Abstract Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.

中文翻译：

---

推理量之间的关系以重组反函数意义：Arya 的案例

摘要 研究人员认为，高中生、大学生、职前教师和在职教师并没有构建生产性反函数意义。在这份报告中，我首先总结了研究学生和教师的反函数含义的文献。然后我提供了我的理论观点，包括我对术语理解和意义的使用以及我对生产反函数意义的操作化。我描述了一个概念分析，学生可以通过推理协变量之间的关系，将他们有限的反函数意义重组为生产意义。然后，我展示了一位职前教师在一个学期的教学实验中的活动，以描述她的定量、协变、双向推理支持她将有限的反函数意义重组为更富有成效的意义。我描述了这种重组如何要求她重建各种相关数学思想的含义。我总结了这项工作的研究和教学意义以及未来研究的方向。