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SONSUZ SAYI KÜMELERİ IŞIĞINDA İLKÖĞRETİM ÖĞRENCİLERİNİN SONSUZLUK ALGI VE YANILGILARININ BELİRLENMESİ

Year 2012, Issue: 33, 122 - 133, 01.12.2012

Abstract

Sonsuzluk, gerçek ve potansiyel sonsuzluk olmak üzere ikiye ayrılmaktadır. Eski zamanlardan beri, potansiyel sonsuzluğun matematiksel sonsuzluk olduğu kabul edilmektedir. Sayı kümeleri ise, matematiksel sonsuzluğu içeren kavramların başında gelir. Bu yüzden, sayı kümeleri, sonsuzluk fikrinin belirlenmesinde kullanılabilir. Diğer taraftan, bu kümelerin tam olarak algılanabilmesi için öğrencilerde sonsuzluk fikrinin bulunması gerekir. Bu bağlamda bu çalışmanın amacı, öğrencilerin sonsuzluk fikrine ne derece sahip olduklarını, okulun öğrencilere sonsuzluk fikrini kazandırıp kazandıramadığını ve öğrencilerin bu konudaki yanılgılarını, sayı kümeleri ile ilişkilendirerek belirlemeye çalışmaktır. Bu amaçla 13-14 yaş arası İlköğretim öğrencilerinden, 131 öğrencinin görüşleri açık uçlu anket ile toplanmış ve bunların 10’u ve 3 ilköğretim öğretmeni ile yarı yapılandırılmış görüşme yapılmıştır. Sonuçta öğrencilerin sahip oldukları sonsuzluk fikirlerinde kişisel deneyimlerin yattığı ve formal eğitimin buna fazla yardım etmediği görülmüştür. Ayrıca öğrencilerin kümelerin sonsuz olup olmamasını belirlemede, sezgisel yollarla, tümevarımsal süreç ve başka bir küme ile karşılaştırma yöntemlerini kullandıkları tespit edilmiştir.  Sonsuz kümelerin karşılaştırılmasında da literatürde bahsedilen bazı yanılgıların bulunduğu anlaşılmıştır.

