The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences

José Mariano Bajo-Benito; Gloria Sánchez-Matamoros García; José María Gavilán-Izquierdo

doi:10.29333/ejmste/11429

Full Text (PDF)

José Mariano Bajo-Benito ¹ ^* , Gloria Sánchez-Matamoros García ¹ , José María Gavilán-Izquierdo ¹

More Detail

¹ Universidad de Sevilla, SPAIN^* Corresponding Author

Abstract

This paper aims to characterise an indicator of the development of the number sequence scheme among students at the level of Compulsory Secondary Education (14-16 years old students). To do so, we use a scheme development proposed by the APOS theory to characterise students’ use of relations between mathematical elements when solving a mathematical task. We use a qualitative methodology and the data collection instruments are a written questionnaire and a semi-structured interview. In this work we show the questionnaire task that provides analytical expressions and ask students to determine which of them numbers sequences are. We find that students’ use of logical implication when solving tasks related to number sequences is an indicator of the development of the scheme. This indicator helps to locate the transition mechanisms between the levels of development of the number sequence scheme. Moreover, our research shows that arithmetic and geometric progressions play a key role as an indicator of the development of the number sequence scheme.

Keywords

number sequences
arithmetic and geometric progressions
APOS theory
scheme
compulsory secondary education

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, Volume 17, Issue 12, December 2021, Article No: em2058

https://doi.org/10.29333/ejmste/11429

Publication date: 17 Dec 2021

Article Views: 1527

Article Downloads: 1042

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

Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer. https://doi.org/10.1007/978-1-4614-7966-6
Asiala, M., Brown, A., DeVries, D. J., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education II, 6, 1-32. https://doi.org/10.1090/cbmath/006/01
Bagni, G. T. (2005). Infinite series from history to mathematics education. International Journal for Mathematics Teaching and Learning https://www.cimt.org.uk/journal/bagni.pdf
Bajo Benito, J. M., Gavilán-Izquierdo, J. M., & Sánchez-Matamoros García, G. (2019). Caracterización del esquema de sucesión numérica en estudiantes de Educación Secundaria Obligatoria [Characterization of the numeric sequence schema among Compulsory Secondary Education students]. Enseñanza de las Ciencias, 37(3), 149-167. https://doi.org/10.5565/rev/ensciencias.2673
Baker, B., Cooley, L., & Trigueros, M. (2000). A calculus graphing schema. Journal for Research in Mathematics Education, 31(5), 557-578. https://doi.org/10.2307/749887
BOE (Boletín Oficial del Estado) (2007). Real Decreto 1631/2006, de 29 de diciembre, por el que se establecen las enseñanzas mínimas correspondientes a la Educación Secundaria Obligatoria [Royal Decree 1631/2006, of December 29, which establishes the minimum education corresponding to Compulsory Secondary Education]. Official State Gazette, 5, 677-773. https://www.boe.es/eli/es/rd/2006/12/29/1631/con
Cañadas, M. (2007). Descripción y caracterización del razonamiento inductivo utilizado por estudiantes de educación secundaria al resolver tareas relacionadas con sucesiones lineales y cuadráticas [Description and characterization of inductive reasoning used by high school students when solving tasks related to linear and quadratic sequences] (Unpublished doctoral thesis) Universidad de Granada.
Codes Valcarce, M., & González-Martín, A. S. (2017). Sucesión de sumas parciales como proceso iterativo infinito: un paso hacia la comprensión de las series numéricas desde el modelo APOS [Sequence of partial sums as an infinite iterative process: A step towards the understanding of numerical series from an APOS perspective]. Enseñanza de las Ciencias, 35(1), 89-110. https://doi.org/10.5565/rev/ensciencias.1927
Codes, M., González Astudillo, M. T., Delgado Martín, M. L., & Monterrubio Pérez, M. C. (2013). Growth in the understanding of infinite numerical series: A glance through the Pirie and Kieren theory. International Journal of Mathematical Education in Science and Technology, 44(5), 652-662. https://doi.org/10.1080/0020739X.2013.781690
Djasuli, M., Sa’dijah, C., Parta, I. N., & Daniel, T. (2017). Students’ reflective abstraction in solving number sequence problems. International Electronic Journal of Mathematics Education, 12(3), 621-632. https://doi.org/10.29333/iejme/638
Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95-126). Kluwer. https://doi.org/10.1007/0-306-47203-1_7
Duval, R. (2006). Un tema crucial en la Educación Matemática: La habilidad para cambiar el registro de representación [A crucial issue in mathematics education: The ability to change the register of representation]. La Gaceta de la Real Sociedad Matemática Española, 9(1), 143-168.
Fuentealba, C, Badillo, E., Sánchez-Matamoros, G., & Trigueros M. (2017). Thematization of derivative schema in university students: nuances in constructing relations between a function's successive derivatives. International Journal of Mathematical Education in Science and Technology, 48(3), 374-392. https://doi.org/10.1080/0020739X.2016.1248508
Fuentealba, C., Badillo, E., & Sánchez-Matamoros, G. (2019a). Identificación y caracterización de los subniveles de desarrollo del esquema de derivada [Identification and characterization of the development sub-levels of the derivative schema]. Enseñanza de las Ciencias, 37(2), 63-84. https://doi.org/10.5565/rev/ensciencias.2518
Fuentealba, C., Badillo, E., Sánchez-Matamoros, G., & Cárcamo, A. (2019b). 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
Gonzalez, J., Medina, P., Vilanova, S., & Astiz, M. (2011). Un aporte para trabajar sucesiones numéricas con Geogebra [A contribution to work numerical sequences with Geogebra]. Revista de Educación Matemática, 26, 1-19.
Mamona-Downs, J. (2001). Letting the intuitive bear on the formal: A didactical approach for the understanding of the limit of a sequence. Educational Studies in Mathematics, 48, 259-288. https://doi.org/10.1023/A:1016004822476.
McDonald, M. A., Mathews, D. M., & Strobel, K. H. (2000). Understanding sequences: A tale of two objects. Research in Collegiate Mathematics Education IV, American Mathematical Society, Providence, Rhode Island, 8, 77-102. https://doi.org/10.1090/cbmath/008/05
Mor, Y., Noss, R., Hoyles, C., Kahn, K., & Simpson, G. (2006). Designing to see and share structure in number sequences. The International Journal for Technology in Mathematics Education, 13(2), 65-78.
Piaget, J. & García, R. (1983). Psicogénesis e historia de la ciencia [Psychogenesis and history of science]. Siglo XXI Editores.
Przenioslo, M. (2006). Conceptions of a sequence formed in secondary schools. International Journal of Mathematical Education in Science and Technology, 37(7), 805-823. https://doi.org/10.1080/00207390600733832
Roa-Fuentes, S. & Oktac, A. (2010). Construcción de una descomposición genética: Análisis teórico del concepto transformación lineal [Constructing a genetic decomposition: Theoretical analysis of the linear transformation concept]. Revista Latinoamericana de Investigación en Matemática Educativa, 13(1), 89-112.
Roh, K.H. (2008). Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69, 217-233. https://doi.org/10.1007/s10649-008-9128-2
Sánchez-Matamoros, G. (2004). Análisis de la comprensión en los alumnos de bachillerato y primer año de universidad sobre la noción matemática de derivada (desarrollo del concepto) [Analysis of the understanding in high school and first year university students about the mathematical notion of derivative (concept development)] (Doctoral thesis). Universidad de Sevilla.
Sánchez-Matamoros, G., García, M., & Llinares, S. (2008). La comprensión de la derivada como objeto de investigación en didáctica de la matemática [The understanding of derivative as the object of investigation in mathematics education]. Revista latinoamericana de investigación en matemática educativa, 11(2), 267-296.
Sánchez-Matamoros, G., García, M., & Llinares, S. (2013). Algunos indicadores del desarrollo del esquema de derivada de una función [Some Indicators of the Development of Derivative Schema]. Bolema, 27(45) 281-302.
Stewart, J., Hernández, R., & Sanmiguel, C. (2007). Introducción al cálculo [Introduction to calculus]. S.A. Ediciones Thomson.
Valls, J., Pons, J., & Llinares, S. (2011). Coordinación de los procesos de aproximación en la comprensión del límite de una función [Coordination of approximations in secondary school students’ understanding of the concept of limit of a function]. Enseñanza de las Ciencias, 29(3), 325-338. https://doi.org/10.5565/rev/ec/v29n3.637
Weigand, H.-G. (2015). Discrete or continuous?– A model for a technology supporteddiscrete approach to calculus. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Conference of the European Society for Research in Mathematics Education (pp. 2580-2586). ERME.

