Mathematical Argumentation in the Middle School is an exploratory project that is working in collaboration with teachers to increase their knowledge of mathematics for teaching in middle school. In addition to geometry and algebra, the professional development is focusing on the role of mathematical argumentation in the middle school and strategies for increasing argumentation. The research component of the project is providing insights into how teachers use their mathematical knowledge to increase argumentation in the classroom and to help students build skills in mathematical argumentation. In addition, the project is studying student outcomes such as reasoning, communication skills, and mathematical knowledge.
The project researchers are using both qualitative and quantitative methodologies to ensure that they have high quality data and analysis that will provide insights into the following research questions: 1) How do teachers use what they learn from the professional development (PD) experiences when they teach in the classroom? 2) To what extent does teachers? use of the project materials and what they learn in the PD result in mathematical argumentation in their classroom discourse? 3) Do students gain conceptual understanding of the mathematics as a result of their participation in argumentation in the classroom? Based on previous research, there is adequate evidence to believe that argumentation in the classroom does increase for participants in the workshop, but the researchers are seeking a better understanding of how teachers use their knowledge and the project materials to enact such an important change in mathematics lessons and in student learning. The professional development uses the dynamic software Geometers' Sketchpad, a carefully-designed, geometry curricular unit, and student materials to help teachers see how to set up a classroom environment that supports mathematical conjectures, arguments, and discussion. The research is done in the classroom and assesses the various components of the professional development in promoting argumentation.
The project is providing insights into how teachers use their mathematical knowledge to implement changes in the classroom. The project is creating effective professional development strategies, a middle school curriculum unit on geometry that emphasizes argumentation, and associated materials. The research is explaining how teachers use their knowledge and the materials and providing information on how students' conceptual knowledge develops through argumentation.
This project brings to the field an approach to professional development (PD) and a set of materials that enable teachers to begin to foster argumentation in their classrooms. Our work is based on the premise that teaching is disciplined improvisation. It improvisational because it emerges collaboratively with students in the classroom, and it is disciplined because it draws on the structure of mathematical argumentation, routines, such as teaching moves that teachers learn with experience, and planning processes that anticipate improvisational moments. We designed a program of PD that incorporated ideas and processes from improvisational theater, specially adapted for use in PD workshops. We used improv games from theater to establish norms of collegiality and to help replace the notion of mathematics as a fixed body of knowledge with the richer sense of mathematics as a creative activity in which teachers can use spontaneously generated ideas as well as facts they recall. We designed and used new teaching games to provide structured opportunities for teachers to develop and practice teaching moves for supporting students’ argumentation. Additionally, teachers engaged in lesson planning facilitated by visualization of classroom discussion, based on research about planning practices of expert and novice teachers, and they participated in improvisational performances for each other incorporating several teaching moves they had learned. Curriculum materials, including computer-based activities, were central to the PD. These supported 10 lessons and explicitly built in phases of argumentation: conjecturing, justifying, and concluding. The mathematics content was about triangles: First, students studied classes of triangles, making the arguments that, for example, a triangle could be both acute and isosceles. Second, students were provided guidance in creating an argument for why the sum of the measures of any triangle is 180 degrees. We provided explicit links to norms for argumentation, such as being a respectful audience, listening closely, and building off of each others’ ideas. Most of the 10 activities in the Bridging curriculum began with an improv game adapted for students in the classroom and a connection to norms for argumentation. Our research addressed three main questions in the classrooms of our four participating teachers: How do the teachers use what they learn from the Bridging PD experiences when they teach in the classroom? To what extent does teachers’ use of the Bridging materials and what they learn in the PD result in mathematical argumentation in their classroom discourse? Do students gain conceptual understanding of the mathematics as a result of their participation in argumentation in the classroom? We used several sources of observation data to develop analyses showing the variety of teaching practices that teachers can use with the materials and give insight into the nature of students’ early experiences in argumentation. The first set of analyses examined two important and related types of teaching moves the teachers made in their classrooms: moves that supported math content learning and moves that supported argumentation. In one classroom we observed that more teaching moves supporting content learning than argumentation. The teaching method in this class was traditional: the teacher asked a question, called on a student for a short answer, and publicly evaluated whether the student was right or not. This limited students’ opportunities for making conjectures and justifications. In another classroom, both argumentation and content were emphasized and as a result, students’ arguments made in a whole-class setting became stronger over the course of the unit, and the quality of paired-student arguments improved; content learning was woven into argumentation. In a third classroom, students engaged extensively in conjecturing, but justification was not extensive, and it appeared that norm setting took precedence over content. In a second set of analyses, we examined how teachers established norms for argumentation inn their classrooms. Each of the teachers made mini-speeches about norms for argumentation, advice for argumentation, and the structure of argumentation. As intended by the program design, teachers supported these speeches using the three posters provided by the project that lay out argumentation norms, as well as the classroom improv games. We were able to track how students took up the norms and advice and worked within the conjecture-justify-conclude structure. While much emphasis in "reform" teaching has been on what to do instead of telling, we definitely saw a role for telling in establishing norms for argumentation. A third set of analyses explored how middle school students engaged with their first experiences of argumentation supported by curriculum, by teaching moves, and by specially designed dynamic geometry software files. Students made tentative steps toward collaborative argumentation and in early lessons in the unit, it could be characterized as "parallel play." Increased test scores indicate that, on average, the students increased both content knowledge and skill in argumentation, although the results are uneven with regard to all students. Curriculum and professional development materials are available on request from Jennifer.knudsen@sri.com
