Constructing a valid geometric proof requires prospective mathematics teachers to coordinate premises, deductive warrants, proof goals, and representations. Although empirical studies have documented multiple proof-related difficulties, less is known about how a breakdown at one point constrains subsequent reasoning. This qualitative descriptive-exploratory study examined four direct proof tasks and task-based semi-structured interviews with six prospective mathematics teachers from an initial sample of 20. Data were analyzed using an adapted Explorative Mathematical Argumentation (EMA) framework composed of initiation, tools, goals, and modalities. A response was classified as a failure when both written and interview data indicated that the participant possessed relevant knowledge but did not use or apply it appropriately. Failures were present in all components of the EMA framework, and were most prevalent in the application of theorems and in the use of deductive logic. Within-participant analysis indicated possible pathways, as the under-integration of a premise limited access to an appropriate warrant; the use of a theorem without support resulted in invalid intermediate assertions, and disrupted the continuity of the argument; and the use of a theorem along with a drawing resulted in a lack of checking the theorem against the conditions. These pathways were not rigid but indicated the interplay of different conceptual, strategic, representational, and communicative demands. The results of the study provide support for a stage-differentiated and functionally interrelated description of proof, and for the construction of pathways of failure as a process-oriented description of gaps across different classifications of failure.
Teachers' beliefs significantly influence the efficacy of mathematics education. The absence of an instrument to assess beliefs about deep learning poses a challenge for evaluating and enhancing mathematics teachers' competence in adopting deep learning. This research aims to develop and test the validity and reliability of an instrument to measure mathematics teachers' beliefs about implementing deep learning. This research involved 108 mathematics teachers from five provinces in Indonesia. The research data were collected using a questionnaire with a three-dimensional scale: mindful learning, meaningful learning, and joyful learning. The collected research data were then analyzed using the Rasch model, assisted by Winstep software version 3.73. The results of the first calibration indicated that the instrument's quality was very good, and the respondents' answers were quite consistent. The analysis was continued by examining each person and item more closely, resulting in the identification of three items as misfits. The second calibration was carried out by eliminating the misfit statement items. In the second calibration, a valid, reliable, and unidimensional instrument was obtained, which did not exhibit bias towards any of the research attributes. The instrument demonstrated very good reliability, and respondents' answers were quite consistent. After repeated calibration, a deep learning belief measurement instrument was obtained, consisting of 17 high-quality items with adequate psychometric properties.
The study examined the effect of neuroeducational methodologies supported by artificial intelligence on logical-mathematical thinking in teachers in training in Basic General Education at the Technical University of Ambato, Ecuador. A quantitative approach was used with a quasi-experimental, longitudinal pre-test-posttest design with non-equivalent groups. The sample was made up of 480 students, distributed in a control group that received mathematics teaching supported by conventional ICT and an experimental group that participated in a 14-week intervention based on neuroeducational methodologies supported by artificial intelligence. Data were collected through a 21-item performance test that evaluated logical-relational reasoning, abstraction and mathematical representation, and strategic problem solving. The internal consistency of the instrument was adequate (Cronbach's alpha = 0.816; McDonald's omega = 0.817). The analysis included descriptive statistics, ANCOVA, and linear regression in Jamovi. After controlling for pre-test scores, the ANCOVA showed a significant effect of group on post-test logical-mathematical thinking F (1,477) = 206, p < 0.001, η²p = 0.302. Linear regression indicated that belonging to the experimental group significantly predicted final performance (B= 0.810, β= 0.882, t= 49.0, p < 0.001). The results suggest that neurodidactic methodologies structured and supported by artificial intelligence can strengthen logical-mathematical thinking in initial teacher training.
Developing mathematical literacy in exponent learning requires students to connect authentic experiences, representations, and mathematical ideas. This study developed and evaluated Augmented Reality (AR)-enhanced instructional materials using local wisdom as an authentic context and AR as a pedagogical mediator. A Research and Development design following the ADDIE model was employed. The materials were validated by 11 mathematics teachers, and the literacy assessment was reviewed by two teachers. A limited implementation involved 72 eighth-grade students from two junior secondary schools in Kupang, Indonesia. Data were collected through validation sheets, questionnaires, observations, and pretest–posttest assessments. Mean validation scores were 3.55 out of 4.00 for content, 3.60 for media, and 3.25 for the assessment instrument. Student and teacher practicality scores were 81.10% and 96%, respectively. Students’ mean mathematical literacy score increased from 58.72 (SD = 12.64) to 82.57 (SD = 11.80), t(71) = 20.63, p < 0.001, with a very large within-group effect size (Cohen’s dz = 2.43) and a moderate N-gain of 0.60. These findings indicate that integrating local wisdom and AR offers a valid, practical, and promising pedagogical mechanism for supporting mathematical literacy in exponent learning.
