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

Problems are central to mathematics, and problem-posing (the generation or reformulation of problems based on a given situation) plays a critical role in mathematical thinking and learning. While prior curricula and research have emphasized problem-posing, the literature lacks systematic classifications of MPP strategies and large-scale empirical evidence on which strategies students prefer and how these preferences develop by grade. Motivated by these gaps, the study constructs an analytical framework distinguishing strategies used to pose problems directly from situations (Category A) and strategies to pose further problems based on existing problems (Category B), and conducts a large-scale survey of Chinese junior high school students to examine: (1) Which strategies junior high school students prefer to use in problem-posing tasks with specific situations; and (2) How these strategies develop as grade level increases.

The study is grounded in Silver’s (1994) definition of MPP and prior work on strategies and processes such as the what-if-not strategy (Brown & Walter), constraint and goal manipulation, symmetry, generalization, systematic variation, and chaining (Silver et al., Dang et al.). Process models describe phases such as orientation/connection/generation/reflection (Cai & Rott) and activities like situation analysis, variation, generation, problem-solving, and evaluation (Baumanns & Rott). Prior studies highlighted that strategies had not been systematically classified nor empirically studied with large samples, especially for specific tasks or grade-level differences. Earlier findings indicate students often pose related/parallel problems and that strategy use can vary by educational level. Building on these, the authors propose a framework: Category A (pose directly from the situation) with A1 accepting given information and A2 changing given information (A21 changing initial information via adding/changing; A22 changing implicit assumptions), and Category B (pose further problems based on existing problems) with B1 symmetry, B2 systematic variation, and B3 chaining.

Design: A paper-and-pencil test presented three open-ended, non-goal-specific problem-posing situations (30 minutes total): (1) Real-life shopping discount scenario; (2) Medical drug concentration over time; (3) Mathematical arrangement of integers. Students were asked to pose as many different mathematical problems as possible. Sample: N=1653 junior high school students (grades 7–9) from three Chinese cities/schools: Kunming (n=932), Lanzhou (n=431), Beijing (n=290); grade counts: Grade 7=556, Grade 8=583, Grade 9=514. Participants were of average academic levels. Coding and scoring: Responses were screened (distinct problems counted; rephrasings with same meaning merged). Problems were grouped into Problem Clusters (PCs): initial problems (Category A) and related problems based on initial problems using B1/B2/B3 (PC1 for symmetry-based related problems; PC2 for systematic variation; PC3 for chaining). Strategies were identified per Table 1 (A1, A2 with A21/A22, B1, B2, B3). Strategy use was scored by frequency (n uses = n points; 0 if unused). Examples illustrated how PCs and strategies were coded and scored. Reliability: A random subset of 165 responses was double-coded by two researchers; inter-coder agreement ~90%. Disagreements were resolved via discussion before formal coding. Analysis: One-way ANOVA and multiple comparisons to test grade-level differences; structural equation modeling (SEM) with AMOS for path analysis of strategy selection dynamics.

