Qualitative analysis of the logistic equation in Mathematics Education
DOI:
https://doi.org/10.15359/ree.29-2.18642Keywords:
Logistic curve, logistic equation, logistic function, mathematical model, literature review, SDG 4, quality education, educational strategiesAbstract
Introduction.University-level mathematics education has had to transform its teaching and learning methods to engage with both academic content and knowledge transmission, as well as the holistic development of individuals, to address contemporary societal issues. The logistic equation enables the modeling of various phenomena across multiple scientific fields and contributes to resolving these issues. Given its broad applicability across scientific fields, this study proposes the need to present a broad and detailed overview of the logistic equation, especially in the context of mathematics teaching and learning, since, in the school settings, some students experience difficulties in understanding it. Objective. This study aims to present a systematic review of research conducted on the logistic equation within the field of mathematics education. Analysis. The review covers studies published between 2000 and 2022 that examine the use of logistics equation in educational contexts. Methodology. Kuckartz’s qualitative text analysis method was employed for data collection and analysis. This method consists of five phases: 1) data preparation, 2) category construction, 3) data coding, 4) analysis of coded data and 5) presentation of results. Results. The main difficulties related to the logistic equation involve the use of different representations in learning it as a differential equation and difficulties in interdisciplinary communication. Didactic proposals addressing these difficulties incorporate different approaches, technological tools, and diverse theoretical frameworks. Conclusions. This approach allows teachers to learn different strategies for teaching the logistics equation and its related concepts. Considering its extensive applicability, mathematical modeling is proposed as one of the primary strategies. With this strategy, students attribute meaning to the mathematical concept and connect mathematics with other sciences.
Downloads
References
*Ang, K. C. (2004). A simple model for a SARS epidemic. Teaching Mathematics and its Applications: An International Journal of the IMA, 23(4), 181-188. https://doi.org/10.1093/teamat/23.4.181
*Barquero, B., Bosch, M., & Gascón, J. (2007). La modelización matemática como instrumento de articulación de las matemáticas del primer ciclo universitario de ciencias. Estudio de la dinámica de poblaciones. En L. Ruiz-Higueras, A. Estepa, & F. J. García (Coords.), Sociedad, escuela y matemáticas: Aportaciones de la teoría antropológica de lo didáctico (TAD) (pp. 573-594). Editorial Universidad de Jaén. https://www.academia.edu/24323370/La_modelizaci%C3%B3n_matem%C3%A1tica_como_instrumento_de_articulaci%C3%B3n_de_las_matem%C3%A1ticas_del_primer_ciclo_universitario_de_Ciencias_Estudio_de_la_din%C3%A1mica_de_poblaciones
*Bejarano, C. A. (2005). Modelos de simulación para el estudio del crecimiento poblacional exponencial. Epsilon, 1(4), 69-81. https://ciencia.lasalle.edu.co/ep/vol1/iss4/23/
*Biembengut, M. S. (2012). Concepções e tendências de modelagem matemática na educação brasileira. Cuadernos de Investigación y Formación en Educación Matemática, 7(10), 195-204. https://revistas.ucr.ac.cr/index.php/cifem/issue/view/1079/292
*Castillo-Garsow, C. (2010). Teaching the Verhulst Model: A teaching experiment in covariational reasoning and exponential growth [Tesis doctoral, Arizona State University]. https://www.proquest.com/openview/0fc4df1fb351865d64104ad5b917c8db/1?pq-origsite=gscholar&cbl=18750
*Cheng, A. K. (2006). Mathematical modelling, technology and H3 mathematics. The Mathematics Educator, 9(2), 33-47. https://math.nie.edu.sg/ame/matheduc/tme/tmeV9_2/Ang%20KC.pdf
*Couoh-Noh, J. R. & Cabañas-Sánchez, M. G. (2013). Un estudio del límite al infinito en el nivel superior bajo el contexto de la resolución de problemas que involucran la función logística. En L. Sosa, J. Hernández y E. Aparicio (Eds.), Memoria de la XVI Escuela de Invierno en Matemática Educativa (pp. 316-323). Red de Cimates. https://core.ac.uk/download/pdf/322888113.pdf
*Díaz Eaton, C., Highlander, H. C., Dahlquist, K. D., Ledder, G., LaMar, M. D., & Schugart, R. C. (2019). A “Rule-of-Five” Framework for models and modeling to unify mathematicians and biologists and improve student learning. Primus, 29(8), 799-829. https://doi.org/10.1080/10511970.2018.1489318
*Ekici, C. & Plyley, C. (2019). Inquiry-based modeling of population dynamics with logistic differential and difference equations. Primus, 29(6), 553-570. https://doi.org/10.1080/10511970.2018.1484399
*Espinosa Flórez, Y. A. & Saavedra Delgado, J. F. (2013). Aprendiendo Winplot para la exploración de la ecuación logística. Revista EJES, 1(1), 64-70. http://funes.uniandes.edu.co/10233/
*Fernández, E. & Geist, K. A. (2011). Flower Power: Sunflowers as a Model for Logistic Growth. Mathematics Teacher, 104(8), 580-585. https://doi.org/10.5951/MT.104.8.0580
García Hernández, A. E. & Reich, D. (2015). Ecuaciones diferenciales: Una nueva visión. E-Book. Grupo Editorial Patria.
