Meaning and Understanding of School Mathematical Concepts by Secondary Students: The Study of Sine and Cosine
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Department of Mathematics Education, Faculty of Education, University of Granada, Campus Universitario Cartuja s/n, 18011, Granada, SPAIN
Online publish date: 2019-07-06
Publish date: 2019-07-06
EURASIA J. Math., Sci Tech. Ed 2019;15(12):em1782
Meaning and understanding are didactic notions appropriate to work on concept comprehension, curricular design, and knowledge assessment. This document aims to delve into the meaning of school mathematical concepts through their semantic analysis. This analysis is used to identify and establish the basic meaning of a mathematical concept and to value its understanding. To illustrate the study, we have chosen the trigonometric notions of sine and cosine of an angle. The work exemplifies some findings of an exploratory study carried out with high school students between 16 and 17 years of age; it collects the variety of emergent notions and elements related to the trigonometric concepts involved when answering on the categories of meaning which have been asked for. We gather the study data through a semantic questionnaire and analyze the responses using an established framework. The subjects provide a diversity of meanings, interpreted and structured by semantic categories. These meanings underline different understandings of the sine and cosine, according to the inferred themes, such as length, ratio, angle and the calculation of a magnitude.
Akkoc, H. (2008). Pre-service mathematics teachers’ concept images of radian. International Journal of Mathematical Education in Science and Technology, 39(7), 857-878.
Allen, H. D. (1977). The teaching of trigonometry in the United States and Canada: A consideration of elementary course content and approach and factors influencing change, 1890-1970 (Unpublished doctoral dissertation), The State University of New Jersey, New Brunswick, NJ.
Biehler, R. (2005). Reconstruction of Meaning as a Didactical Task: The Concept of Function as an Example. In Kilpatrick J., Hoyles C., Skovsmose O., Valero P. (Eds), Meaning in Mathematics Education. Mathematics Education Library (vol 37, pp. 61-81). New York, NY: Springer.
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 The trigonometric connections: Students’ understanding of sine and cosine (Unpublished doctoral dissertation), Illinois State University, Illinois, IL.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. Barcelona, Spain: Gedisa Editorial.
Byers, P. (2010). Investigating trigonometric representations in the transition to college mathematics. College Quarterly, 13, 2, 1–10. Retrieved from
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).
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. Retrieved from
Cohen, L. Manion, L., Morrison, K. (2011). Research Methods in Education, Routledge, London. UK.
Common Core State Standards Initiative. Mission statement [Internet]. 2010 [cited 2015 Oct 5]. Available from: District of Columbia: The Council of Chief State School Officers and National Governors Association Center for Best Practices.
De Kee, S., Mura, R., & Dionne J. (1996). La comprensión des notions de sinus et de cosines chez des élèves du secondaire [Understanding the notions of sine and cosine in high school students]. For the Learning of Mathematics, 16(2), 19-22. Retrieved from
De Villiers, M., & Jugmohan, J. (2012). Learners’ conceptualisation of the sine function during an introductory activity using sketchpad at grade 10 level. Educ. Matem. Pesq., São Paulo, 14(1), 9-30. Retrieved from
DeJarnette, A. F. (2018). Students’ Conceptions of Sine and Cosine Functions When Representing Periodic Motion in a Visual Programming Environment. Journal for Research in Mathematics Education, 49(4), 390-423.
Demir, O. (2012). Students’ concept development and understanding of sine and cosine functions (Unpublished doctoral dissertation), Amsterdan University, Amsterdam, The Netherlands. Retrieved from
Demopoulos, W. (1994). Frege, Hilbert, and the conceptual structure of model theory. History and Philosophy of Logic, 15(2), 211-225.
Dogân, A. (2001). A research on misconceptions and mistakes of the students and their attitudes towards the trigonometry subjects which are teaching in the highschools (Unpublished doctoral dissertation), Selçuk University Institute of Sciences, Konya: Turkey.
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.
Fanning, J. D. (2016). Student responses to instruction in rational trigonometry. In Tim Fukawa-Connelly, Nicole Engelke Infante, Megan Wawro and Stacy Brown (Eds), Proceedings from 19th Annual Conference on Research in Undergraduate Mathematics Education, (pp. 741-751) Pittsburgh: PA.
Fernández-Plaza, J. A., Rico, L., & Ruiz-Hidalgo, J. F. (2013). Concept of finite limit of a function at a point: Meanings and specific terms. International Journal of Mathematical Education in Science and Technology, 44(5), 699-710.
Fi, C. (2003). Preservice secondary school mathematics teachers’ knowledge of trigonometry: subject matter content knowledge, pedagogical content knowledge and envisioned pedagogy (Unpublished doctoral dissertation), University of Iowa, Iowa. Retrieved from
Fraenkel, J. R., & Warren, E. A. (2006). How to Design and Evaluate Research in Education. Newyork: McGraw-Hill.
Frege, G. (1962). Über Sinn und Bedeutung. [About sense and meaning] In G. Patzig (Ed.), Funktion, Begriff, Bedeutung. Fünf logische Studien (pp. 40–65).
Freudenthal, H. (1983). Didactycal phenomenology of mathematical structures. Dordrecht, Holland: Reidel. Retrieved from
Golding, G. A., & Kaput, J. J. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 397-430). Hillsdale, NJ: Erlbaum.
Gur, H. (2009). Trigonometry Learning. New Horizons in Education, 57(1), 67-80. Retrieved from
Hertel, J. T. (2013). Investigating the purpose of trigonometry in the modern sciences (Unpublished doctoral dissertation), Illinois State University, Illinois, IL.
