In this study, we present a mathematical analysis distinguishing two conceptions of equivalence: proportional equivalence and unit equivalence. These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a) the distinction of equivalence gives a unified framework of equal fractions that has not previously been described in the literature; (b) a conceptual understanding of both fraction equivalences is integral to understanding rational numbers; and (c) knowledge of both conceptions of equivalence is important for developing a conceptual understanding of fraction arithmetic. Past research has largely overlooked the distinction between the two types of equivalence. However, this may provide an important foundation for central topics that build on equivalence, and a better understanding of these two types of equivalence may support a more flexible understanding of fractions. Last, we propose future directions for teaching equivalence in mathematics.

在这项研究中，我们提供了一种数学分析，将两个等效概念区分开：比例等效和单位等效。相对于等价分数，这两个概念具有不同的含义：一个以比例为基础，而另一个以相等的整体为基础。我们认为：（a）等价性的区别给出了一个等分的统一框架，这在文献中是从未有过的描述；（b）对两个等价等价物的概念性理解对于理解有理数是必不可少的；（c）对两种等效概念的了解对于发展分数算术的概念性理解非常重要。过去的研究在很大程度上忽略了两种等效形式之间的区别。但是，这可能为以等价为基础的中心主题提供重要的基础，并且对这两种等价类型的更好理解可以支持对分数的更灵活理解。