scholarly journals The Development of Students Geometrical Thinking through Transformational Processes and Interaction Techniques in a Dynamic Geometry Environment

10.28945/3235 ◽  
2008 ◽  
Author(s):  
Stavroula Patsiomitou
Keyword(s):  
Mathematics Classroom ◽  
Mathematical Problem ◽  
Dynamic Geometry ◽  
Research Process ◽  
Geometer's Sketchpad ◽  
Rigorous Proof ◽  
Interaction Techniques ◽  
Dynamic Geometry Environment ◽  
Visual Reaction ◽  
Experimental Use

The paper draws on a didactic experiment conducted in a secondary school mathematics classroom in Greece which aimed to explore a) ways in which students develop problem representations, reasoning and problem-solving, making decisions and receiving feedback about their ideas and strategies in a DGS-supported environment b) ways in which students develop rigourous proof through building linking visual active representations and c) ways to develop students’ van Hiele level. The mathematical problem the students engaged with - either in the Geometer’s Sketchpad dynamic geometry enviroment (Jackiw, 1988) or in the static environment - generated potentially insightful data on the issues focused on the comparison between the experimental and control groups. Initially, three pairs from the experimental group explored the treasure problem within a dynamic geometry environment. The discussions and results of the discussion were videotaped. The problem was then reformulated by the researcher taking into account the research group’s retroaction, and re-explored by both the control and experimental groups in a paper-pencil test. The researcher then (semi) pre-designed multiple-page sketches detailing the sequential phases of the solution to the problem using rigorous proof, and in so doing transferring her classroom reaching style into the software design, drawing on the chain questioning method of Socrates, which aim to stimulate interaction. For this reason, she linked all the software func-tions/actions using the interaction techniques supported /facilitated by the Geometer’s Sketchpad v4 (DGS) environment (Jackiw, 1988) to better allow students to discover solution paths and to reason by rigorous proof. This mode of design and the results of the experimental use of the software with students led to the need to define two new concepts: the meanings of Linking Visual Active Representations (LVAR) and Reflective Visual Reaction (RVR). The researcher observed the students’ actions and thinking processes during the research process and offers a description and analysis of these processes. An analysis of the results of the experimental procedure revealed

scholarly journals Hybrid-dynamic objects: DGS environments and conceptual transformations

2019 ◽  
Vol 1 (1) ◽  
pp. 31
Author(s):  
Stavroula Patsiomitou
Keyword(s):  
Mathematical Thinking ◽  
Dynamic Geometry ◽  
Research Process ◽  
Dynamic Geometry Environment ◽  
Theoretical Perspectives ◽  
Dynamic Objects ◽  
Dynamic Hybrid ◽  
Geometric Processes ◽  
Theoretical Considerations ◽  
Didactics Of Mathematics

A few theoretical perspectives have been taken under consideration the meaning of an object as the result of a process in mathematical thinking. Building on their work, I shall investigate the meaning of ‘object’ in a dynamic geometry environment. Using the recently introduced notions of dynamic-hybrid objects, diagrams and sections which complement our understanding of geometric processes and concepts as we perform actions in the dynamic software, I shall explain what could be considered to be a ‘procept-in-action’. Finally, a few examples will be analyzed through the lenses of hybrid and dynamic objects in terms of how I designed them. A few snapshots of the research process will be presented to reinforce the theoretical considerations. My aim is to contribute to the field of the Didactics of Mathematics using ICT in relation to students’ cognitive development

scholarly journals STUDENTS LEARNING PATHS AS ˜'DYNAMIC ENCEPHALOGRAPHS' OF THEIR COGNITIVE DEVELOPMENT

2013 ◽  
Vol 4 (3) ◽  
pp. 802-806
Author(s):  
Stavroula Patsiomitou
Keyword(s):  
Cognitive Development ◽  
Cognitive Process ◽  
Research Process ◽  
Geometer's Sketchpad ◽  
Interaction Techniques ◽  
Van Hiele ◽  
Learning Path ◽  
Learning Paths ◽  
Phd Thesis

