Skip to main content
Springer Nature Link
Log in

Collective and Individual Mathematical Progress: Layering Explanations in the Case of the Sierpiński Triangle

  • Published:
International Journal of Research in Undergraduate Mathematics Education Aims and scope Submit manuscript

Abstract

This paper presents methodology aimed at developing a rich understanding of the interplay of mathematical progress in the different social settings in which learning in inquiry-oriented classrooms occurs: individually; in small groups; and as a whole class. For this purpose, we enhance a theoretical-methodological approach of coordinating Documenting Collective Activity and the Recognizing-Building-Constructing model of Abstraction in Context that have been developed in earlier studies. We do this using an intact lesson on the area and perimeter of the Sierpiński triangle in a mathematics education master’s level course on Chaos and Fractals. The enhancement of the methodology allows integrating Collective and Individual Mathematical Progress (CIMP) by Layering the Explanations (LE) provided by the two approaches, and thus exhibits the complexity of learning processes in inquiry-oriented classrooms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
€32.70 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Finland)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. In place of taken-as-shared (Yackel & Cobb, 1996).

  2. The original formulation of criterion 3 used multiple arguments across several class sessions. In this report, we expand the use of criterion 3 to include multiple arguments within the same class session. This adaptation is justified based on the facts that (i) in the context of undergraduate mathematics, it is atypical to stay on the same topic for several class sessions; (ii) even Stephan and Rasmussen (2002) evidenced some of the ideas that function-as-if-shared on time scales consistent with what we did here; (iii) in this paper, we focus only on ideas that function-as-if-shared and not on classroom mathematical practices (for which longer time scales would be required).

  3. Turns are numbered in three strands: consecutively throughout the lesson within Group A (A1, A2, …, A758), consecutively within Group B (B1, B2, …, B422), and consecutively within the whole class discussions (W1, W2, …, W195).

  4. With each story we present the part of the transcript relevant for that story. The entire transcript (25 turns) is presented in Appendix Table 4.

References

  • Ben-Zvi, D., & Arcavi, A. (2001). Junior high school students’ construction of global views of data and data representations. Educational Studies in Mathematics, 45(1), 35–65. https://doi.org/10.1023/A:1013809201228

    Article  Google Scholar 

  • Bikner-Ahsbahs, A., & Prediger, S. (2014). Networking as research practices: Methodological lessons learnt from the case studies. In A. Bikner & S. Prediger (Eds.), Networking of theories as a research practice in mathematics education (pp. 235–247). Springer.

    Chapter  Google Scholar 

  • Blumer, H. (1969). Symbolic interactionism: Perspectives and method. Prentice-Hall.

    Google Scholar 

  • Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190. https://doi.org/10.1080/00461520.1996.9653265

    Article  Google Scholar 

  • Cole, R., Becker, N., Towns, M., Sweeney, G., Wawro, M., & Rasmussen, C. (2012). Adapting a methodology from mathematics education research to chemistry education research: Documenting collective activity. International Journal of Science and Mathematics Education, 10(1), 193–211. https://doi.org/10.1007/s10763-011-9284-1

    Article  Google Scholar 

  • Dreyfus, T., Hershkowitz, R., & Schwarz, B. (2015). The nested epistemic actions model for abstraction in context - Theory as methodological tool and methodological tool as theory. In A. Bikner-Ahsbahs, C. Knipping & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education: Examples of methodology and methods (pp. 185–217). Springer, Advances in Mathematics Education series.

  • Freeman, S., Eddy, S. L., Mcdonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 111(23), 8410–8415. https://doi.org/10.1073/pnas.1319030111

    Article  Google Scholar 

  • Hershkowitz, R., Dreyfus, T., & Tabach, M. (2022). Constructing the self-similarity concept. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-022-00173-0

    Article  Google Scholar 

  • Hershkowitz, R., Schwarz, B., & Dreyfus, T. (2001). Abstraction in Context: Epistemic actions. Journal for Research in Mathematics Education, 32(2), 195–222. https://doi.org/10.2307/749673

    Article  Google Scholar 

  • Hershkowitz, R., Tabach, M., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts in a probability classroom – A case study coordinating two methodologies. ZDM - Mathematics Education, 46(3), 363–387. https://doi.org/10.1007/s11858-014-0576-0

    Article  Google Scholar 

  • Kuster, G., Johnson, E., Keene, K., & Andrews-Larson, C. (2018). Inquiry-oriented instruction: A conceptualization of the instructional components and practices. Primus, 28(1), 13–30. https://doi.org/10.1080/10511970.2017.1338807

    Article  Google Scholar 

  • Laursen, S. L., & Rasmussen, C. (2019). I on the prize: Inquiry approaches in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 5(1), 129–146. https://doi.org/10.1007/s40753-019-00085-6

    Article  Google Scholar 

  • Martin, L., Towers, J., & Pirie, S. (2006). Collective mathematical understanding as improvisation. Mathematical Thinking and Learning, 8(2), 149–183. https://doi.org/10.1207/s15327833mtl0802_3

    Article  Google Scholar 

  • Peitgen, H.-O., Jürgens, H., & Saupe, D. (1990/2003). Fractals: An animated discussion. Films for the Humanities & Sciences.

