Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers

Ioannis Rizos; Maria Adam

doi:10.29333/iejme/11977

Full Text (PDF)

Ioannis Rizos ¹ ^* , Maria Adam ²

More Detail

¹ Department of Mathematics, University of Thessaly, Lamia, GREECE² Department of Computer Science & Biomedical Informatics, University of Thessaly, Lamia, GREECE^* Corresponding Author

Abstract

In this paper, we present parts of an online qualitative research project which lasted a whole semester about the understanding of real numbers by sixty first-year Greek mathematics undergraduate students during the COVID-19 pandemic. We focus on the distinction between rational and irrational numbers and their location on the number line using straightedge and compass. The results, came of questionnaires, focused interviews and tasks, disclosed some gaps regarding real numbers that the students had from school and revealed confusion between irrational numbers and their decimal approximation. The results also led us to group students’ conceptions of rational and irrational numbers into five categories related to the number and pattern of decimal digits. Teaching practices and perspectives as well as the, negative, impact of the pandemic on teaching and learning mathematics are discussed.

Keywords

real numbers
number line
geometrical constructions
ICT in mathematics education
categories of conceptions

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, 2022, Volume 17, Issue 3, Article No: em0686

https://doi.org/10.29333/iejme/11977

Publication date: 06 Apr 2022

Article Views: 2173

Article Downloads: 1014

Open Access Disclosures References How to cite this article

Disclosure

Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Adhikari, K. P. (2021). Difficulties and misconceptions of students in learning limit. Interdisciplinary Research in Education, 5(1-2), 15-26. https://doi.org/10.3126/ire.v5i1-2.34731
Adiredja, A. P. (2021). The pancake story and the epsilon-delta definition. PRIMUS, 31(6), 662-677. https://doi.org/10.1080/10511970.2019.1669231
Alibert, D., & Thomas, M. (2002). Research on mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking (pp. 215-230). Springer. https://doi.org/10.1007/0-306-47203-1_13
Apostol, T. M. (1997). Modular functions and dirichlet series in number theory. Springer.
Arcavi, A., Bruckheimer, M., & Ben-Zvi, R. (1987). History of mathematics for teachers: The case of irrational numbers. For the Learning of Mathematics, 7(2), 18-23.
Artigue, M. (1997). Teaching and learning elementary analysis: What can we learn from didactical research and curriculum evolution? In G. Makrides (Ed.), Proceedings of 1^st Mediterranean Conference on Mathematics (pp. 207-219). https://doi.org/10.1007/0-306-47231-7_21
Bampili, A. C., Zachariades, T., & Sakonidis, C. (2019). The transition from high school to university mathematics: The effect of institutional issues on students’ initiation into a new practice of studying mathematics. In Proceedings of the 11^th Congress of the European Society for Research in Mathematics Education. Utrecht University.
Bansilal, S., & Mkhwanazi, T. W. (2021). Pre-service student teachers’ conceptions of the notion of limit. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2020.1864488
Biza, I., Christou, C., & Zachariades, T. (2008). Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean geometry to analysis. Research in Mathematics Education, 10(1), 53-70. https://doi.org/10.1080/14794800801916457
Borba, M. C. (2021). The future of mathematics education since COVID-19: Humans-with-media or humans-with-non-living-things. Educational Studies in Mathematics, 108, 355-400. https://doi.org/10.1007/s10649-021-10043-2
Burgess, R. G. (1984). In the field: An introduction to field research. Routledge.
Chinnappan, M., & Forrester, T. (2014). Generating procedural and conceptual knowledge of fractions by pre-service teachers. Mathematics Education Research Journal, 26(4), 871-896. https://doi.org/10.1007/s13394-014-0131-x
