The Case of Time Axis Fallacy: 11th Grade Students’ Intuitively-based Misconception in Probability and Teachers’ Corresponding Practices

Year 2018, Volume: 6 Issue: 3, 86 - 105, 30.11.2018

Mehmet Fatih Öçal

Abstract

This case study aimed to investigate whether mathematics teachers’ instructional practices were effective in resolving eleventh grade
students’ intuitively-based misconceptions regarding time axis fallacy.
The participants were three mathematics teachers from different high
schools and their students. Students were administered a diagnostic test
comprising questions related to intuitively-based misconceptions in
probability. The test was administered before and after students
received teachers’ instructions for probability subject. Teachers were
interviewed about their knowledge of students’ difficulties and
misconceptions. Teachers’ instructions for probability were observed and
videotaped. Content analysis method was used in the data analysis.
Considering the findings, it was observed that teachers did not give
emphasis on unfamiliar situations related to time axis fallacy.
Comparing the test results, there was slight
increase in the number of students who fell into time axis fallacy.
Based on the findings, it can be asserted that practitioners should be
aware of possible intuitively-based misconceptions in probability and
organize their instructions accordingly.

Keywords

Intuition, misconceptions, probability, time axis fallacy, mathematics teachers

Zaman Etkisi Örneği: 11. Sınıf Öğrencilerinin Olasılıkta Sezgi Temelli Kavram Yanılgısı ve Öğretmenlerin İlgili Uygulamaları

Abstract

Bu durum çalışmasının amacı, matematik öğretmenlerinin öğretim uygulamalarının on birinci sınıf öğrencilerinin sezgi temelli kavram yanılgıları bağlamında olasılık konusunda zaman etkisi yanılgısını gidermede etkili olup olmadığını incelemektir. Bu çalışmanın katılımcılarını farklı liselerde çalışan üç matematik öğretmeni ve bu öğretmenlerin öğrencileri oluşturmaktadır. Öğrencilere olasılıkta sezgi temelli kavram yanılgılarıyla ilgili sorular içeren bir tanı testi uygulanmıştır. Test, öğretmenlerin olasılık konusunun öğretiminin öncesi ve sonrasında uygulanmıştır. Ayrıca öğrencilerin zorlandıkları noktalar ve kavram yanılgılarına yönelik öğretmenler ile mülakatlar yapılmıştır. Öğretmenlerin olasılık öğretimleri sırasında hem gözlem yapılmış hem de dersler video kaydı altına alınmıştır. Verilerin analizinde içerik analiz yöntemi kullanılmıştır. Bulgular göz önüne alındığında, zaman etkisi kavram yanılgısıyla ilgili aşina olunmayan durumlara öğretmenlerin odaklanmadığı gözlemlenmiştir. Test sonuçları karşılaştırıldığında, zaman etkisi yanılgısına düşen öğrenci sayısında az bir artış olduğunu göstermiştir. Bulgular ışığında, uygulayıcıların olasılıkta sezgi temelli kavram yanılgılarından haberdar olmaları ve öğretimlerini bu bağlamda düzenlemeleri gerektiği söylenebilir.

Keywords

Sezgi, kavram yanılgıları, olasılık, zaman etkisi yanılgısı, matematik öğretmenleri