  • Rasmussen, C., Apkarian, N., Tabach, M., & Dreyfus, T. (2020). Ways in which engaging in someone else’s reasoning is productive for one’s own reasoning. Journal of Mathematical Behavior, 58, 100742. https://doi.org/10.1016/j.jmathb.2019.100742

  • Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.). Handbook of design research methods in education: Innovations in science, technology, engineering and mathematics learning and teaching (pp. 195–215). Routledge.

  • Rasmussen, C., & Wawro, M. (2017). Post-calculus research in undergraduate mathematics education. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 551–581). National Council of Teachers of Mathematics.

  • Rasmussen, C., Wawro, M., & Zandieh, M. (2015). Examining individual and collective level mathematical progress. Educational Studies in Mathematics, 88(2), 259–281. https://doi.org/10.1007/s10649-014-9583-x

  • Saxe, G. B. (1999). Cognition, development, and cultural practices. In E. Turiel (Ed.), Culture and development: New directions in child psychology (pp. 19–35). Jossey-Bass.

    Google Scholar 

  • Saxe, G. B., & Esmonde, I. (2005). Studying cognition in flux: A historical treatment of Fu in the shifting structure of Oksapmin mathematics. Mind, Culture, and Activity, 12(3–4), 171–225. https://doi.org/10.1080/10749039.2005.9677810

    Article  Google Scholar 

  • Saxe, G. B., & Farid, A. M. (2022). The interplay between individual and collective activity: An analysis of classroom discussions about the Sierpinski Triangle. The International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-021-00151-y

    Article  Google Scholar 

  • Sfard, A. (2022). Taming fantastic beasts of mathematics: Struggling with incommensurability. The International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-021-00156-7

    Article  Google Scholar 

  • Sierpiński, W. (1915). Sur une courbe dont tout point est un point de ramification. Comptes Rendues Hebdomadaires de l’Académie des Sciences, 160, 302–305. Accessed March 4, 2021, https://gallica.bnf.fr/ark:/12148/bpt6k31131.

    Google Scholar 

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

    Article  Google Scholar 

  • Tabach, M., Hershkowitz, R., Rasmussen, C., & Dreyfus, T. (2014). Knowledge shifts in the classroom – A case study. Journal of Mathematical Behavior, 33, 192–208. https://doi.org/10.1016/j.jmathb.2013.12.001

  • Tabach, M., Rasmussen, C., Dreyfus, T., & Apkarian, N. (2020). Towards an argumentative grammar for networking: A case of coordinating two approaches. Educational Studies in Mathematics, 103(2), 139–155. https://doi.org/10.1007/s10649-020-09934-7

  • Theobald, E. J., Hill, M. J., Tran, E., Agrawal, S., Arroyo, E. N., Behling, S., … Freeman, S. (2020). Active learning narrows achievement gaps for underrepresented students in undergraduate science, technology, engineering, and math. Proceedings of the National Academy of Sciences, 201916903. https://doi.org/10.1073/pnas.1916903117

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

    Google Scholar 

  • Towers, J., & Martin, L. C. (2015). Enactivism and the study of collectivity. ZDM - Mathematics Education, 47(2), 247–256. https://doi.org/10.1007/s11858-014-0643-6

    Article  Google Scholar 

  • Treffers, A. (1993). Wiskobas and Freudenthal - Realistic Mathematics Education. Educational Studies in Mathematics, 25(1–2), 89–108. https://doi.org/10.1007/BF01274104

    Article  Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. https://doi.org/10.5951/jresematheduc.27.4.0458

    Article  Google Scholar 

Download references

Acknowledgements

This research has been partially supported by the Israel Science Foundation under grant #438/15.

Author information

Authors and Affiliations

  1. Department of Mathematics, Science and Technology Education, School of Education, Tel Aviv University, P. O. Box 39040, 69978, Tel Aviv, Israel

    Tommy Dreyfus & Michal Tabach

  2. Arizona State University, Tempe, AZ, USA

    Naneh Apkarian

  3. San Diego State University, San Diego, CA, USA

    Chris Rasmussen

Authors
  1. Tommy Dreyfus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Naneh Apkarian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chris Rasmussen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Michal Tabach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tommy Dreyfus.

Ethics declarations

All authors contributed to the study conception, design, data analysis and writing. All authors read and approved the final manuscript. There are no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Table 4

Table 4 Transcript of whole class discussion 9
Full size table

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dreyfus, T., Apkarian, N., Rasmussen, C. et al. Collective and Individual Mathematical Progress: Layering Explanations in the Case of the Sierpiński Triangle. Int. J. Res. Undergrad. Math. Ed. 9, 694–722 (2023). https://doi.org/10.1007/s40753-022-00211-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40753-022-00211-x

Keywords

Search

Navigation