Christou, K. P., & Vamvakoussi, X. (2021). Natural number bias on evaluations of the effect of multiplication and division: The role of the type of numbers. Mathematics Education Research Journal. https://doi.org/10.1007/s13394-021-00398-3
Chrystal, G. (1889). Algebra, An elementary text-book, part I. A. & C. Black.
Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Springer. https://doi.org/10.1007/0-306-47203-1_10
Denzin, N. K. (2009). The research act: A theoretical introduction to sociological methods. Transaction Publishers.
Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational numbers in high-school students and prospective teachers. Educational Studies in Mathematics, 29, 29-44. https://doi.org/10.1007/BF01273899
Fried, N. M. (2001). Can mathematics education and history of mathematics coexist? Science & Education, 10(4), 391-408. https://doi.org/10.1023/A:1011205014608
Furinghetti, F. (2020). Rethinking history and epistemology in mathematics education. International Journal of Mathematical Education in Science and Technology, 51(6), 967-994. https://doi.org/10.1080/0020739X.2019.1565454
Gellert, U., & Jablonka, E. (2009). The demathematising effect of technology. In P. Ernest, B. Greer, & B. Sriraman (Eds), Critical Issues in Mathematics Education (pp. 19-24). IAP.
Giannakoulias, E., Souyoul, A., & Zachariades, T. (2007). Students’ thinking about fundamental real numbers properties. In Proceedings of the 5^th Congress of the European society for Research in Mathematics Education (pp. 416-425).
Greenwald, S. J. (2016). Popular culture in teaching, scholarship, and outreach: The Simpsons and Futurama. In J. Dewar, P. Hsu, & H. Pollatsek (Eds), Mathematics education (pp. 349-362). Springer. https://doi.org/10.1007/978-3-319-44950-0_24
Gulikers, I., & Blom, K. (2001). “A historical angle”, A survey of recent literature on the use and value of history in geometrical education. Educational Studies in Mathematics, 47, 223-258. https://doi.org/10.1023/A:1014539212782
Guven, B., Cekmez, E., & Karatas, I. (2011). Examining preservice elementary mathematics teachers’ understandings about irrational numbers. PRIMUS, 21(5), 401-416. https://doi.org/10.1080/10511970903256928
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44, 5-23. https://doi.org/10.1023/A:1012737223465
Helfgott, M. (2004). Two examples from the natural sciences and their relationship to the history and pedagogy of mathematics. Mediterranean Journal for Research in Mathematics Education, 3(1-2), 147-164.
Jahnke, H. N., Arcavi, A., Barbin, E., Bekken, O., Furinghetti, F., El Idrissi, A., da Silva, C.M.S., & Weeks, C. (2000). The use of original sources in the mathematics classroom. In J. Fauvel, & J. van Maanen (Eds), History in mathematics education: The ICMI study (pp. 291-328). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47220-1_9
Jankvist, U. T. (2009). A categorization of the “whys” and “hows” of using history in mathematics education. Educational Studies in Mathematics, 71, 235-261. https://doi.org/10.1007/s10649-008-9174-9
Karalis, T., & Raikou, N. (2020). Teaching at the times of COVID-19: Inferences and implications for higher education pedagogy. International Journal of Academic Research in Business and Social Sciences, 10(5), 479-493. https://doi.org/10.6007/IJARBSS/v10-i5/7219
Katz, V. J. (1993). Using the history of calculus to teach calculus. Science & Education, 2, 243-249. https://doi.org/10.1007/BF00490066
Keitel, C. (1989). Mathematics education and technology. For the Learning of Mathematics, 9(1), 7-13.
Kidron, I. (2016). Understanding irrational numbers by means of their representation as non-repeating decimals [Paper presentation]. 1^st Conference of International Network for Didactic Research in University Mathematics, Montpellier, France.
Klein, M. (1972). Mathematical thought from ancient to modern times, vol. 1-3. Oxford University Press.
