An Investigation of Pre-service Middle School Mathematics Teachers’ Understanding of Distribution

Year 2020, , 374 - 398, 01.09.2020

Nadide Yılmaz

https://doi.org/10.9779/pauefd.556836

Cited By: 2

Abstract

In this study, it was aimed to reveal pre-service middle school mathematics teachers’ understandings about the concept of distribution and related concepts. It was adopted, the phenomenographic research method, one of the qualitative research designs, was employed on 66 third-year pre-service teachers. Through the activities developed on the basis of the study conducted by Lee and Meletiou-Mavrotheris (2003), interviews were conducted. The collected data were analyzed within the context of the knowledge and skills intended to be measured. The findings have revealed that the pre-service teachers were more successful in naming histogram axes complying with the context than constructing distribution graphs. They were not able to evaluate real-life situations on the basis of the distribution. They had difficulties in establishing connections between the given graphical display of distribution and variation. When they were asked to work on different contexts, it was observed that the pre-service teachers even arrived at correct answers through incorrect reasoning. Thus, it was suggested that pre-service teachers should be engaged in activities helping them recognize the relationships between the concept of distribution and related concepts and should discuss these concepts.

Keywords

distribution, variation, histogram, pre-service teachers, content knowledge

Ortaokul Matematik Öğretmen Adaylarının Dağılım Kavramına İlişkin Anlamalarının İncelenmesi

Abstract

Bu çalışmada ortaokul matematik öğretmen adaylarının dağılım ve bununla ilişkili kavramlara ilişkin anlamalarını ortaya çıkarmak amaçlanmıştır. Bu amaçla nitel araştırma desenlerinden fenomenografik araştırma yöntemi kullanılmış ve 66 üçüncü sınıf öğretmen adayı ile çalışılmıştır. Lee ve Meletiou-Mavrotheris (2003) çalışmasından yararlanılarak hazırlanan etkinlikler yardımıyla öğretmen adaylarıyla görüşmeler gerçekleştirilmiştir. Elde edilen veriler ölçülmek istenen bilgi ve beceriler kapsamında analiz edilmiştir. Bulgular öğretmen adaylarının bağlama uygun histogramın eksenlerini isimlendirebilmede, dağılım grafiği oluşturabilmeye göre daha başarılı olduklarını ortaya çıkarmıştır. Dağılım grafiğinden yola çıkarak gerçek yaşam durumlarını değerlendirememişlerdir. Verilen dağılımın grafiksel gösterimi ile değişkenlik arasında bağlantı kurmakta zorlanmışlardır. Farklı bağlamlar üzerinde çalışmaları istendiğinde ise öğretmen adaylarının doğru cevaplara bile yanlış akıl yürütmelerle ulaştıkları gözlenmiştir. Burada hareketle öğretmen adaylarının dağılım ve bu kavramla ilişkili kavramlar arasındaki ilişkileri fark etmelerine yönelik etkinliklerle meşgul olmaları ve bu kavramlar üzerine tartışmaları önerisi dile getirilmiştir.

