Conocimiento de un profesor de Álgebra Lineal sobre los errores de los estudiantes y su uso en la enseñanza

Diana  Vasco Mora; Nuria Climent Rodríguez

doi:10.48489/quadrante.23008

Authors

Diana Vasco Mora Universidad Técnica Estatal de Quevedo, Ecuador https://orcid.org/0000-0002-7279-6410
Nuria Climent Rodríguez Universidad de Huelva, Spain https://orcid.org/0000-0002-0064-1452

DOI:

https://doi.org/10.48489/quadrante.23008

Keywords:

tertiary education, case study, knowledge of students’ errors, pedagogical content knowledge, Linear Algebra

Abstract

In the light of the Mathematics Teacher’s Specialised Knowledge (MTSK) model and through a case study we investigate the knowledge of students’ errors of a university lecturer when teaching the content of matrices and determinants. The data were collected through video recordings of classes and semi-structured interviews conducted during two school periods. The information collected and transcribed was analyzed looking for evidence that alluded to the category Strengths and Weaknesses in learning mathematics of the MTSK, and specifically to the lecturer’s knowledge of students’ errors. The results show a lecturer’s knowledge of common errors in learning content that could have different origins, as well as the teacher’s use of that knowledge in teaching and focusing on remediation.

References

Andrews‑Larson, C., Johnson, E., Peterson, V., & Keller, R. (2019). Doing math with mathematicians to support pedagogical reasoning about inquiry‑oriented instruction. Journal of Mathematics Teacher Education. https://doi.org/10.1007/s10857-019-09450-3

Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014). APOS theory. A framework for research and curriculum development in mathematics education. New York, NY: Springer. https://doi.org/10.1007/978-1-4614-7966-6

Artigue, M., Assude, T., Grugeon, B., & Lenfant, A. (2001). Teaching and learning Algebra: Approaching complexity through complementary perspectives. In H. Chick, K. Stacey, & J. Vincent (Eds.), The future of teaching and learning of algebra, Proceedings of the 12th ICMI Study Conference (pp. 21-32). Melbourne, Australia: The University of Melbourne.

Ashlock, R.B. (2010). Errors patterns in computation: Using error patterns to improve instruction. Boston: Allyn & Bacon.

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching. What makes it special? Journal of Teacher Education, 59(5), 389-407.

Bardin, L. (1996). Análisis de contenido. Madrid, España: Ediciones Akal.

Barros, P., Mendes, C., & Fernandes, J. A. (2013). Raciocínios de estudantes do ensino superior na resolução de tarefas sobre matrizes. In J. A. Fernandes, M. H. Martinho, J. Tinoco, & F. Viseu (Orgs.), Atas do XXIV Seminário de Investigação em Educação Matemática (pp. 295-308). Braga: Centro de Investigação em Educação da Universidade do Minho.

Biza, I., Giraldo, V., Hochmuth, R., Khakbaz, A., & Rasmussen, C. (2016). Research on teaching and learning mathematics at the tertiary level: State-of-the-art and looking ahead. Berlin: Springer International Publishing. https://doi.org/10.1007/978-3-319-41814-8_1

Borasi, R. (1994). Capitalizing on errors as “springboards for inquiry”: A teaching experiment. Journal for Research in Mathematics Education, 25(2), 166-208.

Carrillo, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, A., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236-253. https://doi.org/10.1080/14794802.2018.1479981

Delgado-Rebolledo, R., & Zakaryan, D. (2020). Relationships between the knowledge of practices in mathematics and the pedagogical content knowledge of a mathematics lecturer. International Journal of Science and Mathematics Education, 18(3), 567-587.

Dorier, J. L. (2016). Duality between formalism and meaning in the learning of linear algebra. In R. Göller, R. Biehler, R. Hochmuth, & H.-G. Rück (Eds.), Didactics of mathematics in higher education as a scientific discipline. Kassel, Germany: Universitätsbibliothek Kassel. Retrieved from https://archive-ouverte.unige.ch/unige:85576

Dorier, J. L., & Sierpinska, A. (2001). Research into the teaching and learning of Linear Algebra. In D. Holton, M. Artigue, U. Kirchgräber, J. Hillel, M. Niss, & A. Schoenfeld (Eds.), The teaching and learning of mathematics at university level: An ICMI Study (pp. 255-273). Dordrecht, Netherlands: Kluwer Academic Publishers.

Ferro, P. (2011). Significado referencial y evaluado de los conceptos de matriz y determinante en estudiantes preuniversitarios. Un estudio a partir de la práctica instruccional (Disertación doctoral, Universidad de Santiago de Compostela, España). Recuperado de http://dspace.usc.es/bitstream/10347/4035/1/rep_168.pdf

González-López, M. J., Gómez, P., & Restrepo, A. M. (2015). Usos del error en la enseñanza de las matemáticas. Revista de Educación, 370, 71-95. https://doi.org/10.4438/1988-592X-RE-2015-370-297

Harel, G., Fuller, E., & Rabin, J. (2008). Attention to meaning by algebra teachers. The Journal of Mathematical Behavior, 27, 116-127.

Hurtado, S., Eagan, K., Pryor, J. H., Whang, H., & Tran, S. (2012). Undergraduate teaching faculty: The 2010–2011 HERI faculty survey. Recuperado de: https://www.heri.ucla.edu/monographs/HERI-FAC2011-Monograph.pdf

Iannone, P., & Nardi, E. (2005). On the pedagogical insight of mathematicians: ‘Interaction’and ‘transition from the concrete to the abstract’. The Journal of Mathematical Behavior, 24(2), 191–215.

