Omega Position – a specific phase of perceiving the notion of infinity

Jiří Cihlář; Petr Eisenmann; Magdalena Krátká

doi:10.14712/18047106.184

Vol. 6 No. 2 (2015): sciED Research Reports

Omega Position – a specific phase of perceiving the notion of infinity

Jiří Cihlář ¹ , Petr Eisenmann ¹ , Magdalena Krátká ¹

Katedra matematiky PřF UJEP v Ústí nad Labem

DOI: https://doi.org/10.14712/18047106.184

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Cihlář, J., Eisenmann, P., & Krátká, M. (2015). Omega Position – a specific phase of perceiving the notion of infinity. Scientia in Educatione, 6(2), 51-73. https://doi.org/10.14712/18047106.184

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Abstract

This article describes a specific phase of the ontogenetic development of understanding infinity, called the omega position, the identification of which is one of the results of extensive research focusing on the perception of the infinity notion. Some 1 432 Czech pupils and students between the ages of 8 to 20, participated in the first two sections of this research between the years 2008 and 2011. This article describes in more detail, the final qualitative part of the research that focuses on interviews with university students with the aim of diagnosing this phase in their perception of infinity in various contexts. It also describes some possibilities for the identification of the omega position and its
consequences for a successful study of those notions and ideas of mathematics, which are associated with infinity. Further, the article puts the omega position into context with potential or actual infinity, specifies individual developmental phases by means of the notion horizon and explains some possibilities for mutual interference in the abovementioned developmental phases, employing the following two notions: that of primary and secondary intuition.

