This study examined the characteristics of mathematical modeling activities that emerged while pre-service elementary teachers solved a mathematical modeling task. Thirty-two first-year pre-service teachers were organized into six groups and guided through the six stages of the mathematical modeling cycle, comprising understanding, simplifying and structuring, mathematizing, working mathematically, interpreting, and validating. Each group's modeling process was audio-recorded, transcribed, and systematically analyzed using a coding framework derived from the cognitive perspective of the mathematical modeling cycle, with Modeling Transition Diagrams employed to visualize stage specific activity patterns over time. The results indicated that most groups concentrated their activity in the simplifying and structuring stage, while some groups became excessively immersed in real world contextual reasoning without progressing toward formal mathematization. These findings reveal specific bottlenecks in student driven mathematical modeling and suggest that timely pedagogical scaffolding is likely necessary to support smoother transitions across modeling stages.

This study examines how preservice mathematics teachers integrate artificial intelligence into lesson design and how the cognitive levels of the student prompts they propose vary according to their modes of AI utilization. Mock lesson materials created by seven groups enrolled in a mathematics education course were analyzed using open coding, which yielded four distinct types of AI-supported instructional design: using AI as a tool for answer verification (Type A), for delivering content (Type B), for guiding modeling processes (Type C), and for facilitating teacher-student interaction (Type D). The prompts that preservice teachers envisioned students would use were further classified according to the six cognitive levels of the Revised Bloom's Taxonomy. The analysis revealed a systematic alignment between instructional types and prompt levels: Type A was associated with the Remember level; Type B extended to Understand; Type C included Apply; and Type D generated higher-order prompts corresponding to Analyze, Evaluate, and Create. These findings indicate that the ways in which teachers position AI within the instructional process shape the cognitive demand placed on students. In particular, higher-order thinking tended to emerge when AI functioned as a partner within teacher-student-AI interactions rather than as a mere answer-providing tool. This study highlights the importance of preparing teachers to design AI-supported interactions that promote deeper mathematical reasoning and provides implications for preservice mathematics teacher education in the era of generative AI.

This study examines pre-service secondary mathematics teachers' Curriculum Noticing regarding Statistical Problem-Solving and Matrix in the 2022 revised curriculum. Findings reveal a general preference for practical guidelines over theoretical frameworks. Specifically, teachers showed a discrepancy between understanding goals and practical implementation in statistics due to insufficient Pedagogical Content Knowledge (PCK), whereas they demonstrated active interpretation and holistic noticing in matrices by anticipating learning difficulties. This study suggests that teacher education must foster an integrated understanding of curriculum hierarchies and strengthen PCK to bridge the gap between curriculum intent and classroom practice.

Analogical problem solving is a core competency in mathematics learning, particularly in combinatorics, where problems often vary in surface representations while sharing isomorphic structures. This study examines mathematically gifted science high school students and, by analyzing the interplay between a teacher's intentional manipulation of surface similarity in combinatorics task design and the problem-solving processes of 98 students, derives implications for teacher professionality. Findings show that, despite sophisticated task design aimed at probing structural understanding, students relied on familiar yet inefficient strategies for low surface-similarity problems, producing diverse transfer errors. The results indicate that teacher professionality must extend beyond advanced task design to include deep insight into student cognition and diagnostic competence in analyzing errors.

This study investigates the facilitating factors and challenges shaping mathematics teachers' design and enactment of AI-integrated instruction aimed at fostering students' mathematical competency. Guided by the AI-TPACK framework, we conducted semi-structured interviews with five in-service mathematics teachers who developed and implemented AI-integrated lessons, and analyzed the data through thematic analysis. Teachers' adoption was facilitated by anticipated instructional benefits-personalized learning and feedback, deeper learning and competency development, enhanced student motivation and agency, and more efficient lesson management-and supported by contextual factors such as teacher agency and policy- or infrastructure-related reforms. However, teachers reported multifaceted barriers across AI-TPACK components: in AI-TK, difficulties in prompt design and tool development alongside concerns about reliability and usability; in AI-TPK, challenges in mediating students' uptake of AI feedback, maintaining instructional flow, and meeting increased monitoring demands; and in AI-TCK, struggles to select competency-aligned tools and to verify the mathematical accuracy and representational appropriateness of AI-generated outputs. Ethical concerns regarding transparency, fairness, accountability, and inclusiveness-compounded by the opacity of AI systems-further constrained instructional control. We conclude that although teachers show strong willingness to adopt AI for competency-based education, they face barriers extending well beyond technical proficiency. These findings underscore the need for professional development that builds holistic AI-TPACK expertise and ethical competencies, as well as systematic institutional support for sustainable AI integration in mathematics education.