References

• Andra, C. (2011). Pre-service primary school teachers’ intuitive use of representations in uncertain situation. In M. Pytlak, E. Swoboda, & T. Rowland (Eds.), The proceeding of 7th conference of the European Society for research in mathematics education, (pp. 715-724). Rzeszow, Poland: University of Rzeszow.
• Babai, R., Brecher, T., Stavy, R., & Tirosh, D. (2006). Intuitive inference in probabilistic reasoning. International Journal of Science and Mathematics Education, 4, 627-639.
• Bar-Hillen, M., & Falk, R. (1982). Some teachers concerning conditional probabilities. Cognition. 11(2), 109-122.
• Batanero, C., & Sanchez, E. (2005). What is the nature of high school students' conceptions and misconceptions about probability? In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 241-266). New York, NY: Springer.
• Castro, C. S. (1998). Teaching probability for conceptual change. Educational Studies in Mathematics, 35, 233-254.
• Chiese, F., & Primi, C. (2008). Primary school children's and college students' recency effects in a gaming situation. Paper presented at 11th international congress on mathematical education, (July 6-13), Monterrey, Mexico.
• Chiese, F., & Primi, C. (2009). Recency effects in primary-age children and college students. International Electronic Journal of Mathematics Education, 4(3), 259-274.
• Common Core State Standards [CCSS]. (2010). Common core state standards for mathematics. Washington, D. C.: Council of Chief State School Officers and National Governors Association.
• Creswell, J. W. (2013). Research design: Qualitative, quantitative, and mixed methods approaches. Los Angeles, LA: Sage Publications.
• Çelik, D., & Güneş, G. (2007). 7, 8 ve 9. sınıf öğrencilerinin olasılık ile ilgili anlama ve kavram yanılgılarının incelenmesi. Milli Eğitim Dergisi, 173, 361–375.
• Demirci, Ö., Özkaya, M., & Konyalıoğlu, A. C. (2017). The preservice teachers mistake approaches on probability. Erzincan Üniversitesi Eğitim Fakültesi Dergisi, 19(2), 153-172.
• Evans, J. S. B. T. (2006). The heuristic-analytic theory of reasoning: Extension and evaluation. Psychonomic Bulletin & Review, 13(3), 378-395.
• Falk, R. (1979). Revision of probabilities and the time axis. Proceedings of the third international conference for the psychology of mathematics education (pp. 64-66). Warwick, England.
• Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Dordrecht, The Netherlands: Reidel Publishing.
• Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Dordrecht, The Netherlands: Reidel.
• Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic: Intuitively based misconceptions. Journal of Research in Mathematics Education, 28(1), 95-106.
• Fischbein, E., Nello, M. S., & Marino, M. S. (1991). Factors affecting probabilistic judgments in children and adolescents. Educational Studies in Mathematics, 22, 523-549.
• Fraenkel, J. R. ve Wallen, N. E. (2009). The nature of qualitative research. How to design and evaluate research in education (7th ed.). Boston: McGraw-Hill.
• Fox, C. R., & Levav, J. (2000). Familiarity bias and belief reversal in relative likelihood judgment. Organizational Behavior and Human Decision Processes, 82(2), 268-292.
• Fox, C. R., & Levav, J. (2004). Partition-edit-count: Naïve extensional reasoning in judgment of conditional probability. Journal of Experimental Psychology: General, 133(4), 626-642.
• Granberg, D., & Brown, T. A. (1995). The Monty Hall dilemma. Personality and Social Psychology Bulletion, 21(7), 711-729.
• Gürbüz, R. (2008). Olasılık konusunun öğretiminde kullanılabilecek bilgisayar destekli bir materyal. Mehmet Akif Ersoy Üniversitesi Eğitim Fakültesi Dergisi, 15, 41-52.
• Havill, D. E. (1998). Traditional and nontraditional probability contexts: The role of instruction-related intuitions and everyday intuitions in students’ reasoning about sequences of events. (Unpublished doctoral dissertation). University of California, Santa Barbara, CA.
• Jones, G. A., Langrall, C., & Mooney, E. S. (2007). Research in probability: Responding to classroom realities. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 909-955). Greenwich, CT: Information Age Publishing, Inc. and NCTM.
• Kahneman, D., & Tversky, A. (1982). Subjective probability: A judgment of representativeness. In D. Kahneman, P. Slovic, & A. Tversky (Eds.), Judgment under uncertainty: Heuristics and biases (pp. 32-47). Cambridge: Cambridge University Press.
• Kazak, S. (2008). Öğrencilerin olasılık konularındaki kavram yanılgıları ile öğrenme zorlukları. In M. F. Özmantar, E. Bingölbali, & H. Akkoç (Eds.), Matematiksel kavram yanılgıları ve çözüm önerileri. (pp. 121-150). Ankara: PegemA Yayınevi.
• Kennis, J. R. (2006). Probabilistic misconceptions across age and gender (Unpublished doctoral dissertation). Graduate School of Arts and Sciences, Columbia University, New York, NY.
• Köğce, D., & Baki, A. (2009). Matematik öğretmenlerinin yazılı sınav soruları ile ÖSS sınavlarında sorulan matematik sorularının Bloom taksonomisine göre karşılaştırılması. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 26, 70-80.
