12/2019 vol. 15
Stats
CC-BY 4.0
Get citation
RESEARCH PAPER
Meaning and Understanding of School Mathematical Concepts by Secondary Students: The Study of Sine and Cosine
Enrique Martín-Fernández 1
,
 
Juan Francisco Ruiz-Hidalgo 1
,
 
Luis Rico 1
 
 
 
More details
Hide details
1
Department of Mathematics Education, Faculty of Education, University of Granada, Campus Universitario Cartuja s/n, 18011, Granada, SPAIN
 
 
Online publication date: 2019-07-06
 
 
Publication date: 2019-07-06
 
 
EURASIA J. Math., Sci Tech. Ed 2019;15(12):em1782
DOI: https://doi.org/10.29333/ejmste/110490
 
 
Article (PDF)
References (62)
 
KEYWORDS
meaning
non-compulsory secondary education
semantic analysis
understanding of a mathematical concept
 
ABSTRACT
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.
REFERENCES (62)
1.
Akkoc, H. (2008). Pre-service mathematics teachers’ concept images of radian. International Journal of Mathematical Education in Science and Technology, 39(7), 857-878. https://doi.org/10.1080/002073....
CrossRef
 
Google Scholar
 
 
2.
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States and Canada: A consideration of elementary course content and approach and factors influencing change, 1890-1970 (Unpublished doctoral dissertation), The State University of New Jersey, New Brunswick, NJ.
Google Scholar
 
 
3.
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. https://doi.org/10.1007/0-387-....
CrossRef
 
Google Scholar
 
 
4.
Brown, S.A. (2005). The trigonometric connections: Students’ understanding of sine and cosine (Unpublished doctoral dissertation), Illinois State University, Illinois, IL.
Google Scholar
 
 
5.
Bunge, M. (2008). Tratado de Filosofía. Semántica I, Sentido y referencia. Barcelona, Spain: Gedisa Editorial.
Google Scholar
 
 
6.
Byers, P. (2010). Investigating trigonometric representations in the transition to college mathematics. College Quarterly, 13, 2, 1–10. Retrieved from http://collegequarterly.ca/201....
Google Scholar
 
 
7.
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). https://doi.org/10.1007/s10649....
CrossRef
 
Google Scholar
 
 
8.
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 http://wrap.warwick.ac.uk/1935....
Google Scholar
 
 
9.
Cohen, L. Manion, L., Morrison, K. (2011). Research Methods in Education, Routledge, London. UK.
Google Scholar
 
 
10.
Common Core State Standards Initiative. Mission statement [Internet]. 2010 [cited 2015 Oct 5]. Available from: http://www.corestandards.org/. District of Columbia: The Council of Chief State School Officers and National Governors Association Center for Best Practices.
Google Scholar
 
 
11.
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 https://flm-journal.org/Articl....
Google Scholar
 
 
12.
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 https://revistas.pucsp.br/inde....
Google Scholar
 
 
13.
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. https://doi.org/10.5951/jresem....
CrossRef
 
Google Scholar
 
 
14.
Demir, O. (2012). Students’ concept development and understanding of sine and cosine functions (Unpublished doctoral dissertation), Amsterdan University, Amsterdam, The Netherlands. Retrieved from https://esc.fnwi.uva.nl/thesis....
Google Scholar
 
 
15.
Demopoulos, W. (1994). Frege, Hilbert, and the conceptual structure of model theory. History and Philosophy of Logic, 15(2), 211-225. https://doi.org/10.1080/014453....
CrossRef
 
Google Scholar
 
 
16.
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.
Google Scholar
 
 
17.
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/euras....
CrossRef
 
Google Scholar
 
 
18.
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.
Google Scholar
 
 
19.
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. https://doi.org/10.1080/002073....
CrossRef
 
Google Scholar
 
 
20.
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 https://ir.uiowa.edu/cgi/viewc....
Google Scholar
 
 
21.
Fraenkel, J. R., & Warren, E. A. (2006). How to Design and Evaluate Research in Education. Newyork: McGraw-Hill.
Google Scholar
 
 
22.
Frege, G. (1962). Über Sinn und Bedeutung. [About sense and meaning] In G. Patzig (Ed.), Funktion, Begriff, Bedeutung. Fünf logische Studien (pp. 40–65).
Google Scholar
 
 
23.
Freudenthal, H. (1983). Didactycal phenomenology of mathematical structures. Dordrecht, Holland: Reidel. Retrieved from http://gpdmatematica.org.ar/wp....
Google Scholar
 
 
24.
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.
Google Scholar
 
 
25.
Gur, H. (2009). Trigonometry Learning. New Horizons in Education, 57(1), 67-80. Retrieved from https://files.eric.ed.gov/full....
Google Scholar
 