How to cite this article

APA

Bajo-Benito, J. M., Sánchez-Matamoros García, G., & Gavilán-Izquierdo, J. M. (2021). The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), em2058. https://doi.org/10.29333/ejmste/11429

Vancouver

Bajo-Benito JM, Sánchez-Matamoros García G, Gavilán-Izquierdo JM. The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences. EURASIA J Math Sci Tech Ed. 2021;17(12):em2058. https://doi.org/10.29333/ejmste/11429

AMA

Bajo-Benito JM, Sánchez-Matamoros García G, Gavilán-Izquierdo JM. The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences. EURASIA J Math Sci Tech Ed. 2021;17(12), em2058. https://doi.org/10.29333/ejmste/11429

Chicago

Bajo-Benito, José Mariano, Gloria Sánchez-Matamoros García, and José María Gavilán-Izquierdo. "The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences". Eurasia Journal of Mathematics, Science and Technology Education 2021 17 no. 12 (2021): em2058. https://doi.org/10.29333/ejmste/11429

Harvard

Bajo-Benito, J. M., Sánchez-Matamoros García, G., and Gavilán-Izquierdo, J. M. (2021). The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences. Eurasia Journal of Mathematics, Science and Technology Education, 17(12), em2058. https://doi.org/10.29333/ejmste/11429

MLA

Bajo-Benito, José Mariano et al. "The Use of Logical Implication as an Indicator of Understanding the Concept of Number Sequences". Eurasia Journal of Mathematics, Science and Technology Education, vol. 17, no. 12, 2021, em2058. https://doi.org/10.29333/ejmste/11429