This paper presents a proposal for a confluence model of problem-based learning and teaching in different instructional contexts, with particular focus on mathematics instruction. The proposed model aims at bridging the accumulated knowledge of how problem solving occurs in realistic instructional situations and the knowledge of various factors that can support or inhibit problem solving. The model capitalizes on an empirically supported premise that the student solution to a problem can be construed as a sequence of shifts of attention stipulated by three types of resources available to the solver: individual resources, interaction with peers and interaction with a source of knowledge about the solution. In turn, the ways of using the resources are stipulated by the solver's choices of: a challenge to cope with, problem-solving schemata, a mode of interaction, an extent of collaboration, and an agent to learn from. The solver choices are in turn stipulated by the following choices made by the teacher while engineering the instructional situation involving problem solving: the choice of a task space, of declarative knowledge, of didactical contract, of expectations and norms of study, and of an agency space. Furthermore, the teacher choices are stipulated by extents of his or her commitment to the perceived needs of the main stakeholders involved in the educational process: self, students, relevant educational institutions, society and the discipline. The proposed model can be used as an pedagogical tool for informed designing problem-based learning environments as well as as a research tool for documenting, analyzing and explaining different phenomena related to problem-based learning and teaching.