Keywords

dağılım, değişkenlik, histogram, öğretmen adayları, alan bilgisi

References

• Ader, E. (2018). Programlardaki Veri ve Olasılık Öğrenme Alanı İçeriklerine Karşılaştırmalı Bir Bakış, In Özmantar, M. F., Akkoç, H., Kuşdemir Kayıran, B. & Özyurt, M. (Eds.) Ortaokul Matematik Öğretim Programları: Tarihsel Bir İnceleme (pp. 275-306), Ankara: Pegem Yayıncılık.
• Asworth, P. & Lucas, U. (1998). What is ‘world’ of phenomenography?. Scandinavian Journal of Educational Research, 42(4), 415-431.
• Ball, D. L. & McDiarmid, G. W. (1990). The subject-matter preparation of teachers. In W. R. Houston and M. H. J. Sikula (Eds.), Handbook of research on teacher education (pp. 437-449). New York: Macmillan.
• Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389–407.
• Bakker, A. (2004). Reasoning about shape as a pattern in variability. Statistics Education Research Journal, 3(2), 64-83.
• Bakker, A. & Gravemeijer, K. P. E. (2004). Learning to reason about distribution. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 147–168). Dordrecht: Springer Netherlands.
• Ben-Zvi, D. (2004). Reasoning about variability in comparing distributions. Statistics Education Research Journal, 3(2), 42-63.
• Biehler, R. (1994). Probabilistic thinking, statistical reasoning and the search for causes: Do we need a probabilistic revolution after we have taught data analysis? In J. Garfield (Ed.), Research Papers from The Fourth International Conference on Teaching Statistics, Marrakech, 1994. Minneapolis, MN: University of Minnesota.
• Borko, H., Eisenhart, M., Brown, C. A., Underhill R. G., Jones, D., et. al. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23(3), 194-222.
• Bruno, A. & Espinel, M. C. (2009). Construction and evaluation of histograms in teacher training. International Journal of Mathematical Education in Science and Technology, 40(4), 473–493.
• Canada, D. (2004). Elementary preservice teachers’ conceptions of variation. Unpublished doctoral dissertation, Portland State University, Portland.
• Chaphalkar, R. & Leary, C. (2014), “Introductory Statistics Students’ Conceptual Understanding of Variation and Measures of Variation in a Distribution,” in Sustainability in Statistics Education. Proceedings of the Ninth International Conference on Teaching Statistic, eds. K. Makar, B. de Sousa, and R. Gould, International Association for Statistical Education.Cooper, L. L. & Shore, F. S. (2008). Students’ misconceptions in interpreting center and variability of data represented via histograms and stem-and-leaf plots. Journal of Statistics Education, 16(2), 1-13.
• Cooper, L. & Shore, F. (2010), The effects of data and graph type on concepts and visualizations of variability, Journal of Statistics Education, 18, 1–16.
• Creswell, J. W. (2013). Research design: Qualitative, quantitative, and mixed methods approaches. Sage publications.
• delMas, R.C. & Liu, Y. (2005). Exploring Students’ Conceptions of the Standard Deviation. Statistics Education Research Journal, 4(1), 55-82.
• Franklin, C.A., Bargagliotti, A.E., Case, C.A., Kader, G.D., Scheaffer, R.L. & Spangler, D.A. (2015). Statistical Education of Teachers (SET), Alexandria, VA: American Statistical Association.
• Franklin, C., Kader, G., Mewborn, D., Moreno, J., Peck, R., Perry, M. et, al. (2005). Guidelines for assessment and instruction in statistics ducation (GAISE) report: A pre-K-12 curriculum framework. Alexandria, VA: American Statistical Association.
• Friel, S. N., Curcio, F. R., & Bright, G. W. (2001). Making sense of graphs: Criticalfactors influencing comprehension and instructional implications. Journal forResearch in Mathematics Education, 32, 124–158.
• Friel, S. N., Mokros, J. R., & Russell, S. J. (1992). Used numbers. Statistics: Middles, means, and in-betweens. Dale Seymour Publications.
• Garfield, J.B. (2003). Assessing Statistical Reasoning. Statistics Education Research Journal, 2(1), 22-38.
• Garfield, J. & Ben-Zvi, D. (2005). A framework for teaching and assessing reasoning about variability. Statistics Education Research Journal, 4(1), 92–99.
