Dynamic Geometry Environment Research Articles

Games in Mathematics Education: Trends and Methodological Approaches in the Context of the Congress of European Research in Mathematics Education (CERME) Review

This study aimed to examine the studies on digital games presented in The Congress of European Research in Mathematics Education (CERME) using a systematic analysis approach in terms of research method, data collection tool, and how they define digital games. Sixteen studies related to games were selected from the proceedings. The findings of the research revealed that the number of studies on games has increased in recent years, and the interest of the mathematics education community in this subject has gradually increased. It was seen that qualitative research approaches were used in most of these studies. As a data collection tool, a semi-structured or clinical interview form was mainly used. To reach more comprehensive results, it is necessary to give importance to quantitative research approaches and even to prefer the use of both approaches together. It was indicated that the tendency towards mathematical reasoning and puzzle games has continued predominantly in recent years. Although it was seen that the number of studies on digital games has increased in the last five years, only six studies in total were presented at the CERME. Moreover, three digital games developed by using the tools in the context of dynamic geometry environments were found. Another significant finding was that most of these studies did not define games. Only a few studies that defined the game considered the game as a tool used to acquire a mathematical concept or knowledge. Based on the findings, it can be argued that digital games are not among the interests of the mathematics education community. Finally, it was observed that mathematical theorems and approaches were central to the studies reviewed. Games were generally defined as tools within these studies. Therefore, future research should particularly consider fundamental game features such as game mechanics to create more integrated research environments with mathematics education.

EFFECTS OF INSTRUCTIONAL MEDIA ON LEARNERS COMPETENCES IN MATHEMATICS IN PUBLIC SECONDARY SCHOOLS IN RWANDA A CASE OF RULINDO DISTRICT

This study explored the impact of instructional media on mathematics competencies in public secondary schools in Rulindo, Rwanda. Employing a descriptive and correlational design with a mixed approach, the study aimed to identify influential instructional media, assess mathematics competence levels, and determine their relationship. The target population included 616 teachers and 100 head teachers, totaling 716 individuals, with a sample size of 257 determined using the Yamane formula through purposive and random sampling. Data were collected via questionnaires, documentation research, and a pilot study, then analyzed using SPSS version 21.Key findings indicated strong agreement (85.5%-98.6%) on the effectiveness of computer-based tools, graphing calculators, mathematics textbooks, and dynamic geometry environments in teaching mathematics. Significant associations were found (p < 0.05) between competencies in numeracy, use of instructional media, and students comprehension of mathematical subjects. Recommendations included government scholarships to enhance teachers mathematics skills, optimized use of teaching resources, training by MINEDUC, and increased educational support for students. The study highlighted the need for further research to explore instructional materials broader impacts on secondary school students mathematics performance.

Examining undergraduate students' abstraction of conic sections in a dynamic geometry environment

In solid geometry, the concept of conic sections plays an important role in teaching graphs such as parabolas, ellipses, and hyperbolas to undergraduate students in Mathematics Education. It is understood that the abstraction process in mastering conic sections is strongly needed. This study examines the abstraction process of conic sections among third-year undergraduate Mathematics Education students (4 males and 21 females) at Universitas Muhammadiyah Malang (UMM), Indonesia. The data was collected by analyzing students' responses in a 60-minute diagnostic test using the Abstraction in Context (AiC) framework. The test consists of 3 questions, validated by 2 Professors of UMM (average score = 4.14) and 2 lecturers (average score = 4.04). The results showed that 1 male and 11 female students did not reach the construction stage of AiC. Subsequently, a student with a low diagnostic test score and the least completion of AiC stages was observed further through an interview. This student passed through all stages of abstraction with the help of DGE. We also underscored undergraduates' challenges in this process, particularly in visualizing conic section objects, spatial thinking, and employing appropriate mathematical signs. Based on these findings, further research with a broader sample is needed to explore diverse abstraction processes.

Assessing the quality of conceptual knowledge through dynamic constructions

In this contribution, we address the gap that has appeared in mathematics education research and practice with the emergence of dynamic geometry environments and build on the opportunities these environments offer to school geometry. In our qualitative empirical study, we investigate how to elaborate on the general model of conceptual knowledge to make it applicable to dynamic geometry tasks, specifically to tasks including dynamic geometric constructions. We present a design of dynamic constructions of quadrilaterals that comply with Euclidean constructions, derive an assessment instrument based on them, and study what information the instrument can provide about the quality of students’ conceptual knowledge. We present the results in the form of an assessment framework consisting of an example of the assessment instrument and an ordered system of qualitative categories serving as an assessment codebook for interpreting students’ responses in terms of the quality of conceptual knowledge. To clarify the relations between the assessment framework and the general model of conceptual knowledge, we establish a system of subdimensions of conceptual knowledge that indicates how conceptual knowledge can be understood in the context of dynamic geometric constructions and identifies the conceptual knowledge needed to achieve individual categories of the assessment framework.

