中学生几何平移概念图外文翻译资料

Middle-school studentsrsquo; concept images of geometrictranslationsH

Bahadır Yaniklowast;Anadolu University, Faculty of Education, Middle School Mathematics Education, Eskisehir 26470, Turkeya

Keywords: mathematics Geometric translations Concept images

ABSTRACT

This study explored sixth grade studentsrsquo; concept images of geometric translations andthe possible sources of their conceptions in a non-technological environment. The datawere gathered through a written instrument, student and teacher interviews and documentanalyses. Analyses of student responses revealed two major concept images of geometrictranslations: (a) translation as translational motion, and (b) translation as both transla-tional and rotational motion. Students who held these conceptions showed various levelsof understanding, such as conceiving translations as undefined motion, partially-definedmotion, and defined-motion of a single geometric figure on the plane. The findings of thestudy suggested, in general, consistencies between studentsrsquo; concept images and their con-cept definitions. However, most of the studentsrsquo; concept definitions were inconsistent withthe formal concept definition of geometric translations.Data analyses also revealed five interpretations of a translation vector,Furthermore, classroom instruction, math-ematics and science textbooks, real-life examples and everyday language were the majorsources of studentsrsquo; concept images of geometric translations.

1. Introduction Recent curriculum reform movements in Turkey brought new topics into the middle-school mathematics curriculum.Geometric transformations were among the new content areas added to the course of geometry. One of the reasons forincluding geometric transformations in the curriculum was its potential to contribute the development of studentsrsquo; mathe-matical thinking abilities such as reasoning and justification skills. Students may discover patterns, construct generalizations,and develop spatial competencies and critical thinking through studying geometric transformations. Specif-ically, geometric transformations allow “students to develop broad concepts of congruence and similarity and apply themto all figures”. “Mathematical properties from the various branches of geometry can be described in terms of transformations which may be representedthrough several types of manipulative activities” . Several national and international documents advocate the importance of studying transformation geometry. According to NCTMrsquo;sPrinciples and Standards for School Mathematics, “instructional programs from pre-kindergarten through grade 12should enable all students to apply transformations and use symmetry to analyze mathematical situations” .However, describing mathematical understanding can be a challenging task. Concept imagesare one of the conceptual tools for exploring understanding. Learnersrsquo; cognitive structures related to a concept which involverepresentations, mental images, properties and processes can be described through concept images.Learners build understanding using various sources of information such as previous courses, textbooks, classroom discuss-ions, and everyday experiences. Investigating the sources of studentsrsquo; conceptions beside the nature of their understandingwould provide specific recommendations for instructional practices. Students may develop misconceptions even after theyare introduced to formal mathematical concepts. Lack of in-class experiences and focusing on particular examples may sup-port studentsrsquo; misconceptions . Representations shown in textbooks have some limitations whichmay hinder the development of coherent ideas about mathematical concepts . Stud-ies have shown that students have difficulties connectingconcrete activities and representations with the formal mathematical concepts. Exploring student understanding wouldinform us about possible sources of these incomplete ideas. Therefore, it is crucial to investigate both the nature of studentsrsquo;understandings and the possible sources of their conceptions together.For the purpose of this study, in particular, the following research questions were investigated:bull;What are the nature of middle-school studentsrsquo; concept images of geometric translations?bull;To what extent do students expose consistencies between the concept images and concept definitions?bull;What prototypical examples do students rely on when they reason about geometric translations?bull;What are the possible sources of their conceptions about geometric translations?2. Framework for the studyThis study was guided by a cognitive model formulated by several researchers which has been characterized as the theory of concept definition and concept image. Tall and Vinner consider the term concept definition as “to be a form of words used to specify that concept” . Researchers argue that students do not always consult with concept definitions during the problemsolving process. Tall and Vinner proposed that the acquisition of concepts of mathematics depends more on thelearnerrsquo;s concept images rather than concept definition. Many concepts people use are not formally defined and everydaymeaning of many mathematical terms may interfere with the learning of mathematical concepts.Whenworking on a mathematical activity or solving a mathematical task, students rely on their individual mental representationswhich might be influenced by daily experience.“During the mental processes of recalling and manipulating aconcept, many associated processes are brought into play, consciously and unconsciously affecting the meaning and usage”.This study focused on oneexample of rigid geometric transformations called geometric translations. Geometric translations can be interpreted astranslational motion The motion interpretation of geometric transfor-mations includes “the mental or physical manipu

