Exploring University Mexican Students’ Quality of Intra-Mathematical Connections When Solving Tasks About Derivative Concept

Camilo Andrés Rodríguez-Nieto; Flor Monserrat Rodríguez-Vásquez; Javier García-García

doi:10.29333/ejmste/11160

Full Text (PDF)

Camilo Andrés Rodríguez-Nieto ¹ ^* , Flor Monserrat Rodríguez-Vásquez ¹ , Javier García-García ¹

More Detail

¹ Autonomous University of Guerrero, MEXICO^* Corresponding Author

Abstract

The quality of the intra-mathematical connections made by Mexican university students when solving tasks on the derivative was characterized. The typology of mathematical connections and the quality levels of mathematical connections were used. Interviews were conducted based on eight tasks applied to three case studies. The results showed that the quality of the intra-mathematical connections: different representations, procedural, implication, part-whole and meaning of the students, is found mostly at level 2 (consistent and argued mathematical connections) and only one student presented inconsistencies to find the correctness equation of the tangent line to the curve at a point (quality level 0 connection). Likewise, the metaphorical connection was identified when a student mentioned the metaphorical expression “the graph has no holes”, which suggests the conceptual metaphor “the graph is a path” referring to the continuity of a function. We believe that quality level 2 mathematical connections ensure understanding, and the quality level 0 mathematical connections are the reasons why students have difficulty understanding the derivative.

Keywords

quality of mathematical connections
derivative
university students
thematic analysis.

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2021, Volume 17, Issue 9, Article No: em2006

