Skip to main content
SpringerLink
Log in

Development and evolution of instrumented schemes: a case study of learning programming for mathematical investigations

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

Abstract

We are interested in understanding how university students learn to use programming as a tool for “authentic” mathematical investigations (i.e., similar to how some mathematicians use programming in their research work). The theoretical perspective of the instrumental approach offers a way of interpreting this learning in terms of development of schemes by students; these development processes are called instrumental geneses. Nevertheless, how these schemes evolve has not been fully explained. In this paper, we propose to enrich the theoretical frame of the instrumental approach by a model of scheme evolution and to use this new approach to investigate learning to use programming for pure and applied mathematics investigation projects at the university level. We examine the case of one student completing four investigation projects as part of a course workload. We analyze the productive and constructive aspects of the student’s activity and the dynamic aspect of the instrumental geneses by identifying the mobilization and evolution of schemes. We argue that our approach constitutes a new theoretical and methodological contribution deepening the understanding of students’ instrumented learning processes. Identifying instrumented schemes illuminates in particular how mathematical knowledge and programming knowledge are combined. The analysis in terms of scheme evolutions reveals which characteristics of the situations lead to such evolutions and can thus inform the design of teaching.

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

Similar content being viewed by others

Notes

  1. “The quick brown fox jumps over the lazy dog” is a sentence using all the letters in the alphabet.

References

  • Abrahamson, D., Berland, M., Shapiro, B., Unterman, J., & Wilensky, U. (2006). Leveraging epistemological diversity through computer-based argumentation in the domain of probability. For the Learning of Mathematics, 26(3), 19–45. https://www.jstor.org/stable/40248549

  • Balt, K., & Buteau, C. (2020, September 4). Process of using programming for pure/applied mathematics investigations [Video]. YouTube. https://youtu.be/irTlCE-eXhc. Accessed 8 Dec 2021.

  • Béguin, P., & Rabardel, P. (2000). Designing for instrument-mediated activity. Scandinavian Journal of Information Systems, 12(1), Article 1. https://aisel.aisnet.org/sjis/vol12/iss1/1. Accessed 8 Dec 2021.

  • Benton, L., Hoyles, C., Kalas, I., Noss, R. (2017). Bridging primary programming and mathematics: Some findings of design research in England. Digital Experiences in Mathematics Education, 3(2), 115–138.  https://doi.org/10.1007/s40751-017-0028-x

    Article  Google Scholar 

  • Broley, L., Caron, F., Saint-Aubin, Y. (2018). Levels of programming in mathematical research and university mathematics education International Journal of Research in Undergraduate Mathematics Education, 4(1), 38–55. https://doi.org/10.1007/s40753-017-0066-1

    Article  Google Scholar 

  • Buteau, C., Gueudet, G., Muller, E., Mgombelo, J., Sacristán, AI. (2020). University students turning computer programming into an instrument for “authentic” mathematical work. International Journal of  Mathematical Education in Science and Technology, 51(7), 1020–1041. https://doi.org/10.1080/0020739X.2019.1648892

    Article  Google Scholar 

  • Buteau, C., & Muller, E. (2010). Student development process of designing and implementing exploratory and learning objects. 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. 1111–1120). Institut national de recherche pédagogique.

  • Buteau, C., Muller, E. (2017). Assessment in undergraduate programming-based mathematics courses. Digital Experiences in Mathematics Education, 3(2), 97–114. https://doi.org/10.1007/s40751-016-0026-4

    Article  Google Scholar 

  • Buteau, C., Muller, E., & Marshall, N. (2014). Competencies developed by university students in microworld-type core mathematics courses. In C. Nicol, P. Liljedahl, S. Oesterle, & D. Allan (Eds.), Proceedings of the Joint Meeting of the International Group Psychology Mathematics Education (PME 38) (Vol. 2, pp. 209–216). https://eric.ed.gov/?id=ED599720. Accessed 8 Dec 2021.

