Problem formulation: An investigation of preservice mathematics teachers thought processes

In this study, by exploring the pre-service mathematics teachers' problem-formulation processes, it was aimed to describe and justify the use of a theoretical model in integrating problem formulation into mathematics classrooms for learning of new topics and demonstrate how it better captures aspects of problem posing. In this study in which the case study method was adopted, 38 third grade pre-service mathematics teachers were given a paper-and-pencil test consisting of six problems, and they were then asked to formulate as many related problems as possible. Task-based interviews related to these problems were conducted with nine pre-service mathematics teachers who did not take the paper-and-pencil test. The problems posed in the paper-and-pencil test were analyzed according to their mathematical validity and problem-formulation strategy types. The data obtained from task-based interviews were subjected to thematic analysis. The results of the paper-and-pencil test indicated that most of the formulated problems were related to the given problem. In addition, they were mathematically appropriate. However, the problem-formulation strategies were unevenly distributed. The results of the task-based interviews indicated that the pre-service mathematics teachers started to formulate a problem by finding the solution to the given problem, and while formulating their problems, they considered cognitive, metacognitive, and instructional factors, which differ according to the task type. Taking the results of this study, the active learning frameworks, which relates active learning, problem posing, and problem solving for the learning of mathematical concepts developed by Ellerton (2013) and extended by Xie and Masingila (2017), was revised and a Revised Active Learning Framework was created.