Kleiner, I. (2001). History of the infinitely small and the infinitely large in calculus. Educational Studies in Mathematics, 48, 137-174. https://doi.org/10.1023/A:1016090528065
Mamona, J. (1987). Students’ interpretations of some concepts of mathematical analysis [Unpublished PhD thesis]. University of Southampton.
Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M. Limón, & L. Mason (Eds), Reconsidering conceptual change: Issues in theory and practice (pp. 233-258). Kluwer Academic Publishers.
Merton, R. K., & Kendall, P. L. (1946). The focused interview. The American Journal of Sociology, 51(6), 541-557. https://doi.org/10.1086/219886
Mishra, L., Gupta, T., & Shree, A. (2020). Online teaching-learning in higher education during lockdown period of COVID-19 pandemic. International Journal of Educational Research Open, 1, 100012. https://doi.org/10.1016/j.ijedro.2020.100012
Moseley, B. (2005). Students’ early mathematical representation knowledge: The effects of emphasizing single or multiple perspectives of the rational number domain in problem solving. Educational Studies in Mathematics, 60, 37-69. https://doi.org/10.1007/s10649-005-5031-2
Moskal, B. M. & Magone, M. E. (2000). Making sense of what students know: Examining the referents, relationships and modes students displayed in response to a decimal task. Educational Studies in Mathematics, 43(3), 313-335. https://doi.org/10.1023/A:1011983602860
Mulenga, E. M., & Marbán, J. M. (2020). Is COVID-19 the gateway for digital learning in mathematics education? Contemporary Educational Technology, 12(2), ep269. https://doi.org/10.30935/cedtech/7949
NCTM. (2006). Navigating through number and operations in grades 9-12. National Council of Teachers of Mathematics.
Neugebauer, O. (1957). The exact sciences in antiquity. Brown University Press.
O’ Connor, M. C. (2001). Can any fraction be turned into a decimal? A case study of a mathematical group discussion. Educational Studies in Mathematics, 46, 143-185. https://doi.org/10.1023/A:1014041308444
Pantsar, M. (2016). The great gibberish–Mathematics in western popular culture. In B. Larvor (Ed.), Mathematical cultures: The London meetings 2012-2016 (pp. 409-437). Springer. https://doi.org/10.1007/978-3-319-28582-5_23
Patel, P., & Varma, S. (2018). How the abstract becomes concrete: Irrational numbers are understood relative to natural numbers and perfect squares. Cognitive Science, 42(5), 1-35. https://doi.org/10.1111/cogs.12619
Patronis, T., & Thomaidis, Y. (1997). On the arithmetization of school geometry in the setting of modern axiomatics. Science & Education, 6, 273-290. https://doi.org/10.1023/A:1008603828653
Peled, I., & Hershkovitz, S. (1999). Difficulties in knowledge integration: Revisiting Zeno’s paradox with irrational numbers. International Journal of Mathematical Education in Science and Technology, 30(1), 39-46. https://doi.org/10.1080/002073999288094
Pitta-Pantazi, D., Christou, C., & Pittalis, M. (2020). Number teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. https://doi.org/10.1007/978-3-030-15789-0_122
Polya, G. (1945). How to solve it. Princeton University Press. https://doi.org/10.1515/9781400828678
Potari, D., Psycharis, G., Sakonidis, C., & Zachariades, T. (2019). Collaborative design of a reform-oriented mathematics curriculum: Contradictions and boundaries across teaching, research, and policy. Educational Studies in Mathematics, 102, 417-434. https://doi.org/10.1007/s10649-018-9834-3
Rizos, I., & Gkrekas, N. (2022). Teaching and learning sciences within the COVID-19 pandemic era in a Greek university department. U. Porto Journal of Engineering, 8(1), 73-83. https://doi.org/10.24840/2183-6493_008.001_0008
Rizos, I., Patronis, A., & Lappas, D. (2017). “There is one geometry and in each case there is a different formula”. Students’ conceptions and strategies in an attempt of producing a Minkowskian metric on space-time. Science & Education, 26(6), 691-710. https://doi.org/10.1007/s11191-017-9915-1
