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\( {V}_B-{V}_A={\int}_A^Bf(x) dx \). The Use of Integrals in Engineering Programmes: a Praxeological Analysis of Textbooks and Teaching Practices in Strength of Materials and Electricity and Magnetism Courses

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Abstract

Calculus is a required course in most engineering programmes and is intended to provide students with tools they will use throughout the rest of their training. However, research seems to indicate that calculus is not very prevalent in the engineering workplace or in university engineering courses. To shed some light on how calculus is actually used in engineering programmes, we analysed two courses (Strength of Materials and Electricity and Magnetism) to examine their use of integrals. We first conducted a praxeological analysis examining the introduction of two topics (bending moments and electric potential) in the courses’ textbooks before interviewing the teachers of each course to compare their praxeologies with those of the books. Our results show that integrals are used to introduce engineering notions in both courses, with praxeologies that combine elements from mathematics and engineering but that do not call upon calculus techniques to any great extent. Moreover, integrals do not factor highly in students’ assessments in either course, which implies that students may pass these courses without any knowledge of integrals as taught in their calculus courses.

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Notes

  1. In this paper, we use “mathematics” when referring to literature about mathematics courses in general, including calculus. We use “calculus” in reference to results or statements that are specific to calculus content.

  2. The Mathematics Working Group of the European Society for Engineering Education (SEFI) proposed a mathematics curriculum for engineering programmes based on the following eight mathematical competences (Alpers 2013): thinking mathematically; reasoning mathematically; posing and solving mathematical problems; modelling mathematically; representing mathematical entities; handling mathematical symbols and formalism; communicating in, with, and about mathematics; and making use of aids and tools.

  3. We have simplified the notation for the purposes of this paper

  4. This book is a translation of the classic, international reference, Benson (1996).

  5. Other exchanges took place in March 2016, November 2016, and August 2019, to guide us in our analyses and deepen our understanding of bending moments and the way the topic is taught.

  6. An earlier exchange also took place in November 2017.

  7. Faulkner et al. (2020) justify this fact, stating that, historically, calculus was used to develop beam theory nearly a century before the first epsilon-delta limit proof was published: “The ill-behaved functions that make mathematically rigorous definitions worthwhile are not encountered.” (p. 417)

  8. See the supplementary materials (B) for an example.

  9. We note that, in E1’s institution, the notation p(x) is used instead of w(x).

  10. These values are obtained using the technique τSM-1.

  11. ATD distinguishes different levels of didactic codetermination (society, school, pedagogy, discipline, area, sector, theme, question), which each introduce particular restrictions (see Barbé et al. 2005, for more details).

References

  • Alpers, B. (Ed.). (2013). A framework for mathematics curricula in engineering education (a report of the mathematics working group). Brussels: European Society for Engineering Education (SEFI).

    Google Scholar 

  • Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at post-secondary level. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1011–1049). Charlotte: Information Age Publishers.

    Google Scholar 

  • Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice: The case of limits of functions in Spanish high schools. Educational Studies in Mathematics, 59(1), 235–268.

    Article  Google Scholar 

  • Beer, F., Johnston, E. R., DeWolf, T., & Mazurek, D. F. (2012). Mechanics of materials (7th ed.). New Yorks: McGraw-Hill Education.

    Google Scholar 

  • Benson, H. (1996). University physics. New York: John Wiley & Sons Inc..

    Google Scholar 

  • Benson, H., Lachance, M., Séguin, M., Villeneuve, B., Marcheterre, B. (2015). Physique 2. Électricité et magnétisme (5th edition). Montréal: Éditions du Renouveau Pédagogique (ERPI).

  • Castela, C. (2016). When praxeologies move from an institution to another: An epistemological approach to boundary crossing. In R. Göller, R. Biehler, R. Hochmuth, & H.-G. Rück (Eds.), Proceedings of the KHDM conference: Didactics of mathematics in higher education as a scientific discipline (pp. 420–427). Kassel: Universitätsbibliothek Kassel.

    Google Scholar 

  • Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221–266.

    Google Scholar 

  • Christensen, O. R. (2008). Closing the gap between formalism and application—PBL and mathematical skills in engineering. Teaching Mathematics and its Applications, 27(3), 131–139.

    Article  Google Scholar 

  • Engelbrecht, J., Bergsten, C., & Kågesten, O. (2017). Conceptual and procedural approaches to mathematics in the engineering curriculum: Views of qualified engineers. European Journal of Engineering Education, 42(5), 570–586.

    Article  Google Scholar 

  • Faulkner, B. E. (2018). Mathematical maturity for engineering students (Doctoral dissertation, University of Illinois at Urbana-Champaign, Urbana, USA). Retrieved from https://www.ideals.illinois.edu/handle/2142/101539

  • Faulkner, B., Earl, K., & Herman, G. (2019). Mathematical maturity for engineering students. International Journal of Research in Undergraduate Mathematics Education, 5(1), 97–128.

    Article  Google Scholar 

  • Faulkner, B., Johnson-Glauch, N., San Choi, D., & Herman, G. L. (2020). When am I ever going to use this? An investigation of the calculus content of core engineering courses. Journal of Engineering Education, 109(3), 402–423.

