Problem solving in the teaching of single variable differential and integral calculus: Perspective of mathematics teachers
DOI:
https://doi.org/10.15359/ru.32-2.3Keywords:
Teaching single variable differential and integral calculus, problem solving, higher mathematics, mathematics educationAbstract
There is a wide diversity of approaches to solving problems in the teaching of mathematics. In particular, the meaning of “problem solving” differs between theory and practice. In the teaching of higher mathematics, problem solving is frequently used in single variable differential and integral calculus, as indicated by course contents and the number of university programs that include it in their curricula. We therefore investigated the ways in which mathematics teachers use problem solving in the teaching of single variable differential and integral calculus. A questionnaire was applied to teachers with experience in teaching single variable differential and integral calculus from the Universidad de Costa Rica, the Universidad Nacional de Costa Rica, the Instituto Tecnológico Costa Rica, and the Universidad Estatal a Distancia. The results reveal contradictions between teachers' conceptions of what a mathematical problem is and their implementation of problem solving in the classroom.
References
Abarca, N. (2007). La enseñanza del cálculo diferencial e integral mediante la resolución de problemas, una propuesta motivadora (The teaching of differential and integral calculus by solving problems, a motivational proposal). Revista Tecnociencia Universitaria Bolivia [online], 5, 14-20.
Arcavi, A. (2000). Problem-driven research in mathematics education. The Journal of Mathematical Behavior, 19(2), 141-173. doi https://doi.org/10.1016/S0732-3123(00)00042-0
Arcavi, A. and Friedlander, A. (2007). Curriculum developers and problem solving: The case of Israeli elementary school projects. ZDM, 39(5-6), 355-364. Doi https://doi.org/10.1007/s11858-007-0050-3
Artigue, M. (2001). What can we learn from educational research at the university level? In D. Holton (Ed.), The teaching and learning of mathematics at university level: an ICMI study (pp. 207-220). Holland: Kluwer Academic.
Inter-American Development Bank. (2006). Conectando a la mayoría, lineamientos estratégicos para la difusión de las tecnologías de información y comunicación para el desarrollo (Connecting the majority, strategic guidelines for the dissemination of information and communication technologies for development). Accessed at http:// www.iadb.org/sds/doc/conectandoalamayoria.pdf
Benítez, S., and Benítez, L. (2013). La resolución de problemas en la enseñanza y aprendizaje de la matemática (Problem solving in the teaching and learning of mathematics). In E. Rodríguez (Ed.), Memorias del VII Congreso Iberoamericano de Educación Matemática (pp. 3206). Uruguay: SEMUR.
Brousseau, G. (1986). Fundamentos y Métodos de la Didáctica de las Matemáticas (Foundations and Methods of Mathematical Didactics). Recherches en Didactique des Mathématiques, 7(2), 33-115.
Burkhardt, H. and Bell, A. (2007). Problem solving in the United Kingdom. ZDM, 39(5-6), 395-403. Doi https://doi.org/10.1007/s11858-007-0041-4
Cantoral, R., Cordero, F., Farfán, R., e Imaz, C. (1990). Cálculo–análisis. Una revisión de la investigación educativa reciente en México (Calculus analysis. A review of recent educational research in Mexico). In R. Cantoral, F. Cordero, R. Farfán and C. Imaz (Eds.), Memorias del Segundo Simposio Internacional sobre Investigación en Educación Matemática (pp. 55–69). México: Universidad Autónoma del Estado de México.
Cantoral, R. and Mirón, H. (2000). Sobre el estatus de la noción de derivada: De la epistemología de Joseph Louis Lagrange al diseño de una situación didáctica (On the status of the notion of derivative: From the epistemology of Joseph Louis Lagrange to the design of a didactic situation). Revista Latinoamericana de Investigación en Matemática Educativa 3(3), 265–292.
Cuevas, A. and Mejía, H. (2003). Cálculo visual (Visual calculus). México: Oxford University Press.
Cuevas, C. and Pluvinage, F. (2009). Cálculo y tecnología. El cálculo y su enseñanza, 1 (Calculus and technology. Calculus and its teaching, 1). D.F., México: Cinvestav del Instituto Politécnico Nacional.
Devlin, K. (1996). Mathematics: the science of patterns, the search for order in life, mind and the universe. New York: Scientific American Library.
English, L., Lesh, R. and Fennewald, T. (2008). Future directions and perspectives for problema solving research and curriculum development. In Proceedings of the 11th International Congress on Mathematical Education.
Gascón, J. (2001). Incidencia del modelo epistemológico de las matemáticas sobre las prácticas docentes (Impact of the epistemological model of mathematics on teaching practices). Revista Latinoamericana de Investigación en Matemática Educativa, 6(1), 129–159.
Hernández, H., Delgado, J., Fernández, B., Valverde, L., and Rodríguez, T. (2001). Cuestiones de didáctica de la matemática: Conceptos y procedimientos en la educación polimodal y superior (Mathematical Didactics Issues: Concepts and procedures in polymodal and higher education). Argentina: Homo Sapiens Ediciones.
