Jul 2026· International Electronic Journal of Mathematics Education· Vol 21, pp. em0889· 0 citations· 54 references
TL;DR
The findings suggest that integrating computational geometry problems with digital tools can enrich traditional mathematics instruction, highlight the relevance of geometry in authentic contexts, and offer a promising and transferable context for developing CT alongside core geometry content in secondary mathematics education.
Abstract
This paper examines how bichromatic separability problems, a classic topic in computational geometry, can be adapted for secondary mathematics education through the use of BichromaticSolver. The software computes simple or convex polygons that separate two finite sets of points under different optimisation criteria, including maximum area, minimum area, maximum perimeter, or minimum perimeter. Unlike traditional approaches, the number of polygon sides k is not fixed in advance but chosen by the user, enabling the exploration of diverse and potentially more effective configurations. Three classroom tasks were designed in which students alternated between paper-and-pencil methods and digital exploration with the software. This two-phase structure encouraged them to verify constructions, compare alternative outcomes, and refine their strategies. Classroom observations from this exploratory study document how these activities created opportunities for students to express and refine geometric reasoning while making computational thinking (CT)-related practices visible, for example decomposition, abstraction, strategic planning, and comparative evaluation of solutions. The findings suggest that integrating computational geometry problems with digital tools can enrich traditional mathematics instruction, highlight the relevance of geometry in authentic contexts, and offer a promising and transferable context for developing CT alongside core geometry content in secondary mathematics education.
This article reports the experience of implementing a didactic sequence on the study of 3D geometric shapes (polyhedrons and round bodies), grounded in the principles of Ethnomathematics and Critical Mathematics Education. The main objective was to promote the recognition of geometric properties in everyday life and the development of spatial perception, enabling students to distinguish solids and their nets. The adopted methodology was characterized as a literature review, based on the National Common Curricular Base (BNCC) and the Learning Recovery Matrix of SEDUC-TO, combined with participant research, in which the teacher mediated the transition from abstract to concrete thinking. Manipulative materials and the educational support of the Projeto Aprova Brasil were used to study three-dimensional models. The results achieved demonstrate that the tactile and visual approach reduced resistance to mathematical content, allowing students to clearly identify elements such as vertices, faces, and edges, in addition to consolidating their understanding of Euler's Formula in an intuitive and collaborative manner.
P. Silva, Dione Cléia Pereira dos Santos, Adriano Pereira de Miranda· Revista de Estudos Interdisc...· 0 citations
Manipulative materials are widely considered valuable resources in geometry education because they may support visualization, spatial reasoning, and conceptual understanding when implemented within appropriate instructional contexts. However, many existing resources focus on isolated concepts and provide limited opportunities for connecting geometric relationships through transformation and decomposition. This study presents Geomaniguras, a manipulative material designed for upper primary and lower secondary education. The research adopts a design and development approach focused on pedagogical design and expert-based validation rather than measuring student learning outcomes. The study combined prototype construction, expert evaluation, and iterative refinement. Twelve specialists in mathematics education assessed the material using a structured validation guide addressing pedagogical usefulness, conceptual coherence, usability, visual design, and classroom applicability. Results indicate that experts perceived Geomaniguras as a potentially valuable resource for exploring polygon classification, area relationships, geometric decomposition, circumference, and the Pythagorean theorem. The findings provide preliminary evidence regarding the pedagogical plausibility and classroom applicability of the material, although no direct conclusions can yet be drawn about instructional effectiveness. The originality of the proposal lies in its integrated conceptual framework, which connects multiple geometric learning experiences within a single manipulative system.
José A. Núñez-lópez, D. Molina-García, J. L. González-Fernández· Education sciences· 0 citations
Trigonometric functions are usually one of the hardest topics for Brazilian high school students, which is reflected in the low math scores seen in large-scale assessments. This paper is an experience report on using GeoGebra and Maxima to solve Question 180 of the 2015 ENEM exam, which models the seasonal price of an agricultural product through the function P(x) = 8 + 5cos((πx − π)/6). The same question was solved in four different ways: the traditional analytical method (pencil and paper), reading the graph in GeoGebra, a price list built and searched inside GeoGebra itself (using the commands Sequence, Min and IndexOf, plus a button-triggered script), and a small search algorithm written in Maxima. All four reach the same answer, the month of July (x = 7), but each one asks something different from the student: algebraic reasoning, visual reading, data organization, and programming logic. The paper also describes, in detail, a syntax error that came up while building the script in GeoGebra, and how it was fixed, arguing that this kind of stumble is part of learning and should not be hidden from students. The conclusion is that combining both tools with traditional calculation helps students understand the same content from different angles, in line with what the BNCC asks for.
