Araştırma Makalesi

Bu çalışmanın amacı 9. sınıf öğrencilerinin cebirsel kesirli ifadeleri içeren denklemler bağlamında işlem esnekliklerinin gelişiminin incelenmesidir. Bu doğrultuda işlem esnekliği kavramı ele alınmıştır. Araştırmanın yöntemini nitel araştırma yaklaşımları arasında ele alınan “öğretim deneyi” modeli oluşturmaktadır. Ayrıca bunu desteklemek için “klinik görüşmeler” ve “doküman incelemesi” yapılmıştır. Araştırmanın katılımcılarını dört dokuzuncu sınıf öğrencisi oluşturmaktadır. Katılımcıların işlem esnekliklerini geliştirmek amacıyla kısmi çözümlü örnekler kullanılmış ve nedenini belirtmek koşuluyla çözümler arasında tercihlerde bulunmaları istenmiştir. Elde edilen veriler (video kayıtları, alan notları ve uygulama kâğıtları) betimsel ve içerik çözümlemesine tabi tutulmuştur. Bulgular kısmi çözümlü örneklerin ve çözüm tercihlerinin katılımcıların işlem esnekliklerini olumlu yönde etkilediğini göstermiştir. Ayrıca uygulamadan yaklaşık 3 ay sonra gerçekleştirilen kalıcılık klinik görüşme sonuçları bu etkinin kısmi bir azalma olmakla birlikte kalıcı olduğunu göstermektedir.

İşlem esnekliği cebirsel kesirli ifadeler Öğretim deneyi dokuzuncu sınıf öğrencileri

Anadolu Üniversitesi BAP Komisyonu

1603E099 no.lu proje kapsamında desteklenmiştir.

