Türkiye’de Okutulan Ortaokul Matematik Ders Kitaplarının Aritmetik Ortalama Kavramına İlişkin Öğrencilere Sunduğu Öğrenme Fırsatları

Year 2020, Volume: 11 Issue: 1, 157 - 187, 30.04.2020

Suphi Önder Bütüner

https://doi.org/10.16949/turkbilmat.627826

Abstract

Bu çalışmada veri işleme öğrenme alanının önemli kavramlarından biri olan aritmetik ortalama kavramının Türkiye’de okutulan ortaokul matematik ders kitaplarında nasıl sunulduğu, aritmetik ortalama ile ilgili ne tip problemlere yer verildiği, problemlerin hangi temsil biçiminde sorulduğu, problemlerin çözümlerinde hangi temsil biçimleri ve çözüm stratejilerinin kullanıldığı incelenmiştir. Çalışma kapsamında incelenen kitaplar 2019-2020 eğitim öğretim yılında kullanılan ders kitaplarıdır. Çalışmanın araştırma sorularını cevaplandırmak için iki boyutlu bir çerçeve (yatay ve dikey analiz) kullanılmıştır. Analizler iki araştırmacı tarafından yapılmıştır. Denge ve adil-paylaşım modelleri aritmetik ortalamanın kavramsal olarak anlaşılmasında güçlü birer analoji olarak kabul edilmesine karşın, her iki altıncı sınıf ders kitabında bu modellerin ya yeterli ölçüde ya da hiç kullanılmadığı tespit edilmiştir. Buna ek olarak, ders kitaplarında aritmetik ortalamanın bir veri kümesindeki elemanlarla olan ilişkisi ve aritmetik ortalamanın veri kümesini temsil eden bir değer olduğuna ilişkin tartışmalara yer verilmemiştir. Ders kitaplarındaki problemlerin çoğunluğu sözel formda sorulmuş olup, problemler sadece ekle-böl algoritması kullanılarak çözülmüştür. Problemlerin çözümlerinde farklı temsil biçimlerinden yararlanılmamış olup sadece aritmetik formda çözümler yapılmıştır. Ders kitaplarındaki bu içerik öğrencilerin aritmetik ortalama kavramını yüzeysel olarak öğrenmelerine ve muhakeme düzeyindeki sorularda zorlanmalarına neden olabilir. Bu açıdan ders kitaplarında aritmetik ortalama kavramının öğretiminde, adil paylaşım ve denge merkezi düşüncesinden yararlanılmalı, ders kitapları farklı tipteki problemler açısından zenginleştirilmelidir. Bunun yanında, ders kitaplarında aritmetik ortalama problemlerinin çözümlerinde farklı çözüm stratejilerinden ve çoklu temsil biçimlerinden yararlanılmalıdır.  

Keywords

Aritmetik ortalama, öğrenme fırsatı, ders kitabı

The Learning Opportunities Presented by Mathematics Coursebooks Used in Middle Schools in Turkey on the Concept of Arithmetic Mean

Abstract

In this study, it was investigated how the concept of arithmetic mean (an essential concept in the learning domain of data) was presented in middle school coursebooks in Turkey, what type of problems related to the arithmetic mean was covered, the type of representation form through which the problems were presented, and the type of representation forms and solving strategies used for solving problems. For this study, the coursebooks used in the 2019–2020 academic year were analyzed. A 2D framework (horizontal and vertical analysis) was used to answer research questions, and the analyses were conducted by two researchers. Although balance and fair-share models are strong analogies for conceptually understanding arithmetic mean, the two coursebooks of grade 6 did not sufficiently cover these models. Furthermore, the relationship between the arithmetic mean and elements in a data set and the discussions related to arithmetic mean as a value that represents the data set were not covered in coursebooks. Most problems in the coursebooks were verbally developed, and the problems were solved only via the add–divide algorithm. Note that, for solving these problems, different representation forms were not used, and the solutions were only presented in the arithmetic form. In the coursebooks, the content might cause students to superficially learn the concept of arithmetic mean and have difficulties with questions at the evaluation level. Thus, for teaching the concept of arithmetic mean, fair share and balance point should be used via coursebooks, and they should be enriched in terms of problems in various types. Furthermore, different strategies and multiple representation forms should be used for solving arithmetic mean problems in coursebooks.

