Analyzing the Mathematical Thinking Flexibility of Pre-Service Mathematics Teachers in Solving Quadratic Inequalities

Abstract

This study aimed to analyze the mathematical thinking flexibility of pre-service mathematics teachers in solving quadratic inequalities, identify the strategies they employed, and explore the factors contributing to the dominance of procedural approaches. The study adopted a mixed methods approach using an explanatory sequential design. The participants consisted of 22 undergraduate students enrolled in a Real Analysis course in a Mathematics Education program. Data were collected through a mathematical problem-solving test and semi-structured interviews. Quantitative data were analyzed descriptively, while qualitative data were analyzed using the Miles, Huberman, and Saldaña interactive model. The findings revealed that all participants relied exclusively on the interval testing method to solve the given problems. No participants employed alternative strategies, such as factor sign analysis, graphical representation, completing the square, or function property analysis. Furthermore, all participants used only symbolic-algebraic representations and did not demonstrate key indicators of mathematical thinking flexibility, including the ability to generate multiple solution strategies, explain relationships among different strategies, and evaluate the effectiveness of the strategies used. Interview findings indicated that the predominance of procedural approaches was influenced by prior learning experiences, the belief that interval testing is the only correct method, and limited opportunities to explore diverse mathematical representations. These findings suggest that obtaining correct answers does not necessarily reflect an adequate level of mathematical thinking flexibility.