The aim of this study was to show that some of the errors made by students when responding to mathematics assessment items can indicate progress in the development of conceptual understanding. By granting partial credit for specific incorrect responses by early secondary students, estimates of the difficulty of demonstrating full and partial knowledge of skills associated with the development of proportional reasoning were determined using Rasch analysis. The errors were confirmed as indicators of progress, and hence partial knowledge, when the thresholds of achievement followed a logical order: The greater the proficiency of the students, the more likely they were to receive a higher score. Consideration of this partial knowledge can enhance the descriptions of the likely behaviours of students at the various levels of learning progressions and this can be informative for teachers in their planning of learning activities.

本研究的目的是表明学生在回答数学评估项目时所犯的一些错误可以表明概念理解发展的进展。通过对早期中学生的特定错误回答给予部分学分，使用 Rasch 分析确定了展示与发展比例推理相关的技能的全部和部分知识的难度的估计。当成绩的阈值遵循逻辑顺序时，错误被确认为进步的指标，因此是部分知识：学生的熟练程度越高，他们获得更高分数的可能性就越大。