Translating Euclid: Designing a Human-Centered Mathematics

Issues in Informing Science and Information Technology journal. Vol.5 pp.353-393 Published by the Informing Science Institute Santa Rosa, California USA. Available on line http://iisit.org/IssuesVol5.htm, 2008

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 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 paperpencil 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 functions/actions using the interaction techniques supported /facilitated by the Geometer's Sketchpad v4 (DGS) environment to better allow students to discover solution paths and to reason by rigorous proof. This mode of design and the results of the

2010

This thesis describes a research project that studies the use of a Dynamic Geometry Software (DGS) tool in geometry learning and its implications for students" learning experiences. The research is based on an assumption that DGS facilitates the externalization of the internal representations of geometrical concepts, theorems and proofs. The dynamic and interactive medium provides students with the opportunity to share and discuss the emergent visual phenomena. Apart from its implication for students" cognitive development, a DGS tool may have an effect on students" learning experiences in geometry classes from different aspects. I conducted this research study in a geometry class at a secondary school in Azerbaijan to explore the use of a DGS tool in a cooperative learning arrangement. I used GeoGebra as a DGS tool with research questions relating to the aspects of (1) motivation, (2) interactions and discussions, (3) student-centered learning, (4) conceptual understanding, and (5) problem solving strategies. The questions were embedded in an instructional research intervention. The intervention comprised the use of worksheets and applets I developed through GeoGebra. The research data were drawn from the used worksheets, classroom observations, results of pre-and post-test, a questionnaire and interview responses.

This paper presents the results from both; a quantitative and qualitative study investigating students’ opinions about the use of tools of the well known educational software Cabri-Geometry II [1] in their every day classrooms. Specifically, 340 students participated in learning experiments –in several regions of Greece- where the tools of the aforementioned software were used in combination with appropriate lesson plans. Teachers used these tools to construct specific interactive constructions and lesson plans in the context of a wider European teachers’ online community in the frame of the Socrates Comenius 2.1 Project VccSSe – Virtual Community Collaborating Space for Science Education. After the aforementioned experiments, students were asked to complete a questionnaire including both; closed and open questions. The results emerging from this study indicated that the majority of pupils liked the use of these tools in their every day classroom practices and addressed various reasons to support their opinions. Students also proposed the use of technology not only for the learning of Geometry but also for other subjects included in school curricula.

When technology is used as a teaching tool without adapting it to human cognition, the learning becomes focused on the tool rather than on the learner. Hence, the current research seeks to examine the advantages of technological tools in designing research tasks in geometry for teaching teachers, taking into consideration that even teachers have different learning attributes. Adapting geometry tasks to variations between teachers is an art that can be learned from research literature on task design and on designing multimedia tasks using technological tools in the context of human cognition. The current study adapted design principles from the research literature to design geometry research tasks for teaching teachers. THEORETICAL BACKGROUND This section outlines the theoretical concepts for constructing research tasks so as to create a distinct framework for designing geometry tasks to be solved using technological tools. Such a framework can be implemented for teaching teachers. The concepts are presented in the following contexts: the advantages of dynamic computerized tools for solving geometry problems; theories of constructing mathematical tasks; theories of constructing tasks using technological tools while taking the learner's cognitive structure into consideration. Integrating Dynamic Geometry Software (DGS) Research in education constantly seeks methods to improve the quality of teaching. One issue often focused upon is the integration of technology. Dynamic geometry software (DGS) is a technological tool that allows mathematical equations to be represented and mathematical objects to be constructed so as to provide constant feedback to the user (Alakoc, 2003; Martinovic & Manizade, 2013). DGS allows users to create and then manipulate constructions and properties. DGS provides students with different opportunities to engage with geometric objects and their measures and has the potential to help them develop a deeper understanding of properties and theorems (Leung, 2008). Dragging in DGS is one of the pedagogical values of a dynamic geometry environment. Such an environment enables users to express their geometrical thoughts in visual-dynamic ways that can help them construct abstract knowledge. While dragging different points and objects, users can identify properties and relations between them, while generating a multitude of specific examples and hypotheses (NCTM, 2010).

9

2005

The focus of this paper is a software genre usually referred to as 'dynamic geometry' because of the ability of the user to dynamically manipulate geometrical figures created with the software tool. Using data from a longitudinal study of 12-13 students' use of dynamic geometry software, the focus of the analysis is on the interpretations the students make of geometrical

We report on a case study of a teacher’s implementation of a CSCL project to understand how teachers’ pedagogical interventions influence students’ mathematical reasoning in a collaborative, dynamic geometry environment. A high school teacher engaged a class of students in the Virtual Math Teams with GeoGebra environment (VMTwG) to solve geometric tasks that the research team designed to promote collaboration and mathematical justification. The VMTwG allows users to share both GeoGebra and chat windows to engage in joint problem solving. Our analysis of the teacher’s implementation and his students’ interactions in VMTwG shows how his technological and pedagogical content knowledge, revealed through his instructional interventions, shaped his students’ movement among empirical explorations and deductive justifications.

Horizontes, 2018

We draw on the theory of instrumental genesis (RABARDEL; BEGUIN, 2005) and the notion of co-action (HEGEDUS; MORENO-ARMELLA, 2010) to understand how teachers’ instrumentation of dynamic geometry environment (DGE) and how this instrumentation shapes their geometric knowledge. In small groups, six middle and high school mathematics teachers engaged in solving open-ended geometric problems in an online dynamic geometry environment for 15 weeks. Our analysis of their interactions indicates that the co-action between the teachers and the environment helped them appropriate the dragging feature of DGE, which shaped their understanding of geometrical relations, particularly dependencies. Designing tasks that support teachers’ effective appropriation of DGEs requires special attention to the co-active nature of DGEs. This study provides insights into aspects of learners’ collaborative interaction with certain technologies.

Proceedings of the 13th International Congress on Mathematical Education

This is a summary report of the ICME-13 survey on the theme of recent research in geometry education. Based on an analysis of the research literature published since 2008, the survey focuses on seven major research threads. These are the use of theories in geometry education research, the nature of visuospatial reasoning, the role of diagrams and gestures, the role of digital technologies, the teaching and learning of definitions, the teaching and learning of the proving process, and moves beyond traditional Euclidean approaches. Within each theme, there is commentary on promising future directions for research.

International Journal of Education in Mathematics, Science and Technology

Pre-service mathematics teacher (PST) education often addresses within Geometry Classes how to utilize Dynamic Geometric Software (DGS). Other classes may also incorporate teaching pre-service teachers about the history of mathematics. Although research has documented the use of Dynamic Geometric Software (DGS) in teaching the history of mathematics (HoM) (Zengin, 2018), the focus of this research specifically targets the development of proof for pre-service teachers by utilizing DGS to revisit historical proofs with a modern lens. The findings concur with Fujita et.al. (2010), Zengin (2018), and Conners (2007) work on proof. The novelty of this article was the combination of incorporating the history of mathematics (HoM), dynamic geometry software (DGS), and Toulmin’s model of argumentation. A pedagogical approach appeared to emerge: DGS’s dynamic nature allowed PSTs to see several examples of a method to provide them with an illustration that may be used in proofs.