Justification and argumentation: Growing understanding of algebraic reasoning (JAGUAR)

Principal Investigator: 
Co-Investigator: 
Project Overview
Background & Purpose: 

The study has three overarching research questions:

  1. How do teachers’ develop in their understanding of algebraic justification and argumentation as new content knowledge?
  2. How do teachers’ transform new content knowledge (justification and argumentation) to their classroom practice and develop advanced practice?
  3. How does teachers’ advancement in practice affect their students’ capabilities for justification and argumentation? 
Setting: 

The project is a collaborative endeavor across Portland State University (lead institution), Purdue University and the University of Connecticut. Twelve 12 teachers from grades 7 and 8 were enrolled in the study. Several teachers had participated in a prior NSF project (NSF-EHR-0412553).   Teachers were recruited who had at least one year of teaching experience at the middle school level; were fully certified; and had developed a level of sophistication in their pedagogy that demonstrates student discourse that can be classified as ‘strategy-reporting’ or ‘inquiry/argument’ (Wood & Turner-Vorbeck, 2001). In general, these teachers already had experience organizing classroom discourse and were committed to eliciting student thinking. Thus this was a purposive sample of teachers.

Research Design: 

This project has a longitudinal study design (2 years) and is designed to generate descriptive case studies.  To document the teachers’ developing conceptions of argumentation and algebraic justification we administered a justification assessment instrument developed as part of the project. We also administered an assessment from the Learning Mathematics for Teaching (LMT) project to determine changes in teachers’ MKT. In addition, we conducted 5 semi-structured interviews with each teacher (one pre, one post, and 3 during the project), as well as two stimulated recall interviews (one at the end of each year) where teachers were shown video clips from their classroom and responded to questions. A pre-and post-assessment of students’ conceptions of justification was administered to all students in each teachers' focal class at the beginning and end of each academic year. This instrument was developed by the project team, and was 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. To document teachers’ classroom practice, we video recorded the implementation of 4 justification tasks (3 common across all teachers) during each of the two years of the project (total of 8 task implementations per teacher). In addition to videotaping, teachers wrote a pre-reflection and a post-reflection, based on their review of the videotape. Student work samples were also collected. 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, teachers’ conceptions of justification, and the development of teaching practice, while retrospective analysis allows us to bring data sources from the teachers’ summer sessions, classroom observations and teacher work sessions together for connected analysis.

Findings: 

Analyses are ongoing. We are focusing on three areas currently: forms of reasoning used by middle grades students; teachers’ pedagogy of justification; and teachers’ noticing with respect to justification and its relationship to their practice. We have documented teachers’ views of justification and proof, and the purposes they report for justification in their classrooms. (See publications below.)

Publications & Presentations: 
Staples, M., Bartlo, J., Thanheiser, E. (2012). Justification as a teaching and learning practice: Its (potential) multifaceted role in middle grades mathematics classrooms. Journal of Mathematical Behavior, 31, 447-462.
Thanheiser, E., Staples, M., Bartlo, J., Heim, K., & Sitomer, A. (2010). Justification in middle school classrooms: How do middle school teachers define justification and its role in the classroom. Brief Report. In P. Brosnan & D. B. Erchick & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (pp. 860 - 868). Columbus OH: The Ohio State University

Williams, L., & Staples, M. (2010). Justification in middle school classrooms: An analysis of student responses to two justification tasks. In Brosnan, P., Erchick, D. B., & Flevares, L. (Eds.), Proceedings of the thirty-second annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 6 (p. 878). Columbus, OH: The Ohio State University.

Heim, K. (February, 2012). Elementary Pre-service Teachers' Generalizations and Justifications of Figural Patterns (AMTE, Forth Worth, Texas).

 Staples, M., Newton, J., & Brown, C. (2012). Promising classroom practices for supporting mathematical justification. Research Presession of the 2011 Annual Meeting and Exposition of the National Council of Teachers of Mathematics. Philadelphia, PA. (poster) 
Cirillo, M., Bieda, M., Ellis, A., Herbst, P., Smith, M. S., & Staples, M. (2012). The notion of proof in mathematics teaching: Is it changing? Research Presession of the 2011 Annual Meeting and Exposition of the National Council of Teachers of Mathematics, Philadelphia, PA.
Staples, M., & Hennessy, B. (2012). Getting to the "Why?": Teacher Practices that Support Mathematically Sound Student Justifications in Middle Grades Classrooms. Poster Presentation at the 2012 Annual Meeting of the American Educational Research Association, Vancouver, CA.
Marrongelle, K., Staples, M., Thanheiser, E., Larsen, S., McCaffrey, C., Casa, T.,,& Gresalfi, M. S. (2011, April). Learning about mathematical justification and its role in the classrooms. Research Presession of the 2011 Annual Meeting and Exposition of the National Council of Teachers of Mathematics (NCTM), Indianapolis, IN. 
Thanheiser, E., Staples, M., McCaffrey, C., & Heim, K. (2011, February). Middle school teachers’ conceptions of justification: Why use it in the classroom?. Presented at Fifteenth Annual Conference of the Association of Mathematics Teacher Educators (AMTE), Irvine, CA.
Thanheiser, E., Staples, M., Bartlo, J., Heim, K., & Sitomer, A. (2010, October). Justification in middle school classrooms: How do middle school teachers define justification and its role in the classroom. Brief Report. In P. Brosnan & D. B. Erchick & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (pp. 860 - 868). Columbus OH: The Ohio State University
 Heim, K., Sitomer, A., and Thanheiser, E. (2010, September). Middle School Teachers’ Definitions of Justification and its Purpose in the Classroom. Presentation given at the annual meeting of Teacher of Teachers of Mathematics (TOTOM), Monmouth, OR. 
Staples, M. & McCaffrey, C. (2011, October). Collaborative research: Justification and argumentation: Growing understanding of algebraic reasoning. Poster session presented at REESE PI meeting, Washington DC.
Williams, L., & Staples, M. (2010). Justification in middle school classrooms: An analysis of student responses to two justification tasks. In Brosnan, P., Erchick, D. B., & Flevares, L. (Eds.), Proceedings of the thirty-second annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 6 (p. 878). Columbus, OH: The Ohio State University.

Other Products: 

We collaborated with Teachers Development Group (TDG), a nonprofit professional development organization and core partner in the OMLI MSP (NSF-HER-0412553), in developing an online course that mirrored our year 1 summer session with the JAGUAR.