The current study presents excerpts of my PhD thesis in which I developed a 'dynamic' hypothetical learning path (DHLP) with regard to student's cognitive development. The aim was to investigate if they would raise their van Hiele levels during and after participation in the research process. Students interacted using the interaction techniques of the Geometer's Sketchpad software and the transformations of its dynamic representations. Moreover, the role of linking visual active representations (LVARs) in the software and the role of students' instrumental decoding in the cognitive process have been investigated. Finally, examples of students learning paths, like 'dynamic' encephalographs'of their cognitive development, will be presented.

scholarly journals Creativity as a function of problem-solving expertise: posing new problems through investigations

ZDM ◽  
2021 ◽  
Author(s):  
Haim Elgrably ◽  
Roza Leikin
Keyword(s):  
Problem Solving ◽  
Dynamic Geometry ◽  
Problem Posing ◽  
Dynamic Geometry Environment ◽  
Mathematics Majors ◽  
Individual Interviews ◽  
University Mathematics ◽  
Domain Specific ◽  
Mathematics Test ◽  
Different Types

AbstractThis study was inspired by the following question: how is mathematical creativity connected to different kinds of expertise in mathematics? Basing our work on arguments about the domain-specific nature of expertise and creativity, we looked at how participants from two groups with two different types of expertise performed in problem-posing-through-investigations (PPI) in a dynamic geometry environment (DGE). The first type of expertise—MO—involved being a candidate or a member of the Israeli International Mathematical Olympiad team. The second type—MM—was comprised of mathematics majors who excelled in university mathematics. We conducted individual interviews with eight MO participants who were asked to perform PPI in geometry, without previous experience in performing a task of this kind. Eleven MMs tackled the same PPI task during a mathematics test at the end of a 52-h course that integrated PPI. To characterize connections between creativity and expertise, we analyzed participants’ performance on the PPI tasks according to proof skills (i.e., auxiliary constructions, the complexity of posed tasks, and correctness of their proofs) and creativity components (i.e., fluency, flexibility and originality of the discovered properties). Our findings demonstrate significant differences between PPI by MO participants and by MM participants as reflected in the more creative performance and more successful proving processes demonstrated by MO participants. We argue that problem posing and problem solving are inseparable when MO experts are engaged in PPI.

scholarly journals Nature of metacognition in a dynamic geometry environment

2015 ◽  
Vol 3 (5) ◽  
pp. 627-646
Author(s):  
Ana Kuzle
Keyword(s):  
Problem Solving ◽  
Dynamic Geometry ◽  
Dynamic Geometry Software ◽  
Dynamic Geometry Environment ◽  
Mathematics Problem Solving ◽  
Managerial Decisions ◽  
Metacognitive Processes ◽  
Geometry Problem ◽  
Mathematics Problem

This case study examined the metacognitive processes of a preservice teacher when solving a nonroutine geometry problem in a dynamic geometry environment. The main purpose of the study was to uncover and investigate patterns of metacognitive processes and to understand what circumstances, situations, and interactions in a dynamic geometry environment promoted metacognitive behaviors. An adaptation of Schoenfeld’s (1981) model of episodes and executive decisions in mathematics problem solving, and the theory of instrumentation (Rabardel, 2001) was used to identify patterns of metacognitive processes in a dynamic geometry environment. During different phases of problem solving the participant engaged in different metacognitive behaviors whereas the dynamic geometry software supported strategies that are available and/or not available on paper and pen. The effectiveness of solution paths was dependent on the presence of managerial decisions, and well-orchestrated usage of different resources, both knowledge and technology. However, the results of the study call to question to which extent engagement in metacognitive behaviors is necessarily desirable or productive.

scholarly journals Comparison of prospective mathematics teachers’ problem posing abilities in paper-pencil test and on dynamic geometry environment in terms of creativity

2020 ◽  
Vol 9 (3) ◽  
pp. 243
Author(s):  
MEHMET FATIH ÖÇAL ◽  
TUĞRUL KAR ◽  
GÜRSEL GÜLER ◽  
ALI SABRI İPEK
Keyword(s):  
Mathematics Teachers ◽  
Prospective Teachers ◽  
Creative Thinking ◽  
Dynamic Geometry ◽  
Thinking Skills ◽  
Problem Posing ◽  
Dynamic Geometry Environment ◽  
Mathematical Concepts ◽  
Prospective Mathematics Teachers