*García López, G. S. (2017). La modelización de las experiencias de enseñanza y aprendizaje del curso diseño de la forma en el espacio estructural [Tesis doctoral, Universitat Autònoma de Barcelona]. http://hdl.handle.net/10803/406127
*Gordon, S. P. (2008). Comparing the discrete and continuous logistic models. Primus, 18(5), 449-455. https://doi.org/10.1080/10511970701884566
*Gordon, S. P. (2009). Modeling population growth and extinction. Primus, 19(6), 548-560. https://doi.org/10.1080/10511970801953212
*Gualano, L., Graieb, A., Baragatti, E., & Andrini, L. (2017). Matemáticas y crecimiento bacteriano: Un trabajo de laboratorio para el aprendizaje significativo. En Facultad de Ciencias Exactas. (Eds.), 1 Jornadas sobre Enseñanza y Aprendizaje en el Nivel Superior en Ciencias Exactas y Naturales (pp.1-8). UNLP. http://sedici.unlp.edu.ar/handle/10915/76014
Guirao Goris, S. J. A. (2015). Utilidad y tipos de revisión de literatura. Ene, 9(2), 1-17. https://dx.doi.org/10.4321/S1988-348X2015000200002
Habre, S. (2000). Exploring students’ strategies to solve ordinary differential equations in a reformed setting. The Journal of Mathematical Behavior, 18(4), 455-472. https://doi.org/10.1016/S0732-3123(00)00024-9
*Hamilton, A. J. (2005). SLAC: A tool for addressing chaos in the ecology classroom. International Journal of Mathematical Education in Science and Technology, 36(5), 489-496. https://doi.org/10.1080/00207390412331336247
*Kadas, Z. (2018). Discrete population models: Why they belong in a differential equations course. Primus, 28(8), 785-796. https://doi.org/10.1080/10511970.2018.1443532
*Kucharavy, D. & de Guio, R. (2011). Application of S-shaped curves. Procedia Engineering, 9, 559-572. https://doi.org/10.1016/j.proeng.2011.03.142
Kuckartz, U. (2019). Qualitative text analysis: A systematic approach. En G. Kaiser & N. Presmeg (Eds.), Compendium for early career researchers in mathematics education (pp. 181-197). Springer. https://doi.org/10.1007/978-3-030-15636-7
*McCartney, M. & Gibson, S. (2004). On the Road to Chaos. Teaching Mathematics and its Applications: An International Journal of the IMA, 23(2), 89-96. https://doi.org/10.1093/teamat/23.2.89
Meadows, D. H., Meadows, D. L., Randers, J., & Behrens III, W. W. (1972). The limits to growth. Universe Books. https://www.donellameadows.org/wp-content/userfiles/Limits-to-Growth-digital-scan-version.pdf
Medina Mendieta, J. F., Cortés Cortés, M. E., Cortés Iglesias, M., Pérez Fernández, A. del C., & Manzano Cabrera, M. (2020). Estudio sobre modelos predictivos para la COVID-19 en Cuba. Medisur, 18(3), 431-442. https://www.medigraphic.com/pdfs/medisur/msu-2020/msu203n.pdf
Murray, J. D. (2002). Mathematical Biology: I. An introduction (3a ed.). Springer. https://dl.icdst.org/pdfs/files/27f6eba850c27d335ff3f93778d8057f.pdf
*Oropesa Anhalt, C., Cortez, R., & Bennett, A. B. (2018). The emergence of mathematical modeling competencies: An investigation of prospective secondary mathematics teachers. Mathematical Thinking and Learning, 20(3), 202-221. https://doi.org/10.1080/10986065.2018.1474532
*Parra, E., Gordillo, W., & Pinzón, W. J. (2019). Modelos de crecimiento poblacional: Enseñanza-aprendizaje desde las ecuaciones recursivas. Formación Universitaria, 12(1), 25-34. http://dx.doi.org/10.4067/S0718-50062019000100025