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   附录A 外文原文

   Middle-school studentsrsquo; concept images of geometrictranslationsH

   Bahadır Yaniklowast;Anadolu University, Faculty of Education, Middle School Mathematics Education, Eskisehir 26470, Turkeya

   Keywords: mathematics Geometric translations Concept images

   ABSTRACT

   This study explored sixth grade studentsrsquo; concept images of geometric translations andthe possible sources of their conceptions in a non-technological environment. The datawere gathered through a written instrument, student and teacher interviews and documentanalyses. Analyses of student responses revealed two major concept images of geometrictranslations: (a) translation as translational motion, and (b) translation as both transla-tional and rotational motion. Students who held these conceptions showed various levelsof understanding, such as conceiving translations as undefined motion, partially-definedmotion, and defined-motion of a single geometric figure on the plane. The findings of thestudy suggested, in general, consistencies between studentsrsquo; concept images and their con-cept definitions. However, most of the studentsrsquo; concept definitions were inconsistent withthe formal concept definition of geometric translations.Data analyses also revealed five interpretations of a translation vector,Furthermore, classroom instruction, math-ematics and science textbooks, real-life examples and everyday language were the majorsources of studentsrsquo; concept images of geometric translations.

   1. Introduction Recent curriculum reform movements in Turkey brought new topics into the middle-school mathematics curriculum.Geometric transformations were among the new content areas added to the course of geometry. One of the reasons forincluding geometric transformations in the curriculum was its potential to contribute the development of studentsrsquo; mathe-matical thinking abilities such as reasoning and justification skills. Students may discover patterns, construct generalizations,and develop spatial competencies and critical thinking through studying geometric transformations. Specif-ically, geometric transformations allow “students to develop broad concepts of congruence and similarity and apply themto all figures”. “Mathematical properties from the various branches of geometry can be described in terms of transformations which may be representedthrough several types of manipulative activities” . Several national and international documents advocate the importance of studying transformation geometry. According to NCTMrsquo;sPrinciples and Standards for School Mathematics, “instructional programs from pre-kindergarten through grade 12should enable all students to apply transformations and use symmetry to analyze mathematical situations” .However, describing mathematical understanding can be a challenging task. Concept imagesare one of the conceptual tools for exploring understanding. Learnersrsquo; cognitive structures related to a concept which involverepresentations, mental images, properties and processes can be described through concept images.Learners build understanding using various sources of information such as previous courses, textbooks, classroom discuss-ions, and everyday experiences. Investigating the sources of studentsrsquo; conceptions beside the nature of their understandingwould provide specific recommendations for instructional practices. Students may develop misconceptions even after theyare introduced to formal mathematical concepts. Lack of in-class experiences and focusing on particular examples may sup-port studentsrsquo; misconceptions . Representations shown in textbooks have some limitations whichmay hinder the development of coherent ideas about mathematical concepts . Stud-ies have shown that students have difficulties connectingconcrete activities and representations with the formal mathematical concepts. Exploring student understanding wouldinform us about possible sources of these incomplete ideas. Therefore, it is crucial to investigate both the nature of studentsrsquo;understandings and the possible sources of their conceptions together.For the purpose of this study, in particular, the following research questions were investigated:bull;What are the nature of middle-school studentsrsquo; concept images of geometric translations?bull;To what extent do students expose consistencies between the concept images and concept definitions?bull;What prototypical examples do students rely on when they reason about geometric translations?bull;What are the possible sources of their conceptions about geometric translations?2. Framework for the studyThis study was guided by a cognitive model formulated by several researchers which has been characterized as the theory of concept definition and concept image. Tall and Vinner consider the term concept definition as “to be a form of words used to specify that concept” . Researchers argue that students do not always consult with concept definitions during the problemsolving process. Tall and Vinner proposed that the acquisition of concepts of mathematics depends more on thelearnerrsquo;s concept images rather than concept definition. Many concepts people use are not formally defined and everydaymeaning of many mathematical terms may interfere with the learning of mathematical concepts.Whenworking on a mathematical activity or solving a mathematical task, students rely on their individual mental representationswhich might be influenced by daily experience.“During the mental processes of recalling and manipulating aconcept, many associated processes are brought into play, consciously and unconsciously affecting the meaning and usage”.This study focused on oneexample of rigid geometric transformations called geometric translations. Geometric translations can be interpreted astranslational motion The motion interpretation of geometric trans

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