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Türevin Değişim Oranı (Hızı) Yorumunun Kavramsal Analizi ve Matematik Ders Kitaplarının İncelenmesi

Year 2021, Volume: 11 Issue: 3, 545 - 568, 01.12.2021
https://doi.org/10.19126/suje.977200

Abstract

Bu çalışmada türevin değişim oranı yorumunun anlamına yönelik kavramsal analizi nicel muhakeme teorik çerçevesinden detaylı olarak açıklanmaktadır. Daha sonra, nicel muhakeme çerçevesinden elde edilen temel prensipler doğrultusunda Türkiye’de okutulan lise ders kitaplarında türev kavramın nasıl ele alındığı nitel araştırma yöntemlerinden doküman inceleme yöntemiyle analiz edilmiştir. Elde edilen bulgular, incelenen ders kitaplarında türev kavramının anlamının tartışıldığı gerçek hayat bağlam çeşitliliğinin yetersiz olduğu ve hatta kinematik bağlamıyla sınırlı olduğunu göstermektedir. Kullanılan farklı gerçek hayat bağlamlarında birimlerin doğru kullanıldığı fakat kullanılan niceliklerin birimleri ile türevin anlamı üzerine derinlemesine yorumlamaların yeterli olmadığı görülmektedir. İncelenen her üç ders kitabında da türev kavramı ve değişim oranı yorumunun doğru anlaşılması bakımından önemli görülen kovaryasyon ve nispi büyüklük fikirlerinin de yeterince vurgulanmadığı görülmüştür. Ayrıca, türev kavramına yönelik olarak ders kitaplarında güçlendirilmesi gereken yönler vurgulanmıştır. Elde edilen bulgular ışığında, türev kavramının öğretimine yönelik ders kitaplarında yer verilmesi gereken bağlam örnekleri ile bunların nispi büyüklük olarak yorumlaması ile ilgili tartışmalar yapılmıştır.

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Destekleyen kurum bulunmamaktadır.

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Herhangi bir proje kapsamında gerçekleştirilmemiştir.

