Conversions Between Trigonometric Representation Systems by Pre-service Secondary School Teachers
DOI:
https://doi.org/10.30827/pna.v16i3.21957Keywords:
Conversions among representation, Meaning of a school mathematical content, Relational systems, Understanding trigonometric contentsAbstract
Understanding trigonometry relational system is a school mathemat-ics demanding topic. The angle, the unit circle and the trigonometric functions are its foundational notions. Trigonometric contents mean-ing and their understanding involve these three concepts and their re-lationships. This research aims to deepen in the pre-service teachers’ understanding about the angle, the unit circle and the trigonometric function when converting notions between two trigonometric repre-sentation systems based on the unit circle and the trigonometric func-tions. The results indicate that pre-service mathematics teachers’ pre-sent a lack of connections between the goniometric and the analytical representation system.
Downloads
References
Akkoç, H. (2008). Pre-service mathematics teachers’ concept images of radian. International Journal of Mathematical Education in Science and Technology, 39(7), 857-878.
Bell A., Costello J., & Küchemann D. (1983). Research on learning and teaching. A Review of Re¬search in Mathematical Education. NFER- Nelson.
Brown, S.A. (2005). The trigonometric connections: Students’ understanding of sine and cosine (Unpublished doctoral dissertation), Illinois State University, Illinois, IL.
Bunge, M. (2008). Tratado de Filosofía. Semántica I, Sentido y referencia [Treatise on philosophy. Semantic I, sense and reference]. Gedisa Editorial.
Camacho, M., & Depool, R. (2003). Using derive to understand the concept of definite integral. International Journal for Mathematics Teaching and Learning. Retrieved from http://www.cimt.org.uk/journal/matiascamacho.pdf
Castro-Rodríguez, E., Pitta-Pantazi, D., Rico, L., & Gómez, P. (2016). Prospective teachers’ understanding of the multiplicative part-whole relationship of fraction. Educational Studies in Mathematics, 92(1), 129-146. https://doi.org/10.1007/s10649-015-9673-4
Çekmez, E. (2020). What generalizations do students achieve with respect to trigonometric functions in the transition from angles in degrees to real numbers? The Journal of Mathematical Behavior, 58, 100778. https://doi.org/10.1016/j.jmathb.2020.100778
Challenger, M. (2009). From triangles to a concept: a phenomenographic study of A-level students’ development of the concept of trigonometry (Unpublished doctoral dissertation), Warwick University, UK.
Chaar, M. (2015). Secondary Pre-service, In-Service, and Student Teachers’ Noticing of Mathematical Work and Thinking in Trigonometry (Unpublished doctoral dissertation), University of New Hampshire, NH.
Chin, K. E. (2013). Making sense of mathematics: Supportive and Problematic Conceptions with special reference to Trigonometry (Unpublished doctoral dissertation), University of Warwick.
Choquet, G. (1964). L’enseignement de la Géométrie. [The teaching of geometry]. Hermann.
Clements, D. H., & Burns, B. A. (2000). Students' development of strategies for turn and angle measure. Educational Studies in Mathematics, 41(1), 31-45. https://doi.org/10.1023/A:1003938415559
Demir, O. (2012). Students’ concept development and understanding of sine and cosine functions (Unpublished doctoral dissertation), Amsterdam University, Amsterdam, The Netherland. Retrieved from https://esc.fnwi.uva.nl/thesis/centraal/files/f107257570.pdf
Dieudonné, J. (1971). Álgebra Lineal y Geometría Elemental. [Linear Algebra and Elementary Geometry]. Selecciones Científicas.
Dündar, S. (2015). Mathematics Teacher-Candidates’ Performance in Solving Problems with Different Representation Styles: The Trigonometry Example. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1379-1397. https://doi.org/10.12973/eurasia.2015.1396a
Duval, R. (1993). Semiosis y noesis [Semiosis and noesis]. In E. Sánchez & G. Zubieta (Eds.), Lecturas en didáctica de la matemática: Escuela Francesa (pp. 118-144). Sección de Matemática Educativa del CINVESTAV-IPN.
Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational studies in mathematics, 21(6), 521-544. https://doi.org/10.1007/BF00315943
Feferman, S. (1989). The Number Systems. Foundations of Algebra and Analysis. Chelsea Pub. Comp.
Fi, C. (2003). Pre-service secondary school mathematics teachers’ knowledge of trigonometry: subject matter content knowledge, pedagogical content knowledge and envisioned pedagogy (Unpublished doctoral dissertation), University of Iowa, Iowa.
Freudenthal, H. (1973). Mathematics as an educational task. Kluwer Academic Publishers.
Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics. Lawrence Erlbaum Associates.
Hilbert, D. (1991). Fundamentos de la Geometría. [Foundations of Geometry]. Consejo Superior de Investigaciones Científicas.
Kaput, J. J. (1992). Technology and Mathematics Education. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515-516). Macmillan.