Educational role-playing games (RPGs) have been increasingly adopted in mathematics education because of their potential to support students' cognitive, affective, and social learning. However, previous studies have predominantly examined their effectiveness from the perspective of student outcomes, while relatively little is known about how teachers experience and interpret student learning during educational RPG instruction. This study aimed to explore the qualitatively different ways Grades 7–9 mathematics teachers experienced student learning in educational role-playing game environments using a phenomenographic approach. Thirty mathematics teachers participated in the broader educational RPG implementation program, from which five teachers representing diverse teaching experiences were purposively selected for in-depth phenomenographic interviews. Classroom observations, teacher reflection sheets, and student activity sheets were used to contextualize and support the interpretation of interview data. Data were analysed through iterative phenomenographic procedures to construct categories of description and an outcome space. The findings identified four qualitatively different ways teachers experienced student learning: (1) engagement and emotional involvement, (2) strategy use and procedural engagement, (3) conceptual understanding of algebra, and (4) integrated cognitive, emotional, and social development. These categories formed an outcome space representing increasing inclusiveness in teachers' ways of experiencing student learning. The study contributes to educational game research by demonstrating that the pedagogical value of educational RPGs depends not only on game design or student performance but also on how teachers recognise and interpret learning within classroom practice.
Quasi-experimental designs (QEDs) are widely used in educational research, however pre-service mathematics teachers often struggle to justify their methodological choices when ideal experimental conditions cannot be met. This study designed and refined a pedagogical approach to support mathematics education students’ conceptual understanding of QEDs. Employing educational design research, the study involved 41 third-year undergraduates enrolled in an experimental design in education course. The intervention was conducted across two iterative cycles: a teaching experiment (n = 9) and a classroom experiment (n = 32). A Hypothetical Learning Trajectory (HLT) was developed to structure learning activities around the critical analysis of authentic research excerpts. Analysis of students' written design proposals and clinical interviews revealed a conceptual progression from viewing interventions as true experiments to recognizing QEDs as contextually grounded compromises. Specifically, students demonstrated transformed reasoning by shifting from rigidly demanding random assignment to strategically utilizing intact groups to manage confounding variables. Retrospective analysis further highlighted the need for strengthened scaffolding regarding the notion of control. Consequently, the refined HLT yielded a proposed Local Instructional Theory (LIT) comprising four empirically grounded design principles: (1) contextualizing the need for comparison, (2) negotiating intact-group constraints, (3) evaluating threats to validity, and (4) reflecting on methodological compromises. In practice, this study provides mathematics teacher education programs with a structured framework to foster pre-service teachers' critical methodological reasoning, directly enhancing their capacity to conduct context-sensitive educational research.
Strengthening mathematics learning that emphasizes conceptual understanding places teachers at the center of students' learning success. However, the mechanisms through which teacher competence shapes perceptions of conceptual mathematics teaching remain insufficiently explored, particularly among early elementary school teachers. This study investigated the relationships among mathematical pedagogical knowledge, teacher numeracy skills, and perceptions of mathematical conceptual teaching, as well as the mediating role of teacher numeracy skills. A quantitative survey was conducted involving 885 early elementary school teachers from various regions of Indonesia. Data were collected through an online questionnaire and analyzed using Structural Equation Modeling with a two-step approach. The results showed that mathematical pedagogical knowledge had significant positive effects on teacher numeracy Skills (β = 0.698, p < 0.001) and perceptions of mathematical conceptual teaching (β = 0.172, p < 0.001). Teacher numeracy skills also positively influenced perceptions of mathematical conceptual teaching (β = 0.355, p < 0.001) and partially mediated the relationship between mathematical pedagogical knowledge and perceptions of mathematical conceptual teaching (indirect effect β = 0.248, p < 0.001). This study provides a novel contribution by demonstrating that teacher numeracy skills function not merely as an independent professional competency but as an explanatory mechanism through which mathematical pedagogical knowledge is translated into teachers' perceptions of conceptually oriented mathematics instruction. These findings suggest that teachers' perceptions of conceptually oriented mathematics instruction are shaped not only by pedagogical knowledge but also by their ability to apply mathematics meaningfully in contextual situations. The study highlights the importance of strengthening both pedagogical and numeracy competencies in teacher education and professional development programs.