AlgeoMath를 활용한 초등 도형 과제의 도구화 스킴 분석

Recently, there has been increasing interest in the use of Dynamic Geometry Environments (DGE) in elementary geometry learning. This study aims to analyze elementary geometry tasks utilizing DGE from an instrumental perspective and to identify the conceptual and technical difficulties that elementary students may encounter. To achieve this, we analyzed textbooks and previous studies to extract tasks that utilize the Korean DGE, AlgeoMath, and conducted practical exercises to analyze the instrumentation schemes for each task. The analysis revealed that some terms and symbols of the tools were not aligned with the elementary curriculum or contained mathematical errors, and that some schemes requiring an understanding of functional perspectives or basic constructions in secondary mathematics could present conceptual and procedural difficulties for elementary students. Furthermore, we found that precisely implemented transformation tools were not suitable for the informal and intuitive learning of elementary plane figure movements. Finally, it was confirmed that the inherent characteristics of DGE, such as the independence and dependence of objects and the hiding feature, as well as the functional limitations and errors of AlgeoMath, could pose challenges for students. Based on these findings, the study suggests pedagogical implications for elementary mathematics lessons using AlgeoMath and provides insights into the functional composition of DGE for elementary use.

Students’ dynamic communication while transforming mathematical representations in a dynamic geometry environment

In mathematics, students’ abilities to make transformations between mathematical representations are fundamental. The recent implementation of digital technologies, such as Dynamic Geometry Environments (DGEs), have changed students’ access to mathematical representations by providing a variety of different representations, available just by pressing a button. Students use of a DGE may change their mathematical communication to become more dynamic and active. However, it is not clear how the use of DGEs and the change in communication style relate to more epistemological aspects of students’ mathematical work. This study, therefore, investigates the interplay between students’ dynamic mathematical communication and their transformations of mathematical representations when using a DGE from a competency perspective. Based on analyses of instances of lower secondary school students’ dynamic mathematical communication, findings indicate that students’ mathematical communication reflect the new ways of engaging with mathematical representations when using a DGE. Hence, the communication becomes dynamic. Theoretically, the complexity of having “continuous” transformations of mathematical representations may reduce the readiness of the mathematical communication competency. The “continuous” transformation may in fact cause students to unintentionally ascribe dynamic properties to mathematical representation.

Is there room for conjectures in mathematics? The role of dynamic geometry environments

Proof, as a central and integral part of mathematics, is an essential component of mathematical education and is considered as the basic procedure for revealing the truth of mathematical propositions and for teaching productive reasoning as part of human civilization. Is there therefore room for conjectures in mathematics? In this paper after discussing at a theoretical level the concepts of proof and conjecture, both in a paper-and-pencil environment and in a dynamic geometry environment (DGE) as well as how school practice affects them, we fully explain a task involving various mathematical disciplines, which we tackle using elementary mathematics, in a mathematics education context. On the occasion of the Greek educational system we refer to some parameters of the teaching of geometry in school and we propose an activity, within a DGE, that could enable students to be guided in the formulation and exploration of conjectures. Finally, we discuss the teaching implications of this activity and make some suggestions.

The Intuitive Way to Solve Construction Problems in the Dynamic Geometry Environment

The ability to solve geometric construction problems is justly regarded as an essential component of mathematical culture. The dynamic, general nature of objects provided by dynamic geometry systems allows the development of intuitive methods for solving construction problems. The core mathematical concept underlying this approach is the loci method, in which the main element of the required object forms as the intersection of two loci, each one obtained by purposely ignoring part of the conditions of the problem. With GeoGebra, the relevant locus may be visualized and detected in trace mode. Using dynamic geometry this way for problem solving replaces the single object by the infinite locus and is similar to the introduction of variables for solving equations in school algebra. This paper presents some examples of how this approach may be realized along with the results of a small-scale experiment with pre-service mathematics teachers.

