Geometry Learning Research Articles

Learn math through motion: a technology-enhanced embodied approach with augmented reality for geometry learning in K-12 classrooms

ABSTRACT Despite the growing attention to the technology-enhanced embodied learning approach in mathematics education and its pedagogical potential, much remains unexplored regarding its implementation in real-world classroom contexts and student engagement in embodied learning activity. Addressing this gap, we introduce an embodied learning game leveraging computer vision and Augmented Reality (AR) techniques. We implemented the embodied AR game into fourth-grade classrooms to explore its impact on students’ geometry understanding, attitudes, and engagement patterns. Our analysis revealed significant enhancements in students’ geometry understanding from pre- to post-test, whereas their attitudes remained unchanged. A multiple regression analysis on students’ in-game log and hand gesture data showed that only time spent on making-angle activities predicts students’ cognitive learning gain, suggesting that not all hand gestures were meaningful to students’ learning. In addition to demonstrating the pedagogical benefits of cost-effective AR-based embodied learning in classroom settings, our findings highlight the importance of thoughtfully designing embodied actions that are both educational and engaging.

Information Geometry and Manifold Learning: A Novel Framework for Analyzing Alzheimer’s Disease MRI Data

Background: Alzheimer’s disease is a progressive neurological condition marked by a decline in cognitive abilities. Early diagnosis is crucial but challenging due to overlapping symptoms among impairment stages, necessitating non-invasive, reliable diagnostic tools. Methods: We applied information geometry and manifold learning to analyze grayscale MRI scans classified into No Impairment, Very Mild, Mild, and Moderate Impairment. Preprocessed images were reduced via Principal Component Analysis (retaining 95% variance) and converted into statistical manifolds using estimated mean vectors and covariance matrices. Geodesic distances, computed with the Fisher Information metric, quantified class differences. Graph Neural Networks, including Graph Convolutional Networks (GCN), Graph Attention Networks (GAT), and GraphSAGE, were utilized to categorize impairment levels using graph-based representations of the MRI data. Results: Significant differences in covariance structures were observed, with increased variability and stronger feature correlations at higher impairment levels. Geodesic distances between No Impairment and Mild Impairment (58.68, p<0.001) and between Mild and Moderate Impairment (58.28, p<0.001) are statistically significant. GCN and GraphSAGE achieve perfect classification accuracy (precision, recall, F1-Score: 1.0), correctly identifying all instances across classes. GAT attains an overall accuracy of 59.61%, with variable performance across classes. Conclusions: Integrating information geometry, manifold learning, and GNNs effectively differentiates AD impairment stages from MRI data. The strong performance of GCN and GraphSAGE indicates their potential to assist clinicians in the early identification and tracking of Alzheimer’s disease progression.

INTEGRATION OF GEOGEBRA IN TEACHING GEOMETRY TO YEAR 5 PUPILS IN PRIMARY SCHOOL

The TIMSS 2019 results indicated that 55% of Form 2 students in Malaysia struggle with learning geometry concepts. This quasi-experimental research aims to improve students’ geometric concepts by developing a geometry learning module. The module integrates GeoGebra and aligns all five van Hiele Learning Phases in the learning activities. The learning activities in this module focus on building and developing basic concepts in geometric measurement such as perimeter, area, and volume. In this research, 50 Year Five students in Tawau District were divided into two groups: the experimental group (n=25) and the control group (n=25). The experimental group used the module while the control group was taught using a traditional teaching style. Pre- and post-tests were used in measuring their geometric thinking levels by referring to the first three van Hiele’s Geometric Thinking Levels: visualization, analysis, and informal deduction. The data analysis showed a noticeable increase in geometry thinking levels in the experimental group. This showed that the usage of the geometry learning module integrated with GeoGebra was beneficial in helping pupils improve their learning rather than the traditional teaching style. In conclusion, using technology such as GeoGebra in learning activities can help students learn basic geometry concepts effectively.

