The main goal of the project is to understand how teachers 1) develop specific mathematical knowledge (algebraic justification), 2) transform this knowledge to classroom practice, and 3) advance in their pedagogy to promote students? learning. The project will clarify the meaning of justification for school algebra through collaborations among mathematicians, teachers, and mathematics educators. The project focuses on the important and under-explored connection between teachers? mathematical learning in a professional development environment and the subsequent a) changes in their teaching practice and b) continued deepening of their content knowledge in practice. Additionally, by working with a select group of exemplary teachers, the project explores the development of advanced inquiry teaching, namely teaching that supports students? mathematics learning via justification activities. The project design features a coordinated two-pronged approach in which teacher learning is examined in the context of a mathematics content course and the teachers? own classrooms. Data collection activities are designed to facilitate coordinated analyses to investigate connections across these two contexts. The summer course for teachers is modeled on a course that has been implemented successfully as part of an NSF-funded professional development project. Similarly, activities conducted with teachers during the academic year will provide opportunities for the teachers to develop advanced teaching practices and opportunities for the project team to investigate the development process. The project will utilize and extend existing instruments for assessing students? and teachers? understanding of justification. Algebra and algebraic reasoning are core to the discipline of mathematics and have national importance for the development of qualified STEM workforce.
JAGUAR (Justification and Argumentation: Growing Understanding of Algebraic Reasoning) Background The JAGUAR project investigated how 12 exemplary teachers developed advanced knowledge of mathematical justification and advanced in their pedagogy to promote students’ capacity for algebraic justification and argumentation. The JAGUAR project had three overarching goals: (1) to examine the development of JAGUAR teachers’ understanding of argumentation and justification, (2) to investigate the transformation of JAGUAR teachers’ understanding to their classroom practices, and (3) to examine the effect of advancement in practice on their students’ algebraic argumentation and justification. The primary components of the project were aimed at attaining the project goals: First, the teachers participated in a 5-day summer workshop on algebraic justification each of the first two years of the project. In this context, we investigated the teachers’ developing understanding of justification. Second, we investigated the teachers’ transformation of this knowledge to their practice during the academic year, through classroom lesson cycles, teacher working sessions, interviews, and video stimulated interviews. Third, we used a variety of strategies (e.g., work samples, analysis of classroom participation) to investigate the effect of the teachers’ development on their students’ ability to construct justifications and understanding of the nature of justification. Outcomes Teachers’ understanding of and views about mathematical justification The teachers began the JAGUAR project with a variety of views about justification and its role in the classroom. As they worked with us, we observed a number of changes. We will highlight two of these here. While most of the teachers entered the project realizing the limitations of using examples to justify statements, their understanding became more sophisticated. Specifically, the teachers came to understand the usefulness of reasoning about specific examples in a general way (generic examples) for developing valid general arguments. The teachers entered the project aware of the usefulness of tables for helping students to identify patterns. However, most of the teachers were unaware that if a student does not know the underlying reasons why a pattern is appearing in a table of data, any justification based on the pattern alone is an example of empirical reasoning (and typically not valid as a mathematical argument). The teachers came to understand this important point and as a result became consistent in asking their students to make sense of why the patterns they observed were occurring. Teachers’ pedagogy and mathematical justification Because, the teachers began the JAGUAR project with different backgrounds and views related to mathematical justification and teaching, the kinds of changes we observed in the teachers’ pedagogy varied from teacher to teacher. However, we can speak to three important shifts that were observed across multiple teachers. The first shift involved the teachers’ expectations of how students would successfully generalize relationships. Initially, many teachers expected students would be able to generalize if given a large number of varied examples. This type of generalizing often relies on students being able to notice patterns. Teachers later seemed to realize that generalizing might come more readily from a deeper analysis or consideration of a very small number of cases. This new understanding affected the ways teachers designed tasks as well as how they questioned students as the students attended to specific examples. A second change was how teachers responded to student assertions that were not completely correct. Teachers initially tried to prompt students to reconsider their assertions by offering counterexamples. Teachers ultimately found this unproductive. Students either did not consider the counterexample as problematic (e.g., it was irrelevant) or students were left with nothing to build on as they continued their work. Later, teachers avoided challenging incorrect assertions directly, allowing students to work out in their small group what was correct or not (and why). Alternatively, they used a strategy of having the student analyze their argument and to discover the flaw in it. The student could then either revise the argument in accordance, or develop a new approach based on this analysis. A third change, that is perhaps difficult to describe specifically in terms of the changes in teachers’ practice, was that teachers got much clearer about what to expect from students. They started the project not knowing what students could do with particular questions. Through experience in the Summer Workshops, working with their students, and reflecting on their lessons, the teachers seemed to develop a better sense of their students’ developmental trajectories. The teachers gained facility in anticipating the type of arguments students might produce in response to different justification prompts, deciding the types of arguments they (as teachers) would accept, and anticipating the issues that would likely arise along the way. More information about the JAGUAR project and its outcomes can be found on the project website: www.web.pdx.edu/~slarsen/JAGUAR2/index.html
