「學習軌道」在數學教育研究的回顧與展望

Learning Trajectories in Mathematics Education Research: A Literature Review

本文旨在探討「學習軌道」構念在數學教育研究的意義與內涵，以及探討學習軌道的形成方式與作用，描繪一個涵蓋教學、評量、課程發展和教師專業發展面向的概念架構，以提供後續有關學習軌道研究的參考。
我們藉由分析文獻回顧探討以下四項：一、說明學習軌道是由教學目標、學生數學認知發展和教學活動所構成學習路徑的意義和內涵；二、由教學實驗、理論概念分析和課程政策三個面向，探討學習軌道的形成來源和方式；三、以及指出學習軌道具有預期學生學習表現、發展課程與教學架構、評量學生能力和支持教師專業發展的功用；四、以圖形與面積概念的學習軌道做為範例，說明學習軌道在研究與實務的歷程與結果。
最後分別從：整合學生個人情境與課室文化、結合多元研究理論觀點與研究方法、發展特殊情境脈絡的教與學的局部理論、探討教師如何使用學習軌道以及應用於教師專業發展等五個面向，討論學習軌道在數學教育研究的啟示與應用。
本文認為結合實證研究和實務需求，以提升學生數學學習和確保教學品質是目前數學教育研究的趨勢。學習軌道即以實證研究的學生數學認知發展為主軸，發展符合適當課程教材，促進學生的數學學習發展，並進而成為學生數學表現的評量架構，和教師專業發展的工具。顯見學習軌道在數學教育研究的重要性及潛在研究價值。

The purpose of this study was to introduce the learning trajectory to mathematics education research. First, I briefly review the literature, thereby documenting that learning trajectories are learning structures composed of teaching goals, students’ mathematical cognitive development, and teaching activities. Second, I explore the sources of the development of learning trajectories for 3 areas, namely teaching experiments, theoretical and conceptual analysis, and curriculum policy; in addition, the technique for constructing a learning trajectory is illustrated. Third, I demonstrate that there are 4 functions of learning trajectories: anticipating students’ learning, developing curriculum and teaching, assessing students’ mathematical ability, and supporting teachers’ professional development. Fourth, I introduce an example of a learning trajectory to explain how to apply trajectories in practice.
Finally, I discuss the implications of learning trajectories in mathematics education research and offer directions for future research through the following 5 points: coordinating student context personalization, combining multiple theoretical perspectives and research methods, developing local theories of teaching and learning in specific contexts, inquiring into how teachers use learning trajectories, and applying learning trajectories to teachers’ professional development.
In this article, I explain that mathematics education research integrates empirical research and the requirements of instructional practices to facilitate students’ mathematical learning and to ensure teaching quality. The function of learning trajectories is to provide mathematical instruction to students by integrating the cognitive development of students, curriculum materials, and intended instructional goals. Learning trajectories are a crucial and effective tool for mathematics education research.