References

Bauer, L. (2011). Mathematik, Intuition, Formalisierung: eine Untersuchung von Schülerinnen- und Schülervorstellungen zu 0.¯9. Journal für Mathematik-Didaktik, 32(1), 79–102. Search in Google Scholar
Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990. Boston: Kluwer Academic Publishers. Search in Google Scholar
Brousseau, G. & Sarrazy, B. (2002). Glossaire de quelques concepts de la théorie des situations didactiques en mathématiques. Bordeaux: DAEST, Université Bordeaux 2. Search in Google Scholar
Brown, T. & Heywood, D. (2010). Geometry, subjectivity and the seduction of language: the regulation of spatial perception. Educational Studies in Mathematics, 77, 351–367. Search in Google Scholar
Cihlář, J., Eisenmann, P. & Krátká, M. (2013). Forming the infinity concept through overcoming obstacles. Ústí nad Labem: UJEP. Search in Google Scholar
Cihlář, J., Eisenmann, P., Krátká, M. & Vopěnka, P. (2009). Cognitive conflict as a tool of overcoming obstacles in understanding infinity. Teaching Mathematics and Computer Science, 7, 279–295. Search in Google Scholar
Conway, J. H. (2001). On numbers and games. London: Academic Press, 1976. Second edition: A. K. Peters, Wellesley/MA. Search in Google Scholar
Dubinsky, E., Weller, K., McDonald, M. A. & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An APOS-Based Analysis, Part 1. Educational Studies in Mathematics, 58, 335–359. Search in Google Scholar
Dubinsky, E., Weller, K., McDonald, M. A. & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: An APOS-Based Analysis, Part 2. Educational Studies in Mathematics, 60, 253–266. Search in Google Scholar
Edwards, B. S., Dubinsky, E. & McDonald, M. A. (2005). Advanced mathematical thinking. Mathematical Thinking and Learning, 7, 15–25. Search in Google Scholar
Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41, 117–146. Search in Google Scholar
Fischbein, E. (1987). Intuition in science and mathematics. Dodrecht: Reidel. Search in Google Scholar
Fischbein, E. (2001). Tacit models and infinity. Educational Studies in Mathematics, 48, 309–329. Scientia in educatione 71 6(2), 2015, p. 51–73 Search in Google Scholar
Hannula, M., Pehkonen, E., Maijala, H. & Soro, R. (2006). Students’ understanding of infinity. In Teaching mathematics: retrospective and perspectives. Proceedings of the 7th international conference (79–83). Tartu: Eritrea. Search in Google Scholar
Jahnke, H. N. (2001). Cantor’s cardinal and ordinal infinities: An epistemological and didactic view. Educational Studies in Mathematics, 48, 175–197. Search in Google Scholar
Jirotková, D. & Littler, G. (2004). An insight into pupils’ understanding of infinity in a geometrical context. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (97–104), Bergen: Univ. College. Search in Google Scholar
Juter, K. (2006). Limits of functions as they developed through time and as students learn them today. Mathematical Thinking and Learning, 8, 407–431. Search in Google Scholar
Katz, K. U. & Katz, M. G. (2010). Zooming in on infinitesimal 1–.9. . . in a post-triumvirate era. Educational Studies in Mathematics, 74, 259–273. Search in Google Scholar
Katz, V., Dorier, J. L., Bekken, O. & Sierpinska, A. (2000). The role of historical analysis in predicting and interpreting students’ difficulties in mathematics. In J. Fauvel & J. V. Maanen (Eds.), History in Mathematics Education (149–161). Dordrecht: Kluwer Academic Publisher. Search in Google Scholar
Kidron, I. (2011). Tacit models, treasured intuitions and the discrete — continuous interplay. Educational Studies in Mathematics, 78, 109–126. Search in Google Scholar
Kidron, I. & Tall, D. (2015). The roles of visualization and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics, 88, 183–199. Search in Google Scholar
Krátká, M. (2005). A geometrical picture as an obstacle. In Proceedings of SEMT’05 (179–186). Praha: UK PedF. Search in Google Scholar
Krátká, M. (2010). Zdroje epistemologických překážek v porozumění nekonečnu. Scientia in educatione, 1(1), 187–100. Search in Google Scholar
Monaghan, J. (2001). Young people’s ideas of infinity. Educational Studies in Mathematics, 48, 239–257. Search in Google Scholar
Piaget, J. (1977). The development of thought: Equilibration of cognitive structures. New York: The Viking Press. Search in Google Scholar
Radford, L., Boero, P., & Vasco, C. (2000). Historical formation and student understanding of mathematics: Epistemological assumptions framing interpretations of students understanding of mathematics. In J. Fauvel & J. Van Maanen (Eds.), History in mathematics education. Dordrecht: Kluwer Academic Publisher. Search in Google Scholar
Robinson, A. (1996). Nonstandard analysis. Princeton University Press. Search in Google Scholar
Sierpinska, A. (1994). Understanding in mathematics. Washington, D.C.: The Falmer Press London. Search in Google Scholar
Singer, F. M. & Voica, C. (2008). Between perception and intuition: Learning about infinity. The Journal of Mathematical Behavior, 27, 188–205. Search in Google Scholar
Swan, M. (1983). Teaching decimal place value. A comparative study of conflict and positive only approaches. Nottingham: University of Nottingham, Shell Centre for Mathematical Education. Search in Google Scholar
Tall, D. O. (1976). Conflicts and catastrophes in the learning of mathematics, Mathematical Education for Teaching, 2(4), 2–18. Search in Google Scholar
Tall, D. O. (1977). Cognitive conflicts and the learning of mathematics. In Proceedings of PME 1, Utrecht. Search in Google Scholar
Tall, D. O. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Mathematics Education Library: Vol. 11 Advanced Mathematical Thinking (3–21). Dordrecht: Kluwer Academic Publisher. Search in Google Scholar
Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 199–238. Search in Google Scholar
Tall, D. O. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM. The International Journal on Mathematics Education, 41, 481–492. Search in Google Scholar
Tall, D. & Tirosh, D. (2001). Infinity — the never-ending struggle. Educational Studies in Mathematics, 48, 129–136. Search in Google Scholar
Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limit and continuity. Educational Studies in Mathematics, 12, 151–169. Search in Google Scholar
Tsamir, P. (2001). When “The Same” is not perceived as such: The case of infinity sets. Educational Studies in Mathematics, 48, 289–307. Search in Google Scholar
Vamvakoussi, X. & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: a conceptual chase approach. Learning and Instruction, 14, 453–467. Search in Google Scholar
Vopěnka, P. (2011). Velká iluze matematiky XX. století a nové základy. Plzeň: Vydavatelství Západočeské univerzity v Plzni. Search in Google Scholar
Yopp, D. A., Burroughs, E. A. & Lindaman, B. J. (2011). Why it is important for in-service elementary mathematics teachers to understand the equality .999 . . . = 1. Journal of Mathematical Behavior, 30, 304–318. Search in Google Scholar

