The role of metaphors in interpreting students’ difficulties in operating with percentages: A mixed method study based on large scale assessment

Chiara Giberti 1 * , George Santi 2, Camilla Spagnolo 3
More Detail
1 University of Bergamo, Bargamo, ITALY
2 University of Macerata, Macerata, ITALY
3 Free University of Bolzano-Bozen, Bolzano, ITALY
* Corresponding Author
EUR J SCI MATH ED, Volume 11, Issue 2, pp. 297-321. https://doi.org/10.30935/scimath/12642
Published Online: 15 November 2022, Published: 01 April 2023
OPEN ACCESS   1383 Views   864 Downloads
Download Full Text (PDF)

ABSTRACT

The issue of students’ difficulties in processing operations with percentages has been addressed in several international research studies from a qualitative perspective. In this study, we analyze students’ difficulties on this topic, focusing on the transition from middle school to high school with a mixed methods research design. We focus on students’ responses in a specific task belonging to the Italian large-scale assessment analyzed through the Rasch model, and we deepen the task analysis thanks to interviews, which enlightened image schemas and metaphors underlying students’ reasoning. From the qualitative point of view, the Rasch model shows that students’ difficulties in dealing with percentages is a macrophenomenon that involves the higher levels of competences. From the qualitative point of view, the metaphoric approach outlines the image schemas that foster the correct conceptualization of percentage and those that hinder their correct learning and can be one of the possible causes of the emerging aforementioned macrophenomenon.
Keywords: percentages, mixed method, large scale assessment, Rasch analysis, semiotics, sensuous cognition, metaphors

CITATION

Giberti, C., Santi, G., & Spagnolo, C. (2023). The role of metaphors in interpreting students’ difficulties in operating with percentages: A mixed method study based on large scale assessment. European Journal of Science and Mathematics Education, 11(2), 297-321. https://doi.org/10.30935/scimath/12642