Pelinovsky, E., Kurkin, A., Kurkina, O., Kokoulina, M., & Epifanova, A. (2020). Logistic equation and COVID-19. Chaos, Solitons & Fractals, 140, 1-13. https://doi.org/10.1016/j.chaos.2020.110241
*Pereira, J. C. & Conceição, A. C. (2013). F.Tool 2.0: Exploring the Logistic Function in the Classroom. En A. Loja, J. Infante Barbosa & J. A. Rodrigues (Eds.), 1st International Conference on Algebraic and Symbolic Computation. ECCOMAS (pp. 149-158). APMTAC. https://www.researchgate.net/publication/259558512_FTool_20_Exploring_the_Logistic_Function_in_the_Classroom
Rasmussen, C. L. (2001). New directions in differential equations: A framework for interpreting students’ understandings and difficulties. The Journal of Mathematical Behavior, 20(1), 55-87. https://doi.org/10.1016/S0732-3123(01)00062-1
*Rodríguez Carrillo, J. A. & Ulloa Ibarra, J. T. (2017). Alternativa didáctica para el estudio del modelo Gompertz. Investigación e Innovación en Matemática Educativa, 2(1), 98-114. https://revistaiime.org/index.php/IIME/article/view/46/18
*Rotem, S.-H. & Ayalon, M. (2021). Exploring israeli high school graduates’ explanations for the spread of the coronavirus. Educational Studies in Mathematics, 108(1-2), 161-181. https://doi.org/10.1007/s10649-021-10042-3
*Roth, W.-M. & Bowen, G. M. (2001). Professionals read graphs: A semiotic analysis.Journal for Research in Mathematics Education, 32(2), 159-194. https://doi.org/10.2307/749672
Secretaría de Educación Pública (SEP). (2022). Plan de estudio para la educación preescolar, primaria y secundaria. https://www.sep.gob.mx/marcocurricular/
*Scott, P. (2000). Populate or Perish. Logo and the Logistic Equation. Mathematics in School, 29(3), 14-16. https://www.jstor.org/stable/30212339
*Soon, W., Tirtasanjaya Lioe, L., & McInnes, B. (2011). Understanding the difficulties faced by engineering undergraduates in learning mathematical modelling. International Journal of Mathematical Education in Science and Technology, 42(8), 1023-1039. https://doi.org/10.1080/0020739X.2011.573867
*Stephan, M. & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. The Journal of Mathematical Behavior, 21(4), 459-490. https://doi.org/10.1016/S0732-3123(02)00145-1
*Stephens, G. P. (2002). Teaching the logistic function in high school. The Mathematics Teacher, 95(4), 286-294. https://doi.org/10.5951/MT.95.4.0286
Tene, T., Guevara, M., Svozilík, J., Tene-Fernandez, R., & Vacacela Gomez, C. (2021). Analysis of COVID-19 outbreak in Ecuador using the logistic model. Emerging Science Journal, 5(Especial), 105-118. http://dx.doi.org/10.28991/esj-2021-SPER-09
*Ulloa Ibarra, J. T., Arrieta Vera, J. L., & Espino Flores, G. A. (2013). El modelo logístico y su deconstrucción. En R. Flores (Ed.). Acta Latinoamericana de Matemática Educativa (26, pp. 717-724). CLAME. http://funes.uniandes.edu.co/4106/
*Valero Cázarez, M. del S. & Lezama Andalón, J. (2020). Una experiencia didáctica con estudiantes de bachillerato en torno a la modelación de los datos del COVID19 en México. El cálculo y su Enseñanza. Enseñanza de la Ciencia y la Matemática, 15(11), 1-19. https://doi.org/10.61174/recacym.v15i1.57
