This study aimed to explore preservice elementary teachers’ use of a bar model as a heuristic for conceptualising relationships between quantities in situations involving ratio and percentages. As a part of a larger project, we focused on two preservice teachers, Maia and Jane, and investigated their solution paths in ratio and percentage problems and the role of the structural representations (i.e., bar model and percent line-bar model) in their solution paths. We analysed these focus participants’ written solutions and their think-aloud process during interviews using three coding phases with a constructivist grounded theory approach. We observed that the problem text fed the construction of the structural representations that served to make sense of the quantities and the relationships among them. Furthermore, the structural representations enhanced Maia and Jane’s mathematical insights about the problem situations. In addition, we observed that preservice teachers’ solution paths were neither linear nor similar, which indicates a mathematical richness in learning. In those solution paths, the model construction played a role in guiding the mathematical operations of constructing and operating on a unit rate in the ratio problem and identifying and operating on a referent whole unit in the percentage problem. We interpreted that variety and nonlinearity in solution paths suggested mathematical richness in learning, the educational implications of which were further discussed.

本研究旨在探讨职前小学教师使用条形模型作为启发式方法，在涉及比率和百分比的情况下概念化数量之间的关系。作为一个更大项目的一部分，我们专注于两位职前教师 Maia 和 Jane，并研究了他们在比率和百分比问题中的解决路径以及结构表示（即条形模型和百分比线条模型）在他们的问题中的作用。解决路径。我们使用基于建构主义理论方法的三个编码阶段分析了这些焦点参与者的书面解决方案和他们在采访中的大声思考过程。我们观察到问题文本提供了结构表示的构建，这些表示有助于理解数量和它们之间的关系。此外，结构表示增强了 Maia 和 Jane 对问题情况的数学洞察力。此外，我们观察到职前教师的解决路径既不是线性的也不是相似的，这表明学习中的数学丰富性。在这些求解路径中，模型构建起到了指导比率问题中单位率的构造和运算以及百分比问题中所指的整体单位的识别和运算的数学运算的作用。我们解释说，解决方案路径的多样性和非线性表明学习中的数学丰富性，其教育意义得到了进一步讨论。这表明学习中的数学丰富性。在这些求解路径中，模型构建起到了指导比率问题中单位率的构造和运算以及百分比问题中所指的整体单位的识别和运算的数学运算的作用。我们解释说，解决方案路径的多样性和非线性表明学习中的数学丰富性，其教育意义得到了进一步讨论。这表明学习中的数学丰富性。在这些求解路径中，模型的构建起到了指导比率问题中单位率的构造和运算以及百分比问题中所指的整体单位的识别和运算的数学运算的作用。我们解释说，解决方案路径的多样性和非线性表明学习中的数学丰富性，其教育意义得到了进一步讨论。