A Framework for Characterizing Learning Contributions Across the Initial Achievement Spectrum
Conceptual assessments are widely used in physics education research to evaluate changes in student understanding, yet class average measures can obscure how those changes are distributed across students with different levels of initial achievement. We introduce a Learning Contribution Framework that characterizes this distribution through the Learning Contribution Curve (LCC) and Learning Contribution Profile (LCP). For a general contribution measure G, the LCC represents cumulative contribution across students ranked by initial achievement, whereas the LCP describes the local mean contribution relative to the population mean of the individual G values. We apply the framework to the individual Hake normalized gain and develop a statistical model linking LCC and LCP behavior to the joint structure of pretest and posttest scores. Under the central assumption that the conditional mean of the subsequent score is linear in the initial score, we identify a score structure parameter $β$ and a critical score structure parameter $β_c$. Whether $β$ is greater than, less than, or equal to $β_c$ determines whether the expected LCP increases, decreases, or remains constant across initial achievement. Given the pretest distribution, the model further yields analytical predictions for the complete LCP and LCC that closely reproduce simulation results. Application to classroom concept-assessment data illustrates how the framework reveals local and cumulative patterns of learning contribution that are not evident from an overall class average revealed by the traditional Hake's normalized gain.