  • Buteau, C., Muller, E., Marshall, N., Sacristán, AI., Mgombelo, J. (2016). Undergraduate mathematics students appropriating programming as a tool for modelling, simulation, and visualization: A case study. Digital Experiences in Mathematics Education, 2(2), 142–166. https://doi.org/10.1007/s40751-016-0017-5

    Article  Google Scholar 

  • Buteau, C., Muller, E., Mgombelo, J., & Sacristán, A. (2018). Computational thinking in university mathematics education: A theoretical framework. In (Eds.) A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown, Proceedings of the 21st Annual Conference on Research in Undergraduate Mathematics Education (pp.1171–1179), San Diego, California.

  • Buteau, C., Muller, E., & Ralph, B. (2015, June). Integration of programming in the undergraduate mathematics program at Brock University. In Online Proceedings of the Math + Coding Symposium, London, ON, Canada. https://researchideas.ca/coding/docs/ButeauMullerRalph-Coding+MathProceedings-FINAL.pdf. Accessed 8 Dec 2021.

  • Buteau, C., Muller, E., Santacruz Rodriguez, M., Gueudet, G., Mgombelo, J., & Sacristán, A. I. (2020). Instrumental orchestration of using programming for mathematics investigations. In A. Donevska-Todorova, E. Faggiano, J. Trgalova, Z. Lavicza, R. Weinhandl, et al. (Eds.), Proceedings of the Tenth ERME Topic Conference (ETC 10) on Mathematics Education in the Digital Age (MEDA) (pp. 443−450). https://hal.archives-ouvertes.fr/hal-02932218/document#page=456. Accessed 8 Dec 2021.

  • Buteau, C., Sacristán, AI., Muller, E. (2019). Roles and demands in constructionist teaching of computational thinking in university mathematics. Constructivist Foundations, 14(3), 294–309.

  • Coulet, JC. (2019). The organization activity: A foresight approach of theoretical knowledge evolution in management science. Technological Forecasting and Social Change, 140, 160–168. https://doi.org/10.1016/j.techfore.2018.04.009

    Article  Google Scholar 

  • Drijvers, P., Godino, JD., Font, V.,  Trouche, L. (2013). One episode, two lenses. Educational Studies in Mathematics, 82(1), 23–49. https://doi.org/10.1007/s10649-012-9416-8

    Article  Google Scholar 

  • Feurzeig, W., & Lukas, G. (1972). LOGO—A programming language for teaching mathematics. Educational Technology, 12(3), 39–46. http://www.jstor.org/stable/44417811

  • Geraniou, E., Jankvist, UT., (2019). Towards a definition of “mathematical digital competency”. Educational Studies in Mathematics, 102(1), 29−45. https://doi.org/10.1007/s10649-019-09893-8

    Article  Google Scholar 

  • Gueudet, G., Buteau, C., Muller, E., Mgombelo, J., & Sacristán, A. (2020, September). Programming as an artefact: What do we learn about university students’ activity? In T. Hausberger, M. Bosch & F. Chellougui (Eds.), Proceedings of the Third Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2020, 12–19 September 2020) (pp. 443–452). University of Carthage and INDRUM. https://hal.archives-ouvertes.fr/hal-03113851/document. Accessed 8 Dec 2021.

  • Guin, D., Ruthven, K., & Trouche, L. (Eds.). (2005). The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument. Springer. https://doi.org/10.1007/b101602

  • Knuth, DE. (1985). Algorithmic thinking and mathematical thinking. The American Mathematical Monthly, 92(1), 170–181.

    Article  Google Scholar 

  • Lagrange, J.-B. & Rogalski, J. (2017). Savoirs, concepts et situations dans les premiers apprentissages en programmation et en algorithmique. Annales de Didactique et de Sciences Cognitives, 22, 119–158. https://hal.archives-ouvertes.fr/hal-01740442. Accessed 8 Dec 2021.