References

  • Altun, M. H. (2018). Matematik 12. sınıf ders kitabı. Ankara: Tutku Yayıncılık.
  • Ärlebäck, J. B., Doerr, H. M., & O’Neil, A. M. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336.
  • Ärlebäck, J. B., & Doerr, H. M. (2018). Students’ interpretations and reasoning about phenomena with negative rates of change throughout a model development sequence. ZDM–Mathematics Education, 50, 187-200.
  • Bektaşlı, B., & Çakmakçı, G. (2011). Consistency of students’ ideas about the concept of rate across different contexts. Education and Science, 36 (162), 273-287.
  • Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22, 481–497.
  • Bezuidenhout, J. (1998). First year university students’ understanding of rate of change. International Journal of Mathematical Education in Science and Technology, 29(3), 389-399.
  • Byerley, C., & Thompson, P. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. The Journal of Mathematical Behavior, 48, 168-193.
  • Byerley, C. (2019). Calculus students’ fraction and measure schemes and implications for teaching rate of change functions conceptually. The Journal of Mathematical Behavior, 55, 100694.
  • Bingolbali, E. (2008). Türev kavramına ilişkin öğrenme zorlukları ve kavramsal anlama için öneriler. M. F. Özmantar, E. Bingolbali, & H. Akkoç (Eds.), Matematiksel kavram yanılgıları ve çözüm önerileri (pp. 223–255). Ankara: Pegem A.
  • Bingolbali, E. & Bingolbali, F. (2020). An examination of tasks in elementary school mathematics textbooks in terms of multiple outcomes and multiple solution methods. International Journal of Educational Studies in Mathematics, 7(4), 214-235.
  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352-378.
  • Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. L. Mayes, & L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 55–73). Laramie: University of Wyoming Press.
  • Confrey, J. & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 134-165.
  • Cooney, T. J., Beckman, S., & Lloyd, G. M. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics.
  • Doorman, L. M., & Gravemeijer, K. P. E. (2009). Emergent modeling: discrete graphs to support the understanding of change and velocity. ZDM Mathematics Education 41(1-2), 199-211. https://doi.org/10.1007/s11858-008-0130-z
  • Ellis, A. B., Ozgur, Z., Kulow, T., Williams, C. C., & Amidon, J. (2015). Quantifying exponential growth: Three conceptual shifts in coordinating multiplicative and additive growth. The Journal of Mathematical Behavior, 39, 135–155. doi:10.1016/j.jmathb.2015.06.004
  • Emin, A., Gerboğa, A., Güneş, G. & Kayacıer, M. (2020). Matematik 12. sınıf ders kitabı. Ankara: MEB yayınları.
  • Gökçek, T., & Açıkyıldız, G. (2016). Matematik öğretmeni adaylarının türev kavramıyla i̇lgili yaptıkları hatalar [Preservice mathematics teachers’ errors related to derivative]. Turkish Journal of Computer and Mathematics Education, 7(1), 112-141.
  • Hawson, G. (2013). The development of mathematics textbooks: historical reflections from a personal perspective. ZDM Mathematics Education, 45, 647-658.
  • Herbert, S., & Pierce, R. (2011). What is rate? Does context or representation matter? Mathematics Education Research Journal, 23(4), 455-477.
  • Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81(1), 85–101.
  • İncikabı, L. & Tjoe, H. (2013). A comparative analysis of ratio and proportion problems in Turkish and the U.S. middle school mathematics textbooks. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi (KEFAD), 14(1), 1-15.
  • Jones, S. R. (2017). An exploratory study on student understandings of derivatives in real-world, non-kinematics contexts. The Journal of Mathematical Behavior, 45, 95–110.
  • Kemancı, B., Büyükokutan, A., Çelik, S. & Kemancı, Z. (2020). Fen lisesi 12. sınıf matematik ders kitabı. Ankara: MEB yayınları.
  • Kertil, M. (2014). Pre-service elementary mathematics teachers' understanding of derivative through a model development unit (Unpublished doctoral dissertation). Middle East Technical University, Graduate School of Natural and Applied Sciences, Ankara.
  • Kertil, M., Erbaş, A. K., & ve Çetinkaya, B. (2017). İlköğretim matematik öğretmen adaylarının değişim oranı ile ilgili düşünme biçimlerinin bir modelleme etkinliği bağlamında incelenmesi Turkish Journal of Computer and Mathematics Education, 8(1), 188-217.
  • Kertil, M., Erbaş, A. K., & Çetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21, 207-233.
  • Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The case of quadratic functions. Mathematical Thinking and Learning, 14(2), 85–119.
  • MEB (2018). Ortaöğretim Matematik Dersi (9-12. Sınıflar) Öğretim Programı. Ankara: Milli Eğitim Bakanlığı Yayınları.
  • Mkhatshwa, T. P. (2016). Business calculus students’ reasoning about optimization problems: A study of quantitative reasoning in an economic context (Doctoral dissertation). New York, NY: Syracuse University.
  • Mkhatshwa, T. P. (2018). Business calculus students' interpretations of marginal change in economic contexts. In Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.), Proceedings of the 40th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 564-571). Greenville, SC: University of South Carolina & Clemson University.
  • Mkhatshwa, T., & Doerr, H. (2018). Undergraduate students’ quantitative reasoning in economic contexts. Mathematical Thinking and Learning, 20(2), 142-161. https://doi.org/10.1080/10986065.2018.1442642
  • Rowland, D. R., & Jovanoski, Z. (2004). Student interpretation of the terms in first-order ordinary differential equations in modeling contexts. International Journal of Mathematical Education in Science and Technology, 35(4), 505-516.
  • Schmidt, W. H. (2012). Measuring content through textbooks: The cumulative effect of middle-school tracking. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources: Mathematics curriculum materials and teacher development (pp. 143–160). Dordrecht: Springer.
  • Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research. International Journal of Computers for Mathematical Learning, 7(2), 167-215.
  • Teuscher, D. & Reys, R. E. (2012). Rate of change: AP calculus students' understandings and misconceptions after completing different curricular paths. School Science and Mathematics, 112(6), 359–376. https://doi.org/10.1111/j.1949-8594.2012.00150.x
  • Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229-274.
  • Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education (Vol. 1, pp. 33-57). Laramie, WY: University of Wyoming.
  • Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics. Thompson, P. W., Hatfield, N. J., Yoon, H., Joshua, S., & Byerley, C. (2017). Covariational reasoning among U.S. and South Korean secondary mathematics teachers. The Journal of Mathematical Behavior, 48, 95–111.
  • Weber, E. & Dorko, A. (2014). Students’ and experts’ schemes for rate of change and its representations. The Journal of Mathematical Behavior, 34, 14-32.
  • Wilhelm, J. A. & Confrey, J. (2003). Projecting rate of change in the context of motion onto the context of money. International Journal of Mathematical Education in Science and Technology, 34(6), 887-904. https://doi.org/10.1080/00207390310001606660
  • Yavuz, İ. & Baştürk, S. (2011). Ders kitaplarında fonksiyon kavramı: Türkiye ve Fransa örneği. Kastamonu Eğitim Dergisi, 19(1), 199-220. Yıldırım, A. & Şimşek, H. (2011). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin Yayıncılık.
  • Zandieh, M. (2000). A theoretical framework for analyzing students' understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education (vol. IV, pp. 103–127). Providence, RI: American Mathematical Society.
  • Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. The Journal of Mathematical Behavior, 25(1), 1-17. https://doi.org/10.1016/j.jmathb.2005.11.002

