Background: Teaching using manipulatives is emphasised, especially in the early grades, to help learners conceptualise operations on whole numbers. Therefore, teachers’ competencies in using manipulatives is the key in helping learners master these basic operation skills. Aim: Drawing from the literature on using manipulatives to improve learners’ performance in mathematics, this study recounts foundation phase pre-service teachers’ conception of using manipulatives to enhance their competencies and reasoning skills to model the solution in number operations. Setting: Data presented here was collected from 31 participants. These pre-service teachers either passed mathematics or mathematical literacy with 40% at the grade 12 level. Methods: Data was collected from participants’ written work (e.g. classroom tasks, homework, tests and examinations) and during class discussions. Interviews were conducted with some students. We analysed their conception guided by the APOS theory, namely, Action-Process- Object-Schema. Results: We observed improvement in the conception of using manipulatives among pre-service teachers. In the first semester, most students display action conception of using manipulatives to either represent or model a solution. However, in the second semester, most students either display process or object conception as explained in the genetic decomposition. We attributed the improvement to change of instruction in the second semester as we taught in accordance with the APOS theory. Conclusion: It is evident that there are a number of contributing factors to pre-service teachers’ conception of mathematical concepts, and teacher educators need to pay particular attention to these to help pre-service teachers master the concepts they would teach at school.

中文翻译：

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在数字运算中使用运算符的职前数学教师的发展过程

背景：尤其是在小学低年级时，强调了使用手法进行教学，以帮助学习者将整数的操作概念化。因此，教师使用操作技巧的能力是帮助学习者掌握这些基本操作技能的关键。目的：从文献中使用操纵来改善学习者的数学表现，本研究阐述了基础阶段的职前教师关于使用操纵来增强他们的能力和推理技能以对数字运算中的模型进行建模的概念。地点：这里提供的数据是从31名参与者那里收集的。这些职前教师通过了数学或数学读写训练，达到12年级的水平达到40％。方法：从参与者的书面工作中收集数据（例如课堂任务，家庭作业，测试和考试）以及课堂讨论中。采访了一些学生。我们在APOS理论的指导下分析了它们的概念，即动作过程对象模型。结果：我们观察到在职前教师中使用操纵的概念有所改善。在第一学期，大多数学生表现出使用操纵来表示或建模解决方案的动作概念。但是，在第二学期，大多数学生要么表现出过程，要么表现出对象的概念，如遗传分解所解释的那样。根据APOS理论，我们将改进归因于第二学期的教学变化。结论：很明显，许多因素影响了岗前教师的数学概念，