• Garfield, J.B. & Ben-Zvi, D. (2008a). Developing students’ statistical reasoning: Connecting researchand teaching practice. New York: Springer.
• Garfield, J.B. & Ben-Zvi, D. (2008b). Learning to reason about variability. In J.B. Garfield, & D. Ben-Zvi (Eds.), Developing Students’ Statistical Reasoning: Connecting Research and Teaching Practice (pp. 201-214). Springer.
• Garfield, J.B. & Ben-Zvi, D. (2008c). Learning to reason about distribution. In J.B. Garfield, & D. Ben-Zvi (Eds.), Developing Students’ Statistical Reasoning: Connecting Research and Teaching Practice (pp. 165-186). Springer.
• Garfield, J. & Chance, B. (2000). Assessment in Statistics Education: Issues and Challenges. Mathematical Thinking and Learning, 2(1&2), 99-125.
• Garfield, J., delMas, R. & Chance, B. (2007). Using students’ informal notions of variability todevelop an understanding of formal measures of variability. In M. Lovett & P. Shah (Eds.),Thinking with data (pp. 117–147). New York, NY: Lawrence Erlbaum.
• Groth, R. E. & Bergner, J.A. (2006). Preservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 8(1),37-63.
• Groth, R. E. & Meletiou-Mavrotheris, M. (2018). Research on statistics teachers’ cognitive and affective characteristics. In D. Ben-Zvi, K. Makar, & J. Garfield (Eds.), International handbook of research in statistics education (pp. 327–355). Cham, Switzerland: Springer.
• Jacobbe, T. (2012). Elementary school teachers’ understanding of mean and median. International Journal of Science and Mathematics Education, 10, 1143–1161.
• Konold, C. & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33(4), 259–289.
• Kuntze, S. (2014). Teachers’ views related to goals of the statistics classroom – from global to content-specific. In K. Makar, B. de Sousa, & R. Gould (Eds.), Sustainability in statistics education. Proceedings of the 9th International Conference on Teaching Statistics (ICOTS 9, July, 2014), Flagstaff, AZ, USA. Voorburg, The Netherlands: International Statistical Institute.
• Leavy, A. M. (2006). Using data comparison to support a focus on distribution: Examining preservice teacher’s understandings of distribution when engaged in statistical inquiry. Statistics Education Research Journal, 5(2), 89-114.
• Leavy, A. M. & O’Loughlin, N. (2006). Preservice Teachers Understanding of the Mean: Moving Beyond the Arithmetic Average. Journal of Mathematics Teacher Education, 9, 53-90.
• Lee, H. S. & Lee, J. T. (2011). Enhancing prospective teachers’ coordination of center and spread: A window into teacher education material development. The Mathematics Educator, 21(1), 33–47.
• Lee, C. & Meletiou-Mavrotheris, M. (2003). Some difficulties of learning histograms in introductory statistics, Paper presented at the Joint Statistical Meeting Section on Statictical Education, 2326-2333.
• Makar, K. (2004). Developing statistical inquiry: Prospective secondary mathematics and science teachers’ investigations of equity and fairness through analysis of accountability data. Unpublished doctoral dissertation, University of Texas at Austin, Austin.
• Makar, K. & Confrey, J. (2003). Chunks, Clumps and Spread Out: Secondary Pre-service Teachers’ Notions of Variation and Distribution. In C. Lee (Eds.), Proceedings of the Third International Research Forum on Statistical Reasoning, Thinking and Literacy (SRTL-3) (pp.). Mount Pleasant, Michigan: Central Michigan University.
• Makar, K., & Confrey, J. (2004). Secondary teachers’ statistical reasoning in comparing two groups. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 353–374). Boston: Kluwer Academic.
• Makar, K. & Confrey, J. (2005). “Variation-talks”: Articulating Meaning in Statistics. Statistics Education Research Journal, 4(1), 27-54.
• Milli Eğitim Bakanlığı (MEB). (2018). Matematik Dersi Öğretim Programı (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. Sınıflar. Ankara, Türkiye
• Mathews, D. & Clark, J. (2003). Successful students’ conceptions of mean, standard deviation andthe central limit theorem. Unpublished paper.
