Skip to main content
SpringerLink
Log in

Habermas’ construct of rationality in the analysis of the mathematical problem-solving process

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

Abstract

Our study aims to determine how Habermas’ construct of rationality can serve to identify and interpret the difficulties experienced by university students in the mathematical problem-solving process. To this end, a problem which required modelling and solving a differential equation was used. The problem-solving processes of university students were analysed based on rationality components. The findings demonstrated that the problems in epistemic rationality such as predominance of the figure over the definition and/or theorems, use of dogmatic knowledge, intuitive generalizations, lack of prior knowledge, incorrect recognition of the differential equation prevented the choosing and using of an appropriate problem-solving method, leading to problems in teleological rationality. It was determined that the student performance in communicative rationality was negatively affected by problems in epistemic rationality such as using of knowledge acquired by rote and predominance of figure prototype on the definitions and/or theorems. Throughout the analysis, it is required to define two new sub-components, named “geometric” and “algebraic” representation under the modelling requirements of epistemic rationality. It is advised to use the extended version of Habermas’ construct of rationality to examine the performance of students in mathematical activities to get more detailed and accurate results.

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
Fig. 11
Fig. 12

Similar content being viewed by others

Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  • Allen, K. (2006). Students’ participation in a differerential equations class: Parametric reasoning to understand systems. [Unpublished doctoral dissertation]. Department of Mathematics and Computer Science, Purdue University.

  • Arslan, S. (2010). Do students really understand what an ordinary differential equation is? International Journal of Mathematical Education in Science and Technology, 41(7), 873–888.

    Article  Google Scholar 

  • Boero, P. (2006). Habermas’ theory of rationality as a comprehensive frame for conjecturing and proving in school. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. II, pp. 185–192). PME.

  • Boero, P., & Morselli, F. (2009). The use of algebraic language in mathematical modelling and proving in the perspective of Habermas’ theory of rationality. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 964–973). INRP.

  • Boero, P. & Planas, N. (2014). Habermas’ construct of rational behaviour in mathematics education: New advances and research questions. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 1, pp. 205–208). PME.

  • Boero, P., Douek, N., Morselli, F. & Pedemonte, B. (2010). Argumentation and proof: A contribution to theoretical perspectives and their classroom implementation. In M. M. F. Pinto, & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (pp. 179–209). PME.

  • Boero, P., Guala, E. & Morselli, F. (2013). Crossing the borders between mathematical domains: A contribution to frame the choice of suitable tasks in teacher education. In M. A. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (pp. 97–104). PME.

  • Buendia, G., & Cordero, F. (2013). The use of graphs in specific situations of the initial conditions of linear differential equations. International Journal of Mathematical Education in Science and Technology, 44(6), 927–937.

    Article  Google Scholar 

  • Camacho-Machín, M., Perdomo-Díaz, J., & Santos Trigo, M. (2012). An exploration of students’ conceptual knowledge built in a first ordinary differential equations course (Part I). The Teaching of Mathematics, 15(1), 1–20.

    Google Scholar 

  • Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. Kelley & L. Richard (Eds.), Handbook of research Design in Mathematics and Science Education (pp. 341–385). Routledge.

    Google Scholar 

  • Czocher, A. J. (2018). How does validating activity contribute to the modelling process? Educational Studies in Mathematics, 99, 137–159.

    Article  Google Scholar 

  • Dana-Picard, T., & Kidron, I. (2007). Exploring the phase space of a system of differential equations: Different mathematical registers. International Journal of Science and Mathematics Education, 6(4), 695–717.

    Article  Google Scholar 

  • Donovan, E. J. (2002). Students’ understanding of first-order differential equations. [Unpublished doctoral dissertation]. Department of Learning and Instruction, University of New York.

  • Douek, N. (2014). Pragmatic potential and critical issues. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 1, pp. 209–213). PME.

  • Duval, R. (1991). Structure du raisonnement déductif et apprentissage de la demonstration. Educational Studies in Mathematics, 22, 233–261.

    Article  Google Scholar 

  • Fischbein, E. (1987). Intuition in science and mathematics. Reidel.

    Google Scholar 

  • Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24, 139–162.

