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Irene Biza - One of the best experts on this subject based on the ideXlab platform.
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Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space
Technology Knowledge and Learning, 2011Co-Authors: Irene BizaAbstract:In this paper I report a lengthy episode from a teaching experiment in which 15 Year 12 Greek students negotiated their definitions of Tangent Line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of Tangent to a function graph. Its design was based on previous research results on students’ perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students’ knowledge develops in specific social and cultural contexts, students’ construction of the meaning of Tangent Line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students’ personal meanings about Tangent Line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students’ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students’ meanings towards a more sophisticated understanding of tangency was not Linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher.
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First Year Mathematics Undergraduates' Settled Images of Tangent Line.
The Journal of Mathematical Behavior, 2010Co-Authors: Irene Biza, Theodossios ZachariadesAbstract:This study concerns 182 first year mathematics undergraduates’ perspectives on the Tangent Line of function graph in the light of a previous study on Year 12 pupils’ perspectives. The aim was the investigation of tangency images that settle after undergraduates’ distancing from the notion for a few months and after their participation in university admission examination. To this end we related the performances of the undergraduates and the pupils in the same questions of a questionnaire; we classified the undergraduates in distinct groups through Latent Class Analysis; and, we examined this classification according to the Analytical Local, Geometrical Global and Intermediate Local perspectives on tangency we had identified among pupils. The findings suggest that more undergraduates than pupils demonstrated intermediate perspectives on tangency. Also, the undergraduates’ settled images were influenced by persistent images about tangency and their prior experience in the context of preparation for and participation in the examination.
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Models of students’ conceptions about the Tangent Line
2008Co-Authors: Irene BizaAbstract:The results presented in this paper are a part of a doctoral study nearing completion which focuses on students’ conceptions about Tangent Lines. The study explores the ways in which students who have studied Tangents in school in different contexts (Euclidean Geometry, Analytic Geometry and Analysis): (i) perceive properties that are not generally valid as defining conditions of the concept and (ii) create new, often not-valid, properties out of the fusion of information from across the different contexts. Data was collected through a questionnaire administered to 182 first year mathematics undergraduates in Greece at the very beginning of their studies. Here I exemplify from the data analysis in the course of which several models of the Tangent Line have emerged based on the above properties.
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Student perspectives on the relationship between a curve and its Tangent in the transition from Euclidean Geometry to Analysis
Research in Mathematics Education, 2008Co-Authors: Irene Biza, Constantinos Christou, Theodossios ZachariadesAbstract:The Tangent Line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the Tangent Line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the Tangent Line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about Tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis.
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Is the Tangent Line tangible? Students’ intuitive ideas about Tangent Lines
2007Co-Authors: Irene BizaAbstract:The results presented in this paper are a part of a doctoral study nearing completion which focuses on students’ intuitive thinking about Tangent Lines. The study explores the ways in which students who have studied Tangents in school in different contexts (Geometry, Analytic Geometry and Calculus): (i) perceive properties that are not generally valid as defining conditions of the concept and (ii) create new, often notvalid, properties out of the fusion of information from across the different contexts. Data was collected through a questionnaire administered to 182 first year mathematics undergraduates in Greece at the very beginning of their studies. Here I exemplify from the data analysis in the course of which several models of the Tangent Line have emerged based on the above properties. THEORETICAL ASSUMPTIONS Students in secondary school or at university often revisit concepts that they have studied earlier and at a more elementary level. In this case their total cognitive structure associated with a concept (the concept images according to Tall and Vinner (1981)) that has developed during their early studies has to be modified (for example, generalised) in order to be applicable in a broader context. This generalisation is a non trivial step in students’ knowledge acquisition which is not a straightforward accumulative process. According to Harel and Tall (1989) there are different kinds of generalisation such as expansive generalisation, in which the student’s existing cognitive structure is extent, without requiring changes of current ideas and reconstructive generalisation, in which the existing cognitive structure is reconstructed in order to widen its range of applicability. In the reconstructive generalisation the old concept images have to be radically changed so as to be applicable in a broader context. When students cannot achieve this generalisation various misconceptions can arise. Some of these are caused by the student’s effort to assimilate new information in their existing knowledge, although these two are
Tomoatsu Kimura - One of the best experts on this subject based on the ideXlab platform.
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Intraoperative Measurements of Femoral Anterior Tangent (FAT) Line for Determining the Rotational Alignment of Femoral Component of Total Knee Arthroplasty
The Journal of arthroplasty, 2013Co-Authors: Hiroki Watanabe, Ryuichi Gejo, Ayano Tokunaga, Norikazu Hirano, Tomoatsu KimuraAbstract:Abstract Previously, we reported using CT images that the anterior surface of the femur immediately proximal to the trochlea and its Tangent Line (femoral anterior Tangent Line; FAT Line) could be used as a good index of the femoral rotation. In this study, we developed a jig that allowed us to measure the FAT Line during surgery, and we examine the relation between preoperative and intraoperative measurement values. The results indicated that the average intraoperative measurement value of the ‘surgical' FAT Line was 9.8° ± 3.2° internally rotated using surgical transepicondylar axis reference. This value significantly correlated to preoperative FAT Line/clinical transepicondylar axis angle. These findings demonstrated that FAT Line is a useful index for appropriate rotational alignment of femoral component, both before and during TKA.
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Femoral anterior Tangent Line of the osteoarthritic knee for determining rotational alignment of the femoral component in total knee arthroplasty.
