The pedagogical landscape of elementary mathematics education is significantly influenced by the type and quality of problems presented in the classroom. Traditional methodologies, which often emphasize rote learning and procedural mastery, fall short of fostering critical thinking and inquiry, essential components for cultivating mathematical proficiency (Henningsen & Stein, 1997). Recognizing this, the literature advocates a paradigm shift toward integrating problem solving and reasoning as fundamental aspects of mathematics education, thereby enriching students' learning experiences and enhancing their conceptual understanding (e.g., Miranda & Mamede, 2022; National Council of Teachers of Mathematics (NCTM), 2000; Van de Walle et al., 2010).

Problem solving, as described by the NCTM (2000), should not be an isolated segment of the curriculum but an integral part of mathematics learning, integrated into the core of education. NCTM (2000) further notes that problem solving highlights mathematical engagement. In addition, the cognitive and metacognitive dimensions of problem solving underscore the importance of engaging with problems in ways that go beyond mere computation. Jonassen (2000) articulates that the significance of a problem derives from its potential to contribute to “societal, cultural, or intellectual” domains, which requires a solver's engagement in mental representation and manipulation of the problem space (p. 65). This perspective is complemented by Lester and Kehle's definition, which emphasizes problem solving as an active engagement process “using prior knowledge and experience” (cf. Santos-Trigo, 2007, p. 525).

Problem-posing, similar to problem-solving, is an integral part of this pedagogical development. It is recognized as a sophisticated mathematical activity that promotes creativity, flexibility, and deeper understanding (Silver, 1994). It is defined as the ability to formulate, reformulate, and explore problems based on existing mathematical situations or concepts.  It could be described as “one of the highest forms of mathematical knowing and a sure path to gain status in the world of mathematics” (Crespo, 2015, p. 494). NCTM (2000) also points out that students need “to create engaging problems by drawing inspiration from various scenarios encompassing mathematical and non-mathematical contexts” (p. 258). In light of these considerations, this study explores the interplay between problem solving and problem posing in the context of mathematics education for preservice elementary teachers. Specifically, it seeks to examine the nature and quality of problems posed by preservice teachers, the challenges they face in the problem-posing process, and how their problem-solving skills influence their ability to generate meaningful mathematical problems. Through a comprehensive analysis of pre-service teachers' engagement with problem solving and problem posing, this research aims to contribute to how to support the pre-service teachers' skills and the interplay between them to create better learning environments for their students by improving their teaching strategies.

This research will employ a qualitative research design, focusing on understanding and interpreting the nature of pre-service primary school teachers' problem-posing abilities and exploring the challenges they face during the process. The study will also investigate the relationship between prospective teachers' problem-solving and problem-posing abilities in specific mathematics content areas, aiming to examine the nature and quality of problems they pose, the difficulties encountered in problem-posing, and how their problem-solving skills influence their problem-posing capabilities.Qualitative data will be gathered from a convenience sample of 28 primary school pre-service teachers enrolled in a mathematics teaching course during the Spring 2024 semester. This course, a required part of the undergraduate primary school teacher education program at a public university in Turkey, includes weekly three-hour lectures over twelve weeks, focusing on problem-solving processes and integrated problem-posing activities within topics such as early algebra, numbers, and operations. Each weekly session will feature problem-solving and problem-posing tasks based on relevant literature. Data collection will use a two-part instrument: the first part will be a paper-pencil test for each week's content, starting with a problem-solving task followed by a problem-posing task. Students will solve the given problem, then create and solve their own posed problems, identifying any issues in their problem formulation. The second part will involve in-depth think-aloud protocols with a subset of participants to understand their cognitive processes during problem-solving and posing, including their strategies and awareness of problem-posing challenges. Data from the paper-pencil tests and think-aloud protocols will be analyzed qualitatively. The paper-pencil tasks will undergo content analysis using thematic coding procedures based on established frameworks (e.g., Problem-Solving Task Rubric and Problem Posing Task Rubric by Rosli et al., 2015). The think-aloud protocols will be transcribed and analyzed to gain insights into participants’ thought processes during problem-posing and solving, and a comparative analysis will be conducted to explore the nature of their problem-solving and posing abilities and their effectiveness in formulating and solving problems.

Building on the exploratory studies by Grundmeier (2015), Hospesova & Ticha (2015), and insights from Rosli et al. (2015), this research aims to deepen our understanding of the problem-solving and posing skills of pre-service primary school teachers. Grundmeier (2015) observed that practice enhances problem-posing efficiency and creativity among prospective elementary and middle school teachers. Hospesova and Ticha (2015) identified significant knowledge gaps and challenges in problem-posing, despite teachers acknowledging its importance in mathematics education. Complementing these findings, Rosli et al. (2015) revealed middle school preservice teachers’ proficiency in solving more straightforward arithmetic tasks. However, they had difficulties in abstract generalization and algebraic interpretation. Notably, these teachers could formulate fundamental yet meaningful problems. The results suggested the integral role of problem-solving in facilitating effective problem-posing. Aligned with these studies, the current research is expected to uncover similar findings within pre-service primary school teachers' problem-solving and posing competencies. This research will explore the nature and quality of problems posed, the challenges encountered in the problem-posing process, and the interrelation between problem-solving prowess and problem-posing skills. Employing comprehensive data collection and analysis methods inspired by Rosli et al. (2015) and others, this research aims to offer new insights and validate existing findings.

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