Kumar, R. S., Subramaniam, K., & Naik, S. S. (2017). Teachers’ construction of meanings of signed quantities and integer operation. Journal of Mathematics Teacher Education, 20(6), 557-590. https://doi.org/10.1007/s10857-015-9340-9
Lakoff, G., & Núñez, R. E. (2000). Where Mathematics come from: how the embodied mind brings mathematics. Basic Books.
Marchi, D. J. (2012). A Study of Student Understanding of the Sine Function through Representations and the Process and Object Perspectives (Unpublished doctoral dissertation), The Ohio State University, Ohio.
Martín Fernández, E., Ruiz Hidalgo, J. F., & Rico, L., (2016). Significado escolar de las razo¬nes trigonométricas elementales [Student’s notions of elementary trigonometric ratios]. Enseñanza de las Ciencias, 34(3), pp. 51-71. https://doi.org/10.5565/rev/ensciencias.1871.
Martín-Fernández, E., Ruiz-Hidalgo, J. F., & Rico, L. (2019). Meaning and Understanding of School Mathematical Concepts by Secondary Students: The Study of Sine and Cosine. Eurasia Journal of Mathematics, Science and Technology Education, 15 (12), em 1782. https://doi.org/10.29333/ejmste/110490
Martínez-Planell, R., & Delgado, A. C. (2016). The unit circle approach to the construction of the sine and cosine functions and their inverses: An application of APOS theory. The Journal of Mathematical Behavior, 43, 111-133. https://doi.org/10.1016/j.jmathb.2016.06.002
Melike Yiǧit Koyunkaya (2016). Mathematics education graduate students’ understanding of trigonometric ratios. International Journal of Mathematical Education in Science and Technology 47(7), pp. 1028-1047. https://doi.org/10.1080/0020739X.2016.1155774
Moore, K. (2014). Quantitative reasoning and the sine function: The case of Zac. Journal for Research in Mathematics Education, 45(1), 102-138. https://doi.org/10.5951/jresematheduc.45.1.0102
Morgan, C., & Kynigos, C. (2014). Digital artefacts as representations: forging connections between a constructionist and a social semiotic perspective. Educational Studies in Mathematics, 85(3), 357-379. https://doi.org/10.1007/s10649-013-9523-1.
Ministerio de Educación y Ciencia (2007). Real Decreto 1393/2007, de 29 de octubre, por el que se establece la ordenación de las enseñanzas universitarias oficiales. [Royal legislative Decree 1393/2007, by which the organization of official university education is established], BOE, 260, 2007, 30th October, pp. 44037-44048.
Ortiz Galarza, M. L. (2017). The Influence of Multiple Representations on Secondary Students' Understanding of Trigonometric Functions (Unpublished doctoral dissertation), University of Texas at El Paso: Texas.
Reinhardt, F., & Soeder, H. (1984). Atlas de matemáticas, 1. [Atlases of Mathematics, 1]. Alianza.
Real Academia de Ciencias Exactas, Físico-Químicas y Naturales (RAC) (1996). Vocabulario Científico y Técnico. [Scientific and Technical Vocabulary]. Espasa.
Rico, L. (1997). Los organizadores del currículo de matemáticas [The Mathematics Curriculum Organizers]. In L. Rico (Ed.), La Educación Matemática en la enseñanza secundaria (pp. 39-59). ICE-Horsori.
Rico, L. (2009). Sobre las nociones de representación y comprensión en la investigación en educación matemática. [On the notions of representation and understanding in mathematics education research]. PNA, 4(1), 1-14. https://doi.org/10.30827/pna.v4i1.6172
Rico, L. (2012). Aproximación a la Investigación en Didáctica de la Matemática. [Research Approach to Mathematics Education]. Avances de Investigación en Educación Matemática, 1, 39- 63.
Rico, L. & Ruiz Hidalgo, J.F. (2018). Ideas to work for the curriculum change in school mathematics. In Y. Shimizu & R. Vithal (Eds.), Conference proceedings of the twenty-fourth ICMI Study: School Mathematics curriculum reforms: Challenges, changes and opportunities (pp. 301-308). ICMI
Rico, L., Martín-Fernández, E., & Ruiz-Hidalgo. J. F. (2020). Significados y sistemas en trigonometría escolar [Meanings and systems in school trigonometry]. In E. Castro-Rodríguez, E. Castro, I. Segovia & P. Flores (Eds.), Libro homenaje a Enrique Castro (pp. 197-220). Octaedro.
Russell, B. (1912). Problems of Philosophy. Oxford University Press.
Skemp, R. R. (1987). The psychology of learning mathematics. Psychology Press.
Strauss, A., & Corbin, J. (1994). Grounded theory methodology: An Overview. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research, (pp. 273-285). Sage publications.
Strauss, A., & Corbin, J. (1998). Basics of Qualitative Research: Techniques and Procedures for Developing Grounded Theory. Sage Publications.
Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. English & D. Kirshner (Eds.), Handbook of International Research in Mathematics Education (pp. 435-461). Taylor & Francis.
Weber, K. (2005). Student’s understanding of trigonometric functions. Mathematics Education Research Journal, 102(2), 144-147. Retrieved from https://files.eric.ed.gov/fulltext/EJ747914.pdf
