從認知診斷模式探究比例式解題的學習路徑

Constructing the Learning Path of Proportional Reasoning by Using Cognitive Diagnosis Models

本研究旨在探索學生在比例推理中解未知數問題時的學習路徑。根據文獻回顧，本研究之比例推理測驗使用五個特質，建立了三個階層模型：一個非結構化模型和兩個發散模型。然後利用認知診斷模型（CDMs）及其對應的Q矩陣估計學生組型，及各個組型在每個試題的期望值。此外，該測驗根據比值內/比值間和整數/非整數將試題分為四種類型，以了解不同學生組型和題型之間的表現差異。研究結果顯示，三種階層模型與高分組的作答反應具有高度一致性；比較DINA模型與特質階層模型（DINA-AHM）和階層診斷分類模型（HDCM）後發現，飽和HDCM模型可提供較多訊息；透過學生組型之預期分數所構建的學習路徑，可知悉精熟那個特質有助於提高學生的表現；四種題型中以比值內和比值間均為非整數的題型難度最高。

This study aims to explore the learning paths of students in proportional reasoning when solving problems with unknowns. Based on the literature review, the proportional reasoning test employed five attributes and three hierarchical models were established: an unstructured model and two divergent models. Cognitive Diagnosis Models (CDMs) and their corresponding Q-matrices were then utilized to estimate student profiles and the expected values on each item for each profile. Additionally, the test categorizes items into four types based on within/between-ratio and integer/non-integer. Thus, it allowed us to understand performance differences among different student profiles and item types. The results indicated high consistency between three hierarchical models and responses from high-performing groups. Comparing the DINA with Attribute Hierarchy Model (DINA-AHM) and the Hierarchical Diagnostic Classification Model (HDCM), the saturated HDCM model provided more information. The learning path constructed from the expected scores of student profiles allowed us to understand which attribute mastery was beneficial for improving student performance. Among the four item types, those involving non-integer ratios in both within and between ratios are the most difficult.