Extended school-based practica are expected to help preservice teachers connect university coursework with classroom practice, yet little is known about how their mathematics-specific noticing develops week by week. This longitudinal multiple-case study traced three preservice mathematics teachers (PST1-PST3) during a semester-long practicum. The data comprised 108 dated practicum journal episodes written by the participants from Weeks 2-15, supplemented by mentor/supervisor feedback, lesson artifacts, and research-lesson transcripts for contextual and process-tracing purposes. Using the FOCUS framework, each episode was coded for the degree of alignment among mathematical key ideas, anticipated or observed student difficulties, and pedagogical responses (A0-A2), with the final code based on the preservice teacher's own written reflection. Findings show that robust alignment was not a simple linear outcome of time in school. Rather, A2 episodes appeared unevenly when preservice teachers explicitly connected mathematical key ideas, student difficulties, and pedagogical responses in their own journal texts. Early peaks occurred during the assisting and coteaching stage for PST2 and PST3, while selected responsibility and closure episodes also showed robust alignment. The study shows how practicum-semester designs can cultivate productive noticing when feedback and teaching experiences are taken up in preservice teachers' reflections as an explicit connection among mathematics, student thinking, and instructional response.

This study analyzed Korean pre-service teachers' MKT in the specific context of the problem that involves the claim that the cross-section of a cylinder cut by an inclined plane is an ellipse. The problem concerning the cross-section of a cylinder is common, as all six geometry textbooks in the 2022 revised curriculum include this problem. However, none of these textbooks provide a proper explanation of why the cross-section of a cylinder is an ellipse. Accordingly, it is reasonable to infer that pre-service teachers may lack understanding of this concept. Based on this inference, the tasks related to this specific instructional situation for pre-service teachers were developed. These tasks were administered to 21 students at J University, and the characteristics of pre-service teachers' MKT were analyzed. Consequently, an overall absence of PCK and SMK, which are sub-elements of MKT, was found. The pre-service teachers' MKT regarding the cross-section of a cylinder was also described. The results of this study not only reveal gaps in the knowledge of pre-service teachers within a practical context but also suggest that even simple problem situations in textbooks can serve as a foundation for improving teachers' professionalism.

Recent discussions in statistics education emphasize the importance of supporting reasoning about variability and uncertainty when data are interpreted and used for judgment. In such contexts, teachers' statistical knowledge involves not only the correct use of procedures but also the interpretation of variation in relation to instructional goals. These demands are particularly salient in discussions on data science education, where statistical ideas are mobilized to support data interpretation and decision-making under uncertainty. Building on previous research on statistical knowledge for teaching (SKT) and key developmental understanding (KDU), this study proposes a theoretical restructuring of SKT based on variability. Variability is proposed as an anchoring KDU within a broader network of statistical ideas, one that organizes teachers' interpretations of statistical content, student reasoning, and pedagogical judgment. Rather than treating statistical concepts as parallel components of teacher knowledge, the proposed perspective highlights the asymmetrical role of variability in structuring developmental relationships among SKT components. By articulating a variability-centered developmental model of SKT, this study extends existing accounts of teachers' statistical knowledge and clarifies their relevance in instructional contexts, where data interpretation and uncertainty are central. The proposed framework provides a conceptual basis for examining how teachers' statistical knowledge develops and transforms in relation to the goals of K-12 data science education.

As artificial intelligence (AI) becomes increasingly embedded in educational practice, strengthening teachers' professional development (PD) for the pedagogically sound integration of AI in mathematics instruction has become a pressing priority. This study develops a framework for mathematics teacher PD aimed at advancing mathematical competency education in AI-integrated environments. The framework was developed through a two-phase process: a synthesis of relevant literature to derive an initial set of components and hypothesized developmental pathways, followed by two rounds of focus group interviews with in-service mathematics teachers and mathematics teacher educators to revise and further elaborate the preliminary framework. The resulting framework organizes PD around two key themes, focusing on AI tools for mathematics education and on AI-based mathematical competency education. It delineates four interrelated activity strands-Understanding, Investigation, Curriculum Reconstruction, and Application-and foregrounds foundational understandings encompassing the rationale and aims of mathematical competency education, the 2022 revised mathematics curriculum, and AI ethics as well as pertinent policies and legislation. The application strand is specified in terms of lesson design, classroom implementation, and reflective practice, highlighting teachers' capacities to monitor students' learning processes, interpret learning data, and provide personalized instructional support. The implications of the framework for PD design are discussed, and directions for future empirical validation are suggested.