  • Leron, U., Dubinsky, E. (1995). An abstract algebra story. American Mathematical Monthly, 102(3), 227–242. https://doi.org/10.1080/00029890.1995.11990563

    Article  Google Scholar 

  • Lockwood, E., Chenne De, A. (2019). Enriching students’ combinatorial reasoning through the use of loops and conditional statements in Python. International Journal of Research in Undergraduate Mathematics Education, 6, 303–346. https://doi.org/10.1007/s40753-019-00108-2

    Article  Google Scholar 

  • Ludwig, P., Tongen, A., Walton, B. (2018). Two project-based strategies in an interdisciplinary mathematical modeling in biology course. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 28(4), 300–317. https://doi.org/10.1080/10511970.2016.1246495

    Article  Google Scholar 

  • Marshall, N., Buteau, C. (2014). Learning by designing and experimenting with interactive, dynamic mathematics exploratory objects. International Journal for Technology in Mathematics Education, 21(2), 49–64.

    Google Scholar 

  • Mascaró, M., Sacristán, AI., Rufino, MM. (2016). For the love of statistics: Appreciating and learning to apply experimental analysis and statistics through computer programming activities. Teaching Mathematics and Its Applications, 35(2), 74–87. https://doi.org/10.1093/teamat/hrw006

    Article  Google Scholar 

  • Misfeldt, M., & Ejsing-Duun, S. (2015). Learning mathematics through programming: An instrumental approach to potentials and pitfalls. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (CERME9, 4–8 February 2015) (pp. 2524–2530). Charles University in Prague, Faculty of Education and ERME. https://hal.archives-ouvertes.fr/hal-01289367/document. Accessed 8 Dec 2021.

  • Muller, E., Buteau, C., Ralph, B., Mgombelo, J. (2009). Learning mathematics through the design and implementation of exploratory and learning objects. International Journal for Technology in Mathematics Education, 63(2), 63–73.

    Google Scholar 

  • Noss, R., Hoyles, C. (1996). Windows on mathematical meanings: Learning cultures and computers. Kluwer. https://doi.org/10.1007/978-94-009-1696-8

    Article  Google Scholar 

  • Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books.

    Google Scholar 

  • Papert, S. (1991). Situating constructionism. In I. Harel & S. Papert (Eds.), Constructionism: Research reports and essays, 1985–1990 (pp. 1−12). Ablex. http://www.papert.org/articles/SituatingConstructionism.html. Accessed 8 Dec 2021.

  • Piaget J. (1975). L’Équilibration des structures cognitives: Problème central du développement. Presses universitaires de France.

  • Rabardel, P. (2002). People and technology: A cognitive approach to contemporary instruments (H. Wood, Trans.). Université Paris 8.

  • Rabardel, P., Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Interacting with Computers, 15(5), 665–691. https://doi.org/10.1016/S0953-5438(03)00058-4

    Article  Google Scholar 

  • Rabardel, P., Bourmaud, G. (2005). Instruments et systèmes d’instruments. In P. Rabardel & P. Pastré (Eds.), Modèles du sujet pour la conception—Dialectiques activités développement (pp. 211−229). Octarès.

    Google Scholar 

  • Ralph, W. (2001). Mathematics takes an exciting new direction with MICA program. Brock Teaching, 1(1), 1.

    Google Scholar 

  • Rogalski , J. (2015). Psychologie de la programmation, didactique de l'informatique : Déjà une histoire... In G.-L.Baron, E. Bruillard & B. Drot-Delange (Eds.). Informatique en éducation : Perspectives curriculaires et didactiques (pp. 280–305).  Presses Universitaires Blaise-Pascal.

