Dynamic Geometry Environment Research Articles

Preservice Middle School Mathematics Teachers’ Development of Flexibility and Strategy Use by Geometric Thinking in Dynamic Geometry Environments

The current study, aimed to examine the development of flexibility and strategy use through instructional sequence encouraging the preservice middle school mathematics teachers’ geometric thinking and problem-solving performance on dynamic geometry environment. Mixed-method research design was used in the study. The study was conducted with 46 preservice middle school mathematics teachers selected by criterion sampling strategy. The data were collected through the tests including open-ended and multiple-choice questions. In the quantitative part of the study, it was attempted to identify whether the tasks affect problem solving skills and geometric thinking. The said individuals participated in six-week instructional sequence designed by the tasks of geometric constructions performed by GeoGebra. For the qualitative part, document analysis technique was used in order to illustrate the ways of this effect in detail and illustrate the development of flexibility and strategy use. It was observed that tasks by DGE improved the preservice middle school mathematics teachers’ scores of problem-solving and geometric thinking. Also, they solved the problems by representing the properties of higher thinking levels than the levels they had before participating in the instructional sequence enacted by GeoGebra. It is believed that this could encourage the improvement of their flexibility and strategy use.

Mathematical strategies and emergence of socially mediated metacognition within a multi-touch Dynamic Geometry Environment

Mathematical strategies and emergence of socially mediated metacognition within a multi-touch Dynamic Geometry Environment

Pre-Service Mathematics Teachers’ Experience with a Dynamic Geometry Environment Whilst Reasoning in Relation to Locus Problems: A Detailed Look at Strategies

The purpose of this study was to examine how pre-service mathematics teachers (PMTs) integrated a dynamic geometry environment (DGE) into their reasoning process while solving geometric locus problems. Task-based interviews based on the locus problems were conducted with eight PMTs working in pairs. The PMTs could use a computer with access to GeoGebra in addition to plain papers, pencils, graph paper, a ruler, and a compass in the interview environment. They were free to use any of them. The PMTs’ reasoning was analyzed by focusing on their use of DGE as they were producing mathematical arguments during problem-solving. Toulmin’s argumentation model and Hollebrands’ strategies for using DGE were combined to analyze the data. The results revealed that the PMTs used DGE reactively to make claims and proactively to test or justify their claims. Their proactive use of DGE differed depending on whether they included their warrants for this use or not.

Teachers’ Understanding of Draggable Geometric Objects: Variance and Invariance in a Dynamic Geometry Environment

Teachers’ Understanding of Draggable Geometric Objects: Variance and Invariance in a Dynamic Geometry Environment

Discourse analysis on understanding the differential concept of high school students in a dynamic geometry environment

This study investigated the cognitive and social processes through which high school students acquire the differential concepts through communication in a dynamic geometry environment through some cases. Additionally, we observed how a dynamic geometry environment affects these processes. To achieve this objective, eight students were recruited by using the pre-test results and divided into an experimental group and a control group. The learning environment for the two groups was designed to be the same except for jointly manipulating Geogebra on the laptop in the experimental group. Students’ discourse was analyzed using an analysis framework that applied both Piaget’s (1959) linguistic categories and Rutherford’s (2011) neo-Piagetian model. We found that the dynamic geometry environment improved communication among students and their achievement levels. In particular, graphic and dynamic representations created by dragging stimulated students’ interest and increased communication among them. Although the quality of communication was somewhat different in the discourse analysis of the two groups, each participant’s role was confirmed in the co-construction of knowledge among all cases of eight students. We expect that precise verbal information on various representations of mathematical content in the process of understanding the concepts of students could be an opportunity to prepare educational environments corresponding thereto.

Illogical use of the converse of a theorem that can cause an incorrect solution

ABSTRACT We present action research of a problem posed as part of a multi-participant national (Israeli) test checking the mathematical knowledge of high school students at the ages of 16–17, where some of those who solved this problem made an error by using the converse to a well-known theorem, where the converse is not true. In order to examine the danger of using a wrong converse, the problem was posed as field research to a group of pre-service teachers and training teachers in the ‘methods (of teaching mathematics)’ course, where similar failures were discovered. The conclusion is that a large part of the students was not aware that the converse is not always true, and its correctness is to be tested before it can be used. The investigation of the problem was accompanied by a dynamic geometry environment software that allowed us to examine the problem in the general case and so to make conclusions for particular cases.

