Skip to main content

Advertisement

SpringerLink
Log in

Examining individual and collective level mathematical progress

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

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 via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5–43.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. Journal of Mathematical Behavior, 26, 60–82.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Larson, C., & Zandieh, M. (2013). Three interpretations of the matrix equation Ax=b. For the Learning of Mathematics, 33(2), 11–17.

    Google Scholar 

  • Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, 27(1), 24–30.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Rasmussen, C. (2001). New directions in differential equations: A framework for interpreting students’ understandings and difficulties. Journal of Mathematical Behavior, 20, 55–87.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Saxe, G. B. (2002). Children’s developing mathematics in collective practices: A framework for analysis. Journal of the Learning Sciences, 11, 275–300.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Stephan, M., & Cobb, P. (2003). The methodological approach to classroom-based research. Journal for Research in Mathematics Education Monograph, 12, 36–50.

    Google Scholar 

  • 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.

    Google Scholar 

  • Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459–490.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Toulmin, S. (1958). The uses of argument. Cambridge: Cambridge University Press.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Bristol: Falmer Press.

    Book  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.

    Article  Google Scholar 

  • Yackel, E., Gravemeijer, K., & Sfard, A. (Eds.). (2011). A journey in mathematics education research: Insights from the work of Paul Cobb. Dordrecht: Springer.

    Google Scholar 

  • 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.

    Google Scholar 

  • Yackel, E., Rasmussen, C., & King, K. (2000). Social and sociomathematical norms in an advanced undergraduate mathematics course. Journal of Mathematical Behavior, 19, 275–287.

    Article  Google Scholar 

  • Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. Research in Collegiate Mathematics Education, IV, 103–127.

    Google Scholar 

  • 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.

    Article  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA, 92182-7720, USA

    Chris Rasmussen

  2. Department of Mathematics (MC 0123), Virginia Tech, McBryde Hall 438, 225 Stanger St., Blacksburg, VA, 24061, USA

    Megan Wawro

  3. Faculty of Sciences and Mathematics, School of Letters and Sciences, Arizona State University, Polytechnic Campus, Mesa, AZ, 85212, USA

    Michelle Zandieh

Authors
  1. Chris Rasmussen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Megan Wawro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michelle Zandieh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chris Rasmussen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-014-9583-x

Keywords

Search

Navigation