Jonhson, E., & Larsen, S. P. (2012). Teacher listening: The role of knowledge of content and students. The Journal of Mathematical Behavior, 31, 117-129.

Johnson, E., Keller, R., & Fukawa-Connelly, T. (2018). Results from a national survey of abstract algebra instructors: Understanding the choice to (not) lecture. International Journal for Research in Undergraduate Mathematics Education, 4(2), 254–285.

Locia-Espinoza, E., Morales-Carballo, A., & Merino-Cruz, H. (2020). Taylor’s formula, limited development, and development of power Series: A study of the knowledge of university professors in training. International Electronic Journal of Mathematics Education, 15(3), em0585. https://doi.org/10.29333/iejme/7852

Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Children’s Mathematical Behavior, 3(1), 93-166.

McCrory, R., Floden, R., Ferrini-Mundy, J., Reckase, M. D., & Senk, S. L. (2012). Knowledge of Algebra for teaching: A framework of knowledge and practices. Journal for Research in Mathematics Education, 43(5), 584-615.

Nardi, E., Jaworski, B., & Hegedus, S. (2005). A spectrum of pedagogical awareness for undergraduate mathematics: From “tricks” to “techniques”. Journal for Research in Mathematics Education, 36(4), 284–316.

NCTM (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Neuman, W. L. (2014). Social research methods: Qualitative and quantitative approaches. Harlow: Pearson Education Limited.

Policastro, M. S., de Almeida, A. R., Ribeiro, M., & Jakobsen, A. (2020). Kindergarten teacher’s knowledge to support a mathematical discussion with pupils on measurement strategies and procedures. In M. Carlsen, I. Erfjord, & P. S. Hundeland (Eds.), Mathematics education in the early years (pp. 263-279). Cham, Switzerland: Springer.

Ribeiro, M., Mellone, M., & Jakobsen, A. (2016). Interpreting students’ non-standard reasoning: Insights for mathematics teacher education. For the Learning of Mathematics, 36(2), 8-13.

Rico, L. (1998). Errores en el aprendizaje de las matemáticas. In J. Kilpatrick, L. Rico, & P. González (Eds.). Educación matemática (pp. 69-109). México: Grupo Editorial Iberoamericana.

Rojas, N., Flores, P., & Carrillo, J. (2015). Conocimiento especializado de un profesor de matemáticas de educación primaria al enseñar los números racionales. Bolema - Boletim de Educação Matemática, 29(51), 143-166.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281.

Ruano, R. M., Socas, M. M., & Palarea, M. M. (2008). Análisis y clasificación de errores cometidos por los alumnos de secundaria en los procesos de sustitución formal, generalización y modelización en álgebra. PNA, 2(2), 61-74.

Scheiner, T., Montes, M. A., Godino, J. D., Carrillo, J., & Pino-Fan, L. R. (2019). What makes mathematics teacher knowledge specialized? Offering alternative views. International Journal of Science and Mathematics Education, 17(1), 153–172. https://doi.org/10.1007/s10763-017-9859-6

Schoenfeld, A., & Kilpatrick, J. (2008). Toward a theory proficiency in teaching mathematics. In T. Wood & D. Tirosh (Eds.), Tools and processes in mathematics teacher education (pp. 321-354). London: Sense Publishers.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Socas, M. M. (2007). Dificultades y errores en el aprendizaje de las natemáticas. Análisis desde el enfoque lógico semiótico. In M. Camacho, P. Flores, & P. Bolea (Eds.), Investigación en educación matemática XI (pp. 19-52). La Laguna: SEIEM.

Son, J. (2013). How preservice teachers interpret and respond to student errors: ratio and proportion in similar rectangles. Educational Studies in Mathematics 84, 49-70. https://doi.org/10.1007/s10649-013-9475-5

Sosa, L., Flores-Medrano, E., & Carrillo, J. (2015). Conocimiento del profesor acerca de las características de aprendizaje del álgebra en bachillerato. Enseñanza de las Ciencias, 33(2), 173-189.

Speer, N., King, K., & Howell, H. (2014). Definitions of mathematical knowledge for teaching: Using these constructs in research on secondary and college mathematics teachers. Journal of Mathematics Teacher Education, 18(2), 105–122.

Stake, R. E. (2008). Qualitative case studies. In N. Denzin & Y. Lincoln (Eds.), Strategies of qualitative inquiry (pp. 119-149). Thousand Oaks, CA: Sage Publications.

Stewart, S., & Reeder, S. (2017). Algebra underperformances at college level: What are the consequences? In S. Stewart (Ed.), And the rest is just Algebra (pp. 3-18). Switzerland: Springer International Publishing. https://doi.org/10.1007/978-3-319-45053-7_1

Vasco, D. (2015). Conocimiento especializado del profesor de Álgebra Lineal. Un estudio de casos en el nivel universitario (Disertación doctoral, Universidad de Huelva, España). Disponible en http://hdl.handle.net/10272/11901

Vasco, D., Climent, N., Escudero-Ávila, D., Montes, M. A., & Ribeiro, M. (2016). Conocimiento especializado de un profesor de Álgebra Lineal y espacios de trabajo matemático. Bolema, 30(54), 222-239. http://dx.doi.org/10.1590/1980-4415v30n54a11