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[1] Bauer, L. (2011). Mathematik, Intuition, Formalisierung: eine Untersuchung von Schülerinnen- und Schülervorstellungen zu 0.¯9. Journal für Mathematik-Didaktik, 32(1), 79–102. Search in Google Scholar

[2] Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques, 1970–1990. Boston: Kluwer Academic Publishers. Search in Google Scholar

[3] Brousseau, G. & Sarrazy, B. (2002). Glossaire de quelques concepts de la théorie des situations didactiques en mathématiques. Bordeaux: DAEST, Université Bordeaux 2. Search in Google Scholar

[4] Brown, T. & Heywood, D. (2010). Geometry, subjectivity and the seduction of language: the regulation of spatial perception. Educational Studies in Mathematics, 77, 351–367. Search in Google Scholar

[5] Cihlář, J., Eisenmann, P. & Krátká, M. (2013). Forming the infinity concept through overcoming obstacles. Ústí nad Labem: UJEP. Search in Google Scholar

[6] Cihlář, J., Eisenmann, P., Krátká, M. & Vopěnka, P. (2009). Cognitive conflict as a tool of overcoming obstacles in understanding infinity. Teaching Mathematics and Computer Science, 7, 279–295. Search in Google Scholar

[7] Conway, J. H. (2001). On numbers and games. London: Academic Press, 1976. Second edition: A. K. Peters, Wellesley/MA. Search in Google Scholar

[8] Dubinsky, E., Weller, K., McDonald, M. A. & Brown, A. (2005a). Some historical issues and paradoxes regarding the concept of infinity: An APOS-Based Analysis, Part 1. Educational Studies in Mathematics, 58, 335–359. Search in Google Scholar

[9] Dubinsky, E., Weller, K., McDonald, M. A. & Brown, A. (2005b). Some historical issues and paradoxes regarding the concept of infinity: An APOS-Based Analysis, Part 2. Educational Studies in Mathematics, 60, 253–266. Search in Google Scholar

[10] Edwards, B. S., Dubinsky, E. & McDonald, M. A. (2005). Advanced mathematical thinking. Mathematical Thinking and Learning, 7, 15–25. Search in Google Scholar

[11] Ely, R. (2010). Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education, 41, 117–146. Search in Google Scholar

[12] Fischbein, E. (1987). Intuition in science and mathematics. Dodrecht: Reidel. Search in Google Scholar

[13] Fischbein, E. (2001). Tacit models and infinity. Educational Studies in Mathematics, 48, 309–329. Scientia in educatione 71 6(2), 2015, p. 51–73 Search in Google Scholar

[14] Hannula, M., Pehkonen, E., Maijala, H. & Soro, R. (2006). Students’ understanding of infinity. In Teaching mathematics: retrospective and perspectives. Proceedings of the 7th international conference (79–83). Tartu: Eritrea. Search in Google Scholar

[15] Jahnke, H. N. (2001). Cantor’s cardinal and ordinal infinities: An epistemological and didactic view. Educational Studies in Mathematics, 48, 175–197. Search in Google Scholar

[16] Jirotková, D. & Littler, G. (2004). An insight into pupils’ understanding of infinity in a geometrical context. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (97–104), Bergen: Univ. College. Search in Google Scholar

[17] Juter, K. (2006). Limits of functions as they developed through time and as students learn them today. Mathematical Thinking and Learning, 8, 407–431. Search in Google Scholar

[18] Katz, K. U. & Katz, M. G. (2010). Zooming in on infinitesimal 1–.9. . . in a post-triumvirate era. Educational Studies in Mathematics, 74, 259–273. Search in Google Scholar

