We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students’ sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of \( f(x)={x}^{\sqrt{2}} \). The task design and data analysis relied on the concept of sensemaking trajectories, blending the notions of sensemaking and (hypothetical/actual) learning trajectories. The findings present four typical sensemaking trajectories the participants went through while coping with the notion of an irrational exponent, alongside associated reasoning themes that seemed to have guided these trajectories. In addition to irrational exponents, the analysis revealed the participants’ reasoning and sensemaking of related mathematical ideas, such as rational exponents, approximations of irrational numbers by rational numbers, even/odd functions and numbers, and the meaning of exponentiation in general. The findings provide a step towards a better understanding of students’ conceptual development of irrational exponents, which could in turn be used for the refinement of tasks aimed at promoting students’ comprehension of the topic.

我们调查学生如何理解非理性指数。数据包括32次对大学生的访谈，其围绕一项任务进行，该任务旨在检查学生在预测和随后解释\（f（x）= {x} ^ {\ sqrt {2} } \）。任务设计和数据分析依赖于感官轨迹的概念，融合了感官构思和（假设/实际）学习轨迹的概念。研究结果提出了参与者在应对非理性指数概念时经历的四种典型的感官轨迹，以及似乎指导了这些轨迹的相关推理主题。除了非理性指数外，分析还揭示了参与者对相关数学概念的推理和意义理解，例如有理指数，无理数对有理数，奇/偶函数和数的近似以及幂的含义。研究结果为更好地理解学生的非理性指数的概念发展提供了一步，