional figures resulted in participants’ difficulties in decoding and encoding visual information. Some participants had hard times in visualizing the given solids. In addition, it is observed that the visual information dominated participants’ thinking.
Accascina, G. & Rogora, E. (2006). Using Cabri3D diyagrams for teaching, International Journal for Technology in Mathematics Education, 13 (1), 11-22. Ambrose, R. & Kenehan, G. (2009) Children's Evolving Understanding of Polyhedra in the Classroom. Mathematical Thinking and Learning, 11:3, 158-176 Battista, M. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-Based Classroom. Journal for Research in Mathematics Education, 30(4), 417– 448. Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Reston, VA: National Council of Teachers of Mathematics. Battista, M., & Clements, D. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27, 258–292. Bozkurt, A. and Koc, Y. (2012). Investigating first year elementary mathematics teacher education students’ knowledge of prism. Educational Sciences: Theory & Practice, 12(4), 2949-2952. Gökkurt, B., Soylu, Y. (2016). Ortaokul matematik öğretmenlerinin matematiksel alan bilgilerinin incelenmesi: Prizma örneği. Abant Izzet Baysal Universitesi Eğitim Fakültesi Dergisi, 16(2), 451-481. Gutiérrez, A. (1996). Vizualization in 3 -dimensional geometry: In search of a framework. In L. Puig & A. Guttierez (Eds.), Proceedings of the 20th conference of the international group for the psychology of mathematics education, vol. 1 (pp. 3-15). Valencia: Universidad de Valencia. Hershkowitz, R., Bruckheimer, M., & Vinner, S. (1987). Activities with teachers based on cognitive research. In M. Lindquist (Ed.), Learning and teaching geometry, K-12: 1987 NCTM Yearbook (pp. 222–235). Reston, VA: National Council of Teachers of Mathematics. Koester, B. (2003). Prisms and Pyramids: Constructing Three-Dimensional Models to Build Understanding. Teaching Children Mathematics, 9(8), 436–442. Leavy, A. & Hourigan, M. (2015). Budding Architects: Exploring the designs of pyramids and prisms. APMC, 20(3), 17-23. Markopoulos, C., Chaseling, M., Lake, W., & Boyd, W. (2015). Pre-Service teachers’ 3D visualization strategies. Creative Education, 6, 1053-1059. National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA: Author. Parzysz, B. (1988). Problems of the plane representation of space geometry figures. Educational Studies in Mathematics, 19(1), 79–92. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Boston: Kluwer.
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