• Li, J. (2000). Chinese students’ understanding of probability. (Unpublished doctoral dissertation). National Institute of Education, Nanyang Technological University, Singapure.
• Marques, J. F., & McCall, C. (2005). The application of interrater reliability as a solidification instrument in a phenomenological study. The Qualitative Report, 10(3), 439-462.
• Memnun, D. S. (2008). Olasılık kavramının öğrenilmesinde karşılaşılan zorluklar, bu kavramların öğrenilmeme nedenleri ve çözüm önerileri. İnönü Üniversitesi Eğitim Fakültesi Dergisi, 9(15), 89-101.
• Ministry of National Education (MoNE). (2017). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı. Retrieved from http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=343.
• Myers, D. G. (2002). Intuition: Its powers and perils. New Heaven & London: Yale University Press. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Virginia: NCTM.
• Ojeda, A. M. (1999). Training and practice of teachers of probability: An epistemological stance. In Bills, L. (Ed.), The proceedings of the British Society for Research into Learning Mathematics (pp. 55-60). Nottingham, the UK: University of Nottingham.
• Papaieronymou, I. (2009). Recommended knowledge of probability for secondary mathematics teachers. Working Group 3. The proceedings of 6th congress of European research in mathematics education, Lyon, France.
• Polaki, M. V. (2002a). Using instruction to identify key features of Basotho elementary students’ growth in probabilistic thinking. Mathematical Thinking and Learning, 4, 285-314.
• Polaki, M. V. (2002b). Using instruction to identify mathematical practices associated with Basotho elementary students’ growth in probabilistic thinking. Canadian Journal for Science, Mathematics and Technology Education, 2, 357-370.
• Pratt, D. (2000). Making sense of the total of two dice. Journal for Research in Mathematics Education, 31(5), 602-625.
• Riccomini, P. J. (2005). Identification and remediation of systematic error patterns in subtraction. Learning Disability Quarterly, 28, 233-242.
• Rubel, L. H. (2002). Probabilistic misconceptions: Middle and high school students’ mechanisms for judgments under uncertainty. (Unpublished doctoral dissertation). Columbia University, New York.
• Savard, A. (2014). Developing probabilistic thinking: What about people’s conception? In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: Presenting plural perspectives (pp. 283-298), Dordrecht, The Netherlands: Kluwer.
• Shaughnessy, J. M. (1992). Research on probability and statistics: Reflections and directions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465-494). New York: Macmillan Publishing Company.
• Stavy, R., & Tirosh, D. (2000). How students (Mis-)understand science and mathematics: Intuitive rules. New York, NY: Teacher College Press.
• Stohl, H. (2005). Probability in teacher education and development. In G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 345-366). New York: Springer.
• Stohl, H., & Tarr, J. E. (2002). Developing notions of inference with probability simulation tools. Journal of Mathematical Behavior, 21(3), 319-337.
• Şengül, S., & Ekinözü, S. (2004). Permütasyon ve olasılık konusunun öğretiminde canlandırma kullanılmasının öğrenci başarısına ve hatırlama düzeyine etkisi. XIII. Ulusal Eğitim Bilimleri Kurultayı, İnönü Üniversitesi, Eğitim Fakültesi, Malatya, Turkey.
• Tarr, J. E., & Jones, G. A. (1997). A framework for assessing middle school students’ thinking in conditional probability and independence. Mathematics Education Research Journal, 9, 39-59.
• Tarr, J. E., & Lannin, J. K. (2005). How can teachers build notions of conditional probability and independence? In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 215-238). Dordrecht, The Netherlands: Kluwer.
• Tirosh, D., & Stavy, R. (1999a). Intuitive rules: A way to explain and predict students’ reasoning. Educational Studies in Mathematics, 38, 51–66.
• Tirosh, D., & Stavy, R. (1999b). Intuitive rules and comparison tasks. Mathematical Thinking and Learning, 1(3), 179–194.
• Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjuction fallacy in probability judgment. Psychological Review, 90, 293-315.
• Watson, J. M., & Kelly, B. A. (2007). The development of conditional probability reasoning. International Journal of Mathematical Education in Science and Technology, 38(2), 213–235.
• Way, J. (2003). The development of young children’s notions of probability. The proceeding of 3rd conference of the European Society for research in mathematics education, Thematic Group 5. Bellaria, Italy. Retrieved from http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG5/TG5_way_cerme3.pdf.
• Weber, K., & Alcock, L. (2004). Semantic and syntactic proof productions. Educational Studies in Mathematics, 56, 209-234.
• Yıldırım, A., & Şimşek, H. (2006). Sosyal bilimlerde nitel araştırma yöntemleri (5nd ed.). Ankara: Seçkin Yayıncılık.
• Zahner, D. C. (2005). Using clinical interviewing and problem solving protocols to uncover the cognitive processes of probability problem solvers. (Unpublished doctoral dissertation). Columbia University, New York.