In the present study, we examined how pre-service mathematics teachers initiated the problem-formulation process when given another problem as a starting point and what types of considerations they made in the problem-formulation process. Nine pre-service mathematics teachers were given car rental and handshake problems and were asked to formulate new problems based on the given problems. Then, task-based interviews were conducted with the pre-service mathematics teachers, and the data obtained were subjected to thematic analysis. The results indicated that the pre-service mathematics teachers tended 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 differed according to the task type. Taking these factors into account, we developed a new learning framework by reinterpreting the existing frameworks that combine problem formulation, problem solving, and active learning.