Keywords

Arithmetic mean, learning opportunity, coursebook

References

• Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183-198.
• Bektaş, M., Kahraman, S. ve Temel, Y. (2018). Matematik ders kitabı 6. Ankara: Milli Eğitim Bakanlığı.
• Bowen, G. A. (2009). Document analysis as a qualitative research method. Qualitative Research Journal, 9(2), 27-40.
• Bremigan, E. G. (2003). Developing a meaningful understanding of the mean. Mathematics Teaching in the Middle School, 9(1), 22-26.
• Brenner, M. E., Herman, S., Ho, H. Z., & Zimmer, J. M. (1999). Cross-national comparison of representational competence. Journal for Research in Mathematics Education, 30(5), 541-547.
• Cai, J. (1995). Beyond the computational algorithm: Students’ understanding of the arithmetic average concept. In L. Meria, & D. Carraher (Eds.), Proceeding of the 19th Psychology of Mathematics Education Conference (Vol. 3. pp. 144-151). Sao Paulo, Brazil: PME Program Committee.
• Cai, J. (1998). Exploring students' conceptual understanding of the averaging algorithm. School Science and Mathematics, 98(2), 93-98.
• Cai, J. (2000). Understanding and representing the arithmetic averaging algorithm: An analysis and comparison of US and Chinese students’ responses. International Journal of Mathematical Education in Science and Technology, 31(6), 839-855.
• Cai, J., Lo, J., & Watanabe, T. (2002). Intended treatmens of arithmetic average in U.S. and Asian School mathematics textbooks. School Science and Mathematics, 102(8), 391-404.
• Cai, J., & Moyer, J. C. (1995). Middle school students' understanding of average: A problem solving approach. In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting of the North American Chapter of the International Group of the Psychology of Mathematics Education (Vol. 1, pp. 359-364). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
• Charalambous, C. Y., Delaney, S., Hsu, H.Y., & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12(2), 117–151.
• Chatzivasileiou, E., Michalis, I., & Tsaliki, C. (2010). Elementary school students’ understanding of concept of arithmetic mean. In C. Reading (Ed.), Data and context in statistics education: Towards an evidence based society. Proceedings of the Eighth International Conference on Teaching Statistics. The Netherlands: International Statistical Institute.
• Çakmak, Z. T., & Durmuş, S. (2015). Determining the conceptes and subjects in the area of learning statistics and probability that 6-8th grade math students have difficulties. Abant İzzet Baysal University Journal of Education, 15(2), 27-58.
• Çağlayan, N., Dağıstan, A., ve Korkmaz, B., (2018). Ortaokul ve İmam Hatip Ortaokulu Matematik 6 Ders Kitabı. Ankara: Devlet Kitapları.
• Duval, R. (2006). A Cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103-131.
• Enisoğlu, D. (2014). Seventh grade students' possible solution strategies, errors and misinterpretations regarding the concepts of mean, median and mode given in bar graph representations (Unpublished master’s thesis). Middle East Technical University, Institute of Social Sciences, Ankara.
• Flanders, J. (1994). Student opportunities in grade 8 mathematics: Textbook coverage of the SIMS test. In I. Westbury, C. A. Ethington, L. A. Sosniak, & D. P. Baker (Eds.), In search of more effective mathematics education: Examining data from the IEA second international mathematics study (pp. 61–93). Norwood, NJ: Ablex.
• Gattuso, L., & Mary, C. (1998). Development of the concept of weighted average among high-school children. In L. Pereira-Mendoza, L. S. Kea, T. W. Kee, & W. K. Wong (Eds.), Proceedings of the Fifth International Conference on Teaching Statistics (pp. 685-692). Voorburg, The Netherlands: International Statistical Institute.
• Gfeller, M. K., Niess, M. L., & Lederman, N. G. (1999). Preservice teachers’ use of multiple representations in solving arithmetic mean problems. School Science and Mathematics, 99(5), 250–257.
• Ginat, D., & Wolfson, M. (2002, July). On limited views of the mean as point of balance. Paper presented at the 26th Annual Conference of the International Group for the Psychology of Mathematics Education, Norwich, England.
• Goodchild, S. (1988). School pupils’ understanding of average. Teaching Statistics, 10, 77–81.
• Hardiman, P. T., Well, A. D., & Pollatsek, A. (1984). Usefulness of the balance model in understanding the mean. Journal of Educational Psychology, 76(5), 792– 801.
• Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Hong, D. S., & Choi, K. M. (2014). A comparison of Korean and American secondary school textbooks: The case of quadratic equations. Educational Studies in Mathematics, 85(2), 241–263.
• Kaynar, Y.,& Halat, E. (2012, June). Investigation of the "statistical" dimension of the "probability and statistics" sub-learning area of primary education II. level mathematics education. Paper presented at the X. Science and Mathematics Education Congress, Niğde.
• Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, 259-289.
• Koparan, T., & Güven, B. (2014). According to the M3ST model analyze of the statistical thinking levels of middle school students. Education and Science, 39(171), 37-51.
