Collaborative Research: Investigating Issues of the Individual and Collective along a Continuum between Informal and Formal Reasoning

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

The goals of this project are to create theoretical and methodological means for understanding and interpreting mathematical progress along a continuum from informal or intuitive to more formal ways of reasoning. The research seeks to use and coordinate between different lenses on individual learning and on the joint production of meaning. This work extends prior research in differential equations and geometry to the case of linear algebra.

Research Design: 

The research design for this project is longitudinal, and is designed to generate evidence which is descriptive (case study, design research) and associative and/or correlational (analytic essay and interpretive commentary). This project has an intervention, consisting of design research, and multiple enactments on classrooms are used for comparison. This project collects original data using assessments of learning/achievement tests; videography and personal observation; and structured and semi-structured face-to-face interviews. We use primarily qualitative methods. For example, one methodological approach for documenting the collective production of meaning is a three-phase approach that uses Toulmin’s (1969) model of argumentation. Other approaches use grounded theory approach for identifying key conceptual milestones and mechanisms that support and constrain students’ progress along these milestones.

Findings: 

Our project work to date has resulted in forwarding the field of mathematics education research in four different areas. First, we have contributed to the research literature that focuses on how students think about particular mathematical ideas in linear algebra, how these ideas can develop toward more formal notions over time, and hypothetical learning trajectories (including innovate task designs) for supporting students’ mathematical progress from informal to more formal ways of reasoning. Stepping back from the particular design of learning trajectories for the teaching and learning of linear algebra, we have also made a contribution to the field regarding comparison between different theoretical perspectives on the role of models and modeling in support of students’ mathematical progress. Finally, we are making contributions related to various ways for conceptualizing and documenting mathematical progress for individuals and for communities of learners.

Publications & Presentations: 

Rasmussen, C., Zandieh, M., & Wawro, M. (in press). How do you know which way the arrows go? The emergence and brokering of a classroom mathematics practice. In W.-M. Roth (Ed.), Mathematical representations at the interface of the body and culture. Information Age Publishing.

Zandieh, M. & Rasmussen, C. (submitted). A case study of defining: creating a new mathematical reality. Journal of Mathematical Behavior.

Larson, C., Harel, G., Oehrtman, M. Rasmussen, C., Speiser, R., Walter, C., & Zandieh, M. (2007). Modeling perspectives in math education research, Proceedings of the 13th International Conference on Teaching Modeling and Applications. Bloomington, IN.

Larson, C. (2007). Modeling and quantitative reasoning: The summer jobs problem. Proceedings of the 13th International Conference on Teaching Modeling and Applications. Bloomington, IN.

Larson, C., & Smith, M. (2008). Students' Conceptions of Vectors, Span, and Linear Dependence and Independence. Presentation at the Joint AMS-MAA Annual meeting.

Larson, C., Zandieh, M, Rasmussen, C., & Henderson, F. (February, 2009). Student interpretations of the equal sign in matrix equations: The case of Ax=2x. Paper presented at the Twelfth Conference on Research in Undergraduate Mathematics Education, Raleigh, NC.

Rasmussen, C., Zandieh, M., & Wawro, M. (February, 2009). The social production of meaning. Paper presented at the Twelfth Conference on Research in Undergraduate Mathematics Education, Raleigh, NC.

Rasmussen, C. (January, 2009). Classroom norms and habits of mind. Research presentation at the Joint Annual Meeting of the Mathematical Association of America and the American Mathematical Society, San Diego, CA.

Rasmussen, C., & Keene, K. (March, 2008). Gestures and a chain of signification: The case of equilibrium solutions. Paper presented at the Annual Meeting of the American Educational Research Association, New York City, NY.

Rasmussen, C., Zandieh, M., & Sweeney, G. (March, 2008). Communities of practice and practices of the community. Paper presented at the Annual Meeting of the American Educational Research Association, New York City, NY.

Rasmussen, C., Kwon, O., & Marrongelle, K. (February, 2008). A framework for interpreting inquiry-oriented teaching. Paper presented at the Eleventh Conference on Research in Undergraduate Mathematics Education, San Diego, CA.

Rasmussen, C., & Zandieh, M. (February, 2008). The emergence of a complex graphical inscription: The case of a bifurcation diagram. Paper presented at the Eleventh Conference on Research in Undergraduate Mathematics Education, San Diego, CA.

Larson, C., Zandieh, M., & Rasmussen, C. (February, 2008). A trip through eigen land: Where most roads lead to the direction associated with the largest eigenvalue. Paper presented at the Eleventh Conference on Research in Undergraduate Mathematics Education, San Diego, CA.

Neering, S., & Veragi, J. (2008). Students’ Understanding and Use of Representations with Vector Concepts. Paper presented at the Eleventh Conference on Research in Undergraduate Mathematics Education, San Diego, CA.

Zandieh, M., Knapp, J., & Roh K. H., (2008). When Students Prove Statements of the Form (P -> Q) ⇒ (R -> S). Paper presented at the Eleventh Conference on Research in Undergraduate Mathematics Education, San Diego, CA.

Zandieh, M., Larson, C., & Rasmussen, C. (2008). A Hypothetical Learning Trajectory for Eigen-Theory, Paper presented at the 15th Conference on the International Linear Algebra Society, Cancun, Mexico.
 

Other Products: 

We expect there to be research results that build theory within the discipline of mathematics education in three main areas:
student learning along a continuum between more informal and more formal student reasoning, student learning as observable from the four different lenses (two collective and two individual), and student learning specific to linear algebra, this may include student difficulties but would focus more broadly on what does it mean to know, understand or learn specific topics in linear algebra. We also expect there to be two main methodological areas in which we expect to make the greatest contribution: An extension of the Rasmussen and Stephan (in press) method for analyzing collective classroom practices to a more specific method for analyzing collective discipline practices and a method for determining individual acquisition data in a way that can be done efficiently within a classroom teaching experiment without compromising the collective or individual participation data analysis. In terms of instructional design we see as a product a local instructional theory specific to the content of linear algebra.