Promising teaching approaches for developing students’ mathematical competencies include the approach of teaching mathematics through problem solving. Orchestrating a whole-class discussion of students’ ideas is an important aspect of teaching through problem solving. There is a wide consensus within the field that it is very challenging for the teacher to conduct class discussions that both build on student ideas and highlight key mathematical ideas and relationships. Further, fostering argumentation in the class, which is important for students’ participation, is also a grand challenge. Teachers need support in these challenges. The aim of the thesis is to characterize challenges and support for mathematics teachers in orchestrating productive problem-solving whole-class discussions that focus on both mathematical connection-making and argumentation. In particular, it is investigated how Stein et al.’s (2008) model with five practices – anticipating, monitoring, selecting, sequencing and connecting student solutions – can support teachers to handle the challenges and what constitutes the limitations of the research-based and widely-used model. This thesis builds on six papers. The papers are based on three intervention studies and on one study of a mathematics teacher proficient in conducting problem-solving class discussions. Video recordings of observed whole-class discussions as well as audio-recorded teacher interviews and teacher meetings constitute the data that are analyzed. It is concluded in the thesis that the five practices model supports teachers’ preparation before the lesson by the practice of anticipating. However, making detailed anticipations, which is shown to be both challenging and important to foster argumentation in the class, is not explicitly supported by the model. Further, the practice of monitoring supports teachers in using the variety of student solutions to highlight key mathematical ideas and connections. Challenging aspects not supported by the monitoring practice are, however, how to interact with students during their exploration to actually get a variety of different solutions as a basis for argumentation. The challenge of selecting and sequencing student solutions is supported for the purpose of connection-making, but not for the purpose of argumentation. Making mathematical connections can be facilitated by the last practice of connecting, with the help of the previous practices. However, support for distinguishing between different kinds of connections is lacking, as well as support for creating an argumentative classroom culture. Since it is a great challenge to promote argumentation among students, support is needed for this throughout the model. Lastly, despite the importance and challenge of launching a problem productively, it is not supported by the model. Based on the conclusions on challenges and support, developments to the five practices model are suggested. The thesis contributes to research on the theoretical development of tools that support teachers in the challenges of orchestrating productive problem-solving whole-class discussions.