Levels of Controversial Reasoning of The Pre-Service Teachers to Solve Mathematical Problems

Controversial reasoning is very important to learn because controversial problems often occur in everyday life. This study examines controversial reasoning in the context of solving mathematical problems. Therefore, this study aims to develop levels of controversial reasoning in solving mathematical problems. The subjects of this study were 185 mathematics education undergraduate students in the sixth
and eighth semesters as prospective mathematics teachers. Most of them had taken courses in Pedagogy and Mathematics. The students were given three controversial mathematical problems and interviewed indepth.
The students' thought processes in solving problems construct and determine the characteristics of controversial mathematical reasoning.
The study found three controversial mathematical reasoning levels
characteristics:
initial, exploration, and clarification. At the initial level, the subjects can recognize the controversy on an issue but cannot trace the cause of the controversy itself. At the exploration level, the
subjects can explore the components that cause the problem to become controversial but cannot clarify the problem as a solution. At the level of clarification, the subjects can clarify controversial issues using
plausible reasons. All the 185 subjects who answered show the answers that can be classified into three
levels: initial 55 (29.73%), exploration 52 (28.11%), and clarification 78 (42.16%) subjects.