*Vargas Hernández, J., Chaves Escobar, R. F., Monserrate Rodríguez, F., & Jaimes Contreraas, L. A. (2020). Una caracterización de la ecuación diferencial logística para el estudio de su comprensión en estudiantes universitarios. En J. A. Blanco Puentes (Comp.), Diario de campo: Resultados del desarrollo de métodos y técnicas de investigación (Vol. 10, Tomo 2, pp. 211-232). Univerrsidad Colegio Mayor de Cundinamarca. https://www.researchgate.net/publication/344380887_Una_caracterizacion_de_la_ecuacion_diferencial_logistica_para_el_estudio_de_su_comprension_en_estudiantes_universitarios_Grupo_BIOMA_-_Beginning_to_inquire_on_mathematics_and_arts
Villarruel Fuentes, M., & Villarruel López, M.d. L. (2023). La educación superior y la nueva escuela mexicana: Sus desafíos y posibilidades. LATAM Revista Latinoamericana de Ciencias Sociales y Humanidades 4(1), 1088-1100. https://doi.org/10.56712/latam.v4i1.320
Williams, S. R. & Leatham, K. (2017). Journal Quality in Mathematics Education. Journal for Research in Mathematics Education, 48(4), 369-396. https://doi.org/10.5951/jresematheduc.48.4.0369
*Winkel, B. J. (2011). Parameter estimates in differential equation models for population growth. Primus, 21(2), 101-129. https://doi.org/10.1080/10511970.2010.534834
*Winkel, B. J. (2012). Sourcing for parameter estimation and study of logistic differential equation. International Journal of Mathematical Education in Science and Technology, 43(1), 67-83. https://doi.org/10.1080/0020739X.2011.582178
Zill, D. G. & Wright, W. S. (2015). Ecuaciones diferenciales con problemas con valores en la frontera (8ª ed.). Cengage Learning Editores, México, DF.
Downloads
Published
Issue
Section
Categories
License
Copyright (c) 2025 Ingrid Quilantán-Ortega, Flor Monserrat Rodríguez-Vásquez

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License.
1. In case the submitted paper is accepted for publication, the author(s) FREELY, COSTLESS, EXCLUSIVELY AND FOR AN INDEFINITE TERM transfer copyrights and patrimonial rights to Universidad Nacional (UNA, Costa Rica). For more details check the Originality Statement and Copyright Transfer Agreement
2. REUTILIZATION RIGHTS: UNA authorizes authors to use, for any purpose (among them selfarchiving or autoarchiving) and to publish in the Internet in any electronic site, the paper´'s final version, both approved and published (post print), as long as it is done with a non commercial purpose, does not generate derivates without previous consentment and recognizes both publisher's name and authorship.
3. The submission and possible publication of the paper in the Educare Electronic Journal is ruled by the Journal’s editorial policies, the institutional rules of Universidad Nacional and the laws of the Republic of Costa Rica. Additionally, any possible difference of opinion or future dispute shall be settled in accordance with the mechanisms of Alternative Dispute Resolution and the Costa Rican Jurisdiction.
4. In all cases, it is understood that the opinions issued are those of the authors and do not necessarily reflect the position and opinion of Educare, CIDE or Universidad Nacional, Costa Rica. It is also understood that, in the exercise of academic freedom, the authors have carried out a rogorous scientific-academic process of research, reflection and argumentation thar lays within the thematic scope of interest of the Journal.
5. The papers published by Educare Electronic Journal use a Creative Commons License:











The articles published by Educare Electronic Journal can be shared with a Creative Commons License: 