Overall strategy preferences and frequencies (Fig. 5): - Category A usage: 94.68% of students used A1 (accepting given information); 84.27% used A2 (changing information). 81.06% used both A1 and A2. Within A2, 83.63% used A21 (changing initial information: adding/changing), while 13.49% used A22 (changing implicit assumptions). - Average uses per student: A1 M=3.09; A2 M=2.07; A211 M=1.47; A212 M=0.43; A22 M=0.17. A1 was used significantly more often than A2 (t=15.775, p<0.001, 2-tailed). - Category B usage: 77.56% of students used Category B to pose related problems. Within B, B3 (chaining) was most common: B3 used by 64.31% (M=1.17), B2 by 36.78% (M=0.66), and B1 by 24.02% (M=0.29). Multiple comparisons showed significant differences in frequency of use between all pairs: B1 vs B2 (t=-11.11, p<0.001), B1 vs B3 (t=-25.163, p<0.001), and B2 vs B3 (t=-11.591, p<0.001). - Example response patterns indicate students often connect posed problems to recently learned content (e.g., linear functions), but content knowledge per se is not tied to strategy choice. Grade-level differences (Tables 4–6): - Total problems posed (means): Grade 9 M=7.66 (SD=16.599), Grade 7 M=7.36 (SD=13.832), Grade 8 M=7.16 (SD=19.709). - Overall Category A: no significant grade difference, F(2,1650)=1.773, p>0.05. Sub-strategies: • A1: F(2,1650)=3.043, p<0.05; significant difference between Grades 7 and 9 (Grade 9 higher). • A2: F(2,1650)=2.840, p>0.05 (ns). • A21: F(2,1650)=9.730, p<0.05; significant differences between Grades 7 vs 8 and 8 vs 9 (Grade 8 higher). • A22: F(2,1650)=31.121, p<0.05; significant differences between Grades 7 vs 8 and 7 vs 9 (Grade 7 higher than both); also 8 vs 9 differed. - Overall Category B: significant grade differences, F(2,1650)=15.913, p<0.001, η²=0.019 (90% CI [0.009,0.031]); significant for Grades 7 vs 8 and 8 vs 9; 7 vs 9 not significant. • B1: F(2,1650)=8.638, p<0.001; Grade 7 > Grade 8 and Grade 9; 8 vs 9 not significant. • B2: F(2,1650)=4.761, p<0.05; significant for 7 vs 8 and 8 vs 9; 7 vs 9 not significant. • B3: F(2,1650)=14.948, p<0.001; all pairwise differences significant (Grades 9 highest). - Descriptively, Grade 9 students tended to accept given information (A1) more and used chaining (B3) more; Grade 8 most often changed initial information (A21); Grade 7 used changing implicit assumptions (A22) and symmetry (B1) more than other grades. - Strategy nonuse counts: 34 students used no strategies at all (Grade 7=8, Grade 8=16, Grade 9=10); 35 used no Category A (7=9, 8=16, 9=10); 371 used no Category B (7=93, 8=159, 9=119). Mean total strategies used: Grade 9 M=7.58, Grade 7 M=7.26, Grade 8 M=7.05. Path analysis (SEM; Fig. 6): - Overall sample: Primary evolution routes were (i) A1 → A2 → branching to B3 (chaining) or B1 (symmetry), with B1 often followed by B2 (systematic variation); and (ii) direct A1 → B3. This supports the framework that students typically move from posing directly from the situation (Category A) to posing related problems (Category B). - Grade-specific: Grade 7 showed broad inter-strategy connections, indicating diverse routes. Grades 8 and 9 revealed two distinct pathways: (1) A1 → A2 → B3; and (2) B1 → B2 sequentially.

The findings address RQ1 by showing clear preferences: students predominantly accept given information (A1) when posing problems and, when extending from existing problems, favor chaining (B3) over symmetry (B1) and systematic variation (B2). Changing implicit assumptions (A22) was rarely used. This suggests junior high students begin with familiar structures and extend incrementally, aligning with prior observations that students often generate related problem series. For RQ2, analyses reveal stage-specific patterns: Grade 9 emphasizes A1 and B3, Grade 8 emphasizes A21 (modifying initial information), and Grade 7 uses A22 and B1 comparatively more. SEM results indicate typical pathways from Category A to Category B; Grades 8–9 show a split between an A→B3 path and a B1→B2 path, while Grade 7 exhibits more varied interconnections. These results imply that competence with Category B strategies develops later and unevenly, and that instruction might scaffold transitions from accepting to changing information and encourage underused strategies such as symmetry and systematic variation. The framework effectively captures selection dynamics and can inform teaching by highlighting common and difficult pathways.

This study proposes and empirically validates a comprehensive framework of mathematical problem-posing strategies distinguishing between posing directly from a situation (Category A) and posing further problems from existing ones (Category B). Using data from 1,653 junior high school students, the study shows strong preferences for accepting given information (A1) and for chaining (B3) when extending problems, with significant grade-specific differences. Path analysis reveals dominant evolution routes from Category A to Category B and branching after changing information (A2) toward symmetry or chaining. The framework and findings offer guidance for instruction, curriculum design, and assessment of MPP, supporting staged development goals in curriculum standards. Future work should broaden samples beyond China, incorporate think-aloud data to capture processes, and investigate how prompting influences strategy selection and development.

- Sample limited to Chinese junior high schools; generalizability across countries/cultures is unknown. - No think-aloud/voiced protocols; analyses rely on written products, limiting insight into real-time cognitive processes and strategy selection. - Framework focuses on strategies consistent with Silver’s definition; other strategy taxonomies or dimensions may extend or refine it. - Effects of prompts on strategy selection were not examined and warrant further study.