The Dual Pathways of Proportional Reasoning on Mathematics Achievement: The Mediating Role of Whole Number Arithmetic and Latent Profile Analysis
Description
Research Hypothesis: We hypothesized that proportional reasoning (PR) directly supports general math achievement (GMA) and indirectly through whole-number arithmetic (WNA). We also expected to find distinct student subgroups with different performance profiles. Data & Methodology: Data was gathered from 123 second-graders (age ~7.9) in China. Participants completed three tasks: A computer-based proportional reasoning task (choosing equivalent ratios). A paper-and-pencil whole-number arithmetic test. A comprehensive general math achievement test. We used path analysis to test the mediation model and latent profile analysis (LPA) to identify subgroups. Notable Findings: Dual Pathways: Path analysis confirmed our model. PR had a significant direct effect on GMA (β = 0.328, p<.001) and a significant indirect effect via WNA (β = 0.143, p=.005). The model explained 47% of the variance in math achievement. Two Subgroups: LPA identified two distinct profiles: High-Performers (95%): Strong, balanced skills across all areas. Low-Performers (5%): Showed significant difficulties in PR, WNA, and GMA. Interpretation & Use: Proportional reasoning is a foundational skill that boosts math achievement directly and by enhancing arithmetic. The small but distinct low-performing group highlights the need for early identification and intervention to address broad mathematical difficulties, potentially linked to a "whole number bias." Educators should integrate proportional concepts early into instruction using visual and contextualized examples.
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Data Collection Methods 1. Participant Recruitment Sample size was determined via G*Power analysis for the mediation model, using an effect size of 0.15 (Resnick et al., 2023), requiring a minimum of 107 participants for 95% statistical power. A total of 123 second-grade students from Shenzhen, China, were recruited (mean age = 7.91 years, SD = 0.85; 59 boys, 64 girls). All participants were in a transitional stage of mastering whole number arithmetic without formal fraction instruction and had normal or corrected vision. Informed consent was obtained from students, parents, and teachers, and the study was approved by the Research Ethics Committee of Shenzhen University. 2. Instruments Proportional Reasoning Task: Adapted from Boyer and Levine (2012), administered via computer. In a "bear’s watermelon juice" scenario, students selected which of two options matched the target cup’s ratio of watermelon juice (red) to blueberries (blue). Participants were randomly assigned to one of three versions: continuous, discretized, or discrete items. Whole Number Arithmetic Test: Paper-and-pencil assessment with two parts: direct calculation (e.g., 32 ÷ 7, 3 × 6, 1800 - 200) and vertical computation (e.g., 18 ÷ 6, 397 + 432). General Mathematics Achievement Test: Paper-and-pencil test covering five domains: basic concepts (e.g., time units), geometric reasoning (e.g., angle comparison), arithmetic operations, logical reasoning (e.g., pattern detection), and applied problem-solving. Additional hands-on tasks (e.g., drawing geometric figures) assessed spatial reasoning. 3. Data Collection Procedures Assessments were conducted in a classroom group setting, proctored by trained researchers for standardization. Tasks were completed in a fixed order: proportional reasoning task → whole number arithmetic test → general mathematics achievement test. Proportional reasoning data were recorded automatically via computer; paper-and-pencil tests were collected and scored uniformly. 4. Data Analysis Tools Path analysis and latent profile analysis were used. Analyses referenced common tools in similar studies (e.g., Mplus or SPSS).
Institutions
- Shenzhen University