Constructing the concept of irrational number by middle school students: An analysis based on APOS theory

Due to centuries of work by mathematicians, irrational numbers represent one of the most challenging sets of numbers for students to understand during the learning process. Despite various studies focusing on students' understanding of this concept across different periods, common issues persist, highlighting the need for further research to improve its teaching. This study seeks to answer the question of how middle school students can mentally construct the concept of irrational numbers. It specifically investigates the mental structures that eighth-grade students encountering irrational numbers for the first time may develop to understand this concept within the framework of a hypothetical genetic decomposition developed based on APOS theory. The study adopts a qualitative research design, precisely a case study approach. The genetic decomposition was developed by considering the historical development of the concept, the mathematics curriculum, an analysis of middle school mathematics textbooks throughout the history of the Turkish Republic, findings from the literature on students' mathematical thinking, and the researchers' instructional experiences. Based on this genetic decomposition, the ACE teaching cycle (activities, classroom discussions, exercises) was designed and implemented with sixty students. At the end of the process, conceptual understanding questions and semi-structured interviews with twenty-seven students (low, medium, and high achievers) were conducted to refine the genetic decomposition further. The study results reveal a genetic decomposition that supports students' understanding of irrational numbers, explaining the differences in student performance and demonstrating the potential of this genetic decomposition to enhance students' conceptual understanding.