[19] Katz, V., Dorier, J. L., Bekken, O. & Sierpinska, A. (2000). The role of historical analysis in predicting and interpreting students’ difficulties in mathematics. In J. Fauvel & J. V. Maanen (Eds.), History in Mathematics Education (149–161). Dordrecht: Kluwer Academic Publisher. Search in Google Scholar

[20] Kidron, I. (2011). Tacit models, treasured intuitions and the discrete — continuous interplay. Educational Studies in Mathematics, 78, 109–126. Search in Google Scholar

[21] Kidron, I. & Tall, D. (2015). The roles of visualization and symbolism in the potential and actual infinity of the limit process. Educational Studies in Mathematics, 88, 183–199. Search in Google Scholar

[22] Krátká, M. (2005). A geometrical picture as an obstacle. In Proceedings of SEMT’05 (179–186). Praha: UK PedF. Search in Google Scholar

[23] Krátká, M. (2010). Zdroje epistemologických překážek v porozumění nekonečnu. Scientia in educatione, 1(1), 187–100. Search in Google Scholar

[24] Monaghan, J. (2001). Young people’s ideas of infinity. Educational Studies in Mathematics, 48, 239–257. Search in Google Scholar

[25] Piaget, J. (1977). The development of thought: Equilibration of cognitive structures. New York: The Viking Press. Search in Google Scholar

[26] Radford, L., Boero, P., & Vasco, C. (2000). Historical formation and student understanding of mathematics: Epistemological assumptions framing interpretations of students understanding of mathematics. In J. Fauvel & J. Van Maanen (Eds.), History in mathematics education. Dordrecht: Kluwer Academic Publisher. Search in Google Scholar

[27] Robinson, A. (1996). Nonstandard analysis. Princeton University Press. Search in Google Scholar

[28] Sierpinska, A. (1994). Understanding in mathematics. Washington, D.C.: The Falmer Press London. Search in Google Scholar

[29] Singer, F. M. & Voica, C. (2008). Between perception and intuition: Learning about infinity. The Journal of Mathematical Behavior, 27, 188–205. Search in Google Scholar

[30] Swan, M. (1983). Teaching decimal place value. A comparative study of conflict and positive only approaches. Nottingham: University of Nottingham, Shell Centre for Mathematical Education. Search in Google Scholar

[31] Tall, D. O. (1976). Conflicts and catastrophes in the learning of mathematics, Mathematical Education for Teaching, 2(4), 2–18. Search in Google Scholar

[32] Tall, D. O. (1977). Cognitive conflicts and the learning of mathematics. In Proceedings of PME 1, Utrecht. Search in Google Scholar

[33] Tall, D. O. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Mathematics Education Library: Vol. 11 Advanced Mathematical Thinking (3–21). Dordrecht: Kluwer Academic Publisher. Search in Google Scholar

[34] Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 199–238. Search in Google Scholar

[35] Tall, D. O. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM. The International Journal on Mathematics Education, 41, 481–492. Search in Google Scholar

[36] Tall, D. & Tirosh, D. (2001). Infinity — the never-ending struggle. Educational Studies in Mathematics, 48, 129–136. Search in Google Scholar

[37] Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limit and continuity. Educational Studies in Mathematics, 12, 151–169. Search in Google Scholar

[38] Tsamir, P. (2001). When “The Same” is not perceived as such: The case of infinity sets. Educational Studies in Mathematics, 48, 289–307. Search in Google Scholar

[39] Vamvakoussi, X. & Vosniadou, S. (2004). Understanding the structure of the set of rational numbers: a conceptual chase approach. Learning and Instruction, 14, 453–467. Search in Google Scholar

[40] Vopěnka, P. (2011). Velká iluze matematiky XX. století a nové základy. Plzeň: Vydavatelství Západočeské univerzity v Plzni. Search in Google Scholar

[41] Yopp, D. A., Burroughs, E. A. & Lindaman, B. J. (2011). Why it is important for in-service elementary mathematics teachers to understand the equality .999 . . . = 1. Journal of Mathematical Behavior, 30, 304–318. Search in Google Scholar

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