This study aims to identify the profiles of K–12 mathematics teachers’ innovative behavior in China using a latent profile analysis approach. Previous studies have mostly used variable-centered methods to examine factors associated with teachers’ innovative behavior, such as school support, professional knowledge, motivation, rewards, and anxiety. However, these approaches provide limited understanding of how mathematics anxiety and innovative behavior coexist within different groups of teachers. To address this gap, this study adopts a person-centered approach to classify mathematics teachers into distinct profiles based on their mathematics anxiety and innovative behavior. Survey data were collected from 336 mathematics teachers from primary, junior high, and senior high schools in China. Latent profile analysis was conducted to identify the optimal number of teacher profiles, and further analyses were used to examine whether the identified profiles differed in social influence, facilitating conditions, engagement, rewards, and technological-pedagogical-mathematical knowledge. The results identified four distinct teacher profiles. These profiles showed different combinations of mathematics anxiety and innovative behavior, indicating that teachers with similar professional contexts may still differ substantially in their emotional responses and innovation practices. Further comparison across profiles showed differences in social influence, facilitating conditions, engagement, rewards, and technological-pedagogical-mathematical knowledge, suggesting that mathematics teachers’ innovative behavior is shaped by the combined influence of emotional, professional, motivational, and contextual factors. Notably, the presence of Anxious–Innovative Teachers indicates that high innovative behavior can coexist with high anxiety, highlighting that innovation should not be interpreted only as a sign of confidence or psychological readiness.
Critical thinking is an essential 21st-century competency. Many junior high school students still demonstrate low levels of mathematical critical thinking. This study aims to develop ARGEONET, an integrated augmented reality media resource that supports the MathVerse M6 learning model and is designed to improve students' critical thinking skills in mathematics. The research employed a Research and Development (Plomp's educational design framework). The participants were 30 junior high school students in Banyuwangi Regency, East Java, Indonesia, selected through purposive sampling. Data were collected using expert validation sheets, observation sheets, student response questionnaires, and mathematical critical-thinking tests. The results show that ARGEONET obtained very high validity, with scores of 89% and 93% from validators, and was classified as very practical, with scores of 86.06% from students and 91.43% from teachers. Effectiveness analysis showed significant differences in five critical thinking indicators: focus (sig. 0.004), reason (sig. 0.001), inference (sig. 0.003), situation (sig. 0.050), clarity (sig. 0.481), and overview (sig. 0.042). The effect size from each indicator, namely small (clarity), medium (situation and overview), and large (focus, reason, inference). ARGEONET shows potential as an augmented reality media to enhance students' critical thinking skills.
High levels of mathematics anxiety are known to negatively affect students’ learning and academic performance; however, empirical evidence within the Malaysian context particularly among secondary school students in Sabah remains limited. Drawing on prior literature that often reports higher anxiety among female students and a negative relationship between anxiety and achievement, this quantitative study aimed to assess mathematics anxiety levels, examine gender differences, and investigate its relationship with students’ mathematics achievement. A total of 300 lower secondary students from a secondary school in Kota Kinabalu, Sabah participated in the study, and data were collected using a modified Mathematics Anxiety Scale (MAS) questionnaire, while mathematics achievement was measured based on students’ school-based mathematics examination scores. The findings indicated that the overall level of mathematics anxiety was moderate (μ = 3.04, 2.33 < μ < 3.67), suggesting that although anxiety is not at a critical level, it may still hinder optimal performance and confidence in learning mathematics. An independent samples t-test revealed that female students reported slightly higher mathematics anxiety (M = 3.11, SD = 0.07) compared to male students (M = 2.97, SD = 0.06); however, the difference was not statistically significant (p = 0.135, p > 0.05), indicating that gender is not a determining factor. In contrast, a significant difference in mathematics anxiety was found based on students’ mathematics achievement (F(5,294), p < 0.05), suggesting that anxiety levels vary meaningfully across achievement groups. These findings imply that even moderate levels of mathematics anxiety warrant attention, as they may still affect students’ academic outcomes; therefore, teachers should implement supportive instructional strategies and foster a positive learning environment to reduce anxiety and enhance students’ engagement and performance in mathematics.
