An Analysis of High School Students’ Understanding and Reasoning of Average Concept

Year 2021, , 187 - 201, 05.02.2021

Özlem Engin Alaattin Pusmaz

https://doi.org/10.16949/turkbilmat.780680

Abstract

The aim of this study is to identify high school students’ understanding of average concept and the reasoning types they appeal to solve average problems. The case study approach was used in this study and the participants were selected by purposeful sampling. The participants consisted of five 9th grade and four 10th grade students, studying at a high school in Istanbul. In order to identify students’ understanding of average, a test consisting of 5 open-ended problems were used and semi-structured interviews were held with each of the students. The data were analyzed by thematic analysis approach. For data analysis, framework proposed by Mokros and Russel (1995) was used to determine students’ understanding of average and Lithner’s (2008) framework was used to reveal their reasoning types. Results showed that students mostly understood average as mathematical point of balance. Creative mathematically founded reasoning and algorithmic reasoning was used the most. Creative reasoning is effective in reaching the right answer. In solutions where creative reasoning is used, students generally also have the idea of representativeness. The type of problem influences the reasoning process. Inadequacy of students’ prior mathematics knowledge hinders both their understanding of the average and their reasoning skills.

Keywords

Average, Mathematical reasoning, Mathematics education

References

• Adu-Gyamfi, K., & Bossé, M. J. (2014). Processes and reasoning in representations of linear functions. International Journal of Science and Mathematics Education, 12(1), 167-192.
• Australian Education Council. (1991). A national statement on mathematics for Australian schools. Carlton, VIC: Australian Education Council.
• Bergqvist, T., & Lithner, J. (2012). Mathematical reasoning in teachers’ presentations. The Journal of Mathematical Behavior, 31(2), 252-269.
• Bergqvist, T., Lithner, J., & Sumpter, L. (2003). Reasoning characteristics in upper secondary school students’ task solving. Department of Mathematics and Mathematical Statistics, Umeå University, Research Reports in Mathematics Education (1).
• Bergqvist, T., Lithner, J., & Sumpter, L. (2008). Upper secondary students' task reasoning. International Journal of Mathematical Education in Science and Technology, 39(1), 1-12.
• Boesen, J., Lithner, J., & Palm, T. (2010). The relation between types of assessment tasks and the mathematical reasoning students use. Educational Studies in Mathematics, 75(1), 89-105.
• Boyatzis, R. E. (1998). Transforming qualitative information: Thematic analysis and code development. Thousand Oaks: Sage Publications.
• 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, 839-855.
• 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.
• Creswell, J. W., & Poth, C. N. (2018). Qualitative inquiry and research design: Choosing among five approaches (Fourth ed.). Thousand Oaks: Sage Publications.
• Dane, A., Kudu, M., & Balkı, N. (2009). The factors negatıvely effectıng hıgh school students’ mathematical success accordıng to theır perceptıons. Erzincan University Journal of Science and Technology, 2(1), 17-34.
• Ellis, A. B., Ozgur, Z., Kulow, T., Dogan, M. F., & Amidon, J. (2016). An exponential growth learning trajectory: Students’ emerging understanding of exponential growth through covariation. Mathematical Thinking and Learning, 18(3), 151-181.
• English, L. D. (1998). Editorial Mathematical Reasoning: Nature, Form, and Development. Mathematical Cognition, 4(2), 81-83. doi:10.1080/135467998387334.
• Enisoglu, D., & Isiksal-Bostan, M. (2017). Identifying students'possible solution strategies while solving questions regarding the concept of mean. The Eurasia Proceedings of Educational and Social Sciences, 6, 24-30.
• Erdem, E. (2011). An ınvestigation of the seventh grade students’ mathematical and probabilistic reasoning skills. (Unpublished master dissertation, Adıyaman University). Retrieved from https://tez.yok.gov.tr/UlusalTezMerkezi/
• Erdem, E., & Gürbüz, R. (2015). An analysis of seventh-grade students’mathematical reasoning. Cukurova University Faculty of Education Journal, 44(1), 123-142.
• Ersoy, Y. (2003). Teknoloji destekli matematik egitimi-1: Gelişmeler, politikalar ve stratejiler. Elementary Education Online, 2(1), 18-27.
• Farmaki, V., & Paschos, T. (2007). The interaction between intuitive and formal mathematical thinking: a case study. International Journal of Mathematical Education in Science and Technology, 38(3), 353-365.
• Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational Studies in Mathematics, 38(1-3), 11-50.
• Fujita, T., & Jones, K. (2014). Reasoning-and-proving in geometry in school mathematics textbooks in Japan. International Journal of Educational Research, 64, 81-91.
• Gal, I., Rothschild, K., & Wagner, D. A. (1990, April). Statistical concepts and statistical reasoning in school children: Convergence or divergence? In annual meeting of the American Educational Research Association, Boston, MA.
• Güler, H. K. (2013). Türk öğrencilerin PISA'da karşılaştıkları güçlüklerin analizi. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 26(2), 501-522.
• Gürbüz, R., Toprak, Z., Yapıcı, H., & Doğan, S. (2011). Subjects perceived as difficult in secondary mathematics curriculum and their reasons. University of Gaziantep Journal of Social Sciences, 10(4), 1311-1323.
• Herbert, S., Vale, C., Bragg, L. A., Loong, E., & Widjaja, W. (2015). A framework for primary teachers’ perceptions of mathematical reasoning. International Journal of Educational Research, 74, 26-37.
