Construction of linear algebra concepts through 3d hologram-based learning

Linear algebra, which is one of the basic courses of university mathematics, consists of abstract concepts that students have difficulty in learning. While learning the concepts of mathematics, when abstract concepts are not based on concrete foundations in the learning process, which should take place from concrete to abstract, students may focus on operational process rather than conceptual understanding. In a linear algebra course, which is taught with definitions and theorems in the formal world, students focus on the operational process during problem-solving and cannot perceive the nature of the concepts. The conflict between the concepts they have learned before (such as vectors) and the concepts they have just learned, or the perceptions that arise due to the nature of the concept (the feeling that space is infinite) may prevent learning linear algebra concepts. Not including real-world applications of concepts in lessons can create difficulties in reaching formal comprehension and object understanding. One of the reasons for the difficulties experienced is the insufficient use of geometric representations in three-dimensional space applications. When three-dimensional geometric representations are drawn on a two-dimensional planar computer screen or blackboard, students may not be able to perceive figural representations as three-dimensional. Three-dimensional hologram devices, which allow the models prepared on a computer to be created in three dimensions using the diffraction of light, can eliminate these difficulties in a linear algebra course. Linear algebra models that can be created using 3D hologram technologies can help students to have conceptual understanding and object understanding by enabling real-life examples of linear algebra concepts to be seen in three dimensions. The main purpose of the research is to examine the process of creating linear algebra concepts in three-dimensional hologram-supported teaching. The study was designed with three separate case studies. The aim of the first case study was to review the literature on the difficulties experienced in learning linear algebra concepts and to determine the genetic analyses of linear algebra concepts, which provide a roadmap of how students learn a concept, with the validity-reliability study carried out within the framework of the APOS theory and researcher experiences. For the validity and reliability study of genetic analyses and linear algebra problems prepared by the researcher, five students studying at Dokuz Eylul University Elementary Mathematics Teaching Department and taking a linear algebra 2 course were studied. Descriptive analysis was used in the analysis of the data and analyses were carried out with rubrics to evaluate problems. It is seen that the errors in the vector space problems of the students were caused by the deficiencies in the binary operations and set and field diagrams, as well as in the function diagrams in the inner product space problems. It has been observed that in finding the inverse of the linear transformation in linear transformation problems, the inverse of the transformation matrix must be inverse, and its determinant must be different from zero. However, since it was not taught that linear transformation should be defined between the same spaces, it is understood that students focused on matrix representation and did not pay attention to the geometric understanding of the concept. In the problems of eigenvalues and eigenvectors, it was seen that they could not express that the vectors in linear transformation are solids and could not explain their geometric meaning by associating them with algebraic representations. In general, it has been observed that there was difficulty in associating the geometric meanings of the concepts with the solutions of the problems related to these concepts. A needs analysis has been made regarding the three-dimensional hologram applications that were to overcome these difficulties. HoloDEU software and the hologram system were developed to include software and platforms to meet these needs. The pilot study of this HoloDEU software and the hologram system was carried out with students studying at Dokuz Eylul University Elementary Mathematics Teaching Department and taking a linear algebra 2 course. The interview and observation data obtained from this study, which was designed with a case study, were analyzed by content analysis and the data obtained from the vector space problems were analyzed by descriptive analysis. At the end of the research, the weaknesses of HoloDEU software and the hologram system were strengthened, the functions were determined, and the software and hologram system were updated. In addition, pedagogical suggestions were made regarding the use of HoloDEU software and the hologram system in mathematics education and other educational processes. When the answers given to the vector space questions were examined, seven students who showed skills at the object stage according to the APOS theory were selected. The main application working group of the research consists of students studying at Dokuz Eylul University, Department of Elementary Mathematics Education. The activities to be applied in linear algebra teaching have been prepared in accordance with the activity-class discussion and exercises (ACE) teaching cycle framework, which is a part of the APOS theory. At the end of the teaching, open-ended problems prepared by the researcher with input from expert opinions were given to the students. The data obtained were evaluated within the framework of the APOS theory with descriptive analysis. According to the findings, the students exhibited skills at the object stage. It has been seen that some mistakes made were caused by inadequacies in concepts that require prior knowledge in the genetic analysis of concepts. In addition, it was seen that the students explained their problem solutions by using the models they used in hologram supported teaching. In addition, it has been observed that in teaching carried out with HoloDEU software and the hologram system, students actively participated, they were interested, they were able to use geometric expressions consisting of abstract concepts such as linear algebra, and they could solve problems where they had to choose the algebraic solution. More detailed information about the newly researched 3D hologram based learning can be obtained by popularizing the use of 3D Hologram devices, HoloDEU and 3D Hologram software system, using 3D holographic technologies and HoloDEU in teaching other linear algebra concepts such as systems of linear equations, linear independence, mathematical concepts, concepts in science and geography lessons supporting realistic teaching and examining teaching process, achievements, attitudes, motivations…etc. Keywords: 3D Hologram, HoloDEU, APOS Theory, linear algebra, geometric modes.