Enhancing geometry learning through project-based teaching materials: A focus on circumference and area of circles

Enhancing geometry learning through project-based teaching materials: A focus on circumference and area of circles

Motivation and Geometry Achievement in 2nd-Graders

ABSTRACT This study explores the relationship between different types of motivation (autonomous and controlled) for learning and achievements in geometry among 75 Hebrew-speaking and Arabic-speaking 2nd-grade boys and girls. The study is grounded in the Self-Determination Theory and Van Hiele’s model for geometric thinking. The participants completed a motivation questionnaire based on Ryan and Connell’s Self-Regulation Questionnaire, adapted specifically for studying geometry, and a test on geometric thinking level based on Usiskin’s test, adapted for 2nd-graders. Both questionnaires were administered orally by trained researchers at the beginning and end of the school year. In individual sessions, the researchers read the questions aloud to each child separately, who then provided their responses verbally. This study explored whether autonomous and controlled motivation types predict geometry achievements and assess changes in children’s perceptions of these motivations over the school year. The study found that only perceived autonomous motivation predicted end-of-year geometry achievements, while both types of perceived motivation decreased. Geometry achievement increased significantly across all groups. However, changes in perceived autonomous and controlled motivation, beyond gender and sector, were not significantly associated with changes in achievement.

Symplectic Bregman Divergences.

We present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel-Young inequality which relies on the notion of symplectic subdifferentials. The symplectic Fenchel-Young inequality is obtained using the symplectic Fenchel transform which is defined with respect to the symplectic form. Since symplectic forms can be built generically from pairings of dual systems, we obtain a generalization of Bregman divergences in dual systems obtained by equivalent symplectic Bregman divergences. In particular, when the symplectic form is derived from an inner product, we show that the corresponding symplectic Bregman divergences amount to ordinary Bregman divergences with respect to composite inner products. Some potential applications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are touched upon.

ANALYSIS OF THE CREATIVE THINKING ABILITY OF PGSD STUDENTS IN OPEN-ENDED PROBLEM-BASED GEOMETRY LEARNING

The ability to think creatively is an important skill in the world of education, especially for students of the Primary School Teacher Education Study Program (PGSD). In learning mathematics, especially geometry, this ability is crucial for understanding concepts in depth and solving problems innovatively. However, there are still many students who show limitations in generating creative ideas, especially in the aspects of originality and elaboration. Therefore, this research aims to analyze the creative thinking abilities of PGSD students in open problem-based geometry learning, which gives students the freedom to explore various solutions in solving geometric problems. This research uses descriptive qualitative methods with data obtained through interviews, observation and document analysis. Research informants include lecturers, PGSD students, heads of study programs, librarians, and the curriculum development team at STKIP PGRI Trenggalek. The research results show that the open problem-based learning method is effective in improving students' fluency and flexibility abilities. However, in the aspects of originality and elaboration, several obstacles were still found that required more intensive guidance and the use of supporting technology. With the right approach, this method can be a solution to optimize students' creative thinking abilities in learning geometry.

Students’ Research Skills Development by Using Visualization

The analysis of modern methodological systems of teaching mathematics shows that the use of interactive visual models can positively influence the result of mastering mathematical knowledge and the level of students' research skills. The development of IT has made the emergence of many specialized environments and services focused on solving mathematical problems possible. Among such software tools, a separate group of programs should be highlighted that allow interactive user interaction with geometric objects. These computer programs enable the student to independently "discover" geometric facts, which gives reason to consider such programs as a tool for developing their research skills. The study aims to substantiate the positive impact of visual models (models created in interactive mathematical environments) on developing students' research skills and general mastery of the school geometry course. We have presented a methodological scheme for developing students' research skills using GeoGebra (Technique), conducted its expert evaluation, and experimentally tested its effectiveness. The experts noted the potential efficacy of the Technique in terms of the quality of students' geometric knowledge (91.7%) and improving their performance in geometry in general (79.2%). The statistical analysis of the results of the pedagogical experiment confirmed that the student's research skills increased along with the semester grades in learning geometry. The results of the pedagogical experiment showed the effectiveness of the Technique we developed and provided grounds for recommending it for implementation.

