This study explores the impact of automated feedback in GeoGebra on students’ learning of Ruffini’s Rule, a polynomial division algorithm. A quasi-experimental design was implemented with two 10th-grade classes in a private Portuguese school. One group used a GeoGebra application offering immediate feedback, while the control group received traditional instruction. Both groups completed pre- and post-tests, and the experimental group also answered a perception questionnaire. Results show that students who used the feedback-enabled tool demonstrated greater improvement in solving polynomial division problems. Their feedback highlighted the benefits of real-time error correction and increased engagement, although some students noted difficulties with the interface and reduced peer interaction. These findings suggest that automated feedback can enhance procedural fluency and autonomy in learning algebra. The study contributes practical insights for mathematics educators seeking to integrate technology meaningfully in secondary classrooms.
In this extended editorial, I explore some issues around the use of generative AI in mathematics education and mathematics education research. In particular, I provide an overview of how generative AI can be utilised and what is required to use it productively.
The rapid adoption of artificial intelligence in higher education is transforming the teaching of mathematics and raising concerns about the learning assessment validity. The present article is about an inquiry-based experiment that engaged our students in carefully tailored hands-on activities conducted in person and in-class. It was encouraging to witness that our students progressed tremendously and became more efficient in mathematical reasoning and used AI as an aid not a substitute.
This paper presents the generation and visualization of conic sections in three-dimensional space using the virtual reality software NeoTrie VR. The proposed approach enables students to construct conics in an immersive and interactive environment, explore their geometric properties dynamically, and connect classical geometric concepts with modern digital tools. Several methods for generating conics are implemented in this environment, ranging from direct intersections of cones and planes to dynamic constructions based on compasses, loci, and Dandelin’s spheres. The results highlight the educational potential of virtual reality for enhancing spatial reasoning and supporting a deeper conceptual understanding of conic sections.
This paper reports the design and refinement of a generative-AI simulation for mathematics teacher education centred on equal-sign misconceptions. We developed three custom chatbots that simulate middle-school students (11–12 years old) who exhibit distinct operational interpretations of “=” across three research-based tasks: 8 + 4 = + 5, a chained equality (4 + 5 = 9 + 3 = 12), and 37 + 54 = + 55. Each simulated student was paired with a short video depicting the student’s incorrect reasoning and was engineered through persona prompting to sustain “student-like” dialogue (brief initial justifications, resistance to immediate correction, inconsistent improvement, and spontaneous doubts) to preserve teachers’ opportunities to elicit, interpret, and respond to student thinking. In parallel, we designed a mentor chatbot that provides structured formative feedback (strengths, areas for improvement, suggestions) anchored in equal-sign instruction and responsive teaching. We describe iterative development and cross-linguistic adaptation that addressed common failure modes of GenAI-based simulations, such as overly articulate student responses and generic feedback. The paper contributes a practical design account of how prompt constraints, role separation, and task-misconception alignment can make GenAI-based simulations more stable and instructionally useful as approximations to practice.
The integration of artificial intelligence (AI) into mathematics education has sparked discussions regarding its feasibility for automated examination. These considerations stem from ChatGPT’s ability to solve various mathematical problems and read and decipher handwritten text, including mathematical expressions. In contrast, the Halomda system provides various solution-checking options, such as multiple-choice questions, answers including algebraic expressions, textual responses, images, and graphs. In the study, we compare three approaches to assessing exam solutions provided by Halomda and ChatGPT: a) evaluation of handwritten test; b) checking the final answer; c) assessment of predefined solution steps. Overall, the findings show that modern AI-based models, such as ChatGPT, despite their potential, are, in our observed cases, less reliable than specialized electronic assessment systems for assessing mathematics exam results.
Digital tools (DTs) are increasingly available in mathematics education, yet their classroom use often remains limited to presentation tools, and their potential to support inquiry and conceptual understanding is underutilised. Our previous work on computer-based testing (CBT) with dynamic objects showed that such assessment becomes effective only when aligned with DT-based learning activities. Building on this insight, this study develops a set of DT-based tasks for upper secondary mathematics designed to integrate exploration, reasoning, and conceptual understanding. Using dynamic representations and 2D/3D visualisation, we present four case studies illustrating different mathematical activities supported by DTs. We also analyse their roles within the Doing–Learning–Teaching framework and discuss how interaction data generated through these tasks may contribute to more effective formative assessment.