The Journal of arthroplasty, 2010Co-Authors: Hiroki Watanabe, Ryuichi Gejo, Yoshikazu Matsuda, Ichiro Tatsumi, Kazuo Hirakawa, Tomoatsu KimuraAbstract:Several reference axes have been used to establish femoral rotational alignment during total knee arthroplasty. The current study examined the configuration of the anterior surface of the femur immediately proximal to the trochlea as an alternative rotational landmark. An analysis of computed tomographic images of 150 knees with osteoarthritis indicated that the configuration of the surface is mostly flat or slightly depressed, and the Line Tangential to the surface (femoral anterior Tangent Line; FAT Line) was consistently determined to be 12.2° ± 3.6° internally rotated to the transepicondylar axis. This value was relatively constant and as reliable as the femoral anteroposterior axis for determining rotational alignment. In addition, the FAT Line was not affected by the degree of the varus-valgus deformity of the osteoarthritic knees.
Theodossios Zachariades - One of the best experts on this subject based on the ideXlab platform.
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First Year Mathematics Undergraduates' Settled Images of Tangent Line.
The Journal of Mathematical Behavior, 2010Co-Authors: Irene Biza, Theodossios ZachariadesAbstract:This study concerns 182 first year mathematics undergraduates’ perspectives on the Tangent Line of function graph in the light of a previous study on Year 12 pupils’ perspectives. The aim was the investigation of tangency images that settle after undergraduates’ distancing from the notion for a few months and after their participation in university admission examination. To this end we related the performances of the undergraduates and the pupils in the same questions of a questionnaire; we classified the undergraduates in distinct groups through Latent Class Analysis; and, we examined this classification according to the Analytical Local, Geometrical Global and Intermediate Local perspectives on tangency we had identified among pupils. The findings suggest that more undergraduates than pupils demonstrated intermediate perspectives on tangency. Also, the undergraduates’ settled images were influenced by persistent images about tangency and their prior experience in the context of preparation for and participation in the examination.
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Student perspectives on the relationship between a curve and its Tangent in the transition from Euclidean Geometry to Analysis
Research in Mathematics Education, 2008Co-Authors: Irene Biza, Constantinos Christou, Theodossios ZachariadesAbstract:The Tangent Line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the Tangent Line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the Tangent Line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about Tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis.
Hiroki Watanabe - One of the best experts on this subject based on the ideXlab platform.
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Intraoperative Measurements of Femoral Anterior Tangent (FAT) Line for Determining the Rotational Alignment of Femoral Component of Total Knee Arthroplasty
The Journal of arthroplasty, 2013Co-Authors: Hiroki Watanabe, Ryuichi Gejo, Ayano Tokunaga, Norikazu Hirano, Tomoatsu KimuraAbstract:Abstract Previously, we reported using CT images that the anterior surface of the femur immediately proximal to the trochlea and its Tangent Line (femoral anterior Tangent Line; FAT Line) could be used as a good index of the femoral rotation. In this study, we developed a jig that allowed us to measure the FAT Line during surgery, and we examine the relation between preoperative and intraoperative measurement values. The results indicated that the average intraoperative measurement value of the ‘surgical' FAT Line was 9.8° ± 3.2° internally rotated using surgical transepicondylar axis reference. This value significantly correlated to preoperative FAT Line/clinical transepicondylar axis angle. These findings demonstrated that FAT Line is a useful index for appropriate rotational alignment of femoral component, both before and during TKA.
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Femoral anterior Tangent Line of the osteoarthritic knee for determining rotational alignment of the femoral component in total knee arthroplasty.
The Journal of arthroplasty, 2010Co-Authors: Hiroki Watanabe, Ryuichi Gejo, Yoshikazu Matsuda, Ichiro Tatsumi, Kazuo Hirakawa, Tomoatsu KimuraAbstract:Several reference axes have been used to establish femoral rotational alignment during total knee arthroplasty. The current study examined the configuration of the anterior surface of the femur immediately proximal to the trochlea as an alternative rotational landmark. An analysis of computed tomographic images of 150 knees with osteoarthritis indicated that the configuration of the surface is mostly flat or slightly depressed, and the Line Tangential to the surface (femoral anterior Tangent Line; FAT Line) was consistently determined to be 12.2° ± 3.6° internally rotated to the transepicondylar axis. This value was relatively constant and as reliable as the femoral anteroposterior axis for determining rotational alignment. In addition, the FAT Line was not affected by the degree of the varus-valgus deformity of the osteoarthritic knees.
Shiqi Duan - One of the best experts on this subject based on the ideXlab platform.
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MLICOM (2) - Obtaining Ellipse Common Tangent Line Equations by the Rolling Tangent Line Method
Machine Learning and Intelligent Communications, 2018Co-Authors: Naizhang Feng, Teng Jiang, Shiqi DuanAbstract:In the field of image processing and machine vision, it is sometimes necessary to obtain common Tangent Line equations and Tangent point coordinates from ellipses. A rolling Tangent Line method was proposed to obtain the 4 common Tangent Line equations and 8 Tangent point coordinates from two ellipses in this paper. The principle of this method is simple and it is easy to program on a computer. Use this method to process two ellipse targets in an image and the experiment results show that the 4 common Tangent equations and 8 Tangent point coordinates can be obtained in high precision and the maximum execution time is less than 0.1 s.
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obtaining ellipse common Tangent Line equations by the rolling Tangent Line method
International Conference on Machine Learning, 2017Co-Authors: Naizhang Feng, Teng Jiang, Shiqi DuanAbstract:In the field of image processing and machine vision, it is sometimes necessary to obtain common Tangent Line equations and Tangent point coordinates from ellipses. A rolling Tangent Line method was proposed to obtain the 4 common Tangent Line equations and 8 Tangent point coordinates from two ellipses in this paper. The principle of this method is simple and it is easy to program on a computer. Use this method to process two ellipse targets in an image and the experiment results show that the 4 common Tangent equations and 8 Tangent point coordinates can be obtained in high precision and the maximum execution time is less than 0.1 s.