This study examined how an experienced teacher's mathematical knowledge for teaching (MKT) is utilized in the process of task modification that considers the dynamic interplay between divergent and convergent thinking, which are central to mathematical creativity. A qualitative case study was conducted on one experienced middle school teacher's task design and modification practices over a one-year period in Grade 9 mathematics. The analysis integrated a framework of mathematical creativity-continuity, complementarity, and interaction-with the six subdomains of MKT, using MKT as an interpretive lens rather than a measurement tool. The findings showed that although all subdomains of MKT were involved in task modification, their patterns of use varied across phases. Complementarity was primarily associated with knowledge of content and students and knowledge of content and teaching, continuity involved multiple domains including specialized content knowledge, and interaction was mainly supported by knowledge of content and students. These results suggest that experienced teachers' expertise lies in their ability to selectively reorganize and deploy MKT in response to instructional contexts, offering implications for teacher education focused on fostering mathematical creativity.

The capacity to design tasks and lessons for technology-integrated teaching is becoming increasingly important in preservice teacher education. This study investigated the development of secondary preservice teachers' capacity to design technology-integrated lessons by examining the knowledge and skills they proposed in peer feedback for improving lesson designs and micro-teaching, and how the recipients internalized these through self-reflection. For the analysis of written peer feedback and self-reflection reports, we used networking of the Anthropological Theory of the Didactic (ATD) and the TPACK framework, along with thematic analysis to identify the main themes emphasized in lesson improvement. From the ATD perspective, the analysis of peer feedback showed that the preservice teachers were primarily concerned with improving the teacher-led didactical praxeology, while from the TPACK perspective, it was mainly related to PCK, PK, and TPCK components. Analysis of self-reflection reports revealed the levels of internalization of the praxeologies mentioned in the feedback from the ATD perspective, and identified the subcomponents of TPACK that developed through internalization from the TPACK perspective. Finally, we discussed how networking of the two perspectives provided complementary explanations of the preservice teachers' design capacity.

This study investigates how pre-service mathematics teachers interpret the differential symbol dx and how their interpretations manifest as explanation modes in instructional situations. The symbol dx exemplifies the epistemological discontinuity between school mathematics, where it appears as mere notation, and university mathematics, where it is defined as an infinitesimal or a differential form. Data were collected from 21 first- and second-year undergraduate students in mathematics education, and 16 valid responses were analyzed using data from written surveys, supplemented by follow-up interviews. Analysis revealed five interpretation types ranging from symbol-based view to differential form, and five explanation modes ranging from conventional to reconstructed. Results indicate that while some pre-service teachers possessed Advanced Mathematical Knowledge (AMK) regarding dx, this did not automatically transform into Specialized Content Knowledge (SCK) or Horizon Content Knowledge (HCK). These findings suggest that pre-service teacher education should provide opportunities to transform AMK into pedagogically meaningful forms of knowledge for teaching.

As AI technologies, including generative AI, become widespread in education, developing mathematics teachers' expertise for effective AI integration is essential. This study examines South Korean mathematics teachers' educational needs for AI integration using the AI-TPACK framework. We administered an online survey to 426 elementary, middle, and high school teachers and identified priorities using Borich's needs scores and the Locus for Focus model, with subgroup analyses by teacher background variables. AI Ethics Knowledge (AI-EK) emerged as the highest priority, followed by AI-TPK and AI-TPACK, while AI-TK and AI-TCK were relatively lower priorities, indicating a shift from basic tool operation to ethical judgment and pedagogical integration. Needs differed by teacher characteristics: high school teachers reported higher needs, and teachers without AI-related professional development or AI instructional experience showed greater overall needs. These findings suggest that teacher education should prioritize ethical and pedagogical competencies and provide differentiated support by school level and experience.

The rapid adoption of large language models (LLMs) has expanded opportunities for natural language-based programming, lowering barriers to developing digital tools through AI-assisted coding. This study examined how a short-term, project-based course in which preservice secondary mathematics teachers developed Streamlit web applications using GitHub Codespaces and Copilot and designed mathematics lessons incorporating their tools influenced their TPACK-P (Technological Pedagogical Content Knowledge-Programming). Participants were 24 second-year pre-service mathematics teachers at a university in Daejeon, South Korea. Using a pre-post design, changes in eight subdomains (PK, CK, TK, PCK, TCK, TPK, TPACK, and TPACK-P) were analyzed through reliability checks, paired-samples t-tests, K-means clustering of standardized gain scores, conceptual network analysis, and coding of open-ended responses. Findings showed significant increases in all subdomains except PK, with the largest gains in CK, TCK, and TPACK-P. Cluster analysis identified balanced growth, high-growth integrative, and partial-growth profiles, indicating heterogeneous development among participants. Network analysis revealed an integrated growth structure centered on TPACK, with strong links among TK, TCK, and TPACK, and a direct connection between TPACK-P and TCK. Overall, the results suggest that natural language-based programming can help pre-service teachers reconceptualize web application development as a productive component of mathematics lesson design while supporting differentiated growth pathways.