• Marton, F. (1988). Phenomenography: Exploring different concepts of reality. In Fetterman, D. (Ed.), Qualitative Approaches to Evaluation in Education. New York: Praeger.
• Meletiou, M. & Lee, C. (2002). Student understanding of histograms: A stumbling stone to the development of intuitions about variation. Proceedings of the Sixth International Conference on Teaching Statistics. Durban, South Africa.
• Mickelson, W., & Heaton, R. (2004). Primary teachers’ statistical reasoning with data. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 327–352). Dordrecht, The Netherlands: Kluwer.
• Moore, DS. (1990). Uncertainty. On the shoulders of giants: new approaches to numeracy. In LS Steen (Ed.), (pp. 95–137). Washington, DC: National AcademyPress.
• Pfannkuch, M. & Reading, C. (2006). Reasoning about distribution: A complex process. Statistics Education Research Journal, 5(2), 4–9.
• Pfannkuch, M.,& Wild, C. (2004). Towards an understanding of statistical thinking. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning, and thinking, (pp.17-46). Dordrecht, the Netherlands: Kluwer Academic Publishers.
• Reading, C. & Canada, D. (2011). Teachers’ Knowledge of Distribution, C. Batanero, G. Burrill, and C. Reading (Eds.), Teaching Statistics in School Mathematics-Challenges for Teaching and Teacher Education: A Joint ICMI/IASE Study, (pp. 223-234), New York.
• Reading, C., & Reid, J. (2006). An emerging hierarchy of reasoning about distribution: From a variation perspective. Statistics Education Research Journal, 5(2), 46–68.
• Reading, C. & Shaughnessy, M. (2004). Reason about variation. In D. Ben-Zvi, & J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking (pp. 201-226). The Netherlands: Kluwer Academic Publishers.
• Sánchez, E.; Silva, C. B., & Coutinho, C. (2011). Teachers’ Understanding of Variation, C. Batanero, G. Burrill, and C. Reading (eds.), Teaching Statistics in School Mathematics-Challenges for Teaching and Teacher Education: A Joint ICMI/IASE Study, (pp. 211-221), New York.
• Scheaffer, R. L. (2006). Statistics and mathematics: On making a happy marriage. In G. F. Burrill, & P. C. Elliott (Eds.), Thinking and reasoning about data and chance: Sixty eighth year book (pp. 309–322). Reston, VA: NCTM.
• Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester (Ed.), Second handbook of research on the teaching and learning of mathematics (pp. 957-1009). United States of America: Information Age Publishing.
• Stein, M. K., Baxter, J. A., & Leinhardt, G. (1990). Subject-Matter Knowledge and Elementary Instruction: A Case from Functions and Graphing. American Educational Research Journal, 27(4), 639-663.
• Sorto, M. A. (2004). Prospective middle school teachers’ knowledge about data analysis and its application to teaching. Unpublished doctoral dissertation, Michigan State University.
• Trigwell, K. (2006). Phenomenography: An approach to research into geography education. Journal of geography in higher education, 30(2), 367-372.
• Tufte, E. R. (1983). The visual display of quantitative information. Cheshire: Graphics Press.
• Vermette, S. & Savard, A. (2019). Necessary Knowledge forTeachingStatistics: Example of the Concept of Variability. In G. Burrill & D. Ben-Zvi (Eds.), Topics and Trends in Current Statistics Education ICME-13 Monographs, (pp.225-244), Germany.
• Watson, J. (2005). Developing an awareness of distribution. In K. Makar (Ed.),Reasoning about distribution: A collection of current research studies. Proceedings of the Fourth International Research Forum on Statistical Reasoning, Thinking, and Literacy, Auckland, 2-7 July 2005, Brisbane,Australia.
• Watson, J. M. & Moritz, J. B. (1999). The beginnings of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37, 145–168.
• Wild, C. (2006). The Concept of Distribution. Statistics Education Research Journal, 5(2), 10-26.
• Wild, C. J. & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-265.
• Zaidan, A., Z. , Ismail, Y., Yusof, M. & Kashefi. H. (2012). “Misconceptions in Descriptive Statistics Among Postgraduates in Social Sciences.” Procedia - Social and Behavioral Sciences, 46, 3535–3540.