Students’ understanding of geometric figures in the process of constructing a conjecture at the secondary school

Constructing conjectures is a major challenge in learning geometry. However, there are still gaps in our understanding of the knowledge about geometric figures that students use when they construct a conjecture. This article explores the understanding of ninth-grade students, aged 14 to 15, about geometric figures during the process of constructing conjectures. The study's data consists of the students’ responses to two geometry tasks in a questionnaire. The first task does not include an associated drawing, while the second task contains an incomplete drawing. Both tasks aim at the construction of conjectures. The research results highlight the influence of students’ knowledge of geometric figures on the stages of constructing conjectures in geometry. The findings indicate that students’ difficulties arise from inconsistencies in their cognitive structures regarding the figures used in conjecture construction. Students often rely on inappropriate prototypical figures when the task is not accompanied by an associated drawing. Additional findings reveal that informal arguments emerge when drawings are used to support these arguments, which are subsequently validated through experimental verification.

AR-based interactive GeoGebra learning media for optimizing transformation geometry learning in higher education

This study aims to create and develop interactive AR (Augmented Reality) GeoGebra learning tools for teaching transformation geometry to support a blended learning system. This approach employs the ADDIE development model, which consists of five phases. The research subjects include 50 fifth-semester students. The research instruments comprise validated questionnaires for material and media experts, student responses, and pre-test and post-test instruments that are both valid and reliable. The results indicate a high level of validity (90%) and excellent practicality as assessed by experts and users, with an N-gain of 0.39 in the experimental class, which outperformed the control class with an N-gain of 0.02. The study concludes that teaching transformation geometry using interactive GeoGebra learning media is more effective. This research implies leveraging the potential of AR technology integration in mathematics education to create interactive and effective learning experiences.

The Application and Effects of Augmented Reality in Geometry Learning

With the development of new technology, augmented reality (AR) technology is revolutionizing the educational landscape, particularly in geometry. This innovative approach deepens their comprehension of geometric principles and stimulates their motivation to learn. This paper aims to provide a comprehensive introduction to ARs application in the educational field, especially the topic of geometry, and evaluate its effectiveness in improving users in two aspects: motivation and academic results. The study applies the Attention, Relevance, Confidence, and Satisfaction (ARCS) Model to assess the impact of AR in education compared to traditional teaching. It finds that AR boosts student motivation, especially for those with lower academic performance. While AR enhances learning experiences, its effect on improving grades is not immediate. The method that is mainly utilized in this study is reviewing the literature, summarizing, and analyzing critical information from them. And it turns out that short-term grade improvements are not significantly observable, AR technology offers long-term benefits by enhancing engagement and conceptual comprehension through interactive and spatial visualizations.

Designing Cultural-Based Worksheets (LKPD) in Osing Village Culture for Elementary School Geometry Learning

Designing Cultural-Based Worksheets (LKPD) in Osing Village Culture for Elementary School Geometry Learning

Recent research and a critique of theories of early geometry learning: the case of the angle concept

Recent research and a critique of theories of early geometry learning: the case of the angle concept

Finding an Effective Assessment Approach to Enhance the Teaching and Learning of Circle Geometry

Assessment is essential for teaching and learning mathematical concepts. Teachers’ and learners’ understanding of how assessment is to be conducted in mathematics classrooms will guide what is to be taught and learnt and how teaching and learning will be conducted. The researchers’ assertion is that circle geometry lessons require thinking-based instructions. This necessitates that an associated assessment approach that elicits learners’ thinking must be formulated. Circle geometry requires rigorous logical, creative, critical, and reflective thinking, thus an assessment approach that will nurture learners’ thinking is essential. This exploration was a qualitative case study, conducted at a high school in South Africa involving two groups of grade 11 mathematics learners; both the control group and the experimental group had 34 participants. Data were gathered through classroom observations, a standardised test, and activities/investigation tasks. APOS theory (actions, processes, objects, schemas) was adopted as the theoretical framework and also as data analysis tool and circle geometry lessons were carried out in a collaborative classroom setting. Circle geometry lessons were conducted in line with the four APOS mental conception levels to evoke participants’ thinking, namely action lesson, process lesson, object lesson, and schema lesson. This study ascertained that the interactive didactic assessment was appropriate, as it enhanced participants’ metacognitive development and participants’ conceptual understanding and mathematical reasoning skills were developed. The researchers, therefore, recommend that this formulated assessment approach be implemented for the teaching and learning of mathematics, especially circle geometry, in schools in South Africa.

