Research Article
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Development of a Learning Trajectory for Early Proportional Reasoning: A Design Research

Year 2021, Volume 17, Issue 3, 433 - 461, 22.12.2021
https://doi.org/10.17860/mersinefd.949263

Abstract

Proportional reasoning is essential for the learning of numerous mathematical concepts. Proportional reasoning is based on linking composite units and iterating linked composites, rather than repeated addition. Those skills are referred to as early proportional reasoning skills in this study. In a three-year-long design research project, a hypothetical learning trajectory for proportional reasoning was developed, tested, and revised. To this purpose, a classroom design research study aimed at developing a hypothetical learning trajectory and related instructional sequence was conducted. This trajectory includes the big ideas of proportional reasoning, the tools and imageries to support those ideas, and possible mathematical discourse and gesture use. In this particular study we present the first phase of this trajectory, which is related to the development of early proportional reasoning. The data of this study consist of video recordings of classroom sessions conducted in a public school in Ankara and audio recordings of student/teacher and design team meetings. The classroom sessions were analyzed by constructing related argumentation schemes to evaluate meaningful learning of students. The findings of the study showed that the learning trajectory developed to support the big ideas of early proportional reasoning has substantial potential for the development of early proportional reasoning.

References

  • Arıcan, M. (2019). A diagnostic assessment to middle school students’ proportional reasoning. Turkish Journal of Education, 8(4), 237-257.
  • Atabaş, Ş., & Öner, D. (2017). An examination of Turkish middle school students’ proportional reasoning. Boğaziçi Üniversitesi Eğitim Dergisi, 33(1), 63-85.
  • Battista, M. ve Borrow, C. V. A. (1995). A proposed constructive itinerary from iterating composite units to ratio and proportion concepts. Paper presented at the Annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education: Columbus, OH.
  • Beswick, K. (2011). Make your own paint chart: a realistic context for developing proportional reasoning with ratios. Australian Mathematics Teacher, 67(1), 6-11.
  • Cobb, P., Confrey, J., DiSessa, A., Lehrer, R. ve Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.
  • Cramer, K. ve Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342-346.
  • Çelik, A. ve Yetkin-Özdemir, E. (2011). İlköğretim öğrencilerinin orantısal akıl yürütme becerileri ile problem kurma becerileri arasındaki ilişki. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 30(30), 1-11.
  • Duatepe, A., Akkuş-Çıkla, O. ve Kayhan, M. (2005). Orantısal akıl yürütme gerektiren sorularda öğrencilerin kullandıkları çözüm stratejilerinin soru türlerine göre değişiminin incelenmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 28, 73-81.
  • Fischbein, E., Deri, M., Nello, M. S. ve Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 3-17.
  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.
  • Freudenthal, H. (1978). Weeding and sowing: Preface to a science of mathematical education. Dordrecht: Reidel.
  • Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer Academic Publishers.
  • Fuson, K. C. ve Abrahamson, D. (2005). Understanding ratio and proportion as an example of the apprehending zone and conceptual-phase problem-solving models. In J. Campbell (Ed.), Handbook of mathematical cognition (ss. 213–234). New York: Psychology Press.
  • Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht: Freudenthal Institute.
  • Gravemeijer, K. ve Cobb, P. (2006). Design research from a learning design perspective. In J. Van den Akker, K. Gravemeijer, S. McKenney ve N. Nieveen (Eds.), Educational design research (ss. 17–51). London, England: Routledge.
  • Greenes, C. ve Fendell, C. (2000). Groundworks: Algebraic puzzles and problems. Chicago, IL: Creative Publications.
  • Harel, G. ve Confrey, J. (Eds.). (1994). The development of multiplicative reasoning in the learning of mathematics. Albany: State University of New York Press.
  • Hart, K. (1988). Ratio and proportion. In J. Hiebert, & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 198-219). Reston, VA: Lawrence Erlbaum & National Council of Teachers of Mathematics, Inc.
  • Kahraman, H., Kul, E., & İskenderoğlu, T. A. (2019). 7. ve 8. sınıf öğrencilerinin nicel karşılaştırma içeren orantısal akıl yürütme problemlerinde kullandıkları stratejiler. Türk Bilgisayar ve Matematik Eğitimi Dergisi, 10(1), 195-216.
  • Kaplan, A., İşleyen, T. ve Öztürk, M. (2011). 6. sınıf oran orantı konusundaki kavram yanılgıları. Kastamonu Eğitim Dergisi, 19(3), 953-968.
  • Kaput, J. J. ve West, M. M. (1994). Missing-value proportional problems: factors affecting informal reasoning patterns. In G. Harel & J. Confrey, (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235-287). Albany: State University of New York Press.
  • Karplus, R., Pulos, S. ve Stage, E. K. (1983). Early adolescents’ proportional reasoning on “rate” problems. Educational Studies in Mathematics, 14(3), 219-233.
  • Lamon, S. J. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey, (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89-120). Albany: State University of New York Press.
  • Lamon, S. J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 167-198). Albany: State University of New York Press.
  • Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. New York, NY: Routledge.
  • Lesh, R., Post, T. ve Behr, M. (1988). Proportional reasoning. In J. Hiebert ve M.Behr (Eds.), Number concepts and operations in the middle grades (vol. 2, ss. 93-118). Reston, VA: Lawrence Erlbaum.
  • Millî Eğitim Bakanlığı [MEB] (2018). Matematik dersi öğretim programi (ilkokul ve ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar). Ankara: MEB.
  • Misailidou, C. ve Williams, J. (2003). Diagnostic assessment of children’s proportional reasoning. The Journal of Mathematical Behavior, 22(3), 335-368.
  • Özgün-Koca, S. A., & Altay, M. K. (2009). An investigation of proportional reasoning skills of middle school students. Investigations in Mathematics Learning, 2(1), 26-48.
  • Park, J. H. ve Nunes, T. (2001). The development of the concept of multiplication. Cognitive Development, 16(3), 763-773.
  • Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. The Journal of Mathematical Behavior, 23(3), 301-323.
  • Resnick, L. B. ve Singer, J. A. (1993). Protoquantitative origins of ratio reasoning. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 107-130). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
  • Spinillo A. G. ve Bryant P. E. (1999). Proportional reasoning in young children: part–part comparisons about continuous and discontinuous quantity. Mathematical Cognition, 5(2), 181–197.
  • Steffe, L. P. (1994). Children’s multiplying schemes: An overview. In G. Harel ve J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (ss. 3-39). Albany, NY: SUNY Press.
  • Stephan, M. McManus, G., Smith, J. ve Dickey (n.d.). Ratio and rates. 21 Haziran 2019 tarihinde, https://cstem.uncc.edu/sites/cstem.uncc.edu/files/media/Ratio%20T%20Manual.pdf adresinden alınmıştır.
  • Toluk-Uçar, Z. ve Bozkuş, F. (2016). İlkokul ve ortaokul öğrencilerinin orantısal durumları orantısal olmayan durumlardan ayırt edebilme becerileri. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi (KEFAD), 17(3), 281-299.
  • Toulmin, S. E. (1958). The uses of argument. Cambridge, UK: Cambridge University Press.
  • Tourniaire, F. (1986). Proportions in elementary school. Educational Studies in Mathematics, 17(4), 401-412.
  • Tourniaire, F. ve Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.
  • Van Dooren, W., De Bock, D. ve Verschaffel, L. (2010). From addition to multiplication… and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360-381.
  • Vergnaud, G. (1980). Didactics and acquisition of "multiplicative structures" in secondary schools. In W. F. Archenhold, R. H. Driver, A. Orton, & C. Wood-Robinson (Eds.), Cognitive development research in science and mathematics (pp. 190-200). Leeds, UK: University of Leeds.