• Jonsson, B., Norqvist, M., Liljekvist, Y., & Lithner, J. (2014). Learning mathematics through algorithmic and creative reasoning. The Journal of Mathematical Behavior, 36, 20-32.
• Lithner, J. (2000). Mathematical reasoning and familiar procedures. International Journal of Mathematical Education in Science and Technology, 31(1), 83-95.
• Lithner, J. (2003). Students' mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52(1), 29-55.
• Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. Journal of Mathematical Behavior, 23, 405–427.
• Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67(3), 255-276.
• Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. The Journal of Mathematical Behavior, 21(1), 87-116.
• Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, 259 – 289.
• Kramarski, B. A., Mevarech, Z. R., & Lieberman A. (2001). Effects of multilevel versus unilevel metacognitive training on mathematical reasoning. Journal of Educational Research, 94(5), 292-300.
• Miles, M. B., & Huberman, A. M. (1994). An Expanded Sourcebook: Qualitative Data Analysis (Second edition). Thousand Oaks, CA: SAGE Publications, Inc
• Ministry of National Education [MoNE] (1998). İlköğretim okulu matematik dersi öğretim programı: 1-8. sınıflar. İstanbul: Milli Eğitim Basımevi
• Ministry of National Education [MoNE] (2009). İlköğretim matematik dersi 1-5. sınıflar öğretim programı. Ankara: T.C. Milli Eğitim Bakanlığı. Talim ve Terbiye Kurulu Başkanlığı. Retrieved November 11, 2020, from http://talimterbiye.mebnet.net/Ogretim%20Programlari/ilkokul/2010-2011/Matematik1-5.pdf
• Ministry of National Education [MoNE] (2013). Ortaokul matematik dersi (5, 6, 7 ve 8. sınıflar) öğretim programı. Ankara: T.C. Milli Eğitim Bakanlığı. Talim ve Terbiye Kurulu Başkanlığı. Retrieved November 11, 2020, from http://talimterbiye.mebnet.net/Ogretim%20Programlari/ortaokul/ana.html
• Ministry of National Education [MoNE] (2015). İlkokul matematik dersi (1, 2, 3 ve 4. sınıflar) öğretim programı. Ankara, T.C. Milli Eğitim Bakanlığı. Talim ve Terbiye Kurulu Başkanlığı. Retrieved November 11, 2020, from http://matematikogretimi.weebly.com/uploads/2/6/5/4/26548246/matematik1-4_prg.pdf
• Ministry of National Education [MoNE] (2018a). Matematik dersi öğretim programı (ilkokul ve ortaokul 1,2,3,4,5,6,7 ve 8. sınıflar). Ankara, T.C. Milli Eğitim Bakanlığı. Retrieved November 11, 2020, from http://mufredat.meb.gov.tr/Dosyalar/201813017165445-MATEMAT%C4%B0K%20%C3%96%C4%9ERET%C4%B0M%20PROGRAMI%202018v.pdf
• Ministry of National Education [MoNE] (2018b). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı. Ankara, T.C. Milli Eğitim Bakanlığı. Retrieved November 11, 2020, from http://mufredat.meb.gov.tr/Dosyalar/201821102727101-OGM%20MATEMAT%C4%B0K%20PRG%2020.01.2018.pdf
• Merriam, S.B. (2009). Qualitative research: A guide to design and implementation: Revised and expanded from qualitative research and case study applications in education. (3rd ed.), Jossey-Bass, San Francisco, California
• Mokros, J., & Russell, S. J. (1995). Children's concept of average and representativeness. Journal for Research in Mathematics Education, 26, 20 – 39. NCTM (National Council of Teachers of Mathematics) (2000). Principles and standards for school mathematics. Reston, VA. Retrieved November 24, 2019, from http://www.nctm.org/
• NCTM (National Council of Teachers of Mathematics) (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA. Retrieved November 24, 2019, from http://www.nctm.org/
• Patton, M. Q. (2002). Qualitative evaluation and research methods (Third ed.). Newbury Park, CA: Sage Publications.
• Poçan, S., Yaşaroğlu, S., & İlhan, A. (2017). Investigation of secondary 7th and 8th grade students' mathematical reasoning skills in terms of some variable. Journal of International Social Research, 10(52). 808-818.
• Pollatsek, A., Lima, S. & Well, A.D. (1981). Concept or computation: Students’ understanding of the mean, Educational Studies in Mathematics, 12, 191-204. Schoenfeld, A. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. Voss, D. Perkins & J. Segal (Eds.), Informal reasoning and education (pp. 311–344). Hillsdale, NJ: Lawrence Erlbaum Associates.
• Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM – The International Journal on Mathematics Education, 29(3), 75-80.
• Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77(1), 20-26.
• Thompson, P. W., Hatfield, N. J., Yoon, H., Joshua, S., & Byerley, C. (2017). Covariational reasoning among US and South Korean secondary mathematics teachers. The Journal of Mathematical Behavior, 48, 95-111.
• Toluk-Uçar, Z. & Akdoğan, E. N. (2009). Middle school students’ understanding of average. Elementary Education Online, 8(2), 391-400.
• Umay, A. (2003). Mathematıcal reasoning ability. Hacettepe University Journal of Education, 24(24), 234-243.
• Umay, A., & Kaf, Y. (2005). A study on flawed reasoning in mathematics. Hacettepe University Journal of Education, 28(28), 188-195.
• Yeşildere, S., & Türnüklü, E. B. (2007). Examination of students’ mathematical thinking and reasoning processes. Ankara University, Journal of Faculty of Educational Sciences, 40(1), 181-213.
• Watson, M. & Moritz, J. (2000). The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2(1&2), 11-50.
• Watson, J., Chick, H., & Callingham, R. (2014). Average: The juxtaposition of procedure and context. Mathematics Education Research Journal, 26(3), 477-502.
• Wyndhamn, J., & Säljö, R. (1997). Word problems and mathematical reasoning—a study of children's mastery of reference and meaning in textual realities. Learning and Instruction, 7(4), 361-382.