Previous studies have shown that the conceptual complexity of horizontal function translation can pose challenges for students, motivating subsequent research to propose instructional approaches that support students in developing more productive ways of thinking about this transformation. This study investigates how students can use emergent graphical shape thinking (EGST) to construct meanings for horizontal function translation when supported by a dynamic visualization. A case study approach was employed to trace the learning progression of two students as they worked through a digital task incorporating this visualization. Their progression unfolded in three phases. First, they identified a relationship between the original function and its horizontal translation, though the direction of this relationship was the reverse of what was intended. Second, in an effort to refine the relationship between the functions, they established a correspondence relationship between the inputs of the original and translated functions. Finally, they used this correspondence relationship to generalize the functional relationship that characterizes a horizontal translation. How EGST and the dynamic visualization supported this meaning-making are subsequently discussed.
This study investigates how ChatGPT can support metacognitive processes during problem-solving activities. A class of 12 female students participated in two work sessions: in the first, the students interacted with ChatGPT without pretraining; in the second, the chatbot was trained with a structured prompt to act as an educational tutor. A qualitative analysis conducted according to Meijer and colleagues’ taxonomy of metacognitive activities showed that, with untrained chatbots, students mainly activate orientation and planning phases, while evaluation and elaboration rarely emerge. On the contrary, the structured prompt encourages engagement in all metacognitive activities, with longer reflection times and less linear sequencing. The study suggests that ChatGPT can support metacognitive activities when properly trained, highlighting the importance of prompt design and the role of the teacher as an orchestrator of interaction. Educational implications, limitations of the study, and directions for future research are discussed.
Geometric construction is a key component of mathematics education worldwide, supporting the development of spatial reasoning, logical thinking, and geometric proof. Although dynamic geometry software such as GeoGebra and Cinderella has broadened opportunities for digital exploration, virtual compasses may lack the tactile realism and procedural clarity of traditional instruments, which may influence students’ understanding of the construction steps and underlying concepts. To address this issue, we developed an electronic geometric compass that replicates the structure and function of a physical compass while capturing detailed process data. The device replaces the pencil with an electronic pen and equips the needle with a position sensor, enabling both stroke data and needle coordinates to be recorded in InkML format. A nationwide study involving 100 Japanese upper-secondary students who completed 20 classical construction tasks demonstrated that the electronic geometric compass effectively captures learners’ strategies, hesitations, and trial-and-error behaviour. These findings highlight the potential of this hybrid analogue–digital tool to enhance formative assessment, provide deeper insight into learners’ geometric reasoning, and support the development of future automated evaluation and personalised feedback systems.
We investigate how undergraduate mathematics students develop understanding of homeomorphism through theoretical covariational thinking and metaphorical reasoning. Thirty second-year geometry students at a southern Italian university explored stereographic projections using Euclidean and Taxicab metrics, mediated by GeoGebra. Students discovered that while Euclidean stereographic projection constitutes a homeomorphism, the Taxicab counterpart does not, despite both metrics inducing identical topologies. Through collaborative “Geometric Thinking Groups”, students experienced ‘structural covariational surprise’, a productive cognitive conflict arising from unexpected patterns in how metric and topological structures covary. GeoGebra functioned as a cognitive partner enabling dynamic mathematical experimentations. Data analysis reveals how digital tools facilitate coordination of visual-geometric and formal-analytical thinking, leading to understanding of the interplay between metric structure, topological structure, and homeomorphic behaviour. Findings show how theoretical covariation, supported by conceptual metaphors and digital mediation, develops advanced mathematical concepts while revealing distinctions between topological equivalence and metric-dependent homeomorphic behaviours.
The pedagogy developed in mathematics education is steadily changing with the steady development of new technologies. We propose examples of activities (2D geometry, plane curves, such as pedals and hyperbolisms, and space curves) sometimes with a STEAM approach. The audience is in- and pre-service teachers, aimed at acquiring new joint mathematical and technological knowledge, and also a standalone excellent undergraduate student who quickly became a full collaborator, especially for his self-teaching ability. The activities allow students to discover mathematics beyond the traditional syllabus, exploring topics in their Zone of Proximal Development (ZPD) , and requiring also a reflection on how to use technology in case their background lacks some theoretical knowledge. Central tools are software for dynamical geometry and automated commands, and a Computer Algebra System.