Rizos, I., Patronis, T., & Papadopoulou, A. (2021). Difficulties in basic arithmetic and geometry as related to school algebra and the current effect of demathematization. For the Learning of Mathematics, 41(1), 37-39.
Roell, M., Viarouge, A., Houdé, O., & Borst, G. (2017). Inhibitory control and decimal number comparison in school-aged children. PLoS ONE, 12(11), e0188276. https://doi.org/10.1371/journal. pone.0188276
Roell, M., Viarouge, A., Houdé, O., & Borst, G. (2019). Inhibition of the whole number bias in decimal number comparison: A developmental negative priming study. Journal of Experimental Child Psychology, 177, 240-247. https://doi.org/10.1016/j.jecp.2018.08.010
Roh, K. H. (2008). Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics, 69(3), 217-233. https://doi.org/10.1007/s10649-008-9128-2
Siegler, R. S., & Lortie-Forgues, H. (2017). Hard lessons: Why rational number arithmetic is so difficult for so many people. Current Directions in Psychological Science, 26(4), 346-351. https://doi.org/10.1177/0963721417700129
Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994-2004. https://doi.org/10.1037/a0031200
Sierpinska, A. (1989). On 15-17 years old students’ conceptions of functions, iteration of functions and attractive fixed points. Académie des Sciences de Pologne [Polish Academy of Sciences].
Sierpinska, A. (2005). On practical and theoretical thinking and other false dichotomies in mathematics education. In M. Hoffmann, J. Lenhard, & F. Seeger (Eds), Activity and sign: Grounding mathematics education (pp. 117-135). Springer. https://doi.org/10.1007/0-387-24270-8_11
Sirotic, N., & Zazkis, R. (2007). Irrational numbers on the number line–where are they? International Journal of Mathematical Education in Science and Technology, 38(4), 477-488. https://doi.org/10.1080/00207390601151828
Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Kluwer Academic Publishers. https://doi.org/10.1007/978-94-017-3556-8
Smith, C. L., Solomon, G. E. A., & Carey, S. (2005). Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter. Cognitive Psychology, 51, 101-140. https://doi.org/10.1016/j.cogpsych.2005.03.001
Stafylidou, S., & Vosniadou, S. (2004). The development of students’ understanding of the numerical value of fractions. Learning and Instruction, 14, 503-518. https://doi.org/10.1016/j.learninstruc.2004.06.015
Stewart, D. W., & Shamdasani, P. N. (1990). Focus group: Theory and practice. SAGE.
Tall, D. (1992). Students’ difficulties in calculus. In Proceedings of Working Group 3 on Students’ Difficulties in Calculus (pp. 13-28).
Tall, D. (2013). How humans learn to think mathematically. Exploring the three worlds of mathematics. Cambridge University Press. https://doi.org/10.1017/CBO9781139565202
Tian, J., & Siegler, R. S. (2018). Which type of rational numbers should students learn first? Educational Psychology Review, 30(2), 351-372. https://doi.org/10.1007/s10648-017-9417-3
Tirosh, D., Fischbein, E., Graeber, A. O., & Wilson, J. W. (1998). Prospective elementary teachers’ conceptions of rational numbers. http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html
Tzanakis, C., Arcavi, A., Correia de Sa, C., Isoda, M., Lit, C. K., Niss, M., de Carvalho, J. P., Rodriguez, M., & Siu M. K. (2000). Integrating history of mathematics in the classroom: An analytic survey. In J. Fauvel, & J. van Maanen (Eds), History in mathematics education: The ICMI study (pp. 201-240). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47220-1_7
Tzekaki, Μ., Sakonidis, C., & Kaldrimidou, M. (2003). Mathematics education in Greece: A study. In A. Gagatsis, & S. Papastavridis (Eds), Proceedings of the 3^rd Mediterranean Conference on Mathematical Education (pp. 629-639). Hellenic Mathematical Society, Cyprus Mathematical Society.
