An Exploratory Analysis of the Representations of Functions in the University Entrance Exam in Spain and Iran
Vahid Borji 1, 2  
More details
Hide details
Department of Linguistic and Literary Education, and Teaching and Learning of Experimental Sciences and Mathematics, University of Barcelona, Catalonia, SPAIN
Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, IRAN
Online publish date: 2019-04-09
Publish date: 2019-04-09
EURASIA J. Math., Sci Tech. Ed 2019;15(8):em1727
The University Entrance Exam (UEE) has an important impact on how math instructors teach mathematics to prepare students for this exam as well as how their students learn mathematics. Since the UEE is like a bridge between high school and university, its importance becomes twofold. However, research regarding this issue is very limited in mathematics education. The purpose of this research is to conduct an exploratory analysis of mathematics questions from the UEE in Spain and Iran from the point of view of representations of functions (i.e., algebraic, graphical, numerical and explanations with words). For this, the mathematics questions related to derivatives and integrals, which are two of the most important concepts in mathematics, from the last five years of the UEE in these two countries were considered to analyze them from the point of view of different representations. The results showed that most of the derivative and integral questions in the UEE in both countries were related to the algebraic representation. However, a very small percentage of the derivative and integral questions were related to the graphical and numerical representations and the relationships between the representations. At the end, suggestions to the curriculum planners and designers of mathematics questions of the UEEs are given.
Ainsworth, S. (1999). The Functions of Multiple Representations. Computers & Education, 33(2), 131-152,
Badillo, E., Azcárate, C., & Font, V. (2011). Analysis of Mathematics teachers’ level of understanding of the objects
Baker, B., Cooley, L., & Trigueros, M. (2000). A calculus graphing schema. Journal for Research in Mathematics Education, 31(5), 557–578.
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.
Barmby, P., Bilsborough, L., Harries, T. & Higgins, S. (2009). Primary Mathematics: teaching for understanding. Berkshire: Open University Press.
Beltrán-Pellicer, P., Godino, J. D. & Giacomone, B. (2018). Elaboration of Specific Didactical Suitability Criteria in Probability: Application for Reflection on the Teaching Practice. Bolema, 32(61), 526-548.
Bolden, D. S., Barmby, P. W., & Harries, A. V. (2013). A representational approach to developing primary ITT students’ confidence in their mathematics. International Journal of Mathematical Education in Science and Technology, 44(1), 70-83.
Borji, V., Alamolhodaei, H., & Radmehr, F. (2018). Application of the APOS-ACE Theory to improve Students’ Graphical Understanding of Derivative. EURASIA Journal of Mathematics, Science and Technology Education, 14(7), 2947-2967.
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.
Breda, A., Font, V., & Pino-Fan, L. (2018). Evaluative and normative criteria in Didactics of Mathematics: the case of didactical suitability construct. Bolema, 32(60), 255-278.
Breda, A., Pino-Fan, L. R., & 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.
Bruner, J. S. & Kenney, H. J. (1965). Representation and mathematics learning. The Society for Research in Child Development (monographs), 30(1), 50-59.
Cooley, L., Trigueros, M., & Baker B. (2007). Schema thematization: A theoretical framework and an example. Journal for Research in Mathematics Education, 38(4), 370-392.
Davis, R. B. (1984). Learning Mathematics: The Cognitive Approach to Mathematics Education. London: Croom Helm.
Dominguez, A., Barniol, P., & Zavala, G. (2017). Test of Understanding Graphs in Calculus: Test of Students’ Interpretation of Calculus Graphs. EURASIA Journal of Mathematics, Science and Technology Education, 13(10), 6507-6531.
Dreher, A., & Kuntze, S. (2015). Teachers’ professional knowledge and noticing: The case of multiple representations in the mathematics classroom. Educational Studies in Mathematics, 88(1), 89-114.
Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking: Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty First Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Morelos, México, Vol. I, pp. 3-26.
Farrokhi-Khajeh-Pasha, Y., Nedjat, S., Mohammadi, A., Malakan Rad, E., Majdzadeh, R., Monajemi, F., Jamali, E., & Yazdani, S. (2012). The validity of Iran’s national university entrance examination (Konkoor) for predicting medical students’ academic performance. BMC Medical Education, 12(60), 60,
Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 147-164). New York, NY, England: Macmillan Publishing Co, Inc.
Fuentealba, C., Sánchez-Matamoros, G., Badillo, E., & 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.
Goldin, G., & Shteingold, N. (2001). Systems of representation and the development of mathematical concepts. In A. A. Cuoco (Ed.), The roles of representation in school mathematics, NCTM 2001 Yearbook (pp. 1-23). Reston, VA: National Council of Teachers of Mathematics.
Greeno, J. G., & Hall, R. P. (1997). Practicing Representation: Learning with and about representational forms. Phi Delta Kappan, 78(5), 361-368.
Hardy, N. (2009). Students’ perceptions of institutional practices: the case of limits of functions in college level Calculus courses. Educational Studies in Mathematics, 72(3), 341-358,
Kaput, J. (1992). Technology and mathematics education. In D.A. Grouws (Ed.) Handbook of research on mathematics and teaching and learning (pp. 515-556). New York, NY: Macmillan Publishing Company.
