Problem-posing is considered as an important component of mathematics education and a powerful tool for fostering students’ mathematical thinking, creativity, and conceptual understanding. To show its importance in 1938, Einstein and Infeld wrote: “The formulation of a problem is often more essential than its solution, which maybe merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old questions from a new angle, requires creative imagination and marks real advance in science” (p. 92). Despite traditional evaluation tests and approaches, modern literature emphasizes that mathematical ability should be assessed not only in terms of results but also in terms of process (Sriraman, 2005). In this context, problem-posing comes to the forefront as a fundamental tool that makes students’ mental processes visible. In literature, different perspectives have been put forward in defining this concept. Silver (1994) defined mathematical problem-posing as both the generation of new problems and the re-formulation of given problems. Also, Stoyanova and Ellerton (1996) examined problem-posing under the light of instructional situations, and they defined problem-posing as “the process by which, on the basis of mathematical experience, students construct personal interpretations on concrete situations and formulate them as meaningful mathematical problems” (p. 518), and classified as free situation, semi structured situation and structured situations. When we look at the modern synthesis, research indicates that children lacking prior experience in problem posing can nonetheless formulate realistic, inventive, multi-step mathematical questions derived from diverse scenarios (Cai et al., 2015). This renders problem-posing an activity with a high-ceiling and low-floor affording all children the opportunity to make sense of mathematics (Cai & Hwang, 2021). In addition, English (1997) asserted that problem posing provides the opportunity for teachers to gain insight into students’ understanding of mathematical concepts and processes.

Geometry is a field of mathematics utilizing abstract and visual thinking. It holds particular importance in problem-posing exercises (Shriki & Lavy, 2012; Van Harpen & Sriraman, 2013). Research has shown that students often experience conceptual difficulties when posing problems in the context of geometry, but problem-posing activities improve students’ geometric reasoning, and with appropriate guidance, they can create meaningful and creative problems (Kontorovich & Koichu, 2013). In this respect, geometric problem-posing tasks are used as an effective tool distinguishing students’ level of mathematical thinking. Studies conducted with diverse student profiles, including gifted students, have examined how mathematical ability differs; some significant research has even directly compared gifted and non-gifted students. The literature indicates that gifted students, compared to non-gifted students, pose more original, complex, high-cognition problems that maintain logical consistency by making structural transformation rather than superficial changes in geometrical tasks, demonstrating greater success in transforming given situations and creating new mathematical structures (Aydoğdu İskenderoğlu & Yurtbakan, 2023; Espinoza et al., 2022). Compared to problem solving, problem posing represented a neglected area of research, but in recent years the mathematics education community has been increasingly focused on mathematical problem-posing for more than thirty years (Getzels, 1979; Kontorovich et al., 2012). Despite the increasing importance of problem-posing studies, there is a lack of studies in the field comparing gifted children with their peers who are non-identified as gifted, such as semi-structured problem-posing (Espinoza et al., 2022. This study attempts to close the gap by comparing gifted and non-identified students’ problem-posing process, which is under-studied relative to problem solving, using a geometric figure as the mathematical domain in order to contribute internationally. In this respect, the study has the following research question:

How gifted students and their non-identified peers differ in the mathematical problem-posing based on given geometrical tasks?

This case study, as a qualitative approach, aims to investigate how gifted students and their non-identified peers differentiate in problem posing process based on the given geometrical task. In this regard, the participants consisted of four gifted students aged 14-15, two girls and two boys, two in the 8th grade and two in the 9th grade, and four non-identified students, three boys and one girl, in 9th grade studying in Türkiye. One of the gifted 9th-grade students stated that she has been diagnosed since the 3rd grade and love mathematics because of the problem-solving techniques she developed herself, while another stated that he has been diagnosed since the 2nd grade, enjoy listening to and learning mathematics in class, and do not accept anything without questioning it. Two other gifted students in 8th grade stated that they had been diagnosed since 2nd grade and had an interest in mathematics. Mathematics grades of non-identified students were served as a criterion, with two students achieving scores of 70 and two others attaining scores of 80 on their initial math examinations during the fall semester of 2025/2026, signifying moderate academic success, their general attitudes towards mathematics were determined as positive. In order to collect the data, an open-ended geometrical task (Xie & Masingila, 2017, p.116) was used with the item to reveal problem posing skills of the participants. As the students involved in the study were minors, parental consent forms were disseminated to their families prior to data collection, and these forms were subsequently retrieved with the requisite signatures of consent. On the same day, a 50-minute after-school session was used to implement the data collection task for gifted students. For non-identified students, the task was given over a 50-minute period in a classroom setting but outside of class time. The two participant group implementations are separated by one week. Data was analyzed based on “Scoring Rubric for Problem Pose Skills” (Cankoy & Özder, 2016) consisting of the following dimensions: 1) solvability; 2) reasonability; 3) mathematical structure (result-unknown vs start-unknown); 4) context (routine vs non-routine), and 5) language. Also, according to this rubric, the maximum score that can be obtained for each problem is 6. Throughout the data analysis process, both authors deliberated on their perspectives regarding the scoring system, and discussions persisted until they reached a consensus on the final scores and methodologies.

Upon analyzing the problem-posing performance of gifted students, a 9th grade gifted student posed five problems and achieved a total score of 16 points, while another 9th grade student posed three problems and attained 10 points, with the maximum score for each problem being 6 according to the rubric. 8th grade gifted students posed four and five problems, achieving scores of 11 and 13 points, respectively. The findings suggest that the quantity and quality of problem-posing among gifted students do not consistently and concurrently advance. When analyzing the problem-posing performance of non-identified students, it was noted that although the quantity of problems they posed was largely comparable, there were marked discrepancies in their overall rubric scores. A non-identified student posed four problems and attained a total of 9 points, whereas the other students posed four problems and scored 6 and 5 points, respectively. Yet, another non-identified student, performed exceptionally well in the group and received a total of 20 points for posing five problems. These results show that among non-identified students, the number of problems posed is similar but the problems' quality can vary greatly. Besides, the problems posed by non-identified students contain misconceptions and inappropriate use of mathematical language (e.g. "round" instead of "circle" and "diameter of a square" instead of "side length of a square"). Gifted students perform more consistently in terms of quality during the problem-posing process. Moreover, none of the 8 students were able to pose a non-routine problem and to use start-unknown approach as a mathematical structure. This study's findings indicate that problem-posing performance cannot be evaluated solely by the quantity of problems generated, and that qualitative assessment using rubrics uncovers distinctions; thus, future research could investigate structured practices and methodologies to enhance the qualitative aspect of the problem-posing process of gifted and non-identified students.

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