REFERENCES

  • Arzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de Investigación en Matemática Educativa RELIME [Latin American Journal of Research in Educational Mathematics RELIME], 9(Supplement 1), 267-299.
  • Asenova, M. (2022). Non-classical approaches to logic and quantification as a means for analysis of classroom argumentation and proof in mathematics education research. Acta Scientiae, 24(5), 404-428. https://doi.org/10.17648/acta.scientiae.7405
  • Barbaranelli, C., & Natali, E. (2005). I test psicologici. Teorie e modelli psicometrici [Psychological tests. Psychometric theories and models]. Carocci.
  • Bennett, A. B., & Nelson, L. T. (1994). A conceptual model for solving percent problems. Mathematics Teaching in the Middle School, 1(1), 20-25. https://doi.org/10.5951/MTMS.1.1.0020
  • Bolondi, G., Ferretti, F., & Gambini, A. (2017). Il database GESTINV delle prove standardizzate INVALSI: Uno strumento per la ricercar [The GESTINV database of INVALSI standardized tests: A research tool]. In P. Falzetti (Ed.), I dati INVALSI: Uno strumento per la ricerca [INVALSI data: A tool for research] (pp. 33-42). Franco Angeli.
  • 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(2018), 415-435. https://doi.org/10.7358/ecps-2018-018-bolo
  • Boone, W. W. (1959). The word problem. Annals of Mathematics, 70(2), 207-265. https://doi.org/10.2307/1970103
  • Clements, D. H., & Sarama, J. (2009). Learning trajectories in early mathematics–sequences of acquisition and teaching. Encyclopedia of Language and Literacy Development, 7, 1-6.
  • Confrey, J., & Maloney, A. (2014). Linking standards and learning trajectories. In A. P. Maloney, J. Confrey, & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education (pp. 125-160). IAP.
  • Corni, F., & Fuchs, H. (2020). Primary physical science for student teachers at kindergarten and primary school levels: Part I-Foundations of an imaginative approach to physical science. Interchange, 51, 315-343. https://doi.org/10.1007/s10780-019-09382-0
  • Corni, F., Fuchs, H., & Savino, G. (2018). An industrial educational laboratory at Ducati Foundation: Narrative approaches to mechanics based upon continuous physics. International Journal of Science Education, 40, 243-267. https://doi.org/10.1080/09500693.2017.1407886
  • D’Amore, B. (2003). La complexité de la noétique en mathématiques ou les raisons de la dévolution manquée [The complexity of noetic in mathematics or the reasons for failed devolution]. For the Learning of Mathematics, 23(1), 47-51.
  • Den Heuvel-Panhuizen, V. (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.0000005212.03219.dc
  • Duval, R. (1993). Registres de représentations sémiotique et fonctionnement cognitif de la pensée [Registers of semiotic representations and cognitive functioning of thought]. Annales de Didactique et de Sciences Cognitives [Annals of Didactics and Cognitive Sciences], 5(1), 37-65.
  • Duval, R. (1995). Sémiosis et pensée humaine: Registres sémiotiques et apprentissages intellectuels [Semiosis and human thought: Semiotic registers and intellectual learning]. Peter Lang.
  • Duval, R. (2017). Understanding the mathematical way of thinking-The registers of semiotic representations. Springer International Publishing. https://doi.org/10.1007/978-3-319-56910-9
  • Edwards, L. D. (2010). Doctoral students, embodied discourse, and proof. In M. M. F. Pinto, & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (pp. 329-336). PME.
  • Egan, K. (1997). The educated mind. How cognitive tools shape our understanding. University of Chicago Press. https://doi.org/10.7208/chicago/9780226190402.001.0001
  • Egan, K. (2002). Getting it wrong from the beginning: Our progressivist inheritance from Herbert Spencer, John Dewey, and Jean Piaget. Yale University Press.
  • Ernest, P. (2006). A semiotic perspective of mathematical activity: The case of number. Educational Studies in Mathematics, 61(1-2), 67-101. https://doi.org/10.1007/s10649-006-6423-7
  • Ferretti, F., & Giberti, C. (2021). The properties of powers: Didactic contract and gender gap. International Journal of Science and Mathematical Education, 19, 1717-1735. https://doi.org/10.1007/s10763-020-10130-5
  • Ferretti, F., Gambini, A., & Santi, G. (2020). The Gestinv Database: A tool for enhancing teachers professional development within a community of inquiry. In H. Borko & D. Potari (Eds.), Proceedings of the Twenty-fifth ICMI Study School Teachers of mathematics working and learning in collaborative groups (pp.621–628). University of Lisbon.
  • 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., Santi, G., & Bolondi, G. (2022). Interpreting difficulties in the learning of algebraic inequalities, as an emerging macrophenomenon in Large scale Assessment. Research in Mathematics Education. https://doi.org/10.1080/14794802.2021.2010236
  • Gambini, A., Desimoni, M., & Ferretti, F. (2022). Predictive tools for university performance: An explorative study. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2021.2022794