    Article  Google Scholar 

  • Fischbein, E. & Kedem, I. (1982). Proof and certitude in the development of mathematical thinking, In A. Vermandel (Ed.), Proceedings of the Sixth International Conference for the Psychology of Mathematical Education, Universitaire Instelling, Antwerpen.

  • Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. International Journal of Science Education, 20(10), 1193–1211.

    Article  Google Scholar 

  • Habermas, J. (2003). Truth and justification. MIT Press.

    Google Scholar 

  • Habre, S. (2000). Exploring students’ strategies to solve ordinary differential equations in a reformed setting. Journal of Mathematical Behavior, 18(4), 455–472.

    Article  Google Scholar 

  • Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on learning and teaching proof. In F. Lester (Ed.), Handbook of Research on Teaching and Learning Mathematics (pp. 805–842). Information Age Publishing.

    Google Scholar 

  • Hatano, G., & Wertsch, J. V. (2001). Sociocultural approaches to cognitive development: The constitutions of culture in mind. Human Development, 44, 77–83.

    Article  Google Scholar 

  • Ismail, Z. B., Zeynivandnezhad, F., Mohammad Y. B., & David, E. (2014). Computing in differential equations with mathematical thinking approach among engineering students. International Conference on Teaching and Learning in Computing and Engineering, Kuching, Malaysia. https://doi.org/10.1109/LaTiCE.2014.39

  • Laborde, C. (2003). Geometrie-Periode 2000 et après. In D. Coray, F. Furinghetti, H. Gispert, B. R. Hodgson, & G. Schubring (Eds.), One hundred years of L’Enseignement Mathematique: Moments of mathematical education in the twentieth century. Monograph 39 (pp. 133–154). L’Enseignement Mathematique.

  • Martignone, F. & Sabena, C. (2014). Analysis of argumentation processes in strategic interaction problems. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education (Vol. 1, pp. 218–223). PME.

  • Morselli, F. & Boero, P. (2009). Proving as a rational behaviour: Habermas’ construct of rationality as a comprehensive frame for research on the teaching and learning of proof. In V. Durand-Guerrier, S. Soury-Lavergne & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 211–220). CERME.

  • Morselli, F., & Boero, P. (2011). Using Habermas’ theory of rationality to gain insight into student’s understanding of algebraic language. In G. Kaiser & B. Sriraman (Eds.), Early Algebraization, Advances in Mathematics Education (pp. 453–479). Springer.

  • 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., & Keene, K. (2019). Knowing solutions to differential equations with rate of change as a function: Waypoints in the journey. Journal of Mathematical Behavior, 56, 100695.

  • Raychaudhuri, D. (2008). Dynamics of a definition: A framework to analyse student construction of the concept of solution to a differential equation. International Journal of Mathematical Education in Science and Technology, 39(2), 161–177.

  • Rowland, D. R., & Jovanoski, Z. (2004). Student interpretations of the terms in first-order ordinary differential equations in modelling contexts. International Journal of Mathematical Education in Science and Technology, 35(4), 503–516.

  • Sijmkens, E., Scheerlinck, N., De Cock M. & Deprez, J. (2022). Benefits of using context while teaching differential equations. International Journal of Mathematical Education in Science and Technology, Advance online publication.

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

  • Trigueros, M. (2004). Understanding the meaning and representation of straight line solutions of systems of differential equations. In D. McDougall, & J. Ross (Eds.), Proceedings of the twenty-sixth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. (pp. 127–134). Preney Print and Litho Inc.

  • Upton, S. D. (2004). Students’ solution strategies to differerential equations problems in mathematical and nonmathematical contexts. [Unpublished doctoral dissertation]. The Arizona State Universıty.

Download references

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. Department of Mathematics and Science Education, Hacettepe University, Ankara, Turkey

    Selin Urhan & Ali Bülbül

Authors
  1. Selin Urhan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Bülbül
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Selin Urhan.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's note

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

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

Urhan, S., Bülbül, A. Habermas’ construct of rationality in the analysis of the mathematical problem-solving process. Educ Stud Math 112, 175–197 (2023). https://doi.org/10.1007/s10649-022-10188-8

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-022-10188-8

Keywords

Search

Navigation