A challenge in mathematics education research is to coordinate different analyses to develop a more comprehensive account of teaching and learning. We contribute to these efforts by expanding the constructs in Cobb and Yackel’s (Educational Psychologist 31:175–190, 1996) interpretive framework that allow for coordinating social and individual perspectives. This expansion involves four different constructs: disciplinary practices, classroom mathematical practices, individual participation in mathematical activity, and mathematical conceptions that individuals bring to bear in their mathematical work. We illustrate these four constructs for making sense of students’ mathematical progress using data from an undergraduate mathematics course in linear algebra.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Bikner-Ahsbahs, A., & Prediger, S. (2014). Networking of theories as a research practice in mathematics education. Cham, Switzerland: Springer International Publishing.
Blumer, H. (1969). Symbolic interactionism: Perspectives and method. Englewood Cliffs: Prentice-Hall.
Bogomolny, M. (2007). Raising students’ understanding: Linear algebra. In J. H. Woo, H. C. Lew, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 65–72). Seoul: PME.
Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17(1), 25–66.
Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5–43.
Cobb, P. (2000). Conducting classroom teaching experiments in collaboration with teachers. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307–334). Mahwah: Lawrence Erlbaum Associates.
Cobb, P. (2003). Investigating students’ reasoning about linear measurement as a paradigm case of design research. Journal for Research in Mathematics Education Monograph, 12, 1–16.
Cobb, P. (2007). Putting philosophy to work: Coping with multiple theoretical perspectives. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 3–38). Reston: NCTM.
Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190.
Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Retrieved from http://www.corestandards.org
Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacle of formalism in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 85–124). Dordrecht: Kluwer Academic Publishers.
Dreyfus, T., Hillel, J., & Sierpinska, A. (1999). Cabri-based linear algebra: Transformations. In I. Schwank (Ed.), Proceedings of the First Conference on European Society for Research in Mathematics Education (Vol. 1, pp. 209–221). Osnabrück, Germany. Retrieved from http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/papers/g2-dreyfus-et-al.pdf
Glaser, B., & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago: Aldine.
Harel, G. (1997). Linear algebra curriculum study group recommendations: Moving beyond concept definition. In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins, & W. Watkins (Eds.), Resources for teaching linear algebra (pp. 106–126). Washington, DC: The Mathematical Association of America.
Harel, G., Behr, M., Lesh, R., & Post, T. (1994). Invariance of ratio: The case of children’s anticipatory scheme for constancy of taste. Journal for Research in Mathematics Education, 25, 324–345.
Hershkowitz, R., Tabach, M., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts in a probability classroom—A case study involving coordinating two methodologies. ZDM - The International Journal on Mathematics Education, 46(3), 363–387.
Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 191–207). Dordrecht: Kluwer Academic Publishers.
Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. Journal of Mathematical Behavior, 26, 60–82.
Krummheuer, G. (2011). Representation of the notion of “learning-as-participation” in everyday situations in mathematics classes. ZDM - The International Journal on Mathematics Education, 43, 81–90.
Larson, C., & Zandieh, M. (2013). Three interpretations of the matrix equation Ax=b. For the Learning of Mathematics, 33(2), 11–17.
Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, 27(1), 24–30.
Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: First steps towards a conceptual framework. ZDM – International Journal for Mathematics Education, 40, 165–178.
Rasmussen, C. (2001). New directions in differential equations: A framework for interpreting students’ understandings and difficulties. Journal of Mathematical Behavior, 20, 55–87.
Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of innovative design research in science, technology, engineering, mathematics (STEM) education (pp. 195–215). New York: Taylor and Francis.
Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A view of advanced mathematical thinking. Mathematical Thinking and Learning, 7, 51–73.
Rasmussen, C., Zandieh, M., & Wawro, M. (2009). 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 (pp. 171–218). Charlotte: Information Age Publishing.
Saxe, G. B. (2002). Children’s developing mathematics in collective practices: A framework for analysis. Journal of the Learning Sciences, 11, 275–300.
