The use of figures related to the complex integral and Cauchy’s integral theorem in complex analysis textbooks
DOI:
https://doi.org/10.23925/1983-3156.2024v26i3p303-326Keywords:
Reference epistemological model, Textbook content analysis, Figures, Didactic transposition, Complex analysisAbstract
This study addresses a question about the similarities and differences between original mathematical works in complex analysis and contemporary textbooks, regarding the use of figures (conceived as two-dimensional drawings) to address concepts in this branch of mathematics. In order to answer this question, we analyzed the four main textbooks that are referenced in the teachers’ guides of all Spanish public universities that offer a degree in mathematics. Specifically, we present how these four textbooks structure the concept of complex integral and the proof of Cauchy’s integral theorem. To carry out our analysis, we retrieved a reference epistemological model that describes how historical subjects used figures to develop complex analysis from the first quarter of the 19th century to the first half of the 20th century. The study shows that the four textbooks structure these concepts in such a way that they are related to the most contemporary forms in which they were attended in their historical development. Although, we are not against structuring the content of the textbooks in this way, we will argue how the reference epistemological model can serve as an epistemological alternative for the elaboration of didactic material that takes into account a historical development of complex analysis.
Metrics
References
Ahlfors, L. (1979). Complex Analysis. An introduction to the theory of analytic functions of one complex variable. McGraw-Hill.
Bak, J., & Popvassilev, S. (2017). The Evolution of Cauchy’s Closed Curve Theorem and Newman’s Simple Proof. The American Mathematical Monthly, 124(3), 217–231. https://doi.org/10.4169/amer.math.monthly.124.3.217
Bosch, M., & Gascón, J. (2006). Twenty-five years of the didactic transposition. ICMI bulletin, 58(58), 51–65.
Bottazzini, U., & Gray, J. (2013). Hidden Harmony –Geometric Fantasies. The Rise of Complex Function Theory. Springer.
Cantoral, R. (2020). Socioepistemology in mathematics education. In Encyclopedia of mathematics education (pp. 790–797). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-15789-0_100041
Cantoral, R., & Farfán, R. (2004). La sensibilité à la contradiction: logarithmes de nombres négatifs et origine de la variable complexe. Recherches en Didactique des mathématiques, 24(2-3), 137–168.
Cauchy, A. (1825). Mémoire sur les intégrales définies, prises entre des limites imaginaires. Bure frères.
Clark, C.M. (2019). History and pedagogy of mathematics in mathematics education: History of the field, the potential of current examples, and directions for the future. In U.T. Jankvist, M. Van den Heuvel-Panhuizen, & M. Veldhuis, (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (pp. 29–55). Freudenthal Group & Freudenthal Institute, Utrecht University and ERME.
Conway, J. (1973). Functions of one complex variable. Springer.
D’Azevedo-Breda, A., & Dos Santos, J. (2021). Learning complex functions with GeoGebra. Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales, 45(177), 1262– 1276. https://doi.org/10.18257/raccefyn.1504
Danenhower, P. (2000). Teaching and learning complex analysis at two British Columbia Universities [Doctoral dissertation, Simon Fraser University]. https://www.collectionscanada.gc.ca/obj/s4/f2/dsk1/tape3/PQDD_0008/NQ61636.pdf
Dittman, M., Soto-Johnson, H., Dickinson, S., & Harr, T. (2016). Game building with complex-valued functions. Primus, 27(8-9), 869–879. https://doi.org/10.1080/10511970.2016.1234527
Dixon, J. D. (1971). A brief proof of Cauchy’s integral theorem. Proceedings of the American Mathematical Society, 29(3), 625–626.
Fan, L. (2013). Textbook research as scientific research: towards a common ground on issues and methods of research on mathematics textbooks. ZDM Mathematics Education, 45, 765–777. https://doi.org/10.1007/s11858-013-0530-6
Fan, L., Zhu, Y., & Miao, Z. (2013). Textbook research in mathematics education: development status and directions. ZDM Mathematics Education, 45, 633–646. https://doi.org/10.1007/s11858-013-0539-x
Garcia, S., & Ross, W. (2017). Approaching Cauchy’s Theorem. PRIMUS, 27(8-9), 758–765. https://doi.org/10.1080/10511970.2016.1234525
Gascón, J. (2014). Los modelos epistemológicos de referencia como instrumentos de emancipación de la didáctica y la historia de las matemáticas. Educación Matemática, 25 años, 99–123.
Gómez, B. (2011). El análisis de manuales y la identificación de problemas de investigación en Didáctica de las Matemáticas. PNA, 5(2), 49–65. https://doi.org/10.30827/pna.v5i2.6157
Goursat, E. (1884). Démonstration du théoréme de Cauchy: Extrait d’une lettre adressée à M. Hermite. Acta Mathematica, 4, 197–200. https://doi.org/10.1007/BF02418419
Grattan-Guinness, I. (2004). History or Heritage? An Important Disctinction in Mathematicsa and for Mathematics Education. The American Mathematical Monthly, 111 (1), 1–12. https://doi.org/10.1080/00029890.2004.11920041
Gray, J. (2000). Goursat, Pringsheim, Walsh, and the Cauchy integral theorem. The Mathematical Intelligencer, 22(4), 60–66. https://doi.org/10.1007/BF03026773
Hanke, E. (2022). Aspects and images of complex path integrals [Doctoral thesis, University of Bremen]. https://doi.org/10.26092/elib/1964
Larivière, G. (2014). On Cauchy’s Rigorization of Complex Analysis [Master’s thesis, Simon Fraser University]. https://summit.sfu.ca/item/17300
Manning, K. (1975). The emergence of the Weierstrassian approach to complex analysis. Archive for History of Exact Sciences, 14(4), 297–383. https://doi.org/10.1007/BF00327297
Marsden, J., & Hoffman, M. (1999). Basic complex analysis. W. H. Freeman.
