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.

The purpose of this study is to investigate student interactions with simulations, and teacher support of those interactions, within naturalistic high school physics classroom settings. This study focuses on data from two lesson sequences that were conducted in several physics classrooms. The lesson sequences were conducted in a whole class discussion format in approximately half of the class sections and in a hands-on-computer small group format in matched class sections. Analysis used a mixed methods approach where: (1) quantitative methods were used to evaluate pre-post data; (2) open coding and selective coding were used for transcript analysis; and (3) comparative case studies were used to consider the quantitative and qualitative data in light of each other and to suggested possible explanations. Although teachers expressed the expectation that the small group students would learn more, no evidence was found in pre-post analysis for an advantage for the small group sections. Instead, a slight trend was observed in favor of the whole class discussion sections, especially for students in the less advanced sections. In seeking to explain these results, qualitative analyses of transcript and videotape data were conducted, revealing that many more episodes of support for interpreting visual elements of the simulations occurred in the whole class setting than in the matched small group discussions; not only teachers, but, at times, students used more visual support moves in the whole class discussion setting. In addition, concepts that had been identified as key were discussed for longer periods of time in the whole class setting than in the matched small group discussions in six of nine matched sets. For one of the lesson sequences, analysis of student work on in-class activity sheets identified no evidence that any of the Honors or College Preparatory students in the small groups had made use in their thinking of the key features of the sophisticated and popular physics simulation they had used, while such evidence was identified in the work of many of the whole class students. Analysis of the whole class discussions revealed a number of creative teaching strategies in use by the teachers that may have helped offset the advantage of hands-on experience with the simulations and animations enjoyed by the small group students. These results suggest that there may exist whole class teaching strategies for promoting at least some of the active thinking and exploration that has been considered to be the strength of small group work, and appear to offer encouragement to teachers who do not have the resources to allow their classes to engage regularly in small group work at the computer. Furthermore, these examples suggest the somewhat surprising possibility that there may be certain instructional situations where there is an advantage to spending at least part of the time with a simulation or animation in a whole class discussion mode.

While the use of student thinking to help build mathematical understandings in a classroom has been emphasized in best teaching practices, teachers still struggle with this practice and research still lacks a full understanding of how such learning can and should occur. To help understand this complex practice, I analyzed every instance of student thinking and every teacher response to that thinking during a high school geometry teacher's whole class discussion and used these codes as evidence of alignment or misalignment with principles of effective use of student mathematical thinking. I explored the teacher's practice both in small and large grains by considering each of her responses to student thinking, and then considered the larger practice through multiple teacher responses unified under a single topic or theme in the class discussion. From these codes, I moved to an even larger grain to consider how the teacher's practice in general aligned with the principles. These combined coding schemes proved effective in providing a lens to both view and make sense of the complex practice of teachers responding to student thinking. I found that when responding to student thinking the teacher tended to not allow student thinking to be at the forefront of classroom discussion because of misinterpretation of the student thinking or only using the student thinking in a local sense to help advance the discussion as framed by the teacher's thinking. The results showed that allowing student thinking to be at the forefront of classroom discussion is one way to position students as legitimate mathematical thinkers, though this position can be weakened if the teacher makes a move to correct inaccurate or incorrect student thinking. Furthermore, when teachers respond to student thinking students are only able to be involved in sense making if the teacher turns the ideas back to the students in such a way that positions them to make sense of the mathematics. Finally, in order to allow students to collaborate a teacher must turn the mathematics to the students with time and space for them to meaningfully discuss the mathematics. I conclude that the teacher's practice that I analyzed is somewhat aligned with honoring student mathematical thinking and allowing student thinking to be at the forefront of class discussion. On the other hand, the teacher's practice was strongly misaligned with collaboration and sense making. In this teacher's class, then, students were rarely engaged in sense making or collaborating in their mathematical work.

The aim of this study was firstly to evaluate the impact of two interactive-engagement models of instruction, namely Whole Class Discussions (WCD) and Computer Simulations (CS) on first year physics student-teachers’ conceptual understanding of Newtonian mechanics, and on their epistemological beliefs about physics. The force concept inventory was used to evaluate the impact on conceptual understanding while the Epistemological Beliefs About Physical Science questionnaire was used to evaluate the impact on their epistemological beliefs. The findings suggest that interactive engagement models had a positive impact on students’ conceptual understanding of Newtonian mechanics, and on their epistemological beliefs about physics. The study also contributed WCD and CS activities that can be used or adapted with an aim of enhancing conceptual understanding in physics. The study did not show any direct relationship between students’ conceptual understanding of Newtonian mechanics and their epistemological beliefs about physics. Key words: Interactive-engagement, Whole Class Discussion, Computer Simulations, epistemological beliefs about physics. / Physics / M.Sc. (Physics Education)

The aim of this study was firstly to evaluate the impact of two interactive-engagement models of instruction, namely Whole Class Discussions (WCD) and Computer Simulations (CS) on first year physics student-teachers’ conceptual understanding of Newtonian mechanics, and on their epistemological beliefs about physics. The force concept inventory was used to evaluate the impact on conceptual understanding while the Epistemological Beliefs About Physical Science questionnaire was used to evaluate the impact on their epistemological beliefs. The findings suggest that interactive engagement models had a positive impact on students’ conceptual understanding of Newtonian mechanics, and on their epistemological beliefs about physics. The study also contributed WCD and CS activities that can be used or adapted with an aim of enhancing conceptual understanding in physics. The study did not show any direct relationship between students’ conceptual understanding of Newtonian mechanics and their epistemological beliefs about physics. Key words: Interactive-engagement, Whole Class Discussion, Computer Simulations, epistemological beliefs about physics. / Physics / M.Sc. (Physics Education)