Quantitative Reasoning: Reflections on Solving Real-World Problems

Year 2018, Volume: 47 Issue: 1, 60 - 108, 22.04.2018

Dilek Tanışlı , Mehmet Dur

https://doi.org/10.14812/cuefd.398406

Cited By: 1

Abstract

The purpose of this research is to determine the
role of quantitative reasoning in the problem-solving process of the two
students who go in sixth grade of secondary school. In this research, in which
the qualitative research method is adopted, there were five weeks of
instruction carried out with students who go in sixth grade of secondary school
in order to determine how the development of quantitative reasoning led to a
change in problem-solving ability. Clinical interviews were conducted with
students selected as the focus at the beginning and end of the process. Data
from this study were also collected from the data of two focal students who
were interviewed clinically. In clinical interviews, the real world problems
with simple, moderate and high difficulty levels were used in quantitative
difference, complex additive situations, combination of differences and
quantitative ratio. In addition to this, student work sheets, student diaries
and researcher's diary were also used as support data. In the analysis of the
data, the thematic analysis method was used. As a result of the research, it
was observed at the interviews that the quantitative reasoning skills of the
two students, who had low quantitative reasoning ability at the preliminary
talks, changed at the end of the teaching process. It has been determined that
this change is reflected in the problem solving process of the students and
that they are also successful in the problems with medium and high difficulty
level according to the pre-teaching. In this process, it has been observed that
students can solve problems more than one way, choose appropriate strategies
for their problem situations and also increase the use of visual representation
which has an effective place to do quantitative reasoning.

Keywords

Clinical interview, Mathematics ecucation, Quantitative reasoning, Secondary school

Nicel Muhakeme: Gerçek Yaşam Problemlerinin Çözüm Sürecinden Yansımalar

Abstract

Bu araştırmanın amacı, ortaokul altıncı sınıfa
devam eden iki öğrencinin nicel muhakeme gelişimlerinin problem çözme
sürecindeki rolünü belirlemektir. Nitel araştırma yönteminin benimsendiği bu
araştırmada nicel muhakeme gelişiminin problem çözme becerisinde nasıl bir
değişime yol açtığını belirleyebilmek için öncelikle altıncı sınıfa devam eden
öğrenciler ile beş hafta süren bir öğretim gerçekleştirilmiştir. Bu sürecin
başında ve sonunda odak olarak seçilen öğrenciler ile de klinik görüşmeler
yapılmıştır. Bu araştırmanın verileri de klinik görüşme yapılan iki odak
öğrencinin verilerinden toplanmıştır. Klinik görüşmelerde nicel fark, karmaşık
toplamsal durumlar, farkların kombinasyonu ve nicel oran türünde basit, orta ve
yüksek güçlük düzeyine sahip gerçek yaşam problemleri kullanılmıştır. Bunun
yanı sıra öğrenci çalışma yapraklarından, öğrenci günlüklerinden,
araştırmacının günlüğünden destek veri olarak da yararlanılmıştır. Araştırmada
verilerin analizinde tematik analiz yöntemi kullanılmıştır. Araştırma sonucunda
ön görüşmelerde düşük nicel muhakeme becerisine sahip olan iki öğrencinin
öğretim süreci sonunda yapılan görüşmelerde nicel muhakeme becerilerinin gözle
görülür şekilde değiştiği gözlenmiştir. Bu değişimin öğrencilerin problem çözme
süreçlerine yansıdığı ve öğretim öncesine göre özellikle güçlük düzeyi orta ve
yüksek olan problemlerde de başarılı oldukları belirlenmiştir. Bu süreçte
öğrencilerin problemleri birden fazla yolla çözebildikleri, problem durumlarına
uygun stratejileri seçebildikleri gözlenmiş, ayrıca nicel muhakeme yapabilmede
etkin bir yere sahip olan görsel temsil kullanımının da arttığı belirlenmiştir. 