A pedagogical reflection on the interplay between variation and invariant: Variational thinking

This commentary paper is a pedagogical reflection on the interplay between variation and invariant. It begins with a brief discussion on the concept of Unity of Opposites as an ancient philosophical theme. Ancient thinking systems regarded the variation and invariant pair as a Unity of Opposites. Next, the use of variation as a pedagogical approach in mathematics education is briefly examined under Marton's variational theory of learning, Gu's bianshi jiaoxue, and the related research done by the author in the context of Dynamic Geometry Environment (DGE). These lead to the formation of the concept of variational thinking, the main contribution of this paper, which is presented and explained. A DGE task design sequence example is presented to illustrate how variational thinking can be used to frame a process of geometrical reasoning and argumentation.

Computer Experiment in Teaching Mathematics

A mathematical experiment has always been a key source for a mathematical discovery. Over the past 50 years, thanks to digital technologies, its role in mathematical research has grown significantly. Digital technologies have opened up fundamentally new opportunities for experimentation in mathematics education, bringing mathematical education closer to mathematical research for the majority of students. This tendency is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the authors’ work over the past decades on the application of a computer mathematical experiment at different levels of school and university education. Particular attention is paid to dynamic geometry environments. The possibilities of using computer algebra systems are also discussed. A project concerning schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.

Fostering a Student’s Abstraction of the Relationship Between Parallelogram and Trapezoid Within Quadrilateral Hierarchy

ABSTRACT This article provides an analysis of a data set coming from a two-phase qualitative study that focused on fostering primary students’ abstraction of interrelations among quadrilaterals (squares, rectangles, parallelograms, rhombuses, trapezoids). The pilot study consisted of work with eight primary students operating at van Hiele level 2 (e.g., understanding quadrilaterals without interrelations). We benefitted from the teaching experiment methodology to develop a task sequence applied in a dynamic geometry environment and a paper-pencil environment to help each participant develop quadrilateral hierarchy at van Hiele level 3 (e.g., understanding quadrilaterals with their interrelations). The main study used a case study approach to investigate two primary students’ progress (Efe and Ayla, age 10). After the pre-interviews, each participant was taught the developed task sequence individually during seven up-to-one-hour teaching sessions, followed by a post-interview. This article only details Efe’s case as he worked on developing the relationship between parallelogram and trapezoid. We analyzed Efe’s data (from the pre-interview, Teaching Session-7, and the post-interview) to describe how the different parts of the task sequence fostered his abstraction of the interrelation between parallelogram and trapezoid as he moved from van Hiele Level 2 to 3. This article provides initial evidence for the classification process.

COMPUTER EXPERIMENT IN TEACHING MATHEMATICS

Mathematical experiment has always been a key source for mathematical discovery. Over the past 50 years, thanks to digital technologies, its role in mathematical research has grown significantly. Digital technologies have opened up fundamentally new opportunities for experimentation in mathematics education, bringing the majority of students of mathematical education closer to mathematical research. Such an approximation is especially desirable in the modern world, where it becomes possible thanks to digital technologies. The article discusses the results of the work of the authors over the past decades on the application of a computer mathematical experiment at different levels of school and university education. Particular attention is paid to dynamic geometry environments. The possibilities of using computer algebra systems are also considered. The project of schoolchildren’s work on generalizations of Napoleon’s theorem is considered in detail.

Who moved my triangle? Pre- and in-service teachers inquiring in a mathematics lab

ABSTRACT This study examines a Technology, Pedagogy, Content Knowledge (TPACK) training framework for pre- and in-service mathematics teachers that enables them to experience inquiry-based learning as learners in a dynamic geometry environment that serves as a mathematics laboratory. We investigate the effect of this experience on the teachers’ TPACK, based on both their perceptions towards implementing technology into their teaching; and on their TPACK level demonstrated in inquiry-based dynamic geometry activities they designed for instructional purposes. We present a rubric for assessing teachers’ TPACK level through inquiry-based dynamic geometry activities after they practiced in the mathematics lab. Our findings reveal a significant positive change in the participants’ perceived TPACK, from pre-to post-intervention, only for factors that relate to the technology component and its integration with the pedagogy and content components. In the activities designed by participants, technology was more content-oriented than inquiry-oriented. No difference was found between the two participating groups in the ability to design dynamic geometry activities after they practiced in the mathematics lab. The study suggests theoretical, practical, and methodological contributions related to conceptualizing TPACK in the context of a mathematics lab and to designing TPACK training frameworks for pre- and in-service teachers.