Mathematics is a crucial aspect of reasoning and problem-solving. However, mathematics is still considered a difficult subject and tends to be unpopular; therefore, there is a need for examples of enjoyable mathematics learning experiences. This study aims to explore students’ experiences in mathematics learning at a private junior high school in South Tangerang, Indonesia. This research employs a qualitative approach using a case study design, conducted at a junior high school in Indonesia involving 203 students, of whom 66 responded to open-ended questions, and 6 were interviewed in depth. Additional data were obtained from two mathematics teachers and one deputy headteacher for student affairs. Data collection was carried out through semi-structured interviews, classroom observations, and document analysis. The data were analysed thematically using an interactive analysis model supported by triangulation to ensure credibility. The findings of this study indicate that students’ mathematics learning outcomes were excellent across all three student groups. Mathematics became an engaging and enjoyable subject, and positive changes were observed in the students through differentiated learning. Consequently, mathematics teaching at this school can be considered a best practice. Thus, an appropriate approach to mathematics teaching is a prerequisite for stimulating students’ interest in learning mathematics.
Mathematical problem-solving is a core competency in classroom contexts, yet many students experience difficulties when dealing with contextual tasks. This study aims to analyze students’ mathematical problem-solving processes based on learning styles using APOS theory and Newman’s Error Analysis (NEA). A qualitative case study was conducted in an Indonesian senior high school involving 35 eleventh-grade students (N = 35), with three students selected for in-depth analysis. Data were collected through a learning style questionnaire, problem-solving tests, and semi-structured interviews, and analyzed thematically by integrating APOS stages with NEA. The findings reveal varied cognitive pathways across learning styles rather than fixed error patterns. Visual students tended to construct accurate representations but still encountered comprehension difficulties. Auditory students articulated solution strategies clearly, although some experienced challenges in transforming problems into mathematical models. Kinesthetic students benefited from concrete supports such as diagrams, which facilitated progression toward higher APOS stages. All focal students demonstrated development toward the schema stage, indicating diverse but productive learning trajectories. These results highlight the importance of adaptive instructional strategies that align with students’ learning characteristics and support structured progression in mathematical understanding.
The transition from Pedagogical Content Knowledge (PCK) to Technological-Pedagogical-Content Knowledge (TPACK) underscores the need for a more meaningful integration of technology into mathematics teaching. Yet, this paradigm shift has not been accompanied by substantial gains in students' conceptual understanding and mathematical problem-solving skills, which are the central focus of the present study. The purpose of this study is 1) to describe the differences in the increase in students' mathematical ability (conceptual understanding and problem solving) in culture and technology-based learning and conventional learning, and 2) to describe the effects of both types of learning on student knowledge retention. This study is a quasi-experimental study comparing culture- and technology-based learning and conventional learning. The instrument used in this study is a mathematical ability test (pretest and posttest). Data analysis uses a difference-of-means test (N-Gain), t-test, two-way ANOVA, and MANOVA. The results show that the average increase in the culture and technology-based learning group is higher than that in the conventional learning group and is significantly different. There is a significant interaction effect between learning and mathematical ability (p<0.05, effect size λ = 0.996), the main effects of time periods pretest, posttest-1, & posttest-2 were significant (p< 0.05, effect size λ = 0.217), the main effects of mathematical ability on culture and technology based learning and conventional learning were also significant (p< 0.05, effect size λ = 0.215). These findings suggest that integrating technology and culture is an effective approach for providing a positive learning experience that promotes improved mathematical skills and longer-term knowledge retention. This sustained information retention is important for educational practice because it enables students to transfer and apply the concepts they have learned to future learning situations.
