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

Author(s):  
Stavroula Patsiomitou

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

2020 ◽  
Vol 10 (6) ◽  
pp. 164
Author(s):  
Xiangquan Yao ◽  
Azita Manouchehri

The relationship between teacher interventions and students’ mathematical thinking has been the subject of inquiry for quite some time. Using the Pirie–Kieren theory for dynamic growth in mathematical understanding, this study documents teacher interventions that support students’ growth toward developing a general understanding of a mathematical idea in a designed learning environment. By studying the interactions of seven middle school students and the teacher-researcher working on a two-week unit on geometric transformations within a dynamic geometry environment, this study identified nine major categories of teacher interventions that support and extend students’ investigations of mathematical ideas around geometric transformations. The typology of teacher interventions reported in this study provides a cognition-based framework for teacher moves that extend and advance students’ mathematical understanding.


Author(s):  
José Milton Lopes Pinheiro ◽  
Cesar Osvaldo Vásquez Flores ◽  
Giovana Alves ◽  
Juscimar Da Silva Araujo

ResumoEste estudo foca o movimento como fenômeno de pesquisa, explicitando-o a partir de diferentes perspectivas teóricas, quais sejam: a física, a matemática e a fenomenologia, porém, assumindo a terceira para efetuar a análise. Mediante estudo no âmbito dessas áreas e análise de uma atividade desenvolvida em ambiente de Geometria Dinâmica, o objetivo da investigação é apresentar compreensões sobre como a percepção do movimento pode direcionar o pensar e contribuir com a aprendizagem de matemática. Para tanto, realizamos um estudo qualitativo de cunho bibliográfico, que forneceu compreensões que uma vez articuladas com a análise da atividade, permitiram ao estudo o entendimento de que o movimento é correlato a um sujeito que se move, movendo, e o permite conhecer as implicações desse ato materializando-se em seu mundo circundante e em seu corpo, que é o ponto zero do movimento e das percepções que realiza. Assim, o aprender dá-se na unidade movimento-percepção-conhecimento.Palavras-chave: Movimento, Fenomenologia, Educação matemática.AbstractThis study focuses on the movement as a research phenomenon, explaining it from different theoretical perspectives, namely: physics, mathematics, and phenomenology, however, assuming the third one to carry out the analysis. Through study in the scope of these areas and analysis of an activity developed in a dynamic geometry environment, this study aims to present understandings about how the perception of movement can direct thinking and contribute to the learning of mathematics. To this end, a qualitative study of a bibliographic nature is carried out, providing understandings that, once articulated with the analysis of the activity, allowed the study to recognize that the movement is correlated to a subject who moves, moving, and allows it to know the implications of this act materialising in his surrounding world and also in their body, which is the base of the movement and the perceptions it realises. Thus, learning occurs in the movement-perception-knowledge unit.Keywords: Movement, Phenomenology, Mathematical Education.ResumenEste estudio enfoca el movimiento como fenómeno a ser investigado, explicándolo a partir de diferentes perspectivas teóricas, sean: la física, la matemática y la fenomenología, sin embargo, asumiremos la tercera para efectuar nuestro análisis. El objetivo es, mediante el estudio en el ámbito de esas areas y el análisis de una actividad desarrollada en el ambiente de la geometría dinámica, presentar cómo la comprensión de la percepción del movimiento pode direccionar el pensamiento y contribuir al aprendizaje de la matemática. Para ese objetivo, se realiza un estudio cualitativo de carácter bibliográfico, que ofreció comprensiones que una vez articuladas con el análisis de la actividad, permitieron al estudio la comprensión de que el movimiento está relacionado a un sujeto que se mueve, moviéndose, y le permite conocer las implicaciones de ese acto materializándose en su mundo circundante y también en su cuerpo, el cual es el punto cero del movimiento y de las percepciones que realiza. Así, el aprendizaje se da en la unidad movimiento-percepción-conocimiento.Palabras clave: Movimiento, Fenomenología, Educación matemática.


10.28945/3235 ◽  
2008 ◽  
Author(s):  
Stavroula Patsiomitou

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


ZDM ◽  
2021 ◽  
Author(s):  
Haim Elgrably ◽  
Roza Leikin

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.


Author(s):  
Ana Kuzle

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.


2020 ◽  
Vol 9 (3) ◽  
pp. 243
Author(s):  
MEHMET FATIH ÖÇAL ◽  
TUĞRUL KAR ◽  
GÜRSEL GÜLER ◽  
ALI SABRI İPEK

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.


Apertura ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 22-37
Author(s):  
José Orozco-Santiago ◽  
◽  
Carlos Armando Cuevas-Vallejo ◽  

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.


Author(s):  
Ilhan Raman ◽  
Yasemin Yildiz

The chapter examines the relationship between orthography, phonology, and morphology in Turkish and what this means for Turkish-English bilingual language processing. Turkish offers a unique language medium in pitching theoretical perspectives both in linguistics and psycholinguistics against each other because of its properties. Empirical and theoretical considerations are employed from both domains in order to shed light on some of the current challenges. In line with contemporary thought, this chapter is written with the view that bilingual speakers engage a singular language or lexical system characterized by fluid and dynamic processes. Particular focus will be given to English-Turkish speaking bilinguals in the UK, which includes heritage (HL) and non-heritage language speakers. Evidence from monolingual developmental research as well as neuropsychology will be examined to confirm findings of previous studies in other European contexts, and also to raise attention to various challenges which need to be addressed across all contexts.


Author(s):  
Nikos Papastergiadis ◽  
Amelia Barikin ◽  
Xin Gu ◽  
Scott McQuire ◽  
Audrey Yue

This chapter details the case studies that were conducted as part of a five-year research project, which conducted the world’s first real-time cross-cultural exchange via the networking of large public screens located in Melbourne and Seoul. The project linked large screens located in Seoul and Melbourne for three media events: SMS_Origins and <Value>, HELLO, and Dance Battle. The chapter details methodological innovations of the research, which involved the reformulation of the way in which the scholar was embedded in the research and transformed according to the interactive research process. It also elucidates critical insights into the process of cultural exchange, the impact of media technologies on public space, and the transformation of the public sphere in the global era. The empirical research generates fresh insights into public interactions with large screens, providing a prototype for future cross-cultural events and offering new theoretical perspectives on the use of public space.


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