The selections and differences in mathematical problem-posing strategies of junior high school students

Mathematical problem-posing (MPP), the generation of new problems or reformulation of existing ones, is crucial for mathematical advancement and learning. While its importance is increasingly recognized in curricula, research on the specific strategies students employ and how these strategies develop remains limited. Existing literature lacks a systematic classification of MPP strategies and large-scale empirical studies examining strategy selection and development across grade levels. This study addresses these gaps by constructing a theoretical framework for categorizing MPP strategies and conducting a survey of 1653 Chinese junior high school students. The primary research questions are: 1. Which strategies do junior high school students prefer when posing problems in specific situations? 2. How do these strategies evolve as students progress through junior high school?

The study begins by defining problem-posing as the generation of new problems based on given situations or the reformulation of existing problems (Silver, 1994). It then reviews existing literature on MPP strategies and processes, noting a lack of systematic classification and large-scale empirical studies, especially concerning strategy selection and development across grade levels. Several researchers have proposed various strategies and processes, including the "what-if-not" strategy (Brown and Walter, 2005), five stages of problem-posing (Gonzales, 1998), and four thinking processes (Christou et al., 2005). However, these studies often lack a comprehensive framework or large-scale empirical investigation of junior high school students' strategy use. The study emphasizes the need for a framework that analyzes strategies from a developmental perspective and considers the selection and evolution of strategies across different grade levels.

An empirical study was conducted involving 1653 junior high school students (grades 7-9) from three schools in different Chinese cities. A paper-and-pencil test was administered, presenting three problem-posing tasks with diverse situations (real-life, medical, mathematical). Students were given 30 minutes to pose as many different mathematical problems as possible for each situation. The tasks were designed to be open-ended, allowing students to pose problems based on their understanding and mathematical knowledge, without restricting the content to previously learned concepts. Data encoding involved several steps: (1) Identifying problem clusters (PCs) as either initial problems or related problems; (2) determining the strategies used in each PC based on a pre-defined framework (accepting/changing given information, symmetry, systematic variation, chaining); (3) scoring strategies based on the number of times each strategy was used. Inter-rater reliability was assessed using a random sample of 165 students, resulting in a 90% agreement rate. Statistical analyses included one-way ANOVA, multiple comparisons, and path analysis using AMOS software to examine strategy selection and development across grades and the relationships between different strategies.

The study's framework categorized problem-posing strategies into two categories: Category A (posing problems directly from a given situation) and Category B (posing further problems based on existing ones). Within Category A, students could either accept the given information (A1) or change it (A2), with A2 further divided into changing initial information (A21) or implicit assumptions (A22). Category B included symmetry (B1), systematic variation (B2), and chaining (B3). Results indicated that: 1. Students overwhelmingly preferred accepting the given information (A1) over changing it (A2). When changing information, adding information was the most frequent strategy. 2. Among strategies in Category B, chaining (B3) was used most frequently, followed by systematic variation (B2) and then symmetry (B1). 3. Significant grade-level differences were found in strategy selection. Ninth graders tended to use A1 and B3 more frequently, while seventh graders showed greater diversity in strategy use. Eighth graders used A21 more frequently. 4. Path analysis revealed two main paths for strategy evolution: A1 → B3 (direct transition from accepting information to chaining) and A2 → B1/B3 (changing information followed by symmetry or chaining).

The findings demonstrate that junior high school students primarily rely on accepting given information when posing problems, suggesting a preference for direct problem formulation. The significant use of the chaining strategy indicates an ability to generate related problems by building upon previous ones. The observed grade-level differences highlight the developmental aspect of strategy selection and use in MPP, with students at different grade levels exhibiting varying preferences. The path analysis provides insights into the dynamic process of strategy selection, revealing the common pathways students follow when generating mathematical problems. These findings are relevant to mathematics education research and practice, highlighting the importance of explicit instruction on various problem-posing strategies and fostering a deeper understanding of the cognitive processes involved in problem-posing.

This study offers a valuable contribution to the field of mathematical problem-posing by providing a comprehensive framework for analyzing students' strategy selection and a large-scale empirical investigation of these strategies across grade levels. The findings reveal a clear preference for accepting given information and using chaining strategies, while also highlighting the developmental aspect of strategy selection. Future research could explore the impact of different instructional approaches on students' strategy use, investigate cross-cultural differences in MPP strategies, and investigate the relationship between MPP strategies and problem-solving performance. Further work could also refine the framework by incorporating other strategies and exploring the cognitive processes underlying strategy selection more deeply.

The study's limitations include the focus on a sample from China only, which might limit the generalizability of the findings to other cultural contexts. The reliance on written responses might not fully capture students' thought processes during problem-posing, and the framework for categorizing MPP strategies is based solely on Silver's definition. Future research should address these limitations by including samples from different countries and incorporating measures to assess students' cognitive processes during problem-posing, and also considering other definitions and frameworks for categorizing MPP strategies.