Characterizing Tasks for Teaching Mathematics in Dynamic Geometry System and Modelling Environments

With an empirical study, we explore the process of designing modeling tasks for teaching mathematics in a Dynamic Geometry environment. We address the research questions: What are the characteristics of the tasks that pre-service teachers design to teach mathematics in digital and modeling environments? What are the activities like that these tasks encourage in students? By means of a qualitative methodology, we characterize the tasks designed by secondary pre-service mathematics teachers and identify the activities that these tasks encourage in students. By analyzing the process of designing and planning a class, we can determine how the characteristics of the modeling tasks are modified based on the software used. In this context, the software also acts as a medium that reflects aspects of the models involved, the objectives of the tasks, and the activities intended for the students. The results contribute to the discussion regarding task design, where technology is a means of supporting the exploration, understanding, and study of different phenomena.

High school mathematics teachers’ discernment of invariant properties in a dynamic geometry environment

High school mathematics teachers’ discernment of invariant properties in a dynamic geometry environment

Symbolizing lines and planes as linear combinations in a dynamic geometry environment

In this paper, we focus on students’ symbolizing activity and mathematization relating to linear combination and span in the context of a task sequence designed with digital tools. Considering tools and functions of a digital environment (specifically of GeoGebra) with design heuristics of Realistic Mathematics Education, we introduce four tasks (in R2 and R3) and results of pilot studies. Data sources come from six individual task-based interviews with two linear algebra students. We analyzed these with a lens of mathematization, with a particular focus on tool use. We found students’ use of sliders supported their reasoning about individual linear combinations; the trace function and slider animation supported their intuition for all possible linear combinations of vectors. After shifting from two to three dimensions, students symbolized all possible linear combinations with a parametric vector equation and reasoned about how this could represent particular points on the corresponding lines or planes.

Isometries: epistemological insights among secondary students interacting in Dynamic Geometric environment with touches on screen

This study investigates secondary students’ understanding of isometry (symmetry, rotation and translations) using dynamic geometry environment with touchscreen (DGEwT). The research question addresses issues concerning geometrical reasoning of Brazilian learners without previous knowledge regarding isometry. Data came from videotapes, written notes for each proposed task and on the icon sheet, and from screen recording. The analytical process was mainly based on the geometrical reasoning and related procedures. One designed task is shown and analyzed. Applying reflectional symmetry ideas, using procedures based on straight line and performing simultaneous touches on the screen, students achieved the composition of rotation and translation movement naturally and synchronically. The study highlighted the relevance of isometry in both school and teacher training curriculum.

Inquiring and learning with DGS Cui-Rods: a proposal for managing the complexity of how primary-school pupils’ mathematically structure odd even numbers

Inquiry approaches using coloured manipulatives are a fruitful field for the investigation of mathematical concepts, embedded in a re-conceptualized, research-based curriculum. Manipulatives are designed to mediate between a particular mathematical concept and the way pupils learn that concept. Many researchers highlight the advantages of computer manipulatives including DGS manipulatives for teaching and learning. Moreover, the effectiveness of the Cuisenaire–Gattegno approach in the teaching and learning of mathematics in primary schools has been the subject of many math-investigations, supporting positive outcomes. In terms of the present study, it is interesting to mention the introduction of DGS Cui-Rods that I created in the Geometer’s Sketchpad dynamic geometry environment. The main focus of the current study is a fundamental pattern structure of our number system: odd and even numbers. The study will propose a multiple representation approach to aid pupils understanding odd and even numbers. The proposed DGS material can be displayed, inquired and managed through properly set-up tasks, using linking representations. Finally, it is important to continue teaching concepts through activities, tasks and problems that involve children in the inquiring and learning process; it is the best route for them to how to develop, interpret, and make sense of mathematical concepts.

A Comparison of U.S. and Chinese Geometry Standards through the Lens of van Hiele Levels

The present study aims to compare the geometry standards in U.S. Common Core State Standards of Mathematics (CCSSM) and Chinese Compulsory Education Mathematics Curriculum Standards (CMCS) through the lens of van Hiele levels. The study considered a standard unit as one or multiple learning expectation(s) and placed each learning expectation into van Hiele levels. By examining the van Hiele level distributions of the learning expectations and major topics, this study investigated how CCSSM and CMCS propose the students' development of geometric thoughts. The findings reveal different emphases in geometry learning expectations of CCSSM and CMCS. Specifically, (a) Geometry standards in CCSSM are more general, whereas geometry standards in CMCS are more specific; (b) CMCS shows a more consistent progression through grade bands than CCSSM; (c) CCSSM makes more connections between various aspects within a topic and across multiple topics than CMCS; (d) CCSSM highlights Dynamic Geometry Environments (DGEs), whereas CMCS underscores abstract and rigor at the upper level. Implications of this study and suggestions for future revisions for both standards are discussed.