References

  • Duval, R.: (1983), ‘L’obstacle du dedoublement des objects mathematiques’, Educational Studies in Mathematics 14, 385–414.
  • Falk, R., Gassner, D., Ben Zoor, F. ve Ben Simon, K.: (1986), ‘How do children cope with the infinity of numbers?’ Proceedings of the 10th Conference of the International Group for the Psychology of Mathematics Education, London, England, pp. 7–12.
  • Fischbein, E. (1987). Intuition in science and mathematics. Dodrecht, Holland: Reidel.
  • Fischbein, E., Tirosh, D. and Hess, P.: 1979, ‘The intuition of infinity’, Educational Studies in Mathematics 10, 3–40
  • Fischbein, E. (2001). “Tacit Models and Infinity” Educational Studies in Mathematics 48: 309-329
  • Fischbein, E., Tirosh, D. ve Melamed, U.: (1981), ‘Is it possible to measure the intuitive acceptance of a mathematical statement?’ Educational Studies in Mathematics 12, 491–512.
  • Güven, B ve Karataş, İ. (2004). Sonsuz Kümelerin Karşılaştırılması: Öğrencilerin kullandıkları yöntemler, Dokuz Eylül Üniversitesi Buca Eğitim Fakültesi Dergisi 15: 65-73
  • Martin, W. G. ve Wheeler, M. M.: (1987), ‘Infinity concepts among preservice elementary school teachers’, Proceedings of the 11th Conference of the International Group for the Psychology of Mathematics Education, France, pp. 362–368.
  • Monaghan, J. (2001). Young peoples’ ideas of infinity. Educational Studies in Mathematics, Vol. 48 pp 239 – 257.
  • Narli,S. (2011). Is constructivist learning environment really effective on learning and long- term knowledge retention in mathematics? Example of the infinity concept. Educational Research and Reviews, in press
  • Narli, S. & Baser, N. (2008). Cantorian Set Theory and teaching prospective teachers. International Journal of Environmental & Science Education, 3(2), 99-107.
  • Narli, S. & Baser, N. (2010). The effects of constructivist learning environment on prospective mathematics teachers’ opinions. Us-China Education Review, 7(1), 1-16.
  • Narli, S., Delice, A.ve Narli, P. (2009). Secondary School Students’ Concept of infinity: primary and secondary intuitions. In Tzekaki, M., Kaldrimidou, M. & Sakonidis, H. (Eds.). Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 209-216. Thessaloniki, Greece
  • Piaget, J. and Inhelder, B. (1956). The Child’s Conception of Space, Routledge and Kegan Paul, London (originally published in 1948).
  • Singer,M., Voica, C. (2003) “Perception of infinity: does it really help in problem solving” The Mathematics Education into the 21st Century Project Proceedings of the International Conference The Decidable and the Undecidable in Mathematics Education Brno, Czech Republic, September 2003
  • Singer,M., Voica, C. (2008). Between perception and intuition: Learning about infinity, The Journal of Mathematical Behavior, 27, 188–205
  • Tirosh, D. (1991). “The role of students’ intuition of infinity in teaching the Cantorian theory” in D. Tall (ed.) Advanced Mathematical Thinking, Kluwer, Dordrecht, pp. 199-214
  • Tirosh, D. (1999). Finite and infinite sets:definitions and intuitions Int J. Math.Educ.Sci.Technol., Vol.30, No. 3, 341-349
  • Tsamir, P. (1999). The transition from the comparison of finite to the comparison of infinite sets: Teaching
  • prospective teachers. Educational Studies in Mathematics. 38 (1-3), 209-234
  • Tsamir, P. (2001). When “the same” is not perceived as such: The case of infinite sets. Educational Studies in Mathematics, 48, 289-307.
  • Tsamir, P. (2002). Primary And Secondary Intuitions: Prospective Teachers’ Comparisons Of Infinite Sets, inCERME 2 Proceedings.
  • Tsamir, P., Tirosh, D. (1992), ‘Students’ awareness of inconsistent ideas about actual infinity’, Proceedings of the 16th Conference of the International Group for the Psychology of Mathematics Education, Durham, USA, 3, 90–97.
  • Tsamir, P., Tirosh, D. (1994). Comparing infinite sets: intuitions and representations. Proceedings of the 18thAnnual Meeting for the Psychology of Mathematics Education (Vol. IV, pp. 345-352). Lisbon: Portugal.

DETERMINATION OF PRIMARY SCHOOL STUDENTS’ PERCEPTIONS AND MISCONCEPTIONS OF INFINITY USING INFINITE NUMBER SET

Year 2012, Issue: 33, 122 - 133, 01.12.2012

Abstract

Infinity has divided into two componenets, the actual and potential infinity. Potential infinity has been considered as mathematical infinity for a long time. Number sets is one of the concepts involving mathematical infinity. Therefore, numer sets can be used to determine perception of infinity. On the other hand, students must have some perception of infinity to perceive this sets exactly. In this context, the aim of this study is threefold: to investigate primary school students’ perception of infinity concept, to determine to what extent schooling is successful in the attainment of the concept of infinity, and to investigate their misconceptions of the concept of infinity by associating it with number sets. Data collected through open-ended questionnaires administered to 131 primary school students aged 13-14 and semi-structured interviews conducted with ten of these students and three primary mathematics teachers. Results indicated that students’ personal experiences mainly determined their concept of infinity and that formal education had minimal effects. Also, it was determined that there were two methods which intuitively used by students to decide whether a given set is finite or infinite: induction process and comparing the given set with another set. Students’ also displayed some misconceptions about the comparison of infinite sets.

