24 SES 09, Studies in Geometry
Geometry is one of the areas of mathematical knowledge that requires complex cognitive activity because it requests the gesture, language, and look (Duval, 2005). In geometric formation it is necessary to take into account the complexity of geometry because it is an interconnected network of concepts, forms of reasoning and representation systems, used to conceptualize and analyze physical and imaginary space environments (Battista, 2007). Since our birth we are immersed in a 3D world. However, many of the visual materials presented to us are 2D. The central theoretical problem in object recognition concerns how internal representations, which incorporate the 3D structure of visual objects, can arise from 2D stimuli (Cooper, 1990). In this type of thinking, spatial visualization is one of the key elements, since it implies being able to generate mental images of shapes and figures, seeing them from different perspectives and predicting possible results of transformations and movements. These mental images are used in the visualization processes, such as the interpretation of figurative information and the visual processing of information (Bishop, 1989; Gutiérrez, 2006). The spatial visualization requires the mental coordination of the plane and the space and, therefore, it is important the development of the capacity to represent spatial figures in the plane and vice versa (Gómez & Martínez, 2011).
Del Grande (1990) enunciates seven spatial capacities absolutely relevant to the development of spatial visualization and analysis of 2D figures of 3D constructions. These capacities are: visual-motor coordination, figure-context perception, conservation of perception, perception of position in space, perception of spatial relations, visual discrimination and visual memory.
In spatial geometry, the process of understanding the concept underlying a flat representation of a 3D object is very demanding for the younger students, since it is necessary to interpret the flat figure to make it into a 3D mental representation and to convert it into geometric concept under study. Some of the representations presented in the tasks conserve the information of the visual aspect of the solids but lose the corresponding to the hidden part of the solids. Others keep information about the structure of the solids, but lose their visual aspect. In addition, part of the information that is preserved is due to the fact that, in order to make flat representation and to interpret it, certain codes have been shared, so that certain objective data are always interpreted in the same way. The lack of knowledge of these codes, characteristic of each type of flat representation, can lead to a wrong reading of flat representations.
Tasks involving constructions with polycubes by evoking the objects and their representations, help to develop the visualization capacity and expressions for the calculation of volumes. It is assumed that the tasks proposed in the national tests, when proposed to students in class, contribute to influence the teaching-learning process (Boesen, Lithner & Palm, 2010). Therefore, in this study, which presents partial data of a broader investigation that is under development, the data collected were based on geometry tasks of external assessment tests of the 6th grade. In this part of the study geometry tasks involving 2D representations of constructions with polycubes are analyzed. More specifically, we study the evoked mental processes, the representations and mental processes used by the students, and the difficulties revealed.
In this study, were taken into account the investigations carried out by Hirstein (1981), Ben-Chaim, Lappan and Houang (1985), Battista and Clements (1996) and Finesilver (2015) on students' understanding of tasks with parallelepiped constructions with polycubos, where several conceptual structures are described that students produce by enumerating cubes and suggesting mental operations underlying these constructions.
The present research, of a qualitative nature, according to an interpretative approach, focuses on the meaning that individuals confer to the phenomena under study (Bogdan & Biklen, 1994). This study involved 171 students who were finishing their 6th grade of schooling. The selection of these students was intentional and took into account the proximity and trust between the author of this work (also teacher of the students during at least the 6th grade), and the participants, since they would thus be more encouraged to exhibit ideas in a familiar school environment. Their levels of performance were also considered because this could allow access to a greater diversity of resolution strategies and, consequently, to different cognitive processes and errors. The tasks were solved after the geometry unit of the 2nd cycle of Basic Education was read, in relation to the Mathematics Program of Basic Education (ME, 2007), for the students who attended the 6th year until 2013, and the Curricular Mathematics Program and Goals of Basic Education (MEC, 2013), for students who attended the 6th year up to 2015. Data collection for the three tasks under study was done in a school context, based on: (i) document collection (task of the 2010 benchmark test - the national benchmarking tests were intended to provide relevant information to the teachers, schools and educational administration, and have no effect on pupils' school progression - and the final examinations of the 2nd cycle of 2012 and 2013 - the final national exams perform the functions of certification, selection, gauging and regulation, producing effects in the students' progression in school - and resolutions of these tasks by the participating students); and (ii) in semistructured interviews with students, recorded on video. Each of the students interviewed had the examination sheet with the resolution produced by themselves and, after reading each question, explained how they arrived at the answer. The analysis of the data initially involved the organization of the information obtained by the written productions of the students and the subsequent interviews. Although they seem very different in what is asked of students, the tasks under study have some similarities because they evoke, in common, some visual abilities necessary to solve them. From the interviews analysis, a categorization was constructed taking into account the proposals of Battista and Clements (1996) and Finesilver (2015).