  • Gestinv. (2018). Archivio interattivo delle prove INVALSI [Interactive archive of INVALSI tests]. http://www.gestinv.it
  • Giberti, C. (2018). Differenze di genere e misconcezioni nell’operare con le percentuali: evidenze dalle prove INVALSI. CADMO, 2018(2), 97-114. https://doi.org/10.3280/CAD2018-002007
  • Hodnik Čadež, T., & Kolar, V. M. (2017). Monitoring and guiding pupils’ problem solving. Magistra Iadertina, 12(2), 0-139. https://doi.org/10.15291/magistra.1493
  • Johnson, M. (1987). The body in the mind: The bodily basis of meaning, imagination, and reason. University of Chicago Press. https://doi.org/10.7208/chicago/9780226177847.001.0001
  • Johnson, M. (2007). The meaning of the body: Aesthetics of human understanding. University of Chicago Press. https://doi.org/10.7208/chicago/9780226026992.001.0001
  • 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
  • Lakoff, G. (1987). Women, fire, and dangerous things. University of Chicago Press. https://doi.org/10.7208/chicago/9780226471013.001.0001
  • Lakoff, G., & Johnson, M. (1980). Metaphors we live by. University of Chicago Press.
  • Lakoff, G., & Johnson, M. (1999). Philosophy in the flesh. Basic Books.
  • Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3(4), 305-342. https://doi.org/10.1207/s1532690xci0304_1
  • Leont’ev, A. N. (1978). Activity, consciousness, and personality. Prentice-Hall.
  • Lestiana, H. T. (2021). What are the difficulties in learning percentages? An overview of prospective mathematics teachers’ strategies in solving percentage problems. Indonesian Journal of Science and Mathematics Education, 4(3), 260-273. https://doi.org/10.24042/ijsme.v4i3.10132
  • MIUR. (2010). Indicazioni nazionali per i licei [National indications for high schools]. Le Monnier.
  • MIUR. (2012). Indicazioni nazionali per il curricolo della scuola dell’infanzia e del primo ciclo d’istruzione [National guidelines for the nursery school curriculum and the first cycle of education]. Le Monnier.
  • Ningsih, S., Indra Putri, R. I., & Susanti, E. (2017). The use of grid 10×10 in learning the percent. Mediterranean Journal of Social Sciences, 8(2), 113. https://doi.org/10.5901/mjss.2017.v8n2p113
  • Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421-481. https://doi.org/10.3102/00346543065004421
  • Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237-268. https://doi.org/10.1023/A:1017530828058
  • 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/S15327833MTL0501_02
  • Radford, L. (2008). The ethics of being and knowing: Towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 215-234). Sense Publishers. https://doi.org/10.1163/9789087905972_013
  • Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2-7.
  • Radford, L. (2014). Towards an embodied, cultural, and material conception of mathematics cognition. ZDM-Mathematics Education, 46(3), 349-361. https://doi.org/10.1007/s11858-014-0591-1
  • Radford, L. (2021). The theory of objectification: A Vygotskian perspective on knowing and becoming in mathematics teaching and learning. BRILL. https://doi.org/10.1163/9789004459663
  • Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests. Danish Institute for Educational Research.
  • Rianasari, V. F., Budaya, I. K., & Patahudin, S. M. (2012). Supporting students’ understanding of percentage. Journal on Mathematics Education, 3(1), 29-40. https://doi.org/10.22342/jme.3.1.621.29-40
  • Santi, G., Bolondi, G., & Ferretti (2021). Large scale assessment (LASA): A tool for mathematics education research. In P. Falzetti (Ed.), INVALSI data: assessment on teaching and methodologie. IV Seminar “INVALSI data: a research and educational teaching tool (pp. 46-65). Franco Angeli.
  • Santi, G., Ferretti, F., & Martignone, F. (in press). Mathematics teachers specialised knowledge and Gestinv Database. In P. Falzetti (Ed.), INVALSI data: assessment on teaching and methodologie. V Seminar “INVALSI data: a research and educational teaching tool. Franco Angeli.
  • Scaptura, C., Suh, J., & Mahaffey, G. (2007). Masterpieces to mathematics: Using art to teach fraction, decimal, and percent equivalents. Mathematics Teaching in the Middle School, 13(1), 24-28. https://doi.org/10.5951/MTMS.13.1.0024
  • Spagnolo, C., Giglio, R., Tiralongo, S., & Bolondi, G. (2022). Formative assessment in LDL workshop activities: Engaging teachers in a training program. In International Conference on Computer Supported Education (pp. 560-576). Springer Science and Business Media Deutschland GmbH, Cham. https://doi.org/10.1007/978-3-031-14756-2_27
  • Van Galen, M. S., & Reitsma, P. (2008). Developing access to number magnitude: A study of the SNARC effect in 7-to 9-year-olds. Journal of Experimental Child Psychology, 101(2), 99-113. https://doi.org/10.1016/j.jecp.2008.05.001
  • Wartofsky, M. W. (1984). The paradox of painting: Pictorial representation and the dimensionality of visual space. Social Research, 51(4), 863-883.
  • Wittgenstein, L. (1953). Philosophische untersuchugen [Philosophical investigations]. Basil Blackwell.