Conceptual Analysis of Derivative as a Rate of Change and Analysis of the Mathematics Textbooks

Year 2021, Volume: 11 Issue: 3, 545 - 568, 01.12.2021
https://doi.org/10.19126/suje.977200

Abstract

In this study, initially, we provided a conceptual analysis for the rate of change interpretation of derivative from the quantitative reasoning perspective. Then, based upon the principles drawn from the quantitative reasoning perspective, the 12th-grade Turkish mathematics textbooks were analyzed, specifically focusing on the derivative topic, using the document analysis research method. The findigs indicated that mathematics textbooks are insufficient in terms of the diversity of real-life contexts. Even, the derivative as a rate of change concept is generally introduced limited with the kinematic contexts. Units of the quantities are properly used in the real-life contexts that appeared in the textbooks, however, the discussions and interpretations on the derivative concept by using the units of the quantities are not deep enough. Moreover, the covariation and relative-size ideas that are accepted to be prominent for the understanding of the rate of change are not sufficiently emphasized in all of the textbooks. The weak points in the Turkish mathematics textbooks that need to be enriched related to the concept of the derivative as a rate of change were emphasized. In the light of the findings, we also discussed the aspects that need to be developed in the textbooks provided with some samples of the real-life interpretations of the derivative as a rate of change and as relative size.

Project Number

Herhangi bir proje kapsamında gerçekleştirilmemiştir.