Uegatani, Y., Nakawa, N., & Kosaka, M. (2021). Changes to tenth-grade Japanese students’ identities in mathematics learning during the COVID-19 pandemic. International Electronic Journal of Mathematics Education, 16(2), em0638. https://doi.org/10.29333/iejme/10905
Valiron, G. (1971). The origin and evolution of the notion of an analytic function of one variable. In F. Le Lionnais (Ed.), Great currents of mathematical thought (pp. 156-173). Dover.
Vamvakoussi, X., & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: A conceptual change approach. Learning and Instruction, 14, 453-467. https://doi.org/10.1016/j.learninstruc.2004.06.013
Vamvakoussi, X., & Vosniadou, S. (2007). How many numbers are there in a rational number interval? Constraints, synthetic models and the effect of the number line. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds), Reframing the conceptual change approach in learning and instruction (pp. 265-282). Elsevier.
van Hiele, P. M. (1957). De problematiek van het inzicht [The problem of insight] [Unpublished thesis]. Utrecht University.
van Hiele, P. M., & van Hiele-Geldof, D. (1958). A method of initiation into geometry at secondary schools. In H. Freudenthal (Ed.), Report on methods of initiation into geometry. J. B. Wolters.
van Hoof, J., Lijnen, T., Verschaffel, L., & van Dooren, W. (2013). Are secondary school students still hampered by the natural number bias? A reaction time study on fraction comparison tasks. Research in Mathematics Education, 15(2), 154-164. https://doi.org/10.1080/14794802.2013.797747
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Springer. https://doi.org/10.1007/0-306-47203-1_5
von Fritz, K. (1945). The discovery of incommensurability by Hippasus of Metapontum. Annals of Mathematics, 46(2), 242-264. https://doi.org/10.2307/1969021
Voskoglou, M. G., & Kosyvas, G. D. (2012). Analyzing students’ difficulties in understanding real numbers. Journal of Research in Mathematics Education, 1(3), 301-336.
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48, 101-119. https://doi.org/10.1023/A:1015535614355
Williams, E. R. (1979). An investigation of senior high school students’ understanding of mathematical proof [PhD thesis, University of Alberta].
Zachariades, T., Christou, C., & Pitta-Pantazi, D. (2013). Reflective, systemic and analytic thinking in real numbers. Educational Studies in Mathematics, 82, 5-22. https://doi.org/10.1007/s10649-012-9413-y
Zazkis, R., & Sirotic, N. (2004). Making sense of irrational numbers: Focusing on representation. In Proceedings of the 28^th Conference of the International Group for the Psychology of Mathematics Education (pp. 497-504).
Zazkis, R., & Sirotic, N. (2010). Representing and defining irrational numbers: Exposing the missing link. CBMS Issues in Mathematics Education, 16, 1-27. https://doi.org/10.1090/cbmath/016/01

How to cite this article

APA

Rizos, I., & Adam, M. (2022). Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), em0686. https://doi.org/10.29333/iejme/11977

Vancouver

Rizos I, Adam M. Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. INT ELECT J MATH ED. 2022;17(3):em0686. https://doi.org/10.29333/iejme/11977

AMA

Rizos I, Adam M. Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. INT ELECT J MATH ED. 2022;17(3), em0686. https://doi.org/10.29333/iejme/11977

Chicago

Rizos, Ioannis, and Maria Adam. "Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers". International Electronic Journal of Mathematics Education 2022 17 no. 3 (2022): em0686. https://doi.org/10.29333/iejme/11977

Harvard

Rizos, I., and Adam, M. (2022). Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers. International Electronic Journal of Mathematics Education, 17(3), em0686. https://doi.org/10.29333/iejme/11977

MLA

Rizos, Ioannis et al. "Mathematics students’ conceptions and reactions to questions concerning the nature of rational and irrational numbers". International Electronic Journal of Mathematics Education, vol. 17, no. 3, 2022, em0686. https://doi.org/10.29333/iejme/11977