Una orquestación instrumental para un curso en línea a nivel universitario

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 Enhance Specialized Content Knowledge: Pre-Service Mathematics Teachers’ Perceptions

The study aimed to expose how pre-service teachers perceived the value of an inquiry-based geometrical task with dynamic geometry software for increasing and deepening their understanding of the task itself, and for heightening their mathematical, pedagogical, and technological knowledge in the case of reflection and transformation. Twenty-seven pre-service teachers studying to become middle-school (junior high) mathematics teachers were assigned an inquiry-based geometrical task to explore – both by conventional, low-tech means (paper and pencil) and then in a dynamic geometry software (DGS) environment – the properties of objects formed by reflections in polygons (triangles and quadrilaterals) and to offer conjectures and proofs regarding the final polygon’s shape and area concerning the original. The results show that the majority of participants indicated that DGS provided a valuable instrument for revising and expanding their knowledge, that the assignment demonstrated the value that technological tools have in teaching and learning processes, and the importance of adapting learning tasks in geometry to a dynamic geometry environment for expanding and deepening geometrical and pedagogical knowledge and reasoning skills.

Reinventing Geometric Linear Transformations in a Dynamic Geometry Environment: Multimodal Analysis of Student Reasoning

This paper reports the multimodal resources that attached to students’ reasoning in the reinvention of specific geometric linear transformations (like reflections according to the axes, projections onto axes and composition of reflections) in a dynamic geometry environment. Following the design heuristics of Realistic Mathematics Education, we design a task (in ℝ2) referring to specific tools and functions of GeoGebra. Task-based interviews were conducted with a pair of linear algebra students, by way of a computer and a teacher. Data was collected using a video camera observing the students’ working environment, screen recorder software, student production and field notes. The data was analysed according to the multimodal paradigm focusing on all semiotic resources, such as gestures and artefact use, in addition to written signs. According to the findings, the artefact use, verbal and written mathematical expressions all interlaced with the emergence of gestures. The students mostly gestured when they faced a new reflection situation and when describing associated geometric actions. Finally, a shared environment with action, production and communication conveyed student reasoning and they managed to reinvent a number of geometric linear transformations.

Enhancing the Skill of Geometric Prediction Using Dynamic Geometry

This study concerns geometric prediction, a process of anticipation that has been identified as key in mathematical reasoning, and its possible constructive relationship with explorations within a Dynamic Geometry Environment (DGE). We frame this case study within Fischbein’s Theory of Figural Concepts and, to gain insight into a solver’s conceptual control over a geometrical figure, we introduce a set of analytical tools that include: the identification of the solver’s geometric predictions, theoretical and phenomenological evidence that s/he may seek for, and the dragging modalities s/he makes use of in the DGE. We present fine-grained analysis of data collected during a clinical interview as a high school student reasons about a geometrical task, first on paper-and-pencil, and then in a DGE. The results suggest that, indeed, the DGE exploration has the potential of strengthening the solver’s conceptual control, promoting its evolution toward theoretical control.

Dynamic Geometry Environment to Enhance High School Students Spatial Skill: A Study Based on Sex and Gender Diversities Perspective

Dynamic Geometry Environment to Enhance High School Students Spatial Skill: A Study Based on Sex and Gender Diversities Perspective

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

This 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.

Revisiting the French Didactic Tradition through Technological Lenses

There is, in France, a long-term tradition of research in the didactics of mathematics. In this paper, we revisit this tradition using, as a specific lens, the research carried out on the educational use of digital tools and resources for teaching and learning mathematics. We first briefly introduce this tradition and the three main theories at the base of it: the theory of didactical situations, the theory of conceptual fields, and the anthropological theory of the didactic. Then, considering three different technological lenses, i.e., dynamic geometry environments, computer algebra systems, and digital resources, we show how these theories and the long-term connection established in this tradition with the fields of cognitive ergonomics and computer sciences have influenced technological research and its outcomes. We also show that, conversely, didactic technological research has led to original and influential theoretical constructions, such as the instrumental approach and the documentational approach to didactics, and that it has contributed in a substantial way to the opening of this didactic tradition to other didactic cultures and other communities, beyond the didactic one.

A descriptive phase model of problem-solving processes

Complementary to existing normative models, in this paper we suggest a descriptive phase model of problem solving. Real, not ideal, problem-solving processes contain errors, detours, and cycles, and they do not follow a predetermined sequence, as is presumed in normative models. To represent and emphasize the non-linearity of empirical processes, a descriptive model seemed essential. The juxtaposition of models from the literature and our empirical analyses enabled us to generate such a descriptive model of problem-solving processes. For the generation of our model, we reflected on the following questions: (1) Which elements of existing models for problem-solving processes can be used for a descriptive model? (2) Can the model be used to describe and discriminate different types of processes? Our descriptive model allows one not only to capture the idiosyncratic sequencing of real problem-solving processes, but simultaneously to compare different processes, by means of accumulation. In particular, our model allows discrimination between problem-solving and routine processes. Also, successful and unsuccessful problem-solving processes as well as processes in paper-and-pencil versus dynamic-geometry environments can be characterised and compared with our model.