https://doi.org/10.29333/ejmste/11160

Publication date: 23 Aug 2021

Article Views: 1632

Article Downloads: 884

Open Access Disclosures References How to cite this article

Disclosure

Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Adu-Gyamfi, K., Bossé, M. J., & Chandler, K. (2017). Student connections between algebraic and graphical polynomial representations in the context of a polynomial relation. International Journal of Science and Mathematics Education, 15(5), 915-938. https://doi.org/10.1007/s10763-016-9730-1
Aguilar, S., & Barroso, J. (2015). La triangulación de datos como estrategia en investigación educativa [data triangulation as education researching strategy]. Píxel-Bit. Revista de Medios y Educación, 45, 73-88. https://doi.org/10.12795/pixelbit.2015.i47.05
Aguilar-González, Á., Muñoz-Catalán, M. C., & Carrillo, J. (2018). An example of connections between the mathematics teacher’s conceptions and specialised knowledge. EURASIA Journal of Mathematics, Science and Technology Education, 15(2), em1664. https://doi.org/10.29333/ejmste/101598
Armas, T. A. D. (2020). Evaluación de la faceta epistémica del conocimiento didáctico-matemático de futuros profesores de matemáticas en el desarrollo de una clase utilizando funciones. Bolema: Mathematics Education Bulletin, 34, 110-131. https://doi.org/10.1590/1980-4415v34n66a06
Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. (1997). The development of student’s graphical understanding of the derivate. Journal of Mathematical Behavior, 16(4), 399-431. https://doi.org/10.1016/S0732-3123(97)90015-8
Assad, D. A. (2015). Task-based interviews in mathematics: Understanding student strategies and representations through problem solving. International Journal of Education and Social Science, 2(1), 17-26.
Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. Educational Studies in Mathematics, 70(3), 217-241. https://doi.org/10.1007/s10649-008-9145-1
Berry, J. & Nyman, M. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479-495. https://doi.org/10.1016/j.jmathb.2003.09.006
Borji, V., Font, V., Alamolhodaei, H., & Sánchez, A. (2018). Application of the complementarities of two theories, APOS and OSA, for the analysis of the university students’ understanding on the graph of the function and its derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(6), 2301-2315. https://doi.org/10.29333/ejmste/89514
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77-101. https://doi.org/10.1191/1478088706qp063oa
Breda, A. (2020). Características del análisis didáctico realizado por profesores para justificar la mejora en la enseñanza de las matemáticas [Characteristics of the didactic analysis carried out by teachers to justify the improvement of mathematics teaching]. Bolema: Mathematics Education Bulletin, 34(66), 69-88. https://doi.org/10.1590/1980-4415v34n66a04
Breda, A., Font, V., & Pino-Fan, L. (2018). Criterios valorativos y normativos en la Didáctica de las Matemáticas: el caso del constructo idoneidad didáctica [Evaluative and normative criteria in Didactics of Mathematics: the case of didactical suitability construct]. Bolema: Mathematics Education Bulletin, 32(60), 255-278. https://doi.org/10.1590/1980-4415v32n60a13
Breda, A., Hummes, V., Da Silva, R. S., & Sánchez, A. (2021). El papel de la fase de observación de la implementación en la metodología estudio de clases [The role of the observation phase of implementation in the lesson study methodology]. Bolema: Mathematics Education Bulletin, 35, 263-288. https://doi.org/10.1590/1980-4415v35n69a13
Breda, A., Pino-Fan, L., & Font, V. (2017). Meta didactic-mathematical knowledge of teachers: Criteria for the reflection and assessment on teaching practice. EURASIA Journal of Mathematics, Science and Technology Education, 13(6), 1893-1918. https://doi.org/10.12973/eurasia.2017.01207a
Businskas, A. M. (2008). Conversations about connections: How secondary mathematics teachers conceptualize and contend with mathematical connections [Unpublished PhD thesis], Simon Fraser University. Canada.
Campo-Meneses, K., Font, V., García-García, J., & Sánchez, A. (2021). Mathematical Connections Activated in High School Students’ Practice Solving Tasks on the Exponential and Logarithmic Functions. EURASIA Journal of Mathematics, Science and Technology Education, 17(9), em1998. https://doi.org/10.29333/ejmste/11126
Da Fonseca, V. G., & Henriques, A. C. C. B. (2020). Learning with understanding the continuity concept: A teaching experiment with Brazilian pre-service mathematics teachers. International Electronic Journal of Mathematics Education, 15(3), 1-17. https://doi.org/10.29333/iejme/8462
Dolores-Flores, C. Rivera-López, M., & García-García, J. (2019). Exploring mathematical connections of pre-university students through tasks involving rates of change. International Journal of Mathematical Education in Science and Technology, 50(3), 369-389. https://doi.org/10.1080/0020739X.2018.1507050
Dolores-Flores, C., & García-García, J. (2017). Conexiones Intramatemáticas y Extramatemáticas que se producen al Resolver Problemas de Cálculo en Contexto: un Estudio de Casos en el Nivel Superior [Intra-mathematical and extra-mathematical connections that occur when solving Calculus’ problems in context: A case study at a higher level]. Bolema: Mathematics Education Bulletin, 31(57), 158-180. https://doi.org/10.1590/1980-4415v31n57a08
Dolores-Flores, C., & Ibáñez-Dolores, G. (2020). Conceptualizaciones de la Pendiente en Libros de Texto de Matemáticas [Slope Conceptualizations in Mathematics Textbooks]. Bolema: Mathematics Education Bulletin, 34, 825-846. https://doi.org/10.1590/1980-4415v34n67a22
Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2013). Mathematical connections and their relationship to mathematics knowledge for teaching geometry. School Science and Mathematics, 113(3), 120-134. https://doi.org/10.1111/ssm.12009
Evitts, T. (2004). Investigating the mathematical connections that preservice teachers use and develop while solving problems from reform curricula [Unpublished dissertation]. Pennsylvania State University College of Education. EE. UU.
Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning. Understanding limits, derivates and integrals. In E. Dubinsky & J. Kaput (Eds.), Research issues in undergraduate mathematics learning. MMA Notes 33 (pp. 31-45). MMA.
Feudel, F., & Biehler, R. (2020). Students’ understanding of the derivative concept in the context of mathematics for economics. Journal für Mathematik-Didaktik, 42, 273-305. https://doi.org/10.1007/s13138-020-00174-z
Fuentealba, C., Badillo, E., & Sánchez-Matamoros, G. (2018a). Puntos de no-derivabilidad de una función y su importancia en la comprensión del concepto de derivada [The non-derivability points of a function and their importance in the understanding of the derivative concept]. Educação e Pesquisa, 44, 1-20. https://doi.org/10.1590/S1678-4634201844181974