Nemirovsky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the Classroom Floor Becomes the Complex Plane: Addition and Multiplication as Ways of Bodily Navigation. Journal of the Learning Sciences, 21(2), 287–323. https://doi.org/10.1080/10508406.2011.611445
Oehrtman, M., Soto-Johnson, H. & Hancock, B. (2019). Experts’ Construction of Mathematical Meaning for Derivatives and Integrals of Complex-Valued Functions. Int. J. Res. Undergrad. Math. Ed., 5(3), 394–423. https://doi.org/10.1007/s40753-019-00092-7
Panaoura, A., Elia, I., Gagatsis, A., & Giatilis, G.P. (2006). Geometric and algebraic approaches in the concept of complex numbers. International Journal of Mathematical Education in Science and Technology, 37(6), 681–706. https://doi.org/10.1080/00207390600712281
Piña-Aguirre, J. G., & Farfán, R. (2022). Sobre los procesos de demostración y el contexto de producción de la memoria Sur les Intégrales Définies de Augustin-Louis Cauchy. Revista De História Da Educação Matemática, 8, 1–22.
Piña-Aguirre, J. G., & Farfán, R. (2023). What enabled the production of mathematical knowledge in complex analysis?. International Electronic Journal of Mathematics Education, 18(2), em0734. https://doi.org/10.29333/iejme/12996
Ponce, J.C. (2019). The use of phase portraits to visualize and investigate iso-lated singular points of complex functions. International Journal of Mathematical Educationin Science and Technology, 50(7), 999–1010. https://doi.org/10.1080/0020739X.2019.1656829
Pringsheim, A. (1903). Der Cauchy-Goursat’sche Integralsatz und seine Übertragung auf reelle Kurven-lntegrale. Sitzungsberichte der math-phys. Classe der Königliche Akademie der Wissenschaften zu München, 33, 673–682.
Rudin, W. (1987). Real and complex analysis. McGraw-Hill.
Schubring, G., & Fan, L. (2018). Recent advances in mathematics textbook research and development: an overview. ZDM Mathematics Education, 50, 765–771. https://doi.org/10.1007/s11858-018-0979-4
Scott, J. (1990). A matter of record, documentary sources in social research. Polity Press.
Smithies, F. (1997). Cauchy and the creation of complex function theory. Cambridge University press. https://doi.org/10.1017/CBO9780511551697
Soto-Johnson, H., & Hancock, B. (2019). Research to Practice: Developing the Amplitwist Concept. PRIMUS, 29(5), 421–440. https://doi.org/10.1080/10511970.2018.1477889
Soto-Johnson, H., & Oehrtman, M. (2022). Undergraduates’ Exploration of contour integration: What is accumulated? The Journal of Mathematical Behavior, 66, 100963. https://doi.org/10.1016/j.jmathb.2022.100963
Spivak, M. (1994). Calculus. Publish or Perish.
Troup, J., Soto, H. & Kemp, A. (2023). Developing Geometric Reasoning of the Relationship of the Cauchy Riemann Equations and Differentiation. Int. J. Res. Undergrad. Math. Ed. Advanced online publication. https://doi.org/10.1007/s40753-023-00223-1
Van Dormolen, J. (1986). Textual analysis. In B. Christiansen, A. G. Howson & M. Otte (Eds.), Perspectives on Mathematics Education (pp. 141–171). Reidel. https://doi.org/10.1007/978-94-009-4504-3_4
Zill, D., & Shanaha, P. (2013). A first course in complex analysis with applications. Jones and Bartlett Publishers
Downloads
Published
How to Cite
Issue
Section
License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Autores que publicam nesta revista concordam com os seguintes termos:- Autores mantém os direitos autorais e concedem à revista o direito de primeira publicação, com o trabalho simultaneamente licenciado sob a Licença Creative Commons Attribution que permite o compartilhamento do trabalho com reconhecimento da autoria e publicação inicial nesta revista.
- Autores têm autorização para assumir contratos adicionais separadamente, para distribuição não-exclusiva da versão do trabalho publicada nesta revista (ex.: publicar em repositório institucional ou como capítulo de livro), com reconhecimento de autoria e publicação inicial nesta revista.
- Autores têm permissão e são estimulados a publicar e distribuir seu trabalho online (ex.: em repositórios institucionais ou na sua página pessoal) a qualquer ponto antes ou durante o processo editorial, já que isso pode gerar alterações produtivas, bem como aumentar o impacto e a citação do trabalho publicado (Veja O Efeito do Acesso Livre).
