Exploration of Students' Computational Thinking on Geometric Transformations Through AI-Assisted Geogebra

Students’ Computational Thinking Difficulty Pathways in Geometric Transformation after GeoGebra-Assisted Problem-Based Learning

Computational thinking is an important mathematical process in problem solving that encompasses the abilities to analyze situations, develop strategies, recognize patterns, and design systematic procedures. Understanding students’ computational thinking cannot be derived solely from overall scores but also requires an explanation of where their reasoning processes break down. This study aimed to diagnose students’ computational thinking difficulties, analyze how these difficulties interacted and propagated throughout the problem-solving process, and identify the reasoning mechanisms underlying the identified difficulty pathways after GeoGebra-assisted problem-based learning. This descriptive study employed quantitative and qualitative approaches and involved 34 Grade 11 students from a public senior high school in Palembang, Indonesia. Data were collected through three computational thinking test items, students’ written responses, classroom observations, and interviews. The test assessed four indicators: decomposition, pattern recognition, abstraction, and algorithmic thinking. Overall, 19 students, or 55.88%, were in the moderate category. Pattern recognition showed the highest achievement at 62.74%, followed by decomposition at 50.32% and abstraction at 49.34%, while algorithmic thinking was the lowest at 44.44%. Students’ difficulties included incomplete organization of information, failure to connect transformation types with appropriate matrices, inability to select relevant information, errors in operations, and weak result verification. The findings indicate that computational thinking indicators do not operate independently but function as interdependent processes, with matrix representation serving as a critical bridge between conceptual recognition and procedural execution. Practically, the findings provide a basis for diagnostic assessment and targeted remediation according to the source of difficulty, including problem structuring, matrix representation, procedural execution, and verification. This study contributes an indicator-based diagnosis of students’ computational thinking difficulties.

Understanding Students’ Learning Obstacles in Geometric Transformations and Their Implications for Didactical Design

Geometric transformations are one of the essential topics in mathematical learning, as they support students’ spatial reasoning, spatial visualization, and understanding of geometric relationships. Despite its importance, many students still have difficulties in interpreting transformation concepts beyond routine procedural exercises, particularly problems that are presented in unfamiliar or contextual situations. Therefore, this study aimed to explore students' learning obstacles in learning geometric transformations and examine their implications for the development of didactical design. Qualitative approach with a didactical design research framework was employed using a phenomenological perspective. Participants involved were 36 ninth-grade students and one mathematics teacher from a junior high school in West Bandung Regency, Indonesia. Research data were collected through a written test on mathematical spatial ability consisting of four contextual problems, a semi-structured interview, and a praxeological analysis of a textbook. The written-test instrument was validated by two mathematics education experts and one experienced teacher before implementation. The data were analyzed using the Miles and Huberman interactive model, involving data reduction, display, and conclusion drawing, to categorize students’ difficulties into epistemological, ontogenic, and didactical obstacles. The research revealed that students experienced interconnected categories of learning obstacles. Epistemological obstacles appeared in students’ inability to concept generalization, misconceptions of the transformations’ properties, and dependence of procedural examples. Ontogenic obstacles related to limited spatial relations ability, weak understanding of Cartesian coordinates, and difficult to interpret contextual problems. Lastly, didactical obstacles emerged from teacher-centered instructions, limited use of interactive visual media or technology, and emphasis on procedural completion compared to conceptual exploration. These obstacles interacted with one another and influenced students' understanding of geometric transformations.  The study implies that effective didactical design should integrate conceptual understanding, spatial visualization, multiple representations, contextual learning, and dynamic technology to facilitate meaningful learning of geometric transformations. Keywords: learning obstacles, geometric transformation, epistemology, ontogenic, didactic.

Designing to transform: High school students’ reasoning with geometric transformations in an authentic real-world project

Designing Learning Trajectories for Nets of Polyhedrons through Simulation-Based PBL

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.

AI-Integrated Student Worksheets: Scaffolding Spatial and Representational Reasoning in Double Integrals

This study aims to develop and examine the effectiveness of Artificial Intelligence (AI)-integrated student worksheets as scaffolding for students' spatial and representational reasoning in double integral material. Using an explanatory sequential mixed-methods approach, this exploratory study employed a one-group pretest-posttest design involving the entire cohort of 11 third-semester students from the 2023 batch of the Mathematics Education Study Program at Universitas PGRI Ronggolawe. Data were analyzed using the Wilcoxon Signed-Rank test and N-Gain. The qualitative phase involved in-depth observations and semi-structured interviews to explore the mechanisms of AI scaffolding. Expert validation yielded a score of 87.7% (highly valid), while practicality reached 85.5% (highly practical). The effectiveness test showed a significant improvement ( , large effect size) with a mean N-Gain of 0.556 (moderate category), and all students surpassed the minimum mastery score of 75. Qualitative findings revealed three distinct AI scaffolding mechanisms: (1) externalized thinking through Socratic interaction for high-ability students; (2) progressive fading aligned with Vygotsky's Zone of Proximal Development (ZPD) for moderate-ability students; and (3) graphical-to-symbolic representational transfer facilitated by graduated AI prompts for low-ability students. As a preliminary, theoretically informed contribution, this exploratory study proposes an initial framework for differentiating AI-driven scaffolding in multivariable calculus pedagogy, to be further tested in future research with larger samples and controlled designs.

Classroom-based Experiences in Integrating GeoGebra: A Meta-synthesis

This study aimed to synthesize classroom-based experiences of teachers and students in integrating GeoGebra into mathematics instruction. GeoGebra, a dynamic mathematics software, has gained attention for enabling interactive and exploratory learning of mathematical concepts. Using a qualitative meta-synthesis research design, the study systematically analyzed existing qualitative research on high school classroom experiences with GeoGebra integration. Academic databases including Crossref, Google Scholar, PubMed, Scopus, and OpenAlex were searched, and study selection followed the PRISMA guidelines. The Critical Appraisal Skills Programme (CASP) checklist was applied to assess the quality and relevance of the included studies. From an initial pool of 2,736 studies, twelve met the inclusion criteria and were synthesized. Thematic analysis revealed four major themes: (1) classroom-based support in GeoGebra integration, (2) GeoGebra as an explorative and engaging learning environment, (3) enhancement of mathematical thinking skills, and (4) students’ personal growth and teachers’ professional development. These themes were unified under the meta-theme GeoGebra Integration: Transforming Classroom Learning Experiences. Despite implementation challenges, findings indicate strong teacher enthusiasm and a shift toward technology-enhanced pedagogies that foster deeper conceptual understanding. The study highlights GeoGebra’s transformative potential in supporting mathematical thinking, personal growth, and 21st-century skills, and recommends further long-term and mixed-methods research across diverse educational contexts.