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DEEPENING TEACHER KNOWLEDGE AND THEIR STRATEGIC COMPETENCEAlong with inclass activities, the teachers were also given the opportunity to reflect on the problem that they worked on each day. The teacher reflections enabled us to focus on the understanding, reactions, and feelings of the individual teachers. While the posters showed how people in a group approached problem solutions in a variety of ways, the reflections gave us insight into how the individual teachers were feeling about the sessions, about their own competence, and about their classroom practices. Several themes were observed in the majority of the reflections. These included the value of the struggle, the joy of using conceptual thinking, the importance of clarity, the advantage of building, the benefit of collaboration, and recognizing that there are multiple valid ways in which to approach problem solving, which leads to viewing student work with new eyes. Teachers appreciated the value of the struggle for several reasons. Being forced to “figure it out” without reliance on rote procedures or “tricks” gave teachers a chance to think about their own thinking. Valuable discussions with their peers ensued. Understanding of concepts was developed. Several teachers reported “Aha” moments concerning ideas about rational numbers which they had formerly accepted but now actually understood, giving them a feeling of liberation. Teachers experienced frustration, which made them more sensitive to the same feeling in their students; and, teachers saw the value in developing a thoughtful and defendable approach to problem solution. This is a skill which they want to transfer to their students. A teacher wrote, “I wish more classroom teachers fostered an environment where students can struggle with problems and work together to solve problems. Struggling through and listening to strategies of others has really opened up my thinking.” As the teachers’ conceptual knowledge deepened, the teachers began to question their own knowledge and assumptions. Teachers gained such insight and expanded understanding through discussions that they want to incorporate more “talking about it” in their classrooms instead of heading straight for procedural solutions. Classroom discussions of problems and sharing solution strategies are seen as a valuable approach both to clarify problems for our students as well as to develop their conceptual thinking. The cathedral problem provided a poignant example of a question which is easy to misinterpret. The question asked, “What would be the expense of just one of each worker?” Whether the answer to the question is “$10” or “$7 for an artist and $3 for a stonemason” may never be resolved. Even after the class discussed the idea that the question did not ask for the individual rates, some teachers were convinced that the question required the rates for both workers. The important point is that the question was, apparently, open to interpretation. Teachers, both in reflections and in verbal commentary, noted that they learned to be very clear when they write questions. One teacher commented to us that she was going to review all of her assessments to ensure that she did not have any “open to interpretation” questions. She showed some angst in saying that she hoped she had not done that to her students in the past. Teachers reported that the reasoning up and down helped them to break problems into chunks and build on those chunks. They saw how building on known concepts or known quantities gave them a sense of control as opposed to the lost feeling we sometimes experience during the introduction of a completely new idea. The teachers realize that the latter is a source of concern, frustration, and fear in their students. One teacher commented that she never realized how emotional the process could be and that she was gaining a new perspective on her students and how she interacts with them. Another teacher wrote that she would use reasoning up and down to help her students focus on what they already know and then guide them in building on that knowledge. Several teachers remarked on the importance of labeling processes so that students have a clear picture of how the concepts tie together; this leads to the development of conceptual understanding and the internalization of concepts and processes for the students. Teachers appreciated the collaborative nature of the problemsolving process that they engaged in as a part of the institute. No one felt as though they were left to fend for themselves with no help. Struggling through problem solutions with colleagues, analyzing their approaches, questioning their reasoning, and contributing to group efforts were noted by the teachers as being very beneficial to their discoveries during the week. Meetings with colleagues were already planned by several of the teachers to discuss their progress and to choose problems to incorporate the reasoning up and down strategies into their curricula before the start of the school year. 
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