Erken Orantısal Akıl Yürütmeye Yönelik Öğrenme Rotasının Geliştirilmesi: Bir Tasarı Araştırması

Year 2021, Volume 17, Issue 3, 433 - 461, 22.12.2021
https://doi.org/10.17860/mersinefd.949263

Abstract

Orantısal akıl yürütme birçok matematiksel kavramın öğrenilmesi için önem taşır. Orantısal akıl yürütmenin temeli tekrarlı toplamadan ziyade birleşik birimleri bağlama ve bağlı birleşik birimleri yineleme becerilerine dayanmaktadır. Bu becerilere dayanan süreç gruplama yoluyla birleşik birimlerin oluşturulması, bu birleşik birimlerin bağlanması ve bağlı birleşik birimlerin yinelenmesini içerir. Bu çalışmada bu beceriler erken orantısal akıl yürütme becerileri olarak adlandırılmaktadır. Üç yıllık bir tasarı araştırması kapsamında orantısal akıl yürütmenin gelişimi için varsayıma dayalı öğrenme rotası geliştirilmiş, test edilmiş ve düzenlenmiştir. Bu amaç için, Gerçekçi Matematik Eğitimi Teorisi’ne dayanan bir öğrenme rotası ve etkinlik dizisinin geliştirilmesine yönelik bir sınıf içi tasarı araştırması yürütülmüştür. Bu öğrenme rotası, orantısal akıl yürütmeye yönelik anahtar öğrenmeleri, bu anahtar öğrenmeler için kullanılacak araç ve imgeler ile muhtemel sınıf içi söylemler ve jest kullanımlarını içermektedir. Mevcut çalışmada, bu öğrenme rotasının başlangıç kısmını oluşturan erken orantısal akıl yürütmenin geliştirilmesinin hedeflendiği kısmı sınıf içi uygulama örnekleriyle sunulmuştur. Çalışmanın verilerini Ankara’da bir devlet okulundaki sınıf içi öğretimin video kayıtları ve öğretmen/öğrenci ve tasarı ekibi görüşmelerinin sesli kayıtları oluşturmaktadır. Sınıf içi öğretim, öğrencilerin anlamlı öğrenmelerinin ölçülmesi amacıyla argümantasyon şemaları oluşturularak analiz edilmiştir. Çalışmanın bulgularına göre hedeflenen anahtar öğrenmelere yönelik geliştirilen öğrenme rotasının erken orantısal akıl yürütmenin geliştirilmesi için önemli bir potansiyele sahip olduğu görülmektedir.