Embodied instrumentation in a dynamic geometry environment: eleven-year-old students’ dragging schemes

Embodied instrumentation in a dynamic geometry environment: eleven-year-old students’ dragging schemes

Mathematics Teachers’ Geometric Thinking: A Case Study of In-service Teachers’ Constructing, Conjecturing, and Exploring with Dynamic Geometry Software

<p style="text-align:justify">Many research studies have been conducted on students’ or pre-service teachers’ geometric thinking, but there is a lack of studies investigating in-service teachers’ geometric thinking. This paper presents a case study of two high school teachers who attended the dynamic geometry (DG) professional development project for three years. The project focused on the effective use of dynamic geometry software to improve students’ geometry learning. The two teachers were interviewed using a task-based interview protocol about the relationship between two triangles. The interviews, including the teachers' work, were videotaped, transcribed, and analyzed based on the three levels of geometric thinking: recognition, analysis, and deduction. We found that the participating teachers manifested their geometric skills and thinking in constructing, exploring, and conjecturing in the DG environment. The study suggests that the DG environment provides an effective platform for examining teachers' geometric skills, and levels of geometric thinking and encourages inductive explorations and deductive skill development.</p>

Computer-Supported Assessment of Geometric Exploration Using Variation Theory

We report on an innovative design of algorithmic analysis that supports automatic online assessment of students’ exploration of geometry propositions in a dynamic geometry environment. We hypothesized that difficulties with and misuse of terms or logic in conjectures are rooted in the early exploration stages of inquiry. We developed a generic activity format for if–then propositions and implemented the activity on a platform that collects and analyzes students’ work. Finally, we searched for ways to use variation theory to analyze ninth-grade students’ recorded work. We scored and classified data and found correlation between patterns in exploration stages and the conjectures students generated. We demonstrate how automatic identification of mistakes in the early stages is later reflected in the quality of conjectures.

Prediction in Transition: Continuities and Discontinuities Moving from the Paper-and-Pencil to the Dynamic Geometry Environment

Prediction in Transition: Continuities and Discontinuities Moving from the Paper-and-Pencil to the Dynamic Geometry Environment

Invariant Subspaces: An Alternative for Introducing Eigenvectors and Eigenvalues

The concepts of eigenvalue and eigenvector are typically approached algorithmically in introductory linear algebra courses. However, a more conceptual orientation involves connecting these notions to the concept of one-dimensional invariant subspace, which allows for the introduction of eigenvectors prior to eigenvalues. In this study, we present data collected from interviews with two linear algebra instructors as they worked with a specific linear transformation in both paper-and-pencil and dynamic geometry environments. The data were analyzed using the perspectives of APOS theory and the theory of Mathematical Working Spaces in a complementary manner. The results indicate that dynamic representations facilitate the establishment of relationships between eigenvectors, eigenvalues, and invariant subspaces. This approach proves to have potential for developing a deeper understanding of the related concepts.

Applying Dynamic Geometry Environment Software as a Visualization Tool for Teaching Planimetry Construction Tasks

The present study addresses a notable gap in the current pedagogical research by rigorously examining the impact of Augmented Reality (AR) on mathematics education, specifically focusing on two-dimensional (2D) plane geometry construction tasks. Our primary aim is to discern and articulate the unique contributions of AR as a pedagogical tool in this domain. Employing a quasi-experimental design, we engaged with 141 high school students, dividing them into two distinct groups: one receiving instruction through AR tools and the other via traditional teaching methods. Performance metrics were derived from pre- and post-construction tests as well as an attitudinal survey gauging students’ perceptions of AR. Our findings reveal that the AR-facilitated group showcased a marked improvement in post-test scores and exhibited a more positive attitude towards AR technology compared to the traditionally taught cohort. This point towards AR’s potential to significantly elevate comprehension and engagement levels in 2D geometry construction. The study further highlights the transformative potential of AR in bolstering students’ enthusiasm and curiosity in mathematical endeavors. The results presented herein emphasize the pressing need for extended research to gauge AR’s broader and sustained effects in mathematics education and its applicability across various mathematical disciplines.

Characterizing Learners’ Construction of Geometry Diagrams in Dynamic Geometry Environments

As a distinct mode of mathematical representation and communication, geometry diagrams are both signs that represent abstract geometric concepts and visual images that offer graphical-spatial properties. Dynamic geometry environment offers a wide range of tools for its users to create and interact with geometry diagrams. In this paper, I propose a framework to characterize geometry diagrams that learners create in DGEs. The framework considers a learner’s approach to creating a geometry diagram (i.e., perceptualbased, measurement-based, construction-based, and transformation-based), the driving force that guides the learner’s specific actions in a DGE (i.e., tool-driven, and property-driven), and the constraints in the resulting diagram (i.e., drawing, underconstrained, overconstrained, and appropriately constrained). Examples of student work on two geometry construction problems are used to illustrate the use of the framework. These examples show that the proposed framework provides a useful tool to characterize geometry diagrams in DGEs.