• Leavy, A. M. (2001). Elementary and middle grade students’ understanding of distribution (Unpublished doctoral dissertation). Arizona State University, the USA.
• Leavy, A., & O’Loughlin, N. (2006). Preservice teachers understanding of the mean: Moving beyond the arithmetic average. Journal of Mathematics Teacher Education, 9, 53–90.
• Leon, M. R., & Zawojewski, J. S. (1991). Use of the arithmetic mean: An investigation of four properties, issues and preliminary results. In D. Vere-Jones (Ed.), Proceedings of the Third International Conference on Teaching Statistics (Vol. I, pp. 302–306). Voorburg, The Netherlands: International Statistical Institute.
• Marnich, M. A. (2008). A knowledge structure for the arithmetic mean: Relationships between statistical conceptualizations and mathematical concepts (Unpublished doctoral dissertation). University of Pittsburgh, the USA.
• Mevarech, Z. R. (1983). A deep structure model of students’ statistical misconceptions. Educational Studies in Mathematics, 14, 415–429.
• Ministry of National Education [MoNE]. (2018). Elementary curricula programs: Mathematics curricula program for middle grades. Retrieved 01.09. 2019 from http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=329
• Mokros, J., & Russell, S. J. (1995). Children’s concepts of a average and representativeness. Journal for Research in Mathematics Education, 26(1), 20-39.
• Mullis, I. V. S., Martin, M. O., Foy, P., & Arora, A. (2012). TIMSS 2011 international results in mathematics. Retrieved 01.09. 2019 from https://timssandpirls.bc.edu/timss2011/downloads/T11_IR_Mathematics_FullBook.pdf adresinden alınmıştır.
• Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). TIMSS 2015 international results in mathematics. Retrieved 01.01.2019 from http://timssandpirls.bc.edu/timss2015/internationalresults/
• National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
• Oğan, A. K., & Öztürk, S. (2019). Matematik 7. Sınıf ders kitabı. Ankara: Milli Eğitim Yayınları.
• Pollatsek, S. J., Lima, S., & Well, A. D. (1981). Concept or understanding: Students’ understanding of the mean. Educational Studies in Mathematics, 12, 191–204.
• Rau, M. A., Aleven, V., & Rummel, N. (2009). Intelligent tutoring systems with multiple representations and self-explanation prompts support learning of fractions. In V. Dimitrova, R. Mizoguchi, & B. du Boulay (Eds.), Proceedings of the 2009 conference on Artificial Intelligence in Education: Building Learning Systems that Care: From Knowledge Representation to Affective Modelling (pp. 441-448). Amsterdam, The Netherlands: IOS Press.
• Russell, S. J., & Mokros, J. (1996). What do children understand about average? Teaching Children, 2(6), 360-364.
• Schnotz, W., & Bannert, M. (2003). Construction and interference in learning from multiple representation. Learning and Instruction, 13(2), 141-156.
• Son, J. W., & Senk, S. L. (2010). How reform curricula in the USA and Korea present multiplication and division of fractions. Educational Studies in Mathematics, 74(2), 117–142.
• Stein, M., Remillard, J., & Smith, M. (2007). How curriculum influences students’ learning. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning Charlotte (pp. 557–628). Greenwich: Information Age.
• Strauss, S., & Bichler, E. (1988). The development of children’s concept of the arithmetic average. Journal for Research in Mathematics Education, 19(1), 64-80.
• Tall, D. (1988). Concept image and concept definition. In J. de Lange, & M. Doorman (Ed.), Senior secondary mathematics education (pp. 37-41). Utrecht: OW & OC.
• Uccellini, J.C. (1996). Teaching the mean meaningfully. Mathematics Teaching in Middle School, 2(3), 112-115.
• Uçar, T. Z., & Akdoğan, N. E. (2009). Middle school students’ understanding of average. Elementary Education Online, 8(2), 391–400.
• Ünlü, M. (2008). The effect of cooperative learning method on the academic success and recall levels of 8th grade students in 'permutation and probability' subject (Unpublished master’s thesis). Gazi University, Institute of Educational Sciences, Ankara.
• Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2013). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Upper Saddle River, NJ: Pearson.
• Vincent, J., & Stacey, K. (2008). Do mathematics textbooks cultivate shallow teaching? Applying the TIMSS video study criteria to Australian eighth-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82-107.
• Watson, J. M., & Moritz, J. B. (2000). The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2(1&2), 11–50.
• Watson, J. M. (2007). The role of cognitive conflict in developing student’s understanding of average. Educational Studies in Mathematics, 65, 21-47.
• Weinberg, A., Wiesner, E., Benesh, B., & Boester, T. (2012). Undergraduate students’ self-reported use of mathematics textbooks. PRIMUS, 22(2), 152-175.
• Zazkis, D. (2013). On students’ conceptions of arithmetic average: The case of inference from a fixed total. International Journal of Mathematical Education in Science and Technology, 44(2), 204-213.