References

  • Arıcan, M. (2019). A diagnostic assessment to middle school students’ proportional reasoning. Turkish Journal of Education, 8(4), 237-257.
  • Atabaş, Ş., & Öner, D. (2017). An examination of Turkish middle school students’ proportional reasoning. Boğaziçi Üniversitesi Eğitim Dergisi, 33(1), 63-85.
  • Battista, M. ve Borrow, C. V. A. (1995). A proposed constructive itinerary from iterating composite units to ratio and proportion concepts. Paper presented at the Annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education: Columbus, OH.
  • Beswick, K. (2011). Make your own paint chart: a realistic context for developing proportional reasoning with ratios. Australian Mathematics Teacher, 67(1), 6-11.
  • Cobb, P., Confrey, J., DiSessa, A., Lehrer, R. ve Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.
  • Cramer, K. ve Post, T. (1993). Making connections: A case for proportionality. Arithmetic Teacher, 60(6), 342-346.
  • Çelik, A. ve Yetkin-Özdemir, E. (2011). İlköğretim öğrencilerinin orantısal akıl yürütme becerileri ile problem kurma becerileri arasındaki ilişki. Pamukkale Üniversitesi Eğitim Fakültesi Dergisi, 30(30), 1-11.
  • Duatepe, A., Akkuş-Çıkla, O. ve Kayhan, M. (2005). Orantısal akıl yürütme gerektiren sorularda öğrencilerin kullandıkları çözüm stratejilerinin soru türlerine göre değişiminin incelenmesi. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 28, 73-81.
  • Fischbein, E., Deri, M., Nello, M. S. ve Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 3-17.
  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.
  • Freudenthal, H. (1978). Weeding and sowing: Preface to a science of mathematical education. Dordrecht: Reidel.
  • Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer Academic Publishers.
  • Fuson, K. C. ve Abrahamson, D. (2005). Understanding ratio and proportion as an example of the apprehending zone and conceptual-phase problem-solving models. In J. Campbell (Ed.), Handbook of mathematical cognition (ss. 213–234). New York: Psychology Press.
  • Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht: Freudenthal Institute.
  • Gravemeijer, K. ve Cobb, P. (2006). Design research from a learning design perspective. In J. Van den Akker, K. Gravemeijer, S. McKenney ve N. Nieveen (Eds.), Educational design research (ss. 17–51). London, England: Routledge.
  • Greenes, C. ve Fendell, C. (2000). Groundworks: Algebraic puzzles and problems. Chicago, IL: Creative Publications.
  • Harel, G. ve Confrey, J. (Eds.). (1994). The development of multiplicative reasoning in the learning of mathematics. Albany: State University of New York Press.
  • Hart, K. (1988). Ratio and proportion. In J. Hiebert, & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 198-219). Reston, VA: Lawrence Erlbaum & National Council of Teachers of Mathematics, Inc.
  • Kahraman, H., Kul, E., & İskenderoğlu, T. A. (2019). 7. ve 8. sınıf öğrencilerinin nicel karşılaştırma içeren orantısal akıl yürütme problemlerinde kullandıkları stratejiler. Türk Bilgisayar ve Matematik Eğitimi Dergisi, 10(1), 195-216.
  • Kaplan, A., İşleyen, T. ve Öztürk, M. (2011). 6. sınıf oran orantı konusundaki kavram yanılgıları. Kastamonu Eğitim Dergisi, 19(3), 953-968.
  • Kaput, J. J. ve West, M. M. (1994). Missing-value proportional problems: factors affecting informal reasoning patterns. In G. Harel & J. Confrey, (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 235-287). Albany: State University of New York Press.
  • Karplus, R., Pulos, S. ve Stage, E. K. (1983). Early adolescents’ proportional reasoning on “rate” problems. Educational Studies in Mathematics, 14(3), 219-233.
  • Lamon, S. J. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey, (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89-120). Albany: State University of New York Press.