- Alibali, M. W., Phillips, K. M. O., & Fischer, A. D. (2009). Learning new problem solving strategies leads to changes in problem representation. Cognitive Development, 24, 89–101. doi:10.1016/j.cogdev. 2008.12.005
- Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1–33). Mahwah, NJ: Erlbaum
- Baroody, A. J., Feil, Y., and Johnson, A. R. (2007). An Alternative Reconceptualization of Procedural and Conceptual Knowledge. Journal for Research in Mathematics Education, 38, 115–131.
- Beilock, S. L., & DeCaro, M. S. (2007). From poor performance to success under stress: Working memory, strategy selection, and mathematical problem solving under pressure. Journal of Experimental Psychology: Learning, Memory, and Cognition, 33, 983–998.
- Berk, D., Taber, S., Gorowara, C., & Poetzl, C. (2009). Developing prospective elementary teachers’ flexibility in the domain of proportional reasoning. Mathematical Thinking and Learning, 11, 113–135. Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93, 627–638.
- Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural knowledge. In J. Heibert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp.113–132). Hillside, NJ: Erlbaum
- De Jong, T., & Ferguson-Hessler, M.G.M. (1996). Types and qualities of knowledge. Educational Psychologist, 31, 105–113
- Durkin,K., Rittle-Johnson, B., Star, J. R & Loehr, A., (2021). Comparing and Discussing Multiple Strategies: An Approach to Improving Algebra Instruction, The Journal of Experimental Education, DOI: 10.1080/00220973.2021.1903377-https://doi.org/10.1080/00220973.2021.1903377
- Haapasalo, L., and Kadjievich, D.(2000).Two types of mathematical knowledge and their relation. Journal für Mathematik Didaktik, 21(2),139-157.
- Hattie, J. (2009). Visible learning: A synthesis of over 800 metaanalysis relating to achievement. New York, NY: Routledge.
- Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM Mathematics Education, 41, 535–540. doi:10.1007/s11858-009- 0214-4
- Hiebert, J. and 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.
- Hilbert, T. S., Schworm, S., & Renkl, A. (2004). Learning from worked-out examples: The transition from instructional explanations to self-explanation prompts. In P. Gerjets, J. Elen, R. Joiner, & P. Kirschner (Eds.), Instructional design for effective and enjoyable computer-supported learning (pp. 184–192). Tübingen: Knowledge Media Research Center.
- Lewis, C. C. (1981). The effects of parental firm control: A reinterpretation of findings. Psychological Bulletin, 90, 547/563.
- Lynch K. and Star, J., R. (2014). Views of struggling students on ınstruction ıncorporating multiple strategies in algebraı: An exploratory study. Journal for Research in Mathematics Education, Vol. 45, No. 1 (January 2014), pp.6-18.
- McNeil, N. M., & Alibali, M. W. (2004). You’ll see what you mean: Students encode equations based on their knowledge of arithmetic. Cognitive Science, 28, 451–466. doi:10.1016/j.cogsci.2003.11.002
- McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76, 883–899
- Mwangi, W., & Sweller, J. (1998). Learning to solve compare word problems: The effect of example format and generating self-explanations. Cognition and Instruction, 16, 173–199. https://doi.org/10.1207/s1532690xci1602.
- National Center for Education Statistics. (2020). NAEP report card: 2019 NAEP mathematics assessment. The Nation’s Report Card. https://www.nationsreportcard.gov/highlights/mathematics/2019
- Paas, F., Renkl, A. & Sweller, J. (2003). Cognitive load theory and instructional design: Recent developments, Educational Psychologist 38: 1–4.
- Patton, M.Q. (1997). How to use qualitative methods in evaluation. Newbury park, CA: SAGE Publications.
- Post, T.R. and Cramer, K.A. (1989). “Knowledge, representation, and quantitative thinking”. In Knowledge Base for the Beginning Teacher, Edited by: Reynolds, MC. 221–232. New York, NY: Pergamon. [Google Scholar]
- Renkl, A. (2014). Learning From Worked Examples: How to Prepare Students for Meaningful Problem Solving Applying Science of Learning in Education: Infusing Psychological Science into the Curriculum. Washington, DC: American Psychological Association.
- Retnowati, E., Ayres, P., & Sweller, J. (2010). Worked example effects in individual and group work settings. Educational Psychology, 30, 349–367. https://doi.org/10.1080/01443411003659960.
- Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.
- Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529–544. https://doi.org/10.1037/a0014224 [Crossref], [Web of Science ®]
- Rittle-Johnson, B. Star, J.,R. and Durkin, K. (2012). Developing procedural flexibility: Are novices prepared to learn from comparing procedures? British Journal of Educational Psychology (2012), 82, 436–455.
- Rittle-Johnson, B., Schneider, M and Star, J.,R. (2015). Not a one-way street: bidirectional relations between procedural and conceptual knowledge of mathematics. Educ Psychol Rev. (2015) 27:587–597. DOI 10.1007/s10648-015- 9302-x.
- Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47, 1525–1538. doi:10.1037/a0024997
- Siegler, R. S. (1994). Cognitive variability: A key to understanding cognitive development. Current Directions in Psychological Science, 3, 1–5.
- Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.
- Skemp, R. R. (1978). Relational and İnstrumental Understanding. Arithmetic Teacher, 26, 9-15.
- Star, J. R. (2001). Re-conceptualizing Procedural Knowledge: Innovation and Flexibility in Equation Solving. Unpublished doctoral dissertation, University of Michigan, Ann Arbor.
- Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.
- Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132–135.
- Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102, 408– 426. https://doi.org/10.1016/j.jecp.2008.11.004.
- Star J. R. & Stylianides, G. J. (2013) Procedural and conceptual knowledge: exploring the gap between knowledge type and knowledge quality. Can J Sci Math Technol Educ 13, 169– 181.
- Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280–300.
- Steffe, L. P. (1991). The constructivist teaching experiment: Illustration sand implications. In E. Von Glasers feld (Ed.), Radical constructivism in mathematics education (pp. 177-194). Boston, MA: Kluwer Academic Press. 173
- Sweller, J., van Merrienboer, J. J. G., & Paas, F. (1998). Cognitive architecture and instructional design.Educational Psychology Review, 10, 251–296. https://doi.org/10.1023/A:1022193728205.
- Verschaffel L, Greer B, De Corte E (2007). Whole number concepts and operations. In: Lester FK (ed) Second handbook of research on mathematics teaching and learning. Information Age, Greenwich, pp 557–628
- Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335–359. doi:10.1007/BF03174765
- Van Loon-Hillen, N., van Gog, T., & Brand-Gruwel, S. (2012). Effects of worked examples in a primary school mathematics curriculum. Interactive Learning Environments, 20, 89–99.
- Yanık, H. B. (2016). Kavramsal ve işlemsel anlama. In E. Bingölbali, S. Arslan, & Zembat, İ. Ö. (Eds.), Matematik eğitiminde teoriler (sf. 101-116). Ankara, Turkey: Pegem Akademi.
- Yıldırım, A. ve Şimşek, H. (2008). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin yayıncılık

The aim of this study is to examine the development of 9th grade students' operation flexibility in the context of equations containing algebraic fractional expressions. In this direction, the concept of transaction flexibility is discussed. The method of the research is the "teaching experiment" model, which is considered among the qualitative research approaches. In addition, "clinical interviews" and "document review" were conducted to support this. The participants of the research are four ninth grade students. Partially solved examples were used in order to improve the transaction flexibility of the participants and were asked to make choices among the solutions provided that they indicate the reason. The data obtained (video recordings, field notes and practice papers) were subjected to descriptive and content analysis. Findings showed that partially solved examples and solution preferences positively affected the participants' transaction flexibility. In addition, the results of the permanence clinical interview performed approximately 3 months after the application show that this effect is permanent, although there is a partial decrease.

algebraic fractional expressions transaction flexibility ninth grade students teaching experiment

1603E099 no.lu proje kapsamında desteklenmiştir.