Keywords

Klinik görüşme, Matematik eğitimi, Nicel muhakeme, Ortaokul öğrencileri

References

• Bayazıt, İ. (2013). İlköğretim 7. ve 8. sınıf öğrencilerinin gerçek-yaşam problemlerini çözerken sergiledikleri yaklaşımlar ve kullandıkları strateji ve modellerin incelenmesi. Kuram ve Uygulamada Eğitim Bilimleri, 13(3), 1903-1927. Bednarz, N., Radford, L., Janvier, B. & Lepage, A. (1992). Arithmetic and algebraic thinking in problem-solving. In W. Geeslin & Graham (Eds.), Proceedings of the 16th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 65-72). Durham, New Hampshire: program Committee.Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Heinemann, NH.Cai, J., Ng, S. F., & Moyer, J. C. (2011). Developing students’ algebraic thinking in earlier grades: Lessons from China and Singapore. In J. Cai ve E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives. (pp. 25-41). Springer Berlin Heidelberg.Çelik, D. & Güler, M. (2003). İlköğretim 6. sınıf öğrencilerinin gerçek yaşam problemlerini çözme becerilerinin incelenmesi. Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi, 20, 180-195.Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly, & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education, (pp. 547-589). London: Lawrence Erlbaum Associates, Publishers.Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for Research in Mathematics Education, 14(2), 83-94.Diezmann, C., & L. English. (2001). Promoting the use of diagrams as tools for thinking. In A.A. Cuoco and F. R. Curcio (Eds.), The role of representation in school mathematics. Reston, VA: National Council of Teachers of Mathematics, 77-89.Dwyer, C. A., Gallagher, A., Levin, J. & Morley, M. E. (2003). What is quantitative reasoning? Defining the construct for assessment purposes. Research Report (RR-03-30), Education Testing Service. Ellis, A. B. (2007). The influence of reasoning with emergent quantities on students’ generalizations. Cognition and Instruction, 25(4), 439–478.Herscovics N. & Kieran C. (1980). Constructing meaning for the concept of equation. Mathematics Teacher, 80, 572-580.Jonassen, D. (2003). Using cognitive tools to represent problems. Journal of Research on Technology in Education, 35(3), 362-381.Kabael, T., & Akın, A. (2016). Problem solving strategies and quantitative reasoning skills in solving algebraic verbal problems of seventh grade students. Kastamonu Education Journal, 24(2), 875-894.Kamal, A. & Ramzi, N. (2000). The Role of Presentation and Response Format in Understanding, Preconceptions and Alternative Concept in Algebra Word Problems. ERIC Document Reproduction Service No. ED 438 174.Koedinger, K.R., & Nathan, M.J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of The Learning Sciences, 13(2), 129-164.Liamputtong, P. (2009) Qualitative research methods, 3rd edition. Melbourne: Oxford University Press.Lochhead, J. (1988). Some Pieces of the Puzzle. In Constructivism in the Computer Age, edited by G. Forman and P. Pufall., Hillsdale , N.J.:Lawrence Erlbaum Associates.Mayer, R. E., Lewis, A. B., & Hegarty, M. (1992). Mathematical misunderstandings: Qualitative reasoning about quantitative problems. Advances in Psychology, 91, 137-153. Milli Eğitim Bakanlığı (2017). Matematik dersi öğretim programı. (İlkokul ve Ortaokul 1, 2, 3, 4, 5, 6, 7, ve 8 Sınıflar) Ankara: MEB Yayınları.Miles M., & Huberman, M. (1994). An expanded sourcebook qualitative data analysis. Second Edition. California: Sage Publications.Moore, K. C., Carlson, M. P., & Oehrtman, M. (2009). The role of quantitative reasoning in solving applied precalculus problems. Paper presented at the Twelfth Annual Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Educati¬on (SIGMAA on RUME) Conference, Raleigh, NC: North Carolina State University.Moore, C.K. (2010). The Role of Quantitative Reasoning in Precalculus Students Learning Central Concepts of Trigonometry. (Unpublished doctoral dissertion) Arizona State University, ABD. Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. The Journal of Mathematical Behavior, 31(1), 48-59.National Council of Teachers of Mathematics (2000). Principles and standarts for school mathematics. Reston, Va.: NCTM.Polya, G. (1945). How to Solve It: A New Aspect of Mathematical Method: A New Aspect of Mathematical Method. Princeton university press.Schoenfeld, A. H. (1985). Mathematical Problem Solving. Orlando, FL: Academic Press.Stacey, K., & MacGregor, M. (1999). Learning the algebraic method of solving problems. The Journal of Mathematical Behavior, 18(2), 149-167.Smith, J., & Thompson, P. (2007). Quantitative reasoning and the development of algebraic reaso¬ning. In J. Kaput & D. Carraher (Eds.), Algebra in the early grades (pp. 95-132). New York, NY: Lawrence Erlbaum Associates. Sowder, L. (1988). Children’s solutions of story problems. Journal of Mathematical Behavior, 7, 227-238.Şener, Z.T. & Bulut, N. (2015). 8. Sınıf öğrencilerinin matematik derslerinde problem çözme sürecinde karşılaştıkları güçlükler. Gazi Üniversitesi Gazi Eğitim Fakültesi Dergisi, 35(3), 637-661.Steffe, L. P. (1991). The constructivist teaching experiment: Illustrations and implications. In E. vonGlasersfeld (Ed.), Radical constructivism in mathematics education (pp. 177-194). Boston,MA: Kluwer Academic Press.Tambychik, T. & Meerah, T.S.M. (2010). Students’ Difficulties in Mathematics Problem-Solving: What do they Say? Procedia Social and Behavioral Sciences, 8, 142–151. Thompson P. W. (1988). Quantitative concepts as a foundation for algebraic reasoning: sufficiency, necessity, and cognitive obstacles. M. Behr, C. Lacampagne & M. Wheeler (Eds.), Proceedings of the Annual Conference of the International Group for the Psychology of Mathematics Education, 163-170. Thompson P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25 (3), 165-208.Thompson P. W. (2011). Quantitative reasoning and mathematical modeling. L. L. Hatfield, S. Chamberlain & S. Belbase (Eds.), New Perspectives and Directions for Collabrative Research in Mathematics Education içinde (s. 33-57). Laramie, WY: University of Wyoming.Türnüklü, E.B. & Yeşildere, S. (2005). Problem, problem çözme ve eleştirel düşünme. GÜ, Gazi Eğitim Fakültesi Dergisi, 25(3), 107-123. Yıldırım, A. & Simsek, H. (2011). Qualitative research methods in the social sciences. Ankara: Seckin Publisher.Verschaffel, L., De Corte, E., & Vierstraete, H. (1999). Upper elementary school pupils' difficulties in modeling and solving nonstandard additive word problems involving ordinal numbers. Journal for Research in Mathematics Education, 265-285.Verschaffel, L., De Corte, E., Lasure, S., Vaerenbergh, G. V., Bogaerts, H., ve Ratinckx, E. (1999). Learning to solve mathematical application problems: a design experiment with fifth graders. Mathematical Thinking and Learning, 1(3), 195-229.Van Amerom, B. (2002). Reinvention of Early Algebra: Developmental Research on the Transition from Arithmetic to Algebra, Doktora Tezi, University of Utrecht, The Netherlands.Van de Walle,. J.,A., Karp, K. S., Bay-Williams, J.M. (2016). İlkokul ve Ortaokul Matematiği, Gelişimsel Yaklaşımla Öğretim. (Çev. Edit. Soner Durmuş), 7. Baskıdan Çeviri. Nobel Yayınları, Ankara.