The increasing demand for twenty-first-century competencies requires mathematics learning environments that not only support conceptual understanding but also foster students' broader competencies, such as critical thinking, creativity, communication, collaboration, character, and citizenship (6C). However, classroom practices in geometry often emphasize procedural knowledge, limiting students' opportunities to construct mathematical meaning through exploration and contextual reasoning. This study aimed to develop an inquiry–realistic digital module, namely E-Miracle, using Scratch to support the development of students' 6C competencies in learning about quadrilaterals. The study employed an educational design research approach consisting of three iterative phases: preliminary design, teaching experiment, and retrospective analysis. During the preliminary phase, the structure of the digital module and the learning trajectory were designed by integrating inquiry-based learning principles with contextual problems inspired by Realistic Mathematics Education. The prototype was subsequently refined through expert validation and small-scale implementation. The results indicate that the developed module achieved high validity according to expert judgment. Furthermore, the module's practicality was demonstrated through classroom implementation, where both teachers and students used it effectively during learning activities. Students actively engaged in collaborative exploration and contextual problem solving through Scratch-assisted tasks. The findings suggest that the E-Miracle module provides a learning environment that supports the development of students' 6C competencies, particularly in collaborative exploration, the communication of ideas, and creative problem-solving. These results imply that integrating inquiry-based learning, realistic mathematical contexts, and digital technology can provide meaningful learning experiences that foster deeper mathematical understanding.
Learning geometry involves using visual manipulatives, mathematical formulas, and verbal explanations to foster critical thinking, deductive reasoning, and problem-solving skills, thereby ensuring understanding of mathematical concepts. Despite this relevance, secondary school students' limited knowledge in plane geometry, especially in concepts such as polygons and circles, creates an epistemological barrier that hinders their engagement with more advanced geometric ideas. Thus, this study investigates the effectiveness of the Van Hiele Group Guided-Discovery Instructional Approach (VHGGDIA) on student engagement in learning plane geometry, employing a mixed-methods research approach with a Convergent Parallel design. It uses a quasi-experimental, non-equivalent control-group pre-posttest design. Two comparable secondary schools from a total of 11 schools were purposively selected and randomly assigned to the experimental and control groups. Honesty, independence, perseverance, attention, interaction, and collaboration — each of which demonstrates dedication, kindness, and teamwork — impact student engagement. The analysis incorporated data from interviews, observations, and a Likert scale questionnaire. Quantitative data analysis used descriptive statistics, independent and paired-samples tests, and ANCOVA, whereas qualitative data were assessed thematically. The study found that students using the VHGGDIA showed greater engagement in plane geometry than those taught via traditional methods. Further, the study concludes that VHGGDIA significantly enhances secondary school students' engagement in this subject.
Mathematics is often perceived as culturally neutral and detached from its social and historical foundations, leading to the neglect of cultural heritage rich in mathematical ideas. This study investigates how the concept of sets is reflected in the spatial layout of the Prambanan Temple Complex through an ethnomathematics perspective. By uncovering these embedded structures, the research demonstrates the potential of cultural heritage as a meaningful context for mathematics education, promoting both conceptual understanding and appreciation of cultural identity. Adopting an ethnomathematics perspective, this study employs ethnographic methods to analyze the spatial layout of the 9th-century Prambanan Temple. The investigation focuses on identifying mathematical principles within the temple’s spatial classification system, particularly in relation to grouping, membership, and separation across the hierarchical zones of Bhurloka, Bhuvarloka, and Swarloka. The analysis reveals that ancient Javanese society implicitly can be interpreted as set-theoretical structures in the spatial layout of Prambanan Temple. The hierarchical spatial divisions demonstrate systematic applications of grouping, inclusion, and exclusion principles, reflecting mathematical thought embedded in cultural practices. These insights suggest that cultural artifacts such as temple architecture can be modeled through mathematical frameworks and provide authentic contexts for mathematics learning. This study contributes to the field of multicultural mathematics education by illustrating how cultural heritage can serve as a valuable resource in teaching through the Ethno-Realistic Mathematics Education approach. Situating mathematical concepts in culturally relevant contexts deepens conceptual understanding and strengthens cultural identity, offering practical implications for educators and curriculum developers seeking to embed multicultural perspectives in mathematics education.
