Teachers come across errors not only in tests but also in their mathematics classrooms virtually every day. When they respond to learners’ errors in their classrooms, during or after teaching, teachers are actively carrying out formative assessment. In South Africa the Annual National Assessment, a written test under the auspices of the Department of Basic Education, requires that teachers use learner data diagnostically. This places a new and complex cognitive demand on teachers’ pedagogical content knowledge. We argue that teachers’ involvement in, and application of, error analysis is an integral aspect of teacher knowledge. The Data Informed Practice Improvement Project was one of the first attempts in South Africa to include teachers in a systematic process of interpretation of learners’ performance data. In this article we analyse video data of teachers’ engagement with errors during interactions with learners in their classrooms and in one-on-one interviews with learners (17 lessons and 13 interviews). The schema of teachers’ knowledge of error analysis and the complexity of its application are discussed in relation to Ball’s domains of knowledge and Hugo’s explanation of the relation between cognitive and pedagogical loads. The analysis suggests that diagnostic assessment requires teachers to focus their attention on the germane load of the task and this in turn requires awareness of error and the use of specific probing questions in relation to learners’ diagnostic reasoning. Quantitative and qualitative data findings show the difficulty of this activity. For the 62 teachers who took part in this project, the demands made by diagnostic assessment exceeded their capacity, resulting in many instances (mainly in the classroom) where teachers ignored learners’ errors or dealt with them partially.

中文翻译：

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教学数学时要面对学习者的错误

老师几乎每天都会在测试中以及数学教室中遇到错误。当他们在课堂上，教学中或教学后对学习者的错误做出回应时，教师正在积极地进行形成性评估。在南非，“年度国家评估”是基础教育部主持下的一项笔试，要求教师诊断性地使用学习者数据。这对教师的教学内容知识提出了新的复杂的认知要求。我们认为，教师对错误分析的参与和应用是教师知识的组成部分。数据知情实践改进项目是南非首次尝试将教师纳入对学习者表现数据的系统解释过程中的尝试之一。在本文中，我们分析了在课堂上与学习者互动以及与学习者进行一对一访谈（17节课和13次访谈）时，教师参与错误的视频数据。结合鲍尔的知识领域和雨果对认知与教学负荷之间关系的解释，讨论了教师错误分析知识的图式及其应用的复杂性。分析表明，诊断评估需要教师将注意力集中在任务的紧要任务上，而这反过来又需要对错误的认识以及与学习者诊断推理有关的特定探测问题的使用。定量和定性的数据发现表明了这项活动的难度。对于参与该项目的62位老师，