Modern Educational Approaches
R. Afkhami; N. Asghary; A. Medghalchi
Abstract
Background and Objectives: The figural patterns have a unique capacity to enhance functional thinking. The patterns generalization in school mathematics is considered as a way to promote functional thinking. Variable is one of the concepts in patterns generalization. Paying attention to figural patterns ...
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Background and Objectives: The figural patterns have a unique capacity to enhance functional thinking. The patterns generalization in school mathematics is considered as a way to promote functional thinking. Variable is one of the concepts in patterns generalization. Paying attention to figural patterns provides an opportunity for students to understand the meaning of variable and how to use it. Reference is also a central concept in patterns generalization. The number of variables is one of the characteristics that has been proposed in the pattern generalization tasks, but all the research has been related to one variable, linear and quadratic patterns. The aim of this study was to identifying the prior schemas in generalization of two-variable figural patterns. As regard to the concept of two variables, understanding three-dimensional space is a prerequisite for understanding and generalizing two-variable patterns. In these patterns, instead of one independent variable, there are two independent variables that change simultaneously and affect the dependent variable. Understanding these patterns requires the development of the R2 space scheme to R3 space, which is not a cognitively complex step and does not require the reconstruction of the existing scheme. Methods: The present research is part of a broad research which is done using quantitative-qualitative (mixed) research method. The research framework is APOS theory and based on the use of ACE (Activities, Class discussions and Exercises) teaching cycles. This research was conducted in three steps. In the first step, initial genetic decomposition for generalization of two-variable figural patterns was designed using the background, self-concept analysis and researchers’ experiences. It includes the prior schemas for generalization. In the second step, from the total 493 students of Malekan city (in East Azerbaijan) as the statistical population of research, a sample of 220, 7th grade students were selected based on the Cochran formula for determination of sample size. Then, a test that includes 7 tasks was designed based on APOS framework. The validity of the test was confirmed by three experts in mathematics education and four experienced teachers. Internal consistency of questions was estimated with Cronbach’s alpha and reported to be 0.69. Students responded the test at 90 minutes. The third step of research began with 19 students, with permission from the education and training office of Malekan, and school principals and parents of students. This step is done in three cycles. Findings: Using the analysis of students' responses to this test based on the APOS framework and doing three cycles of the research were conducted with the teaching method of Activity-Class Discussion-Exercise (ACE) with 19 students; genetic decomposition was finalized in this way, and defects of students in reference schema, R3 schema and variables schema as prior schemas in generalization of two-variable figural patterns were identified and encoded. Most of students had a good understanding of working with two variables. However in the context of generalization of two-variable figural pattern revealed many difficulties at the naming of variables, and using independent and dependent variables in proper positionConclusion: Identifying the mental constructs of students in generalizing patterns eases better teaching and learning. Conclusion: By identifying the mental structures of students in generalizing patterns, the path of teaching and learning will be smoother.
Education technology -training course
E. Reyhani; Z. Sharifi
Abstract
Background and Objectives:One of the most important concepts in mathematics, which has always been difficult for students to understand, is the concept of limit. Due to the connection of this concept to many other concepts such as infinitely large and infinitesimally small, continuity, derivative and ...
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Background and Objectives:One of the most important concepts in mathematics, which has always been difficult for students to understand, is the concept of limit. Due to the connection of this concept to many other concepts such as infinitely large and infinitesimally small, continuity, derivative and integral, its correct understanding and comprehension is of particular importance and this has led to its teaching and learning by math educators. Although this concept has been explored many times in educational research by researchers, it is still difficult for students to understand. There are several ways to identify problems in understanding concepts, including the concept of limit. One of these methods is to study how concepts and structures are formed that students create to learn concepts in their minds. The aim of this study is to assess students, understanding of the concept of limit in the third year of secondary school based on the APOS theory. APOS theory is a theory of learning that is used in academic mathematics. The theory categories students’ understanding of concept across the levels, and is able to models mental structures that person to understand of the concept. Method and Materials: This research is a descriptive study using survey. The sample of this research is 234 students in third grade from Qarchak city who have been randomly selected. The instrument is a researcher-made questionnaire with six questions. The reliability of the test was estimated by Cronbach’s alpha and is approved in the amount of 0.82. Findings:The results show that most students do not have a good understanding of the concept of limit and they mostly can do the limit problems correctly, if they have a routine way to solve them. The weak structures affect not only their understanding of the concept, but also depend on the understanding of the concepts such as continuity. Conclusion: When introducing the concept of limit, the teacher can prevent the construction of correct schemas of the concept of limit by using slang and giving the initial idea. Because the role of the teacher in constructing a concept of limit is very important, if the teacher teaches in an inappropriate way, it may prevent the student from absorbing the concept of limit. Another reason for stopping the growth and development of the concept is the continuous evaluations in educational environments that do not emphasize the need for conceptual understanding of the concepts and are limited to routine methods to achieve better results. For this reason, doing research in the field of teaching and learning any concept in mathematics, such as the concept of limit, can lead to more effective teaching strategies.