Fuentealba, C., Badillo, E., Sánchez-Matamoros, G., & Cárcamo, A. (2018b). The understanding of the derivative concept in higher education. EURASIA Journal of Mathematics, Science and Technology Education, 15(2), em1662. https://doi.org/10.29333/ejmste/100640
Fuentealba, C., Sánchez-Matamoros, G., & Badillo, E. (2015). Análisis de tareas que pueden promover el desarrollo de la comprensión de la derivada [Analysis of tasks that can promote the development of understanding of the derivative]. Uno: Revista de didáctica de las matematicas, 71, 72-78.
Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24(5), 645-657. https://doi.org/10.1080/0144341042000262953
García-García, J. (2019). Escenarios de exploración de conexiones matemáticas [Scenarios for exploring mathematical connections]. Números: Revista de didáctica de las matemáticas, 100, 129-133.
García-García, J., & Dolores-Flores, C. (2018). Intra-mathematical connections made by high school students in performing Calculus tasks. International Journal of Mathematical Education in Science and Technology, 49(2), 227-252. https://doi.org/10.1080/0020739X.2017.1355994
García-García, J., & Dolores-Flores, C. (2021a). Pre-university students' mathematical connections when sketching the graph of derivative and antiderivative functions. Mathematics Education Research Journal, 33, 1-22. https://doi.org/10.1007/s13394-019-00286-x
García-García, J., & Dolores-Flores, C. (2021b). Exploring pre-university students’ mathematical connections when solving Calculus application problems, International Journal of Mathematical Education in Science and Technology, 52(6), 912-936. https://doi.org/10.1080/0020739X.2020.1729429
Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517-545). Lawrence Erlbaum Associates.
Hashemi, N., Abu, M. S., Kashefi, H., & Rahimi, K. (2014). Undergraduate students’ difficulties in conceptual understanding of derivation. Procedia-Social and Behavioral Sciences, 143, 358-366. https://doi.org/10.1016/j.sbspro.2014.07.495
Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of research of mathematics teaching and learning (pp. 65-79). Macmillan.
Lyublinskaya, I. (2006). Making connections: Science experiments for algebra using TI Technology. Eurasia Journal of Mathematics, Science and Technology Education, 2(3), 144-157. https://doi.org/10.12973/ejmste/75471
Mhlolo, M. K. (2012). Mathematical connections of a higher cognitive level: A tool we may use to identify these in practice. African Journal of Research in Mathematics, Science and Technology Education, 16(2), 176-191. https://doi.org/10.1080/10288457.2012.10740738
Mhlolo, M.K., Venkat, H., & Schäfer, M. (2012). The nature and quality of the mathematical connections teachers make. Pythagoras, 33(1), 1-9. https://doi.org/10.4102/pythagoras.v33i1.22
Muzangwa, J., & Chifamba, P. (2012). Analysis of errors and misconceptions in the learning of calculus by undergraduate students. Acta Didactica Napocensia, 5(2), 1-10.
National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
Nurwahyu, B., Tinungki, G. M., & Mustangin (2020). Students’ concept image and its impact on reasoning towards the concept of the derivative. European Journal of Educational Research, 9(4), 1723-1734. https://doi.org/10.12973/eu-jer.9.4.1723
Padgett, D. K. (2016). Qualitative methods in social work research (Vol. 36). Sage publications.
Pino-Fan, L., Godino, J. D., & Font, V. (2018). Assessing key epistemic features of didactic mathematical knowledge of prospective teachers: The case of the derivative. Journal of Mathematics Teacher Education, 21, 63-94. https://doi.org/10.1007/s10857-016-9349-8
Pino-Fan, L., Guzmán, I., Font, V., & Duval, R. (2017). Analysis of the underlying cognitive activity in the resolution of a task on derivability of the absolute-value function: Two theoretical perspectives. PNA, 11(2), 97-124. https://doi.org/10.30827/pna.v11i2.6076
Rodríguez-Nieto, C. (2020). Explorando las conexiones entre sistemas de medidas usados en prácticas cotidianas en el municipio de Baranoa [Exploring the connections between measurement systems used in daily practices in the municipality of Baranoa]. IE Revista de Investigación Educativa de la REDIECH, 11, e-857. https://doi.org/10.33010/ie_rie_rediech. v11i0.857
Rodríguez-Nieto, C. (2021). Conexiones etnomatemáticas entre conceptos geométricos en la elaboración de las tortillas de Chilpancingo, México [Ethnomatematical connections between geometric concepts in the making of tortillas from Chilpancingo, Mexico]. Revista de investigación desarrollo e innovación, 11 (2), 273-296. https://doi.org/10.19053/20278306.v11.n2.2021.12756
Rodríguez-Nieto, C., Font, V., Borji, V., & Rodríguez-Vásquez, F. M. (2021). Mathematical connections from a networking theory between Extended Theory of Mathematical connections and Onto-semiotic Approach. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2021.1875071
Rodríguez-Nieto, C., Rodríguez-Vásquez, F. M., & Font, V. (2020). A new view about connections. The mathematical connections established by a teacher when teaching the derivative. International Journal of Mathematical Education in Science and Technology, https://doi.org/10.1080/0020739X.2020.1799254
Rodríguez-Nieto, C., Rodríguez-Vásquez, F. M., & García-García, J. (2021). Pre-service mathematics teachers’ mathematical connections in the context of problem-solving about the derivative. Turkish Journal of Computer and Mathematics Education, 12(1), 202-220. https://doi.org/10.16949/turkbilmat.797182
Rodríguez-Nieto, C., Rodríguez-Vásquez, F. M., Font, V., & Morales-Carballo, A. (2021). Una visión desde el networking TAC-EOS sobre el papel de las conexiones matemáticas en la comprensión de la derivada [A view from the ETC-OSA networking of theories on the role of mathematical connections in understanding the derivative]. Revemop, 3, e202115. https://doi.org/10.33532/revemop.e202115
Sánchez-Matamoros, G., García, M., & Llinares, S. (2008). La comprensión de la derivada como objeto de investigación en didáctica de la matemática [The understanding the derivative as the object of investigation in mathematics education]. Revista Latinoamericana de Investigación en Matemática Educativa, 11(2), 267-296.
Sari, P., Hadiyan, A., & Antari, D. (2018). Exploring derivatives by means of GeoGebra. International Journal on Emerging Mathematics Education, 2(1), 65-78. https://doi.org/10.12928/ijeme.v2i1.8670
Vargas, M. F., Fernández-Plaza, J. A., & Ruiz-Hidalgo, J. F. (2020). Significado de derivada en las tareas de los libros de 1° de Bachillerato [Meaning of derivative in the book tasks of 1st of “Bachillerato”]. Bolema: Mathematics Education Bulletin, 34, 911-933. https://doi.org/10.1590/1980-4415v34n68a04
Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. The Journal of Mathematical Behavior, 25(1), 1-17. https://doi.org/10.1016/j.jmathb.2005.11.002