- Alibali, M. W., Phillips, K. M. O., & Fischer, A. D. (2009). Learning new problem solving strategies leads to changes in problem representation. Cognitive Development, 24, 89–101. doi:10.1016/j.cogdev. 2008.12.005
- Baroody, A. J. (2003). The development of adaptive expertise and flexibility: The integration of conceptual and procedural knowledge. In A. J. Baroody & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1–33). Mahwah, NJ: Erlbaum
- Baroody, A. J., Feil, Y., and Johnson, A. R. (2007). An Alternative Reconceptualization of Procedural and Conceptual Knowledge. Journal for Research in Mathematics Education, 38, 115–131.
- Beilock, S. L., & DeCaro, M. S. (2007). From poor performance to success under stress: Working memory, strategy selection, and mathematical problem solving under pressure. Journal of Experimental Psychology: Learning, Memory, and Cognition, 33, 983–998.
- Berk, D., Taber, S., Gorowara, C., & Poetzl, C. (2009). Developing prospective elementary teachers’ flexibility in the domain of proportional reasoning. Mathematical Thinking and Learning, 11, 113–135. Blöte, A. W., Van der Burg, E., & Klein, A. S. (2001). Students’ flexibility in solving two-digit addition and subtraction problems: Instruction effects. Journal of Educational Psychology, 93, 627–638.
- Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural knowledge. In J. Heibert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp.113–132). Hillside, NJ: Erlbaum
- De Jong, T., & Ferguson-Hessler, M.G.M. (1996). Types and qualities of knowledge. Educational Psychologist, 31, 105–113
- Durkin,K., Rittle-Johnson, B., Star, J. R & Loehr, A., (2021). Comparing and Discussing Multiple Strategies: An Approach to Improving Algebra Instruction, The Journal of Experimental Education, DOI: 10.1080/00220973.2021.1903377-https://doi.org/10.1080/00220973.2021.1903377
- Haapasalo, L., and Kadjievich, D.(2000).Two types of mathematical knowledge and their relation. Journal für Mathematik Didaktik, 21(2),139-157.
- Hattie, J. (2009). Visible learning: A synthesis of over 800 metaanalysis relating to achievement. New York, NY: Routledge.
- Heinze, A., Star, J. R., & Verschaffel, L. (2009). Flexible and adaptive use of strategies and representations in mathematics education. ZDM Mathematics Education, 41, 535–540. doi:10.1007/s11858-009- 0214-4
- Hiebert, J. and 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.
- Hilbert, T. S., Schworm, S., & Renkl, A. (2004). Learning from worked-out examples: The transition from instructional explanations to self-explanation prompts. In P. Gerjets, J. Elen, R. Joiner, & P. Kirschner (Eds.), Instructional design for effective and enjoyable computer-supported learning (pp. 184–192). Tübingen: Knowledge Media Research Center.
- Lewis, C. C. (1981). The effects of parental firm control: A reinterpretation of findings. Psychological Bulletin, 90, 547/563.
- Lynch K. and Star, J., R. (2014). Views of struggling students on ınstruction ıncorporating multiple strategies in algebraı: An exploratory study. Journal for Research in Mathematics Education, Vol. 45, No. 1 (January 2014), pp.6-18.
- McNeil, N. M., & Alibali, M. W. (2004). You’ll see what you mean: Students encode equations based on their knowledge of arithmetic. Cognitive Science, 28, 451–466. doi:10.1016/j.cogsci.2003.11.002
- McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76, 883–899
- Mwangi, W., & Sweller, J. (1998). Learning to solve compare word problems: The effect of example format and generating self-explanations. Cognition and Instruction, 16, 173–199. https://doi.org/10.1207/s1532690xci1602.
- National Center for Education Statistics. (2020). NAEP report card: 2019 NAEP mathematics assessment. The Nation’s Report Card. https://www.nationsreportcard.gov/highlights/mathematics/2019
- Paas, F., Renkl, A. & Sweller, J. (2003). Cognitive load theory and instructional design: Recent developments, Educational Psychologist 38: 1–4.
- Patton, M.Q. (1997). How to use qualitative methods in evaluation. Newbury park, CA: SAGE Publications.
- Post, T.R. and Cramer, K.A. (1989). “Knowledge, representation, and quantitative thinking”. In Knowledge Base for the Beginning Teacher, Edited by: Reynolds, MC. 221–232. New York, NY: Pergamon. [Google Scholar]