References

  • Duval, R.: (1983), ‘L’obstacle du dedoublement des objects mathematiques’, Educational Studies in Mathematics 14, 385–414.
  • Falk, R., Gassner, D., Ben Zoor, F. ve Ben Simon, K.: (1986), ‘How do children cope with the infinity of numbers?’ Proceedings of the 10th Conference of the International Group for the Psychology of Mathematics Education, London, England, pp. 7–12.
  • Fischbein, E. (1987). Intuition in science and mathematics. Dodrecht, Holland: Reidel.
  • Fischbein, E., Tirosh, D. and Hess, P.: 1979, ‘The intuition of infinity’, Educational Studies in Mathematics 10, 3–40
  • Fischbein, E. (2001). “Tacit Models and Infinity” Educational Studies in Mathematics 48: 309-329
  • Fischbein, E., Tirosh, D. ve Melamed, U.: (1981), ‘Is it possible to measure the intuitive acceptance of a mathematical statement?’ Educational Studies in Mathematics 12, 491–512.
  • Güven, B ve Karataş, İ. (2004). Sonsuz Kümelerin Karşılaştırılması: Öğrencilerin kullandıkları yöntemler, Dokuz Eylül Üniversitesi Buca Eğitim Fakültesi Dergisi 15: 65-73
  • Martin, W. G. ve Wheeler, M. M.: (1987), ‘Infinity concepts among preservice elementary school teachers’, Proceedings of the 11th Conference of the International Group for the Psychology of Mathematics Education, France, pp. 362–368.
  • Monaghan, J. (2001). Young peoples’ ideas of infinity. Educational Studies in Mathematics, Vol. 48 pp 239 – 257.
  • Narli,S. (2011). Is constructivist learning environment really effective on learning and long- term knowledge retention in mathematics? Example of the infinity concept. Educational Research and Reviews, in press
  • Narli, S. & Baser, N. (2008). Cantorian Set Theory and teaching prospective teachers. International Journal of Environmental & Science Education, 3(2), 99-107.
  • Narli, S. & Baser, N. (2010). The effects of constructivist learning environment on prospective mathematics teachers’ opinions. Us-China Education Review, 7(1), 1-16.
  • Narli, S., Delice, A.ve Narli, P. (2009). Secondary School Students’ Concept of infinity: primary and secondary intuitions. In Tzekaki, M., Kaldrimidou, M. & Sakonidis, H. (Eds.). Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 209-216. Thessaloniki, Greece
  • Piaget, J. and Inhelder, B. (1956). The Child’s Conception of Space, Routledge and Kegan Paul, London (originally published in 1948).
  • Singer,M., Voica, C. (2003) “Perception of infinity: does it really help in problem solving” The Mathematics Education into the 21st Century Project Proceedings of the International Conference The Decidable and the Undecidable in Mathematics Education Brno, Czech Republic, September 2003
  • Singer,M., Voica, C. (2008). Between perception and intuition: Learning about infinity, The Journal of Mathematical Behavior, 27, 188–205
  • Tirosh, D. (1991). “The role of students’ intuition of infinity in teaching the Cantorian theory” in D. Tall (ed.) Advanced Mathematical Thinking, Kluwer, Dordrecht, pp. 199-214
  • Tirosh, D. (1999). Finite and infinite sets:definitions and intuitions Int J. Math.Educ.Sci.Technol., Vol.30, No. 3, 341-349
  • Tsamir, P. (1999). The transition from the comparison of finite to the comparison of infinite sets: Teaching
  • prospective teachers. Educational Studies in Mathematics. 38 (1-3), 209-234
  • Tsamir, P. (2001). When “the same” is not perceived as such: The case of infinite sets. Educational Studies in Mathematics, 48, 289-307.
  • Tsamir, P. (2002). Primary And Secondary Intuitions: Prospective Teachers’ Comparisons Of Infinite Sets, inCERME 2 Proceedings.
  • Tsamir, P., Tirosh, D. (1992), ‘Students’ awareness of inconsistent ideas about actual infinity’, Proceedings of the 16th Conference of the International Group for the Psychology of Mathematics Education, Durham, USA, 3, 90–97.
  • Tsamir, P., Tirosh, D. (1994). Comparing infinite sets: intuitions and representations. Proceedings of the 18thAnnual Meeting for the Psychology of Mathematics Education (Vol. IV, pp. 345-352). Lisbon: Portugal.
There are 24 citations in total.

Details

Other ID JA44VV63DH
Journal Section Articles
Authors

Serkan Narlı This is me

Pınar Narlı This is me

Publication Date December 1, 2012
Published in Issue Year 2012 Issue: 33

Cite

APA Narlı, S., & Narlı, P. (2012). SONSUZ SAYI KÜMELERİ IŞIĞINDA İLKÖĞRETİM ÖĞRENCİLERİNİN SONSUZLUK ALGI VE YANILGILARININ BELİRLENMESİ. Dokuz Eylül Üniversitesi Buca Eğitim Fakültesi Dergisi(33), 122-133.