Justification in Algebra: Growing Understanding of Algebraic Reasoning (JAGUAR)

Principal Investigator: 
Project Overview
Background & Purpose: 

The study has three overarching goals: (1) to examine the development of in-service 7th- and 8th-grade pre-algebra and algebra teachers’ understanding of algebraic argumentation and justification; (2) to investigate the transformation of teachers’ algebraic understanding to their classroom practices; and (3) to examine the effect of advancement in practice on their students’ algebraic argumentation and justification.


The project is based in Portland, Oregon at Portland State University. Places studied will include pre-algebra and algebra classrooms in various middles schools throughout Oregon.

The target population is in-service 7th-and 8th-grade pre-algebra and algebra teachers. The study sample will include 16 exemplary pre-algebra and algebra teachers from grades 7-8, 12 coming from an existing project (NSF-EHR-0412553), prioritizing having 2 teachers per school. Selection of the exemplary teachers for participation will be based on a 3-step process: (1) Teachers must have full certification, major in mathematics or mathematics education, and 3 or more years of teaching experience; (2) teachers must have developed a level of sophistication in their pedagogy that demonstrates student discourse that can be classified as ‘strategy-reporting’ or ‘inquiry/argument’; and (3) teachers will be selected who teach students from diverse ethnic and socio-economic backgrounds.

Research Design: 

This project has a longitudinal study design and is designed to generate descriptive [case study, design research, observational] and associative and/or correlational [quasi-experimental] evidence. Original data is collected through survey research [self-completed questionnaires, semi-structured interviews, focus groups] observation [personal, videographic] and assessments of learning or achievement tests, diaries/journals/records kept by study subjects.

To document the teachers’ developing conceptions of argumentation and algebraic justification we will administer a justification assessment instrument developed in the project planning sessions. We will also administer the Knowledge of Algebra for Teaching (KAT) (NSF REC 0337595 and NSF REC 0106709; Ferrini-Mundy, Burrill, Floden, Sandow, & Allen, 2005) to assess teachers’ knowledge of algebra.

During the Year 1 AJA Course, 2-4 interviews will be conducted with each teacher using interview protocols developed in the project planning sessions. Interviews will generally focus on teachers’ developing knowledge of and about algebraic justification and argumentation.

A pre-and post-assessment of students’ conceptions of justification will be administered to all students at the beginning and end of each academic year. This instrument will be an adaptation of one developed by Healy and Hoyles (2000) for the purposes of exploring grade 7-8 students’ conceptions of what makes an acceptable justification.

The study’s analytic plan includes two qualitative approaches: (1) constant comparison and (2) ‘retrospective’ data analysis. The constant comparison design and procedures allow for continual feedback about the development of notions of justification in algebra, teachers’ conceptions of justification in algebra, the development of teaching practice, and students’ learning, while retrospective analysis allows us to bring all data sources from the teachers’ summer sessions, classroom observations and teacher work sessions together for connected analysis. A quantitative quasi-experimental design is used to evaluate student achievement in a pre and posttest analysis, and teacher growth in algebra and justification is assessed by the KAT (described above) and an instrument to be designed by our research team.

Initial analysis of videotaped student discourse will use the instrument (Weaver et al., 2005; Weaver & Dick, 2006) developed for a related research project. Additional mathematical analysis of justifications will use codes developed in project planning sessions. Analysis of transcribed justification videotaped lessons will be coded using Wood et al. (1999) coding categories (developed for NSF RED 9254939) for interaction patterns to identify teaching practices. Simulated recall interviews will be analyzed using a coding rubric developed in project planning sessions for analysis of justification discourse and shifts in pedagogy and teachers’ conceptions of argumentation and justification as disciplinary and learning practice.


The results of this study will be applicable to classroom practice beginning in Fall 2010. Dissemination will occur on two levels: (1) dissemination to the community of educators and in particular to mathematics education researchers and; (2) dissemination to practitioners through the professional development community.

Other Products: 

We will collaborate with Teachers Development Group (TDG), a nonprofit professional development organization and core partner in the OMLI MSP (NSF-HER-0412553), based in Portland, OR. TDG collaborates with school districts across the U.S. to create comprehensive professional development plans that focus on improving all students' mathematical achievement, and organizes national seminars and symposia on mathematics professional development. We will work with TDG to develop new courses and other professional development opportunities based on our findings.