Fraction Conceptual and Procedural Knowledge: Validity of Equivalence Judgment as an Indicator, Mediating Role of Procedural Knowledge, and Latent Profile Classification of Students
Description
Research Hypotheses: Fraction equivalence judgment serves as a valid measure of Fraction Conceptual Knowledge (FCK) Fraction Procedural Knowledge (FPK) mediates the relationship between FCK and General Mathematics Achievement (GMA) Students can be classified into distinct profiles based on FCK, FPK, and GMA Data and Methods: 282 students (grades 6-8) completed: Computer-based FCK tasks (number line estimation, fraction comparison, equivalence judgment) Paper-based FPK test (fraction arithmetic operations) Standardized mathematics exam for GMA Key Findings: Confirmatory Factor Analysis showed good model fit when including equivalence tasks in FCK measurement (CFI=1, TLI=1.013) Structural Equation Modeling revealed partial mediation: Direct effect of FCK on GMA: β=0.525 Indirect effect through FPK: β=0.108 Latent Profile Analysis identified three classes: Low-achieving (4.6%): weak in all three areas High-achieving (62.4%): strong in all areas Concept-strong (33.0%): high FCK but moderate FPK and low GMA Interpretation and Application: Understanding fraction equivalence is crucial for conceptual knowledge Procedural knowledge enables transformation of conceptual understanding into mathematical achievement Differentiated instruction needed: Low-achievers need both conceptual and procedural support Concept-strong students require procedural skill development Data Value: This multi-method study reveals the complex structure of fraction knowledge and provides empirical evidence for personalized mathematics instruction based on students' knowledge profiles.
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Data Collection Protocol: Participants: 282 students (Grades 6-8) provided consent and were tested in a quiet room. Instruments & Tasks: Fraction Conceptual Knowledge (FCK): Administered via computer. *0-1 & 0-5 Number Line Estimation:* Adapted from Bailey et al. (2015). Participants estimated locations of fractions on a line. Primary measure: Percent Absolute Error (PAE). Fraction Magnitude Comparison: Participants selected the larger of two fractions. Measure: Accuracy. Fraction Equivalence Judgment: Adapted from Fitzsimmons et al. (2020). Participants judged if equations (e.g., 3/9=1/3) were true/false. Measure: Accuracy. Fraction Procedural Knowledge (FPK): Paper-and-pencil test. Fraction Arithmetic: 12 problems covering all operations (addition, subtraction, multiplication, division). Two versions (A/B) with reversed orders controlled for sequence effects. Measure: Accuracy. General Mathematics Achievement (GMA): Measure: End-of-semester exam scores based on the standard math curriculum. Workflow: Computer tasks (randomized order) were completed first (~15 min), without scratch paper. The arithmetic task was completed next (~10 min). Total time: ~25 minutes per participant. Software & Analysis: Statistical Analysis: Mplus version 8.3 was used for all major analyses. Analytical Methods: Confirmatory Factor Analysis (CFA) to validate the FCK measurement model, Structural Equation Modeling (SEM) to test mediation, and Latent Profile Analysis (LPA) to identify student subgroups. Maximum Likelihood estimation was used for CFA/SEM with bootstrapping (5,000 samples) for mediation effect confidence intervals.