Abstract Many studies show that prospective teachers often have misconceptions about fractions. In this case study, we report on some of the mal–rules used by a group of 60 prospective South African primary school teachers. The students’ written responses to two items focusing on addition and multiplication of fractions which formed part of an assessment, were analyzed. Semi-structured interviews were also used to elicit the reasoning used in the students’ calculations. Less than half of the participants completed both items correctly, and many of the other students displayed various mal–rules. To interpret the pre–service teachers’ misconceptions, we studied the rules used by the participants, and expressed them as theorems–in–action. An interesting mal–rule governing the multiplication of fractions was the widespread ‘cross multiplication’ rule which after some mutations led to other mal–rules, illustrating how students’ misconceptions can persist many years after their initial learning.

中文翻译：

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职前教师在分数运算中使用交叉乘法和其他错误规则

摘要 许多研究表明，未来的教师往往对分数有误解。在这个案例研究中，我们报告了一组 60 位未来的南非小学教师使用的一些错误规则。分析了学生对两个项目的书面回答，这些项目侧重于构成评估一部分的分数的加法和乘法。半结构化访谈也被用来引出学生计算中使用的推理。不到一半的参与者正确完成了这两个项目，而许多其他学生则表现出各种错误规则。为了解释职前教师的误解，我们研究了参与者使用的规则，并将它们表达为实际中的定理。