Kendal, M., & Stacey, K. (2003). Tracing learning of three representations with the Differentiation Competency Framework. Mathematics Education Research Journal, 15(1), 22-41.
Kim, C. W., & Dembo M. H. (2000). Social- cognitive factors influencing success on college entrance exams in South Korea. Social Psychology of Education, 4(2), 95-115,
Koçkar, A. İ., & Gençöz, T. (2004). Personality, Social Support, and Anxiety among Adolescents Preparing for University Entrance Examinations in Turkey. Current Psychology, 23(2), 138-146.
Konečný, T., Basl, J., Mysliveček, J., & Simonová, N. (2012). Alternative models of entrance exams and access to higher education: The case of the Czech Republic. Higher Education, 63(2), 219-235. https://doi:10.1007/s10734-011....
Kusayanagi, C. (2013). Constructing and Understanding an Incident as a Social Problem: A Case Study of University Entrance Exam Cheating in Japan. Human Studies, 36(1), 133-148.
Lesh, R., Landau, M., & Hamilton, E. (1983). Conceptual models and applied mathematical problem-solving research. In R. Lesh and M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 263-343). Orlando, FL: Academic Press.
Ley Orgánica 2/2006, de 3 de mayo, de Educación. Boletín Oficial del Estado, 4 de mayo de 2006, num. 106, pp. 17158¬–17207. Retrieved from
Mao, Y., White, T., Sadler, P. M., & Sonnert, G. (2017). The association of precollege use of calculators with student performance in college calculus. Educational Studies in Mathematics, 94(1), 69-83,
McAteer, M. (2012). Improving Primary Mathematics Teaching and Learning. England: Open University Press.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM, Reston VA.
National Organization of Educational Testing. (n.d). Retrieved from
National Statistics Institute (2012). Boletín informativo del Instituto Nacional de Estadística. Panorámica de la educación universitaria. Retrieved from
Orden ECD/42/2018, de 25 de enero, por la que se determinan las características, el diseño y el contenido de la evaluación de Bachillerato para el acceso a la Universidad, las fechas máximas de realización y de resolución de los procedimientos de revisión de las calificaciones obtenidas, para el curso 2017/2018. Boletín Oficial del Estado, 26 de enero de 2018, num. 23, pp. 9757–9789. Retrieved from
Özmantar M. F., Akkoç, H., Bingölbali, E., Demir, S., & Ergene B. (2010). Pre-Service Mathematics Teachers’ Use of Multiple Representations in Technology-Rich Environments. Eurasia Journal of Mathematics, Science & Technology Education, 6(1), 19-36.
Pape, S. J., & Tchoshanov, M. (2001). The role of representation(s) in developing mathematical understanding. Theory into Practice, 40(2), 118-127.
Park, J. (2016). Communicational approach to study textbook discourse on the derivative. Educational Studies in Mathematics, 91(3), 395–421.
Post, T., & Cramer, K. A. (1989). Knowledge, representation, and quantitative thinking: A special AACTE publication. In M. Reynolds, & W. Gardner (Eds.), Knowledge base for beginning teachers: A special AACTE publication (pp. 221-232). Oxford, UK: Pergamon Press.
Real Decreto 1892/2008, de 14 de noviembre, por el que se regulan las condiciones para el acceso a las enseñanzas universitarias oficiales de grado y los procedimientos de admisión a las universidades públicas españolas. Boletín Oficial del Estado, 24 de noviembre de 2008, num. 283, pp. 46932–46946. Retrieved from
Real Decreto 558/2010, de 7 de mayo, por el que se modifica el Real Decreto 1892/2008, de 14 de noviembre, por el que se regulan las condiciones para el acceso a las enseñanzas universitarias oficiales de grado y los procedimientos de admisión a las universidades públicas españolas. Boletín Oficial del Estado, 8 de mayo de 2010, num. 113, pp. 40784–40788. Retrieved from
Real Decreto 310/2016, de 29 de julio, por el que se regulan las evaluaciones finales de Educación Secundaria Obligatoria y de Bachillerato. Boletín Oficial del Estado, 30 de julio de 2016, num. 183, pp. 53049–53065. Retrieved from
Research and Educational Planning Organization. (2013). Secretariat of designing and producing the mathematics curriculum of Islamic Republic of Iran [In Persian].
Ronda, E. (2015). Growth points in linking representations of function: a research-based framework. Educational Studies in Mathematics, 90(3), 303-319.
Sahin, Z., Erbas, A. K., & Yenmez, A. A. (2015). Relational Understanding of the Derivative Concept through Mathematical Modeling: A Case Study. EURASIA Journal of Mathematics, Science & Technology Education, 11(1), 177-188.
Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational Researcher, 15(2), 4-14.
Spangler, D. B. (2010). Strategies for teaching whole number computation: using error analysis for intervention and assessment. Thousand Oaks, CA: Corwin Press.
Stewart, J. (2010). Calculus, 7th Edition. Brooks/Cole Cengage Learning, Mason.
Wood, D. (1999). Editorial: Representing, learning and understanding. Computers & Education, 33(2-3), 83-90.
Zhang, Y. (2016). National College Entrance Exam in China: Perspectives on Education Quality and Equity. The Netherlands: Springer Briefs in Education.