This study examines the implementation of the paid version of Socrative platform as a formative assessment (FA) tool in first-year mathematics courses. A key feature of the software is that it provides immediate, individualized feedback designed to support student engagement and learning. Employing a convergent mixed-method design, the study employed statistical analysis to compare assessment outcomes between control and experimental groups. Additionally, we conducted qualitative analyses of selected assessment items to gain deeper insight into students’ learning processes. Findings reveal both anticipated and unanticipated effects of formative assessment on student performance. The quantitative data did not show any significant effect of FA on students’ achievement. The qualitative data revealed multiple levels of influence on learning, ranging from no to substantial improvement, as well as instances of inconsistent outcomes. In this article, we will discuss the outcomes in relation to pedagogical implications, and we will provide some recommendations for future research.
This case study investigates how mathematics teacher education students develop statistical reasoning about the concept of center within a Statistical Reasoning Learning Environment using TinkerPlots. The researchers developed and implemented instructional tasks to foster reasoning about measures of central tendency and their interplay with data variability and distribution. The study involved 12 mathematics teacher education students from a public university. Through two activities, students explored real-world datasets and used TinkerPlots to visualize, analyze, and discuss statistical ideas. Results indicated that technology integration enhanced understanding of key statistical ideas of center, including the effects of extreme values on central measures and the importance of considering center and spread together. While students demonstrated proficiency in some areas, challenges persisted, particularly regarding interpreting the median and its relationship to data distribution. The findings of this study highlight the potential of TinkerPlots to deepen statistical reasoning regarding the idea of the center and recommend an instructional pathway. It provides insights to optimize technology-integrated learning environments in statistics education.
The aim of this case study was to determine whether ChatGPT supports students in building and transforming representations of calculus concepts and to examine whether this support differs according to students' academic performance. We purposefully selected two university students studying in the Mathematics Education program and focused on the collective argumentation between ChatGPT and each student on a mathematical task involving calculus concepts. The performances of ChatGPT and students in building and transforming shared representations during collective argumentation were analyzed using Duval's Theory of Registers of Semiotic Representation. The results showed that ChatGPT mainly built verbal and algebraic representations during collective argumentation and performed treatment and conversion. However, in some parts of the argumentation, the transformations performed by ChatGPT were faulty in mathematical context. In these parts of the argumentation, ChatGPT could not support the students in terms of building and transforming shared representations. Nevertheless, during the collective argumentation, ChatGPT made students' mathematical thinking more observable, particularly in the context of their use of representations of mathematical concepts. ChatGPT can be used by teachers and researchers as a useful mediator to reveal the students' mathematical thinking in the context of use of representations.
This paper presents the Technology-Based Geometric Thinking Framework for analyzing students' geometric thinking within dynamic geometry environments (DGEs). The framework identifies four interconnected dimensions: engaging in exploration, developing and applying definitions, investigating invariance, and generating explanations, justifications, and proofs. By synthesizing established geometric theories with DGE-specific interactions, this framework provides researchers a structured approach for examining how technology mediates geometric thinking, illustrated through the analysis of student interactions with a dynamic geometry task. This framework has the potential to inform curriculum design, teacher practice, and the development of digital tools by offering a structured lens for interpreting students' geometric reasoning in technology-rich contexts. While this study draws on a single task with a small sample of students, the framework lays groundwork for broader application and further validation.
The paper discusses ways to enhance students' mathematical training, focusing on their ability to apply mathematics in practical, academic, or scientific contexts. We reviewed different scientific approaches, which either emphasise the development of conceptual understanding of mathematics or procedural knowledge. We concluded that university-level mathematics education should be based on the conceptual and procedural triad, which comprises conceptual understanding, procedural fluency, and problem-solving ability. We proposed teaching practices based on the educational strategy of Inquiry-based learning (IBL). These practices include (i) determining learning objectives to focus students' attention and motivate them to achieve them; (ii) solving problems with high cognitive demands that encourage students to search, research, solve problems, and make connections; (iii) engaging students in productive work on knowledge construction and fostering meaningful mathematical discourse; (iv) procedural fluency training based on conceptual understanding. The effectiveness of this approach has been tested experimentally. Special emphasis was placed on implementing an integrated conceptual and procedural approach using the Moodle LMS. This digital platform provided a flexible and convenient blend of various educational activities, which is especially beneficial for blended learning models. Moodle facilitated the organisation of ongoing interaction between lecturers and students, supported autonomous learning, and enabled feedback, all of which contributed to the overall effectiveness of mathematics education.
