Rational numbers can be represented in multiple formats (e.g., fractions, decimals, and percentages), and a rational number notation can be used to express different concepts in different contexts. The present study investigated the distribution of the multiple concepts expressed by these different rational number notations in real-world contexts as well as the semantic alignment between entity type (discrete vs. continuous) and rational number format (decimal vs. fraction) in the Chinese context. Textbook analysis and two paper-and-pencil experiments yielded the following four major findings: (1) Decimals were more likely used to represent numerical magnitudes, while fractions were more likely used to represent relations between two numerical magnitudes. (2) Decimals were more often used to represent continuous entities while fractions were preferred to represent discrete entities. (3) The strength of the association between different formats of rational numbers and their preferred conceptual meanings seemed more pronounced than the semantic alignment between number type and entity type. (4) Percentages were used in a way more similar to fractions than decimals in terms of the concepts they express. These findings indicate that different formats of rational numbers differ dramatically in their use in real-world contexts both in terms of the conceptual meanings they express and the entities they model. Educational implications of this study are discussed.

中文翻译：

---

跨有理数表示的概念区别和优先对齐

有理数可以用多种格式表示（例如，分数、小数和百分比），并且有理数符号可以用于在不同的上下文中表达不同的概念。本研究调查了这些不同有理数符号在现实世界中表达的多个概念的分布以及实​​体类型（离散与连续）和有理数格式（小数与分数）之间的语义对齐。语境。教科书分析和两次纸笔实验得出以下四个主要发现：（1）小数更可能用于表示数值量级，而分数更可能用于表示两个数值量级之间的关系。(2) 小数更常用于表示连续实体，而分数更倾向于表示离散实体。(3) 不同形式的有理数与其优选概念含义之间的关联强度似乎比数类型和实体类型之间的语义对齐更明显。(4) 就其表达的概念而言，百分比的使用方式更类似于分数而不是小数。这些发现表明，不同格式的有理数在现实世界中的使用在它们表达的概念意义和它们建模的实体方面存在巨大差异。讨论了这项研究的教育意义。(3) 不同形式的有理数与其优选概念含义之间的关联强度似乎比数类型和实体类型之间的语义对齐更明显。(4) 就其表达的概念而言，百分比的使用方式更类似于分数而不是小数。这些发现表明，不同格式的有理数在现实世界中的使用在它们表达的概念意义和它们建模的实体方面存在巨大差异。讨论了这项研究的教育意义。(3) 不同形式的有理数与其优选概念含义之间的关联强度似乎比数类型和实体类型之间的语义对齐更明显。(4) 就其表达的概念而言，百分比的使用方式更类似于分数而不是小数。这些发现表明，不同格式的有理数在现实世界中的使用在它们表达的概念意义和它们建模的实体方面存在巨大差异。讨论了这项研究的教育意义。这些发现表明，不同格式的有理数在现实世界中的使用在它们表达的概念意义和它们建模的实体方面存在巨大差异。讨论了这项研究的教育意义。这些发现表明，不同格式的有理数在现实世界中的使用在它们表达的概念意义和它们建模的实体方面存在巨大差异。讨论了这项研究的教育意义。