 
26.
Hertel, J. T. (2013). Investigating the purpose of trigonometry in the modern sciences (Unpublished doctoral dissertation), Illinois State University, Illinois, IL.
Google Scholar
 
 
27.
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. https://doi.org/10.2307/749496.
CrossRef
 
Google Scholar
 
 
28.
Kamber, D., & Takaci, D. (2018). On problematic aspects in learning trigonometry. International Journal of Mathematical Education in Science and Technology, 49(2), 161-175. https://doi.org/10.1080/002073....
CrossRef
 
Google Scholar
 
 
29.
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 http://repository.upenn.edu/as....
Google Scholar
 
 
30.
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.
Google Scholar
 
 
31.
Maor, E. (1998). Trigonometric delights. Princeton, NJ: Princeton University Press.
Google Scholar
 
 
32.
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 https://etd.ohiolink.edu/!etd.....
Google Scholar
 
 
33.
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. https://doi.org/10.5565/rev/en....
CrossRef
 
Google Scholar
 
 
34.
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.
Google Scholar
 
 
35.
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. https://doi.org/10.1590/1980-4....
CrossRef
 
Google Scholar
 
 
36.
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 http://www.uwyo.edu/wisdome/_f....
Google Scholar
 
 
37.
Moore, K. C., & LaForest, K. R. (2014). The circle approach to trigonometry. Mathematics Teacher, 107(8), 616-623. https://doi.org/10.5951/mathte....
CrossRef
 
Google Scholar
 
 
38.
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. https://doi.org/10.1007/s10649....
CrossRef
 
Google Scholar
 
 
39.
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....
CrossRef
 
Google Scholar
 
 
40.
Orhun, N. (2004). Students’ mistakes and misconceptions on teaching of trigonometry. Retrieved from http://dipmat.math.unipa.it/~g....
Google Scholar
 
 
41.
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.
Google Scholar
 
 
42.
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. https://doi.org/10.1207/S15327....
CrossRef
 
Google Scholar
 
 
43.
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 http://funes.uniandes.edu.co/6....
Google Scholar
 
 
44.
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 https://dialnet.unirioja.es/se....
Google Scholar
 
 
45.
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. https://doi.org/10.1016/S0732-....
CrossRef
 
Google Scholar
 
 
46.
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. https://doi.org/10.1080/147948....
CrossRef
 
Google Scholar
 
 
47.
Shapiro, S. (1997). Philosophy of mathematics: Structure and ontology. Oxford, Uk: Oxford University Press.
Google Scholar
 
 
48.
Sickle, J. V. (2011). A history of trigonometry Education in the United States 1776-1990 (Unpublished doctoral dissertation), Combia Uniluversity, Nueva York, NY. https://doi.org/10.7916/D8G166....
CrossRef
 
Google Scholar
 
 
49.
Skemp, R. (1987). The psychology of learning mathematics, Hillsdale, NJ: Erlbaum Associates.
Google Scholar
 
 
50.
Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers. Journal of Mathematics Teacher Education, 1(2), 157-189. https://doi.org/10.1023/A:1009....
CrossRef
 
Google Scholar
 
 
51.
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. https://doi.org/10.1007/s10649....
CrossRef
 
Google Scholar
 
 
52.
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 http://200.24.17.68:8080/jspui....
Google Scholar
 
 
53.
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.
Google Scholar
 
 
54.
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. https://doi.org/10.1007/s10857....
CrossRef
 
Google Scholar
 
 
55.
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. https://doi.org/10.1023/B:EDUC....
CrossRef
 
Google Scholar
 
 
56.
Van den Heuvel-Panhuizen, M. (2014). Didactical phenomenology (Freudenthal). In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 174-176). Heidelberg: Springer. https://doi.org/10.1007/978-94....
CrossRef
 
Google Scholar
 
 
57.
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.
Google Scholar
 
 
58.
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.
Google Scholar
 
 
59.
Weber, K. (2005). Student’s understanding of trigonometric functions. Mathematics Education Research Journal, 102(2), 144-147. Retrieved from https://files.eric.ed.gov/full....
Google Scholar
 
 
60.
Weber, K. (2008). Teaching trigonometric functions: Lessons learned from research. Mathematics Teacher, 17(3), 91-112. https://doi.org/10.1007/BF0321....
CrossRef
 
Google Scholar
 
 
61.
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. https://doi.org/10.1080/002073....
CrossRef
 
Google Scholar
 
 
62.
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.
Google Scholar
 
 
Share
Send by email
RELATED ARTICLE
Exploring Geometry Teaching Model: Polygon Pieces and Dictionary Tools for the Model
Indexes
 
Keywords index
Topics index
Authors index
eISSN:1305-8223
ISSN:1305-8215
Journals System - logo
© 2006-2024 Journal hosting platform by Bentus
Scroll to top