Noticing Students’ Geometry Learning Using Van Hiele’s Geometry Thinking Model at the Senior High School level in Builsa South District of Upper East Region

Noticing Students’ Geometry Learning Using Van Hiele’s Geometry Thinking Model at the Senior High School level in Builsa South District of Upper East Region

Geometric neural operators (gnps) for data-driven deep learning in non-euclidean settings

Abstract We introduce Geometric Neural Operators (GNPs) for data-driven deep learning of geometric features for tasks in non-euclidean settings. We present a formulation for accounting for geometric contributions along with practical neural network architectures and factorizations for training. We then demonstrate how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures of surfaces, (ii) to approximate solutions of geometric partial differential equations on manifolds, and (iii) to solve Bayesian inverse problems for identifying manifold shapes. These results show a few ways GNPs can be used for incorporating the roles of geometry in the data-driven learning of operators.

Culturally responsive pedagogy for the promotion of understanding Mathematics: The case of rural situated primary schools in Post-apartheid South Africa

The language of instruction, awareness, and creativity are at the centre of the realities faced by both mathematics teachers and learners, shaped by a socio-political history whose impact and legacy transcend generations in South Africa. Consequently, the dire state of mathematics education in South Africa remains a cause for concern. Recognising the need to conceptualise and develop culturally responsive pedagogy, this paper presents the results from the third phase of a three-year longitudinal study focused on teachers' journeys in infusing indigenous knowledge when teaching geometry to senior primary learners in rural schools. This qualitative study, which followed a case study design, explored several approaches to discussing, teaching, and considering an immediate, perpetual, operative, and discursive approach as culturally disruptive pedagogy used as an indigenous way to promote geometric understanding. Observations and semi-structured interviews were conducted with two purposefully selected participants after their lesson presentations to allow them to clarify, elaborate, and introduce more detail, enriching their explanations of the artefacts used. Results indicated that teachers’ references to several artefacts and indigenous activities, such as using straws, wool, and staplers, promoted the understanding of the construction of prisms and pyramids. Recommendations include training mathematics teachers to prepare lessons that incorporate indigenous knowledge and to explore ethnomathematics as a culturally responsive teaching strategy, particularly in contextualising mathematical geometry learning, so that this concept can have relevance and meaning for rural students.

Adaptation of retinal discriminability to natural scenes.

Sensory systems discriminate between stimuli to direct behavioral choices, a process governed by two distinct properties - neural sensitivity to specific stimuli, and neural variability that importantly includes correlations between neurons. Two questions that have received extensive investigation and debate are whether visual systems are optimized for natural scenes, and whether correlated neural variability contributes to this optimization. However, the lack of sufficient computational models has made these questions inaccessible in the context of the normal function of the visual system, which is to discriminate between natural stimuli. Here we take a direct approach to analyze discriminability under natural scenes for a population of salamander retinal ganglion cells using a model of the retinal neural code that captures both sensitivity and variability. Using methods of information geometry and generative machine learning, we analyzed the manifolds of natural stimuli and neural responses, finding that discriminability in the ganglion cell population adapts to enhance information transmission about natural scenes, in particular about localized motion. Contrary to previous proposals, correlated noise reduces information transmission and arises simply as a natural consequence of the shared circuitry that generates changing spatiotemporal visual sensitivity. These results address a long-standing debate as to the role of retinal correlations in the encoding of natural stimuli and reveal how the highly nonlinear receptive fields of the retina adapt dynamically to increase information transmission under natural scenes by performing the important ethological function of local motion discrimination.

Design and evaluation of a geometry learning application for visually impaired students

Design and evaluation of a geometry learning application for visually impaired students

Dynamic object removal by fusing deep learning and multiview geometry

Dynamic object removal by fusing deep learning and multiview geometry