The quality of Calculus learning in higher education remains a challenge, as instructional practices tend to be predominantly procedure-oriented, thereby limiting opportunities for meaningful learning interactions and for concept visualization that supports conceptual understanding. This study aims to examine content validity and learning effectiveness in Calculus instruction through TPACK-integrated e-learning. The research employed a Research and Development (R&D) approach using the ADDIE model (Analyze, Design, Develop, Implement, and Evaluate. The developed instrument consisted of 20 items encompassing interactivity aspects across three dimensions and accessibility aspects based on four principles. Content validation was conducted by three experts using Aiken’s V index. The results indicated that all instrument items were content-valid and were categorized as relevant to highly relevant. Furthermore, the effectiveness test results showed that the proportion of students achieving classical mastery exceeded the established threshold, and students’ learning improvement fell into the high category. Additionally, the effect size analysis shows that implementing TPACK-integrated e-learning has a very strong impact on students’ understanding. These findings confirm that TPACK-integrated e-learning is not only valid and effective but also has a substantial impact on enhancing the quality of Calculus learning in higher education.
Malaysian students have experienced considerable challenges in mastering calculus in recent years, with over 40% failing to attain a credit grade in mathematics. While technology like Desmos can enhance understanding, educators need insights into how students develop conceptual knowledge in technology-based environments. Therefore, this study investigated the pre-university science students’ understanding of differential calculus using the Action-Process-Object-Schema (APOS) theory through a Desmos instructional lesson. A design-based research (DBR) strategy was employed to facilitate the real-time design, implementation, and refinement of the intervention. Classroom observations and document analysis revealed that most students achieved the Process phase of understanding based on APOS. The students’ interviews also indicated that they found the Desmos-based lesson engaging and valued the opportunity for group discussion. Nevertheless, some students still encountered difficulties entering equations in Desmos. Further refinements of the instructional design were then conducted through a comparative retrospective analysis between the hypothetical learning trajectory and actual learning outcomes. Overall, this study contributes to the existing literature by providing detailed, stage-specific analyses of the design, implementation through a teaching experiment, and retrospective analysis and refinement processes within the DBR framework, which are often underreported in previous research.
Numeracy literacy in this study refers to students' capacity to interpret quantitative information in context, mathematize situations, use representations, justify their reasoning, select strategies, and employ symbols and tools to support defensible decisions. To address Indonesia's low performance on PISA tasks, we developed an operational profile of 9th-grade students using OECD indicators adapted to the Indonesian junior high school geometry curriculum through Candi Jiwa-based items. Using a descriptive qualitative design with quantitative support, we administered seven open-ended geometry problems to students in Karawang, Purwakarta, and Subang. Test scores were summarized to classify achievement levels. At the same time, classroom observations and semi-structured interviews with students and teachers were used to diagnose reasoning and strategy tendencies and recurring error patterns. Results show that 85.71% of students scored below 60 (low), 14.28% scored 60–80 (medium), and none exceeded 80 (high). Students' strongest area was written communication, mainly descriptive or procedural (restating givens and listing steps). Weaknesses clustered in mathematization, reasoning and argumentation, representation, strategy selection, and use of formal symbols and mathematical tools, indicating procedural competence with limited transfer to contextual modeling. Candi Jiwa serves as a place-based anchor that can lower entry barriers and guide culturally grounded, cognitively manageable task design aligned with OECD processes.
Prior studies on students' mathematical reasoning abilities have shown that multimodal representations and local culture can bridge the gap between abstract concepts and students' understanding. To address this gap, this study developed and evaluated the interactive website Hy MAI, grounded in the Ethno-STEM approach, as a pedagogical innovation and theoretical contribution to design-based mathematics education. This study used Research and Development methods within an educational design research framework. Participants included expert validators (n = 4), one mathematics teacher, and eleventh-grade students (n = 74; 37 experimental, 37 control). Data were collected through expert validation sheets, teacher and student questionnaires, and pre- and post-tests. Expert validation yielded an average score of 3.40 (Good), confirming the accuracy of the content, interactivity, and cultural integration. Teachers rated practicality at 4.00 (Very Good), while students rated it at 3.03 (Practical). Effectiveness analysis showed significant improvements in reasoning skills. A paired-sample t-test yielded t(36) = –19.60, p < 0.001, and an independent-sample t-test revealed a significant post-test difference, t(72) = 8.77, p < 0.001. The N-Gain analysis showed 62.35% for the experimental group versus 16.08% for the control group. Overall, Hy MAI proved valid, practical, and effective, while also making theoretical contributions by demonstrating the systematic integration of technology, STEM principles, and local culture in mathematics education.
