The proposed study develops a mathematical knowledge base for teaching and learning generalization in basic algebra at the middle school level in an urban context. It involves analyzing children's generalizing processes based on individual and social activity. Classroom teaching experiments shall be developed, implemented, tested, and revised with the goal of articulating models of generalization in children. The project shall also address relevant instructional strategies and factors that will enable children to learn about the generalizing process, including ways the factors can all be coordinated properly so that teachers are able to provide their students with successful experiences. The study shall start with a class of 5th graders in a school in California whose thinking processes will be closely monitored for one semester. If it is possible, the same group of 5th graders will be investigated longitudinally for a period of four years.
The overall theoretical framework of the study is drawn from an emergent perspective in which children's strategies of generalization are viewed as being constructed from individual and social processes. Further, the instructional theory of realistic mathematics education (RME), developed at the Freudenthal Institute, will be used in the design of classroom teaching experiments. The study has a developmental research focus. Children will perform activities that will allow them to engage in complex chains of reasoning and signification leading to the emergence of models for generalization. Also, classroom teaching experiments are developed using a development cycle. This cycle assumes a symbiotic relationship between instructional theory and actual practice. Data sources include video recordings of all classroom and small group discussions, audiotape recordings of project meetings and debriefing sessions with participating teachers, copies of student's written work, and individual clinical interviews. Data analyses of individual and collective learning in a classroom community shall use a method developed by Cobb, Stephan, McClain, and Gravemeijer (2001).
Broader Impact The integrated research and education plan involves developmental research activities. The first activity concerns the development of structures of generalization among urban middle school children, including usable instructional theories that middle school teachers in such contexts may be able to adapt in their own classrooms. The five-year project is school-based and will be conducted in a collaborative partnership with teachers and their students who agree to participate in the study. Classroom teaching experiments will be designed, tested, and revised with a particular emphasis on how instructional resources and the classroom social context can be coordinated so that students succeed in algebra-related tasks involving generalization. The middle school mathematics curriculum provides the important formal foundation in algebra that needs to be strengthened at least in the area of formulating generalizations so that students are able to successfully transition to high school algebra. In the second activity, an ongoing circular process of describing, analyzing, and testing of strategies that children employ to generalize will be conducted in order to establish developmental models. Research-based and classroom-tested instructional materials that contain lessons and analyses drawn from the two activities shall be prepared and disseminated for public use by way of: presentations in various research and teachers. conferences; presentations in faculty in-service programs in urban middle schools; incorporation of research findings in various mathematics education courses in the university and in professional in-service programs with the aim of developing urban teachers whose instructional practices have been drawn from research; journal articles for teachers and researchers; CDs; and a website that will include activities, think pieces, and relevant publications in order to provide all teachers, students, and concerned stakeholders with ways in which to assist all middle-school students develop effective generalization skills. The proposed project shall also benefit from a diverse panel of experts whose expert knowledge has been drawn from having extensively worked with children in local, statewide, and international contexts.
The five-year NSF Career grant basically investigated the development of generalization and structural thinking among a cohort of middle school students over the course of three years. The initial theoretical framework employed real life contexts in helping middle schools students search for patterns and establish structures. Results of the initial three-year study have been published in peer-reviewd academic and professional journals, books, and monographs. Overall the longitudinal study has empirically established a progressive structural development of pattern generalization drawn mostly from intense and detailed qualitative investigations with the participants. In the remaining years of the project, findings from the study with middle school students have been empirically tested on elementary school children from first through third grades. Findings from this recent work shows fundamental cognitive actions in pattern generalization between younger and older children, with younger children performing abduction-induction strategies (i.e. generating and testing structures) and older children performing deduction (i.e. stating a rule from the known cases in order to predict future cases). The NSF funded project has been situated in urban schools in Northern California. Most of the students who participated in the project were of Hispanic and Southeast Asian origins. There was a small number of Caucasian and African American participants and students from South Asia. The project was initially conceptualized with urban children as the focus of the investigations. One extension of the NSF grant is the international work on pattern generalization among first grade students in Hongkong, Germany, and the US. A comparative analysis of their ability to generalize shows more similarities in performance among students in Germany and the US than their counterparts in Hongkong and the cause might be curricular in nature. Students in Germany and the US expose children to early patterning activity, which is oftentimes framed as extensions of counting activity. Students in Hongkong are not exposed to patterning activity until sixth grade. Consequently, German and US students tend to search for patterns and structures numerically, while Hongkong students do it visually. Results of the study are now being replicated on a larger scale internationally. The findings are also cited in very recent work on pattern generalization and are used as a theoretical framework by researchers within and outside the US. There is also evidence to show that the findings have now been incorporated in recent textbooks with chapters on patterns. Results of the NSF grant have also been successfully shared with several urban school districts in Northern California. Teachers from first through eighth grade have been provided with intensive professional development on algebraic thinking based on how students process them developmentally.
