Geometric Reasoning to Reinventing Quadratic Formula: The Learning Trajectory on Realistic Mathematics Education Principles

This study aims to establish a knowledge base on how to support students in learning. We develop an initial hypothetical learning trajectory by formulating learning activities and predicting the development of students' thinking and understanding. The methodological framework employed in this study is design research, which seeks to generate actionable knowledge for achieving various educational goals through design. The design process incorporates the Hypothetical Learning Trajectory (HLT) thought experiment and the teaching experiment based on the six Realistic Mathematics Education (RME) principles. By integrating individual learning activities with the context and principles of the RME approach, students can construct their knowledge and rediscover the quadratic equation formula. Through comparing the HLT with observed learning outcomes, we redesign the process, revise our HLT, and provide answers to research questions regarding the attainment of specific learning objectives. Similar to our retrospective data analysis, we create a revised HLT to reinforce the concept of the completing the square and reinvent the quadratic formula.