This study aims to investigate the similarities and differences between prospective mathematics teachers’ creative thinking skills in paper-pencil test and on a Geogebra-supported environment in terms of problem-posing. This case study used purposive sampling method for determining the participants. Findings revealed that the activities carried out in the GeoGebra-supported environment were insufficient to produce creative problems, and GeoGebra’s main utility to prospective teachers was in identifying their mistakes related to mathematical concepts and discrepancies among numerical values of the problems posed. The reasons for the low achievement in posing problem were discussed: These were; (i) lack of problem-posing experience, (ii) the structure of problem-posing activity, and (iii) prospective teachers’ mathematical content knowledge.

scholarly journals An instrumental orchestration for online course at the university level

Apertura ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 22-37
Author(s):  
José Orozco-Santiago ◽  
◽  
Carlos Armando Cuevas-Vallejo ◽  
Keyword(s):  
Mathematics Education ◽  
Engineering Students ◽  
Dynamic Geometry ◽  
Dynamic Geometry Environment ◽  
Synchronous Mode ◽  
Instrumental Orchestration ◽  
Video And Audio ◽  
The University ◽  
Eigenvalue And Eigenvector ◽  
Online Mathematics

In this article, we present a proposal for instrumental orchestration that organizes the use of technological environments in online mathematics education, in the synchronous mode for the concepts of eigenvalue and eigenvector of a first linear algebra course with engineering students. We used the instrumental orchestration approach as a theoretical framework to plan and organize the artefacts involved in the environment (didactic configuration) and the ways in which they are implemented (exploitation modes). The activities were designed using interactive virtual didactic scenarios, in a dynamic geometry environment, guided exploration worksheets with video and audio recordings of the work of the students, individually or in pairs. The results obtained are presented and the orchestrations of a pedagogical sequence to introduce the concepts of eigenvalue and eigenvector are briefly discussed. This work allowed us to identify new instrumental orchestrations for online mathematics education.

Using Dynamic Geometry Software to Engage Students in the Standards for Mathematical Practice

2015 ◽  
pp. 227-256
Author(s):  
Milan Sherman ◽  
Carolyn McCaffrey James ◽  
Amy Hillen ◽  
Charity Cayton
Keyword(s):  
Mathematics Classroom ◽  
Dynamic Geometry ◽  
Mathematical Practice ◽  
Dynamic Geometry Software ◽  
Teaching Mathematics ◽  
Instructional Technologies ◽  
Mathematical Relationships ◽  
Case Based

This case provides readers with an opportunity to consider issues pertaining to the use of instructional technologies in the mathematics classroom. As a narrative case based on a lesson observed in a real classroom, the case reflects the complexities of this context, yet was written to highlight certain themes relevant to teaching mathematics with technology. In particular, how students use dynamic geometry software to explore mathematical relationships, how they engage with the Standards for Mathematical Practice, and the important role of the teacher in this process are prominent themes in the lesson.

Bringing Dynamic Geometry to Three Dimensions

2015 ◽  
pp. 68-99
Author(s):  
Nicholas H. Wasserman
Keyword(s):  
Teaching And Learning ◽  
Spatial Reasoning ◽  
Mathematics Classroom ◽  
Three Dimensional ◽  
Dynamic Geometry ◽  
State Standards ◽  
Three Dimensions ◽  
Two Dimensional ◽  
The Common ◽  
K 12

Contemporary technologies have impacted the teaching and learning of mathematics in significant ways, particularly through the incorporation of dynamic software and applets. Interactive geometry software such as Geometers Sketchpad (GSP) and GeoGebra has transformed students' ability to interact with the geometry of plane figures, helping visualize and verify conjectures. Similar to what GSP and GeoGebra have done for two-dimensional geometry in mathematics education, SketchUp™ has the potential to do for aspects of three-dimensional geometry. This chapter provides example cases, aligned with the Common Core State Standards in mathematics, for how the dynamic and unique features of SketchUp™ can be integrated into the K-12 mathematics classroom to support and aid students' spatial reasoning and knowledge of three-dimensional figures.

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