    Article  Google Scholar 

  • Flegg, J., Mallet, D., & Lupton, M. (2011). Students’ perceptions of the relevance of mathematics in engineering. International Journal of Mathematical Education in Science and Technology, 43(6), 717–732.

    Article  Google Scholar 

  • González-Martín, A. S., Gueudet, G., Barquero, B., & Romo-Vázquez, A. (2021). Mathematics and other disciplines. In V. Durand-Guerrier, R. Hochmut, E. Nardi, & C. Winsløw (Eds.), Research and Development in University Mathematics Education (pp. 169–189). Routledge ERME Series: New Perspectives on Research in Mathematics Education.

  • González-Martín, A. S., & Hernandes Gomes, G. (2017). How are Calculus notions used in engineering? An example with integrals and bending moments. In T. Dooley & G. Gueudet (Eds.), Proceedings of the 10th congress of the European Society for Research in mathematics education (pp. 2073–2080). Dublin: DCU Institute of Education and ERME.

    Google Scholar 

  • González-Martín, A. S., & Hernandes-Gomes, G. (2019). How engineers use integrals: The cases of mechanics of materials and electromagnetism. In M. Graven, H. Venkat, A. Essien, & P. Vale (Eds.), Proceedings of the 43rd conference of the International Group for the Psychology of mathematics education (Vol. 2, pp. 280–287). Pretoria: PME.

    Google Scholar 

  • González-Martín, A. S., & Hernandes-Gomes, G. (2020). Mathematics in engineering programs: What teachers with different academic and professional backgrounds bring to the table. An institutional analysis. Research in Mathematics Education, 22(1), 67–86.

    Article  Google Scholar 

  • Kent, P., & Noss, R. (2003). Mathematics in the University Education of Engineers (a report to the Ove Arup Foundation). Retrieved from Ove Arup Foundation website: https://www.ovearupfoundation.org/public/data/chalk/file/a/5/Kent-Noss-report.pdf

  • Legrand, M. (2001). Scientific debate in mathematics courses. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 127–135). The Netherlands: Kluwer Academic Publishers.

    Google Scholar 

  • Loch, B., & Lamborn, J. (2016). How to make mathematics relevant to first-year engineering students: Perceptions of students on student-produced resources. International Journal of Mathematical Education in Science and Technology, 47(1), 29–44.

    Article  Google Scholar 

  • Mesa, V., & Griffiths, B. (2012). Textbook mediation of teaching: An example from tertiary mathematics instructors. Educational Studies in Mathematics, 79(1), 85–107.

    Article  Google Scholar 

  • Noss, R. (2002). Mathematical epistemologies at work. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th conference of the International Group for the Psychology of mathematics education (Vol. 1, pp. 47–63). Norwich: PME.

    Google Scholar 

  • Peters, J., Hochmuth, R., & Schreiber, S. (2016). Applying an extended praxeological ATD-model for analyzing different mathematical discourses in higher engineering courses. In R. Göller, R. Biehler, R. Hochmuth, & H.-G. Rück (Eds.), Proceedings of the KHDM conference: Didactics of mathematics in higher education as a scientific discipline (pp. 173–179). Kassel: Universitätsbibliothek Kassel.

  • Quéré, P.-V. (2019). Les mathématiques dans la formation des ingénieurs et sur leur lieu de travail: études et propositions (cas de la France). (Doctoral dissertation, Université de Bretagne Occidentale, Rennes, France). Retrieved from https://www.theses.fr/2019BRES0041

  • Saldaña, J. (2013). The coding manual for qualitative researchers (2nd ed.). Thousand Oaks: Sage.

    Google Scholar 

  • Stewart, J. (2012). Calculus (7th ed.). Belmont: Cengage Learning.

    Google Scholar 

  • Van Dyken, J., & Benson, L.C. (2019). Precalculus as a death sentence for engineering majors: A case study of how one student survived. International Journal of Research in Education and Science, 5(1), 355–373.

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Acknowledgments

The author wishes to thank E1 and E2 for their participation and for sharing their experience. He also wishes to thank Gisela Hernandes Gomes for her collaboration with the interviews and in the data analysis. This research was funded by Grant 435-2016-0526 of the Social Sciences and Humanities Research Council (SSHRC) through Canada’s Insight programme.

The author states that there is no conflict of interest.

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  1. Département de Didactique, Faculté des Sciences de l’Éducation, Université de Montréal, Montréal, QC, H3C 3J7, Canada

    Alejandro S. González-Martín

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Correspondence to Alejandro S. González-Martín.

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González-Martín, A.S. \( {V}_B-{V}_A={\int}_A^Bf(x) dx \). The Use of Integrals in Engineering Programmes: a Praxeological Analysis of Textbooks and Teaching Practices in Strength of Materials and Electricity and Magnetism Courses. Int. J. Res. Undergrad. Math. Ed. 7, 211–234 (2021). https://doi.org/10.1007/s40753-021-00135-y

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