References

  • Altun, M. H. (2018). Matematik 12. sınıf ders kitabı. Ankara: Tutku Yayıncılık.
  • Ärlebäck, J. B., Doerr, H. M., & O’Neil, A. M. (2013). A modeling perspective on interpreting rates of change in context. Mathematical Thinking and Learning, 15(4), 314–336.
  • Ärlebäck, J. B., & Doerr, H. M. (2018). Students’ interpretations and reasoning about phenomena with negative rates of change throughout a model development sequence. ZDM–Mathematics Education, 50, 187-200.
  • Bektaşlı, B., & Çakmakçı, G. (2011). Consistency of students’ ideas about the concept of rate across different contexts. Education and Science, 36 (162), 273-287.
  • Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22, 481–497.
  • Bezuidenhout, J. (1998). First year university students’ understanding of rate of change. International Journal of Mathematical Education in Science and Technology, 29(3), 389-399.
  • Byerley, C., & Thompson, P. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. The Journal of Mathematical Behavior, 48, 168-193.
  • Byerley, C. (2019). Calculus students’ fraction and measure schemes and implications for teaching rate of change functions conceptually. The Journal of Mathematical Behavior, 55, 100694.
  • Bingolbali, E. (2008). Türev kavramına ilişkin öğrenme zorlukları ve kavramsal anlama için öneriler. M. F. Özmantar, E. Bingolbali, & H. Akkoç (Eds.), Matematiksel kavram yanılgıları ve çözüm önerileri (pp. 223–255). Ankara: Pegem A.
  • Bingolbali, E. & Bingolbali, F. (2020). An examination of tasks in elementary school mathematics textbooks in terms of multiple outcomes and multiple solution methods. International Journal of Educational Studies in Mathematics, 7(4), 214-235.
  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events. Journal for Research in Mathematics Education, 33(5), 352-378.
  • Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. L. Mayes, & L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 55–73). Laramie: University of Wyoming Press.
  • Confrey, J. & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 134-165.
  • Cooney, T. J., Beckman, S., & Lloyd, G. M. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics.
  • Doorman, L. M., & Gravemeijer, K. P. E. (2009). Emergent modeling: discrete graphs to support the understanding of change and velocity. ZDM Mathematics Education 41(1-2), 199-211. https://doi.org/10.1007/s11858-008-0130-z
  • Ellis, A. B., Ozgur, Z., Kulow, T., Williams, C. C., & Amidon, J. (2015). Quantifying exponential growth: Three conceptual shifts in coordinating multiplicative and additive growth. The Journal of Mathematical Behavior, 39, 135–155. doi:10.1016/j.jmathb.2015.06.004
  • Emin, A., Gerboğa, A., Güneş, G. & Kayacıer, M. (2020). Matematik 12. sınıf ders kitabı. Ankara: MEB yayınları.
  • Gökçek, T., & Açıkyıldız, G. (2016). Matematik öğretmeni adaylarının türev kavramıyla i̇lgili yaptıkları hatalar [Preservice mathematics teachers’ errors related to derivative]. Turkish Journal of Computer and Mathematics Education, 7(1), 112-141.
  • Hawson, G. (2013). The development of mathematics textbooks: historical reflections from a personal perspective. ZDM Mathematics Education, 45, 647-658.
  • Herbert, S., & Pierce, R. (2011). What is rate? Does context or representation matter? Mathematics Education Research Journal, 23(4), 455-477.
  • Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81(1), 85–101.
  • İncikabı, L. & Tjoe, H. (2013). A comparative analysis of ratio and proportion problems in Turkish and the U.S. middle school mathematics textbooks. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi (KEFAD), 14(1), 1-15.
  • Jones, S. R. (2017). An exploratory study on student understandings of derivatives in real-world, non-kinematics contexts. The Journal of Mathematical Behavior, 45, 95–110.
  • Kemancı, B., Büyükokutan, A., Çelik, S. & Kemancı, Z. (2020). Fen lisesi 12. sınıf matematik ders kitabı. Ankara: MEB yayınları.
  • Kertil, M. (2014). Pre-service elementary mathematics teachers' understanding of derivative through a model development unit (Unpublished doctoral dissertation). Middle East Technical University, Graduate School of Natural and Applied Sciences, Ankara.