- Renkl, A. (2014). Learning From Worked Examples: How to Prepare Students for Meaningful Problem Solving Applying Science of Learning in Education: Infusing Psychological Science into the Curriculum. Washington, DC: American Psychological Association.
- Retnowati, E., Ayres, P., & Sweller, J. (2010). Worked example effects in individual and group work settings. Educational Psychology, 30, 349–367. https://doi.org/10.1080/01443411003659960.
- Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.
- Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529–544. https://doi.org/10.1037/a0014224 [Crossref], [Web of Science ®]
- Rittle-Johnson, B. Star, J.,R. and Durkin, K. (2012). Developing procedural flexibility: Are novices prepared to learn from comparing procedures? British Journal of Educational Psychology (2012), 82, 436–455.
- Rittle-Johnson, B., Schneider, M and Star, J.,R. (2015). Not a one-way street: bidirectional relations between procedural and conceptual knowledge of mathematics. Educ Psychol Rev. (2015) 27:587–597. DOI 10.1007/s10648-015- 9302-x.
- Schneider, M., Rittle-Johnson, B., & Star, J. R. (2011). Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge. Developmental Psychology, 47, 1525–1538. doi:10.1037/a0024997
- Siegler, R. S. (1994). Cognitive variability: A key to understanding cognitive development. Current Directions in Psychological Science, 3, 1–5.
- Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Strawhun, B. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom. Journal of Mathematical Behavior, 24, 287–301.
- Skemp, R. R. (1978). Relational and İnstrumental Understanding. Arithmetic Teacher, 26, 9-15.
- Star, J. R. (2001). Re-conceptualizing Procedural Knowledge: Innovation and Flexibility in Equation Solving. Unpublished doctoral dissertation, University of Michigan, Ann Arbor.
- Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36, 404–411.
- Star, J. R. (2007). Foregrounding procedural knowledge. Journal for Research in Mathematics Education, 38(2), 132–135.
- Star, J. R., & Rittle-Johnson, B. (2009). It pays to compare: An experimental study on computational estimation. Journal of Experimental Child Psychology, 102, 408– 426. https://doi.org/10.1016/j.jecp.2008.11.004.
- Star J. R. & Stylianides, G. J. (2013) Procedural and conceptual knowledge: exploring the gap between knowledge type and knowledge quality. Can J Sci Math Technol Educ 13, 169– 181.
- Star, J. R., & Seifert, C. (2006). The development of flexibility in equation solving. Contemporary Educational Psychology, 31, 280–300.
- Steffe, L. P. (1991). The constructivist teaching experiment: Illustration sand implications. In E. Von Glasers feld (Ed.), Radical constructivism in mathematics education (pp. 177-194). Boston, MA: Kluwer Academic Press. 173
- Sweller, J., van Merrienboer, J. J. G., & Paas, F. (1998). Cognitive architecture and instructional design.Educational Psychology Review, 10, 251–296. https://doi.org/10.1023/A:1022193728205.
- Verschaffel L, Greer B, De Corte E (2007). Whole number concepts and operations. In: Lester FK (ed) Second handbook of research on mathematics teaching and learning. Information Age, Greenwich, pp 557–628
- Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335–359. doi:10.1007/BF03174765
- Van Loon-Hillen, N., van Gog, T., & Brand-Gruwel, S. (2012). Effects of worked examples in a primary school mathematics curriculum. Interactive Learning Environments, 20, 89–99.
- Yanık, H. B. (2016). Kavramsal ve işlemsel anlama. In E. Bingölbali, S. Arslan, & Zembat, İ. Ö. (Eds.), Matematik eğitiminde teoriler (sf. 101-116). Ankara, Turkey: Pegem Akademi.
- Yıldırım, A. ve Şimşek, H. (2008). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin yayıncılık

Toplam 45 adet kaynakça vardır.

Birincil Dil | Türkçe |
---|---|

Konular | Alan Eğitimleri |

Bölüm | Araştırma Makalesi |

Yazarlar | |

Proje Numarası | 1603E099 no.lu proje kapsamında desteklenmiştir. |

Erken Görünüm Tarihi | 14 Aralık 2021 |

Yayımlanma Tarihi | 14 Aralık 2021 |

Yayımlandığı Sayı | Yıl 2021 Cilt: 8 Sayı: 4 |