The learning of algebra represents a difficulty within the mathematics learning process. Precisely because of their epistemological and ontological nature, semiotics provides a good key to understanding the main difficulties that hinder the learning of algebraic objects. The answers to a large scale assessments task that requires a treatment in the sense of Duval of sizeable sample of upper secondary school student is analysed. Statistical analysis allows the nationwide data to be considered according to students’ ability levels. The study confirms and quantifies the distance between personal meaning and the cultural meaning attributed to algebraic objects. The difficulties of the manipulation of algebraic expressions is deepened and characterized.

• Arcavi, A. (1995). Teaching and Learning Algebra: Past, Present, and Future. Journal of Mathematical Behavior, 14(1), 145-62. https://doi.org/10.1016/0732-3123(95)90033-0
• Bell, A. (1995). Purpose in school algebra. The Journal of Mathematical Behavior, 14(1), 41-73. https://doi.org/10.1016/0732-3123(95)90023-3
• Bolondi, G., Ferretti, F., & Giberti, C. (2018). Didactic Contract as a Key to Interpreting Gender Differences in Maths. Journal of Educational, Cultural and Psychological Studies (ECPS Journal), (18), 415-435. https://doi.org/10.7358/ecps-2018-018-bolo
• Callingham, R., & Bond, T. (2006). Research in mathematics education and Rasch measurement. Mathematics Education Research Journal, 18(2), 1-10. https://doi.org/10.1007/BF03217432
• De Lange, J. (2007). Large-Scale Assessment and Mathematics Education. In Frank K. Lester, Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 1111-1142). Charlotte, NC: National Council of Teachers of Mathematics (NCTM).
• Duval, R. (1988). Pour une approche cognitive des problèmes de géométrie en termes de congruences. Annales de Didactique et de Sciences Cognitives, 1, 57–74. Strasbourg, France: IREM et Université Louis Pasteur.
• Duval, R. (1993). Registres de représentations sémiotique et fonctionnement cognitif de la pensée. Annales de Didactique et de Sciences Cognitives, 5, 37–65. Strasbourg, France: IREM et Université Louis Pasteur.
• Duval, R. (1995). Sémiosis et pensée humaine: Registres sémiotiques et apprentissages intellectuels. Berne: Peter Lang.
• Duval, R. (2008). Eight problems for a semiotic approach in mathematics education. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 39–61). Rotterdam: Sense Publishers.
• Ernest, P. (1991). The philosophy of mathematics education. London, Falmer Press.
• Ferretti, F., Giberti, C., & Lemmo, A. (2018). The Didactic Contract to interpret some statistical evidence in mathematics standardized assessment tests. EURASIA Journal of Mathematics, Science and Technology Education, 14(7), 2895-2906. https://doi.org/10.29333/ejmste/90988
• Ferretti, F., Lemmo, A. & Martignone, F. (2018). Attained curriculum and external assessment in Italy: how to reflect on them?. In Y. Shimizu and R. Vithal (Eds.), Proceedings of the Twenty-fourth ICMI Study School Mathematics Curriculum Reforms: Challenges, Changes and Opportunities (pp.381–388). Tsukuba, Japan: ICMI.
• Gestinv 2.0. Interactive archive of the INVALSI tests. http://www.gestinv.it (ver. 15.04.2019).
• Godino J., Batanero C. (1994). Significado institucional y personal de los objectos matematicos. Recherches en Didactique des Mathématiques. 14(3), 325-355.
• Johnson, R. B., & Onwuegbuzie, A. J. (2004). Mixed methods research: A research paradigm whose time has come. Educational Researcher, 33(7), 14-26. https://doi.org/10.3102/0013189X033007014
• Hart, L.C., Smith, S.Z., Swars, S. L., Smith, M.E. (2009). An Examination of Research Methods in Mathematics Education (1995-2005). Journal of Mixed Methods Research 3(1), 26-41. https://doi.org/10.1177/1558689808325771
• Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (ed.), Handbook of research on mathematics teaching and learning. Macmillan, New York.
• Kieran, C. (1995). A new look at school algebra-Past, present, and future. The Journal of Mathematical Behavior, 14(1), 7-12. https://doi.org/10.1016/0732-3123(95)90020-9
• Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. Second handbook of research on mathematics teaching and learning, 2, 707-762.
• Kieran, C. (2014). Algebra teaching and learning. Encyclopedia of Mathematics Education, 27-32. https://doi.org/10.1007/978-94-007-4978-8_6
• Radford, L. (2002). The seen, the spoken and the written: A semiotic approach to the problem of objectification of mathematical knowledge. For the learning of mathematics, 22(2), 14-23.
• Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In Semiotics in mathematics education (pp. 215-234). Rotterdam: Sense Publishers
• Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective. Research in Mathematics Education, 12(1), 1-19. https://doi.org/10.1080/14794800903569741
• Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Denmarks Paedagogiske Institut.
• Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives. The Journal of Mathematical Behavior, 14(1), 15-3. https://doi.org/10.1016/0732-3123(95)90022-5

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