  • Kertil, M., Erbaş, A. K., & ve Çetinkaya, B. (2017). İlköğretim matematik öğretmen adaylarının değişim oranı ile ilgili düşünme biçimlerinin bir modelleme etkinliği bağlamında incelenmesi Turkish Journal of Computer and Mathematics Education, 8(1), 188-217.
  • Kertil, M., Erbaş, A. K., & Çetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21, 207-233.
  • Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The case of quadratic functions. Mathematical Thinking and Learning, 14(2), 85–119.
  • MEB (2018). Ortaöğretim Matematik Dersi (9-12. Sınıflar) Öğretim Programı. Ankara: Milli Eğitim Bakanlığı Yayınları.
  • Mkhatshwa, T. P. (2016). Business calculus students’ reasoning about optimization problems: A study of quantitative reasoning in an economic context (Doctoral dissertation). New York, NY: Syracuse University.
  • Mkhatshwa, T. P. (2018). Business calculus students' interpretations of marginal change in economic contexts. In Hodges, T.E., Roy, G. J., & Tyminski, A. M. (Eds.), Proceedings of the 40th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 564-571). Greenville, SC: University of South Carolina & Clemson University.
  • Mkhatshwa, T., & Doerr, H. (2018). Undergraduate students’ quantitative reasoning in economic contexts. Mathematical Thinking and Learning, 20(2), 142-161. https://doi.org/10.1080/10986065.2018.1442642
  • Rowland, D. R., & Jovanoski, Z. (2004). Student interpretation of the terms in first-order ordinary differential equations in modeling contexts. International Journal of Mathematical Education in Science and Technology, 35(4), 505-516.
  • Schmidt, W. H. (2012). Measuring content through textbooks: The cumulative effect of middle-school tracking. In G. Gueudet, B. Pepin, & L. Trouche (Eds.), From text to ‘lived’ resources: Mathematics curriculum materials and teacher development (pp. 143–160). Dordrecht: Springer.
  • Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research. International Journal of Computers for Mathematical Learning, 7(2), 167-215.
  • Teuscher, D. & Reys, R. E. (2012). Rate of change: AP calculus students' understandings and misconceptions after completing different curricular paths. School Science and Mathematics, 112(6), 359–376. https://doi.org/10.1111/j.1949-8594.2012.00150.x
  • Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26, 229-274.
  • Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education (Vol. 1, pp. 33-57). Laramie, WY: University of Wyoming.
  • Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics. Thompson, P. W., Hatfield, N. J., Yoon, H., Joshua, S., & Byerley, C. (2017). Covariational reasoning among U.S. and South Korean secondary mathematics teachers. The Journal of Mathematical Behavior, 48, 95–111.
  • Weber, E. & Dorko, A. (2014). Students’ and experts’ schemes for rate of change and its representations. The Journal of Mathematical Behavior, 34, 14-32.
  • Wilhelm, J. A. & Confrey, J. (2003). Projecting rate of change in the context of motion onto the context of money. International Journal of Mathematical Education in Science and Technology, 34(6), 887-904. https://doi.org/10.1080/00207390310001606660
  • Yavuz, İ. & Baştürk, S. (2011). Ders kitaplarında fonksiyon kavramı: Türkiye ve Fransa örneği. Kastamonu Eğitim Dergisi, 19(1), 199-220. Yıldırım, A. & Şimşek, H. (2011). Sosyal bilimlerde nitel araştırma yöntemleri. Ankara: Seçkin Yayıncılık.
  • Zandieh, M. (2000). A theoretical framework for analyzing students' understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education (vol. IV, pp. 103–127). Providence, RI: American Mathematical Society.
  • Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. The Journal of Mathematical Behavior, 25(1), 1-17. https://doi.org/10.1016/j.jmathb.2005.11.002
There are 44 citations in total.

Details

Primary Language English
Subjects Other Fields of Education
Journal Section Articles
Authors

Mahmut Kertil 0000-0002-0633-7144

Project Number Herhangi bir proje kapsamında gerçekleştirilmemiştir.
Publication Date December 1, 2021
Published in Issue Year 2021 Volume: 11 Issue: 3

Cite

APA Kertil, M. (2021). Conceptual Analysis of Derivative as a Rate of Change and Analysis of the Mathematics Textbooks. Sakarya University Journal of Education, 11(3), 545-568. https://doi.org/10.19126/suje.977200