How to cite this article

APA

Rodríguez-Nieto, C. A., Rodríguez-Vásquez, F. M., & García-García, J. (2021). Exploring University Mexican Students’ Quality of Intra-Mathematical Connections When Solving Tasks About Derivative Concept. Eurasia Journal of Mathematics, Science and Technology Education, 17(9), em2006. https://doi.org/10.29333/ejmste/11160

Vancouver

Rodríguez-Nieto CA, Rodríguez-Vásquez FM, García-García J. Exploring University Mexican Students’ Quality of Intra-Mathematical Connections When Solving Tasks About Derivative Concept. EURASIA J Math Sci Tech Ed. 2021;17(9):em2006. https://doi.org/10.29333/ejmste/11160

AMA

Rodríguez-Nieto CA, Rodríguez-Vásquez FM, García-García J. Exploring University Mexican Students’ Quality of Intra-Mathematical Connections When Solving Tasks About Derivative Concept. EURASIA J Math Sci Tech Ed. 2021;17(9), em2006. https://doi.org/10.29333/ejmste/11160

Chicago

Rodríguez-Nieto, Camilo Andrés, Flor Monserrat Rodríguez-Vásquez, and Javier García-García. "Exploring University Mexican Students’ Quality of Intra-Mathematical Connections When Solving Tasks About Derivative Concept". Eurasia Journal of Mathematics, Science and Technology Education 2021 17 no. 9 (2021): em2006. https://doi.org/10.29333/ejmste/11160

Harvard

Rodríguez-Nieto, C. A., Rodríguez-Vásquez, F. M., and García-García, J. (2021). Exploring University Mexican Students’ Quality of Intra-Mathematical Connections When Solving Tasks About Derivative Concept. Eurasia Journal of Mathematics, Science and Technology Education, 17(9), em2006. https://doi.org/10.29333/ejmste/11160

MLA

Rodríguez-Nieto, Camilo Andrés et al. "Exploring University Mexican Students’ Quality of Intra-Mathematical Connections When Solving Tasks About Derivative Concept". Eurasia Journal of Mathematics, Science and Technology Education, vol. 17, no. 9, 2021, em2006. https://doi.org/10.29333/ejmste/11160