Patrick Corrêa dos Santos· Revista de Estudos Interdisc...· 0 citations
Students frequently experience difficulties in learning nets of polyhedrons because they must coordinate two-dimensional representations with three-dimensional objects and mentally anticipate folding processes. This study aimed to develop and refine an empirically grounded learning trajectory for nets of polyhedrons by integrating Problem-Based Learning (PBL), contextual simulation, animated videos, and student worksheets. A design research approach was employed through three phases: preliminary design, teaching experiment, and retrospective analysis. The pilot teaching experiment involved six ninth-grade students representing high, moderate, and low levels of mathematical achievement, while the large-scale implementation involved 28 ninth-grade students in a regular classroom. Data were collected through classroom observations, semi-structured interviews, students’ written work, worksheets, learning outcome tasks, and instructional documentation, and were analyzed by comparing predicted responses in the Hypothetical Learning Trajectory with students’ actual learning processes. The findings showed a progressive development from recognizing familiar packaging patterns to visualizing folding processes, distinguishing valid and invalid nets, analyzing relationships among faces and edges, constructing alternative nets, and communicating mathematical justifications. Animated visualization and contextual simulation helped students connect informal experiences with formal geometric concepts, while collaborative investigation and reflection supported the transition from perceptual judgments to structural reasoning. Retrospective analysis produced a refined eight-stage learning trajectory: gift-shop simulation, activation of prior knowledge, animated visualization, identification of valid and invalid nets, collaborative investigation, alternative net construction, presentation and mathematical justification, and reflection and generalization. The trajectory offers a practical and theoretically informed design for supporting conceptual understanding and spatial visualization, although further scaffolding is needed to strengthen students’ written mathematical communication.
Computational thinking (CT) is a key competence in 21st-century mathematics learning, particularly in geometry, which involves spatial reasoning and problem solving. This study aims to explore how students’ CT develop when solving geometric transformation problems through artificial intelligence (AI)-assisted GeoGebra learning. The study employed an Educational Design Research (EDR) approach involving 18 junior high school ninth-grade students. The learning design followed a hypothetical learning trajectory with three increasingly complex challenges: translation with fixed orientation, translation with varied positions, and a combination of translation and rotation. Data were collected from student artifacts, including worksheets, AI prompts, GeoGebra commands and screenshots, teacher observations, and exit tickets, and analyzed qualitatively using deductive coding based on four CT components. The results show that students demonstrated strong abilities in decomposition and pattern recognition, while abstraction and algorithmic thinking remained the main challenges, especially in complex transformations. GeoGebra and AI primarily functioned as tools for reflection and debugging, rather than as direct solution providers. This study highlights the importance of scaffolded, challenge-based learning to support CT development in geometry. However, the findings are limited to a single learning session with a small sample size.
This article examines the didactic potential of applying Rhino and AutoCAD
graphic software in the teaching process of descriptive geometry and analyzes their effectiveness
in solving positional problems. The study focuses on methods for determining intersection lines
of surfaces, particularly the construction features of intersections between second-order surfaces
and planes in general position. Special attention is given to the possibilities of visual modeling
of complex spatial problems, accurate geometric constructions based on orthogonal projections,
and mastering algorithmic approaches to defining intersection lines using graphic software tools.
The article presents step-by-step methods for solving positional tasks in Rhino and AutoCAD
environments and highlights their role in developing students’ spatial thinking, graphic literacy,
and independent cognitive activity. In addition, the influence of modern digital technologies on
improving the quality of knowledge acquisition, forming engineering and graphic competencies,
and increasing the effectiveness of the educational process is scientifically and methodologically
substantiated. The research results demonstrate that graphic software serves as an effective tool
for integrating theoretical knowledge with practical skills and contributes to the modernization of
descriptive geometry teaching in the context of digital transformation in education.
N. Tashimov· Рroblems of engineering and...· 0 citations