Lesh, R., Doerr, H., Cramer, K., Post, T., and Zawojewski, J. (2003). Model Development Sequences. In R. Lesh and H. Doerr (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics teaching, learning, and problem solving (pp. 211-233). Mahwah, N.J.: Lawrence Erlbaum Associates.
Lesh, R. and Zawojewski, J. S. (2007). Problem solving and modeling. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763-804). Charlotte, NC: Information Age Publishing.
Lois, A. E. and Milevicich, L. M. (2008). La enseñanza y aprendizaje del cálculo integral desde la perspectiva del nuevo paradigma de la sociedad del conocimiento (Teaching and learning integral calculus from the perspective of the new paradigm of the knowledge society). Revista Iberoamericana de Educación, 47(5), 1-15.
Mason, J. (2016). When is a problem…? “When” is actually the problem! In P. Felmer, E. Pehkonen and J. Kilpatrick (Eds.), Posing and solving mathematical problems: Advances and new perspectives (pp. 263-286). Springer: International Publishing. Doi https://doi.org/10.1007/978-3-319-28023-3_16
Morales-López, Y. (2017). Costa Rica: The Preparation of Mathematics Teachers. In A. Ruiz (Ed.), Mathematics Teacher Preparation in Central America and the Caribbean (pp. 39–56). Springer International Publishing. Accessed at http://link.springer.com/chapter/10.1007/978-3-319-44177-1_3. doi https://doi.org/10.1007/978-3-319-44177-1_3.
Moreno-Armella, L. and Santos-Trigo, M. (2008). Democratic access and use of powerful mathematics in an emerging country. In L. English (Ed.), Handbook of international research in mathematics education (p. 319). New York: Routledge.
National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics (Vol.1). Reston, VA: National Council of Teachers of Mathematics.
National Research Council. (1999). Improving Student Learning: A strategic plan for education research and its utilization. Washington, DC: National Academic Press.
Puig, L. (1998). Réplica a elementos de resolución de problemas, cinco años después de Ma Luz Callejo y José Carrillo (Reply to problem solving elements, five years after Ma Luz Callejo and José Carrillo). In J. R. Pascual (Ed.), Actas del Segundo Simposio de la Sociedad Española de Investigación en Educación Matemática, 107-112. Pamplona: Universidad Pública Navarra.
Salinas, P., and Alanís, J. A. (2009). Hacia un nuevo paradigma en la enseñanza del cálculo dentro de una institución educativa (Towards a new paradigm in the teaching of calculus in an educational institution). Revista Latinoamericana de Investigación en Matemática Educativa, 12(3), 355-382.
Santos, L. M. (2008). La resolución de problemas matemáticos: Avances y perspectivas en la construcción de una agenda de investigación y práctica (The resolution of mathematical problems: Advances and perspectives in the construction of a research and practice agenda). Investigación en Educación Matemática, xii, 159-192.
Santos-Trigo, M. (2007). Mathematical problem solving: an evolving research and practice domain. ZDM, 39(5- 6), 523-536. Doi https://doi.org/10.1007/s11858-007-0057-9.
Santos-Trigo, M. (2016). Problem solving in mathematics education. In P. Liljedahl, M. Santos-Trigo, U. Malaspina and R. Bruder (Eds.), Memorias del 13th. International Congress on Mathematical Education (pp. 19-30). Alemania: Springer Open.
Schoenfeld, A. H. (1985). Mathematical Problem Solving. New York: Academic Press.
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grows (Ed.), Handbook of Research on Mathematics Teaching and Learning, 334-370. NY: Macmillan.
Stanic, G. and Kilpatrick, J. (1989). Historical perspectives on problem solving in the mathematics curriculum. The teaching and assessing of mathematical problem solving. 3, 1-22.
Steen, L. A. (2003). Analysis 2000: challenges and opportunities. In D. Coray, F. Furinghetti, H. Gispert, B. R. Hodgson and G. Schubring (Eds.), One hundred years of I'enseignement mathématique: Moments of mathematics education in the twentieth century (Monografía No. 39, pp. 191–210). Genova, Italia: L'Enseignement Mathématique.
Törner, G., Schoenfeld, A. H., and Reiss, K. M. (2007). Problem solving around the World: Summing up the state of the art. ZDM, 39(5-6), 353-353. Doi https://doi.org/10.1007/s11858-007-0053-0.
Trouche, L. (2009). Recursos para procesar, aprender, enseñar el cálculo: Nuevos modos de concepción y difusión (Resources for processing, learning, and teaching calculus: New modes of conceptualization and diffusion). Revista El Cálculo y su enseñanza, 2. D.F., México: Cinvestav del Instituto Politécnico Nacional.
Valverde, J., and Garrido, M. (1999). El impacto de las tecnologías de la información y la comunicación en los roles docentes universitarios (The impact of information and communication technologies on university teaching roles). Revista electrónica interuniversitaria de formación del profesorado, 2(1), 543-554.
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