In general, the resolution of the three tasks evokes the simultaneous need for some visualization abilities, mainly the capacity of conservation of perception and the perception of spatial relations. In addition, it requires the construction and manipulation of mental representations of 3D objects and the perception of an object from different perspectives. The results of the present study suggest that some students, in general terms, have revealed important flaws in the knowledge involved in the visualization of constructions with two-dimensionally represented polycubes. We can verify that the students had different levels of dexterity in the use of the processes and visual capacities. The resolution of the tasks also shows the simultaneous need for spatial structuring and organized strategies of counting cubes in a construction with polycubes. Some students have found difficulties in relating perspective representations with the solids they represent, as well as in the visualization of hidden parts of graphically presented objects, because the spatial structuring they have of the construction is incorrect. The root of this erroneous spatial structuring seems to be the inability to coordinate and integrate the views of a construction to form a coherent and unique mental model. Solving the second task, which required greater cognitive demand, in addition to the aforementioned capabilities, requires students to master organized cub count strategies. However, the representation presented in perspective and the constitution of the three constructions with polybonds, apparently, facilitated the counting of the cubes. This task proved to be extremely rich in information about the nature of the variety of enumerative strategies in use. From the analysis of the interviews, a categorization was constructed taking into account the strategies that the interviewed students used in counting the cubes.
Battista, M. (2007). The Development of Geometric and Spatial Thinking. Em F. Lester, The Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-907). Reston VA: NCTM. Battista, M. T., & Clements, D. H. (1998). Students’ understanding of three-dimensional cube arrays: Findings from a research and curriculum development project. Em R. Lehrer, & D. Chazan(Eds.), Designing learning environments for developing understanding (pp. 227-248). London, UK: Routledge. Battista, M., & Clements, D. H. (1996). Students' understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27 (3), 258-292. Ben-Chaim, D., Lappan, G., & Houang, R. T. (1985). Visualizing rectangular solides made of small cubes: Analyzing and effecting studens' performance. Educational Studies in Mathematics, 16, 389-409. Bishop, A. (1989). Review of research on visualization in mathematics education. Focus on Learning Problem Solving in Mathematics, 11(1), 7-16. Boesen, J., Lithner, J., & Palm, T. (2010). The relation between types of assessment tasks and the mathematical reasoning students use. Educational Studies in Mathematics, 105, 75-89. Bogdan, & Biklen. (1994). Investigação Qualitativa em Educação. Porto: Porto Editora. Cooper, L. (1990). Mental representation of three-dimensional objects in visual problem solving and recignition. Journal of Experimental Psychology: Learning, Memoty and Cognition, 16 (6), 1097-1106. Del Grande, J. (1990). Spatial sense. Arithmetic Teacher, 37(6), 14-20. Duval, R. (2005). Les Conditions Cognitives de L'apprentissage de la Géométrie: Développement de la visualisation, différenciation des raisonnements et coordination de leurs fonctionnements. Annales de Didatctique et sciences cognitive. 10, pp. 5-53. Strasbourg: Irem. Finesilver, C. (2015). Spatial structuring, enumeration and errors of S.E.N. students working with 3D arrays. Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education, (pp. 252-258). Praga. Gómez, J. L., & Martínez, P. F. (2011). Sentido espacial. Em I. S. Alex, & L. R. (Coords.), Matemáticas para maestros de Educación Primaria (pp. 329-349). Madrid: Pirámide. Gutiérrez, A. (2006). La investigación sobre enseñanza y aprendizaje de la geometria. Em P. Flores, F. Ruiz, & M. Fuente, Geometría para el siglo XXI (pp. 14-58). Andaluzia: Federación Española de Sociedades de Profesores de Matemáticas e Sociedad Andaluza de Educación Matemática Thales. Hirstein, J. J. (1981). The second national assessment in mathematics: area and volume. Mathematics Teacher, 74, 704-708. ME. (2007). Programa de Matemática do Ensino Básico. Lisboa: Ministério da Educação. MEC. (2013). Programa e Metas Curriculares de Matemática do Ensino Básico. Lisboa: Ministério da Educação e Ciência.
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