Abstract This paper introduces a quadratic growth learning trajectory, a series of transitions in students’ ways of thinking (WoT) and ways of understanding (WoU) quadratic growth in response to instructional supports emphasizing change in linked quantities. We studied middle grade (ages 12–13) students’ conceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic growth as phenomenon of constantly-changing rate of change. We elaborate the duality, necessity, repeated reasoning framework, and methods of creating learning trajectories. We report five WoT: Variation, Early Coordinated Change, Explicitly Quantified Coordinated Change, Dependency Relations of Change, and Correspondence. We also articulate instructional supports that engendered transitions across these WoT: teacher moves, norms, and task design features. Our integration of instructional supports and transitions in students’ WoT extend current research on quadratic function. A visual metaphor is leveraged to discuss the role of learning trajectories research in unifying research on teaching and learning.

中文翻译：

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二次增长学习轨迹

摘要 本文介绍了二次增长学习轨迹、一系列学生思维方式 (WoT) 和理解方式 (WoU) 的二次增长转变，以响应强调相关数量变化的教学支持。我们在一个小规模的教学实验中研究了中年级（12-13 岁）学生的概念，旨在培养对二次增长作为不断变化的变化率现象的理解。我们详细阐述了创建学习轨迹的二元性、必要性、重复推理框架和方法。我们报告了五个 WoT：变异、早期协调变化、明确量化的协调变化、变化的依赖关系和对应关系。我们还阐明了在这些 WoT 之间产生转变的教学支持：教师动作、规范、和任务设计功能。我们在学生 WoT 中整合教学支持和转换扩展了当前对二次函数的研究。利用视觉隐喻来讨论学习轨迹研究在统一教与学研究中的作用。