Saxe, G. B., Gearhart, M., Shaughnessy, M., Earnest, D., Cremer, S., Sitabkhan, Y., et al. (2009). A methodological framework and empirical techniques for studying the travel of ideas in classroom communities. In B. B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 203–222). London: Routledge.
Selinski, N., Rasmussen, C., Zandieh, M., & Wawro, M. (2014). A method for using adjacency matrices to analyze the connections students make within and between concepts: The case of linear algebra. Journal for Research in Mathematics Education, 45(5), 550–583.
Sfard, A. (1998). On two metaphors for learning and on the danger of choosing just one. Educational Researcher, 27, 4–13.
Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 209–246). Dordrecht: Kluwer Academic Publishers.
Stephan, M., & Akyuz, D. (2012). A proposed instructional theory for integer addition and subtraction. Journal for Research in Mathematics Education, 43(4), 428–464.
Stephan, M., & Cobb, P. (2003). The methodological approach to classroom-based research. Journal for Research in Mathematics Education Monograph, 12, 36–50.
Stephan, M., Cobb, C., & Gravemeijer, K. (2003). Coordinating social and individual analyses: Learning as participation in mathematical practices. Journal for Research in Mathematics Education Monograph, 12, 67–102.
Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459–490.
Stewart, S., & Thomas, M. O. J. (2009). A framework for mathematical thinking: The case of linear algebra. International Journal of Mathematical Education in Science and Technology, 40(7), 951–961.
Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181–234). Albany: State University of New York Press.
Toulmin, S. (1958). The uses of argument. Cambridge: Cambridge University Press.
Trigueros, M., & Possani, E. (2013). Using an economics model for teaching linear algebra. Linear Algebra and Its Applications, 438(4), 1779–1792. doi:10.1016/j.laa.2011.04.009.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Bristol: Falmer Press.
Wawro, M. J. (2011). Individual and collective analyses of the genesis of student reasoning regarding the Invertible Matrix Theorem in linear algebra. (Doctoral dissertation). Available from ProQuest Dissertations and Theses database. (Order No. 3466728)
Wawro, M., & Plaxco, P. (2013). Utilizing types of mathematical activities to facilitate characterizing student understanding of span and linear independence. In S. Brown, G. Karakok, K. H. Roh, and M. Oehrtman (Eds.), Proceedings of the 16th Annual Conference on Research in Undergraduate Mathematics Education, Volume I (pp. 1–15), Denver, Colorado.
Wawro, M., Rasmussen, C., Zandieh, M., Larson, C., & Sweeney, G. (2012). An inquiry-oriented approach to span and linear independence: The case of the magic carpet ride sequence. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 22(8), 577–599. doi:10.1080/10511970.2012.667516
Wawro, M., Rasmussen, C., Zandieh, M., & Larson, C. (2013). Design research within undergraduate mathematics education: An example from introductory linear algebra. In T. Plomp & N. Nieveen (Eds.), Educational design research—Part B: Illustrative cases (pp. 905–925). Enschede: SLO.
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.
Yackel, E., Gravemeijer, K., & Sfard, A. (Eds.). (2011). A journey in mathematics education research: Insights from the work of Paul Cobb. Dordrecht: Springer.
Yackel, E., & Rasmussen, C. (2002). Beliefs and norms in the mathematics classroom. In G. Leder, E. Pehkonen, & G. Toerner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 313–330). Dordrecht: Kluwer.
Yackel, E., Rasmussen, C., & King, K. (2000). Social and sociomathematical norms in an advanced undergraduate mathematics course. Journal of Mathematical Behavior, 19, 275–287.
Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. Research in Collegiate Mathematics Education, IV, 103–127.
Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning. Journal of Mathematical Behavior, 29, 57–75.
This material is based upon the work supported by the National Science Foundation under collaborative grants DRL 0634099 and DRL 0634074 and collaborative grants DUE 1245673, DUE 1245796, and DUE 1246083. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
About this article
Cite this article
Rasmussen, C., Wawro, M. & Zandieh, M. Examining individual and collective level mathematical progress. Educ Stud Math 88, 259–281 (2015). https://doi.org/10.1007/s10649-014-9583-x
- Individual and collective
- Emergent perspective
- Linear algebra