  • Schmidt, K., & Winsløw, C. (2018). Task design for engineering mathematics: Process, principles and products. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild & N.M Hogstad (Eds.), Proceedings of the Second Conference of the International Network for Didactic Research in University Mathematics (INDRUM 2018, 5–7 April 2018) (pp. 165–174). Kristiansand, University of Agder and INDRUM. https://orbit.dtu.dk/en/publications/task-design-for-engineering-mathematics-process-principles-and-pr

  • Trouche, L. (2004). Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 9(3), Article 281. https://doi.org/10.1007/s10758-004-3468-5

  • Vergnaud, G. (2009). The theory of conceptual fields. Human Development, 52(2), 83−94. https://doi.org/10.1159/000202727

    Article  Google Scholar 

  • Vermersch, P. (2006). Les fonctions des questions. Expliciter, 65, 1–6. https://expliciter.fr/IMG/pdf/Les_fonctions_des_questions.pdf. Accessed 8 Dec 2021.

  • Vygotsky, L. (1978). Mind in society. Harvard University Press.

    Google Scholar 

  • Wilensky, U. (1995 ). Paradox, programming, and learning probability: A case study in a connected mathematics framework. The Journal of Mathematical Behavior, 14(2), 253–280. https://doi.org/10.1016/0732-3123(95)90010-1

    Article  Google Scholar 

  • Wing, J. M. (2014, January 10). Computational thinking benefits society. Social Issues in Computing. http://socialissues.cs.toronto.edu/index.html%3Fp=279.html

Download references

Acknowledgements

This work is funded by the Social Sciences and Humanities Research Council of Canada (#435-2017-0367) and has received ethics clearance from the Research Ethics Board at Brock University (REB #17-088). We thank all of the research assistants involved in our project for their valuable work toward our research, in particular Kirstin Dreise who significantly contributed by creating Jim’s Excel table of schemes.

Author information

Authors and Affiliations

  1. Etudes sur les Sciences et les Techniques, Université Paris-Saclay, Orsay, 91400, France

    Ghislaine Gueudet

  2. Department of Mathematics and Statistics, Brock University, St. Catharines, ON, Canada

    Chantal Buteau & Eric Muller

  3. Brock University, St. Catharines, ON, Canada

    Joyce Mgombelo

  4. Cinvestav, Mexico City, Mexico

    Ana Isabel Sacristán

  5. Universidad del Valle, Cali, Colombia

    Marisol Santacruz Rodriguez

Authors
  1. Ghislaine Gueudet
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chantal Buteau
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eric Muller
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joyce Mgombelo
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ana Isabel Sacristán
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Marisol Santacruz Rodriguez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ghislaine Gueudet.

Additional information

Publisher's Note

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

Electronic supplementary material

Appendices

Appendix 1 Interview question template for MICA projects (i.e., programming-based pure or applied mathematics investigation projects) and associated parts of the development process model

figure a

Appendix 2 Tool developed and used for the analysis of Jim’s mobilization and evolution of schemes

Figure 4 presents a table illustrating the process used to analyze Jim’s schemes throughout his MICA I experience. It is a screenshot of the “articulating” scheme (goal in the green column A) and its rules-of-action, concepts-in-action, and theorems-in-action components (green column E) after Jim’s first labs, its evolution (green column G) during project 1, and then its mobilization (empty green columns I, K, M, O, Q) in his subsequent labs and projects, progressing in time (along MICA I course activities) from column C to R. These were interpreted from evidence in Jim’s data of rules-of-action (blue cells in columns H, L, P) and operational invariants (yellow cells in columns F, J) with their associated strategy and perception codes listed in column B in blue and yellow cells, respectively, used to identify scheme components and also from inferred evidence in Jim’s data of scheme mobilization (dark green cell in columns N and R, with code in column B). Finally, columns C-D (pre-MICA I) are hidden as they were blank representing no evidence of the scheme.

Fig. 4
figure 4

Analysis of Jim’s scheme of “articulating a mathematical process in the programming language” throughout his MICA I experience

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gueudet, G., Buteau, C., Muller, E. et al. Development and evolution of instrumented schemes: a case study of learning programming for mathematical investigations. Educ Stud Math 110, 353–377 (2022). https://doi.org/10.1007/s10649-021-10133-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-021-10133-1

Keywords

Search

Navigation