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D.A. Grouws (Eds.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 65-98). New York: Macmillan.
Kamber, D., & Takaci, D. (2018). On problematic aspects in learning trigonometry. International Journal of Mathematical Education in Science and Technology, 49(2), 161-175.
Krippendorff, K. (1989). Content analysis. In E. Barnouw, G. Gerbner, W. Schramm, T. L. Worth, & L. Gross (Eds.), International encyclopedia of communication (Vol. 1, pp. 403-407). New York, NY: Oxford University Press. Retrieved from
Lamon, S. J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J.T. Sowder & B.P. Schapelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 167-198). Albany, NY: State University of New York.
Maor, E. (1998). Trigonometric delights. Princeton, NJ: Princeton University Press.
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. Retrieved from!etd.....
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), 51-71.
Ministerio de Educación y Ciencia (2007). Real Decreto 1467/2007, por el que se establece la estructura del bachillerato y se fijan sus enseñanzas mínimas. [Royal legilastive Decree 1467/2007, by which the structure of the baccalaureate is established and its minimum teachings are set], BOE, 266, 2007, 2nd November, pp. 45381-45477.
Montiel Espinosa, G., & Jácome Cortés, G. (2014). Significado trigonométrico en el professor [Trigonometric meaning in the teacher]. Boletim de Educação Matemática, 28(50), 1193-1216.
Moore, K. C. (2012). Coherence, quantitative reasoning, and the trigonometry of students. Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context, 2, 75-92. Retrieved from
Moore, K. C., & LaForest, K. R. (2014). The circle approach to trigonometry. Mathematics Teacher, 107(8), 616-623.
Moore, K. C., LaForest, K. R., & Kim, H. J. (2016). Putting the unit in pre-service secondary teachers’ unit circle. Educational Studies in Mathematics, 92(2), 221-241.
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.
Orhun, N. (2004). Students’ mistakes and misconceptions on teaching of trigonometry. Retrieved from
Paoletti, T., Stevens, I. E., Hobson, N. L., Moore, K. C., & LaForest, K. R. (2015). Pre-service teachers’ inverse function meanings. In T. Fukawa-Connelly, N. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the Eighteenth Annual Conference on Research in Undergraduate Mathematics Education (pp. 853-867). Pittsburgh, PA: West Virginia.
Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical thinking and learning, 5(1), 37-70.
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. Retrieved from
Rico, L., & Moreno, A. (2016). Elementos de didáctica de la matemática para el profesor de secundaria [Mathematics didactic elements for the secondary school teacher]. Madrid, Spain: Ediciones Pirámide. Retrieved from
Sáenz-Ludlow, A. (2003). A collective chain of signification in conceptualizing fractions: A case of a fourth-grade class. The Journal of Mathematical Behavior, 22(2), 181-211.
Sevimli, E., & Delice, A. (2012). The relationship between students’ mathematical thinking types and representation preferences in definite integral problems. Research in Mathematics Education, 14(3), 295-296.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford, Uk: Oxford University Press.
Sickle, J. V. (2011). A history of trigonometry Education in the United States 1776-1990 (Unpublished doctoral dissertation), Combia Uniluversity, Nueva York, NY.
Skemp, R. (1987). The psychology of learning mathematics, Hillsdale, NJ: Erlbaum Associates.
Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers. Journal of Mathematics Teacher Education, 1(2), 157-189.
Steinbring, H. (2006). What makes a sign a mathematical sign? –An epistemological perspective on mathematical interaction. Educational Studies in Mathematics, 61(1-2), 133-162.
Tavera, F., & Villa Ochoa, J. A. (2016). Sobre las razones y las funciones trigonométricas: ¿Qué tratamiento hacen los libros de texto? [On the ratios and trigonometric functions: What treatment do textbooks do?]. Acta Latinoamericana de Matemática Educativa. Retrieved from
Thompson, K. (2007). Student’s understanding of trigonometry enhanced through the use of a real world problem: improving the instructional sequence (Unpublished doctoral dissertation), Illinois State Uni¬versity, Illinois, IL.
Thompson, P. W., Carlson, M. P., & Silverman, J. (2007). The design of tasks in support of teachers’ development of coherent mathematical meanings. Journal of Mathematics Teacher Education, 10(4-6), 415-432.
Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational studies in Mathematics, 54(1), 9-35.
Van den Heuvel-Panhuizen, M. (2014). Didactical phenomenology (Freudenthal). In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 174-176). Heidelberg: Springer.
Vergnaud G. (1990). La théorie des champs conceptuels, Recherches en Didactique des Mathématiques [Conceptuals Fields Theory, Research in Mathematical Didactics], 10(2.3), 133-170.
Vizmanos, J. R., Alcaide, F., Hernández, J., Moreno, M., & Serrano, E. (2008). Matemáticas 1º Bachillerato (Ciencias y Tecnología) [Mathematics 1st Baccalaureate (Science and Technology)]. Madrid, Spain: S.M.
Weber, K. (2005). Student’s understanding of trigonometric functions. Mathematics Education Research Journal, 102(2), 144-147. Retrieved from
Weber, K. (2008). Teaching trigonometric functions: Lessons learned from research. Mathematics Teacher, 17(3), 91-112.
Yiǧit Koyunkaya, M. (2016). Mathematics education graduate students’ understanding of trigonometric ratios. International Journal of Mathematical Education in Science and Technology, 47(7), 1028-1047.
Zhang, Y., & Wildemuth, B. M. (2009). Qualitative analysis of content. In B. Wildemuth (Ed.), Applications of social research methods to questions in information and library science (pp. 222–231). Westport CT: Libraries Unlimited.