  • Lamon, S. J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J. T. Sowder & B. P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades (pp. 167-198). Albany: State University of New York Press.
  • Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. New York, NY: Routledge.
  • Lesh, R., Post, T. ve Behr, M. (1988). Proportional reasoning. In J. Hiebert ve M.Behr (Eds.), Number concepts and operations in the middle grades (vol. 2, ss. 93-118). Reston, VA: Lawrence Erlbaum.
  • Millî Eğitim Bakanlığı [MEB] (2018). Matematik dersi öğretim programi (ilkokul ve ortaokul 1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar). Ankara: MEB.
  • Misailidou, C. ve Williams, J. (2003). Diagnostic assessment of children’s proportional reasoning. The Journal of Mathematical Behavior, 22(3), 335-368.
  • Özgün-Koca, S. A., & Altay, M. K. (2009). An investigation of proportional reasoning skills of middle school students. Investigations in Mathematics Learning, 2(1), 26-48.
  • Park, J. H. ve Nunes, T. (2001). The development of the concept of multiplication. Cognitive Development, 16(3), 763-773.
  • Rasmussen, C., Stephan, M., & Allen, K. (2004). Classroom mathematical practices and gesturing. The Journal of Mathematical Behavior, 23(3), 301-323.
  • Resnick, L. B. ve Singer, J. A. (1993). Protoquantitative origins of ratio reasoning. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 107-130). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
  • Spinillo A. G. ve Bryant P. E. (1999). Proportional reasoning in young children: part–part comparisons about continuous and discontinuous quantity. Mathematical Cognition, 5(2), 181–197.
  • Steffe, L. P. (1994). Children’s multiplying schemes: An overview. In G. Harel ve J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (ss. 3-39). Albany, NY: SUNY Press.
  • Stephan, M. McManus, G., Smith, J. ve Dickey (n.d.). Ratio and rates. 21 Haziran 2019 tarihinde, https://cstem.uncc.edu/sites/cstem.uncc.edu/files/media/Ratio%20T%20Manual.pdf adresinden alınmıştır.
  • Toluk-Uçar, Z. ve Bozkuş, F. (2016). İlkokul ve ortaokul öğrencilerinin orantısal durumları orantısal olmayan durumlardan ayırt edebilme becerileri. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi (KEFAD), 17(3), 281-299.
  • Toulmin, S. E. (1958). The uses of argument. Cambridge, UK: Cambridge University Press.
  • Tourniaire, F. (1986). Proportions in elementary school. Educational Studies in Mathematics, 17(4), 401-412.
  • Tourniaire, F. ve Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.
  • Van Dooren, W., De Bock, D. ve Verschaffel, L. (2010). From addition to multiplication… and back: The development of students’ additive and multiplicative reasoning skills. Cognition and Instruction, 28(3), 360-381.
  • Vergnaud, G. (1980). Didactics and acquisition of "multiplicative structures" in secondary schools. In W. F. Archenhold, R. H. Driver, A. Orton, & C. Wood-Robinson (Eds.), Cognitive development research in science and mathematics (pp. 190-200). Leeds, UK: University of Leeds.

Details

Primary Language Turkish
Subjects Education and Educational Research
Journal Section Makaleler
Authors

Rukiye AYAN CİVAK (Primary Author)
İZMİR DEMOKRASİ ÜNİVERSİTESİ
0000-0002-1278-0257
Türkiye


Mine IŞIKSAL
ORTA DOĞU TEKNİK ÜNİVERSİTESİ
0000-0001-7619-1390
Türkiye


Seçil YEMEN KARPUZCU
KÜTAHYA DUMLUPINAR ÜNİVERSİTESİ
0000-0002-2150-000X
Türkiye

Supporting Institution Türkiye Bilimsel ve Teknolojik Araştırma Kurumu (TÜBİTAK)
Project Number 217K430
Publication Date December 22, 2021
Published in Issue Year 2021, Volume 17, Issue 3

Cite

APA Ayan Civak, R. , Işıksal, M. & Yemen Karpuzcu, S. (2021). Erken Orantısal Akıl Yürütmeye Yönelik Öğrenme Rotasının Geliştirilmesi: Bir Tasarı Araştırması . Mersin Üniversitesi Eğitim Fakültesi Dergisi , 17 (3) , 433-461 . DOI: 10.17860/mersinefd.949263

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