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Prospective Teachers’ Graphical Representations for Two Simultaneously Changing Quantities in Dynamic Functional Situations

Year 2020, Volume: 8 Issue: 2, 462 - 488, 30.04.2020

Abstract

This study aims to reveal how prospective teachers express the relationships between variables through graphical representations when interpreting dynamic functional situations involving two simultaneously changing quantities. 100 prospective middle school mathematics teachers participated to this case study. The data consisted of prospective teachers’ written responses to the task involving filling bottles with water and the graphs of volume as a function of height and clinical interviews were used to examine their covariational reasoning and graphing abilities. The findings showed that only six prospective teachers' graphical representations were correct for both dynamic functional situations. The most significant and common problems in the graphical representations were found such as (i) inability to coordinate slopes for linear relationships between variables, (ii) representing nonlinear relations of variables as linear relations, (iii) reversing the roles of dependent and independent variables, (iv) representing the relationship between variables as decreasing rather than increasing and (v) representing the relationships between variables to include more or less partitions to the graph than required by the given dynamic functional situation.

References

  • Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. In J. J. Kaput, E. Dubinsky, & A. H. Schoenfeld (Eds.), Research in collegiate mathematics education, III. issues in mathematics education (Vol. 7, pp. 115–162). Washington, DC: American Mathematical Society.
  • 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.
  • Carlson, M., Larsen, S., & Lesh, R. (2003). Integrating models and modeling perspective with existing research and practice. In R. Lesh, & H. Doerr (Eds.), Beyond constructivism: A models and modeling perspective (pp. 465–478). Mahwah, NJ: Erlbaum.
  • Confrey, J., & Smith, E. (1995). Splitting, covariation and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86.
  • Cooney, T. J., Beckmann, S., Lloyd, G. M., Wilson, P. S., & Zbiek, R. 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.
  • Ellis, A. B., Ozgur, Z., Kulow, T., Williams, 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
  • Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1-3), 111–129.
  • Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81, 85–101.
  • Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Mathematical Thinking and Learning, 17(1), 64–90.
  • Johnson, H. L., McClintock, E., & Hornbein, P. (2017). Ferris wheels and filling bottles: A case of a student’s transfer of covariational reasoning across tasks with different backgrounds and features. ZDM–Mathematics Education, 49(6), 851–864. doi:10.1007/s11858-017-0866-4
  • 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. doi:10.1016/j.jmathb.2016.11.002
  • Kertil, M. (2020). Matematik öğretmen adaylarının kovaryasyonel düşünme becerileri: Dinamik animasyonlar nasıl etkiliyor?. Turkish Journal of Computer and Mathematics Education. Advance online publication. doi:10.16949/turkbilmat.652481.
  • Kertil, M., Erbas, A. K., & Cetinkaya, B. (2017). Pre-service elementary mathematics teachers’ ways of thinking about rate of change in the context of a modeling activity. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 8(1), 188–217.
  • Kertil, M., Erbas, A. K., & Cetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21(3), 207–233. doi:10.1080/10986065.2019.1576001
  • Koklu, O., & Jakubowski, E. (2010). From interpretations to graphical representations: A case study investigation of covariational reasoning. Eurasian Journal of Educational Research, 40, 151–170.
  • Milli Eğitim Bakanlığı [MEB] (2018). Ortaokul matematik dersi (5, 6, 7, ve 8. sınıflar) öğretim programı. Ankara, Turkey.
  • Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel, & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA Notes (Vol. 25, pp. 175–193). Washington, DC: Mathematical Association of America.
  • Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In E. Dubinsky, A. H. Shoenfeld & J. J. Kaput (Eds.), Research in collegiate mathematics education I (pp. 139–168). Providence, RI: American Mathematical Society.
  • 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. doi:10.1016/j.jmathb.2011.09.001.
  • Moore, K. C., & Thompson, P. W. (2015, February). Shape thinking and students’ graphing activity. In Proceedings of the 18th Meeting of the MAA Special Interest Group on Research in Undergraduate Mathematics Education (pp. 782–789). Pittsburgh, PA: RUME.
  • National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author. Paoletti, T., & Moore, K. C. (2017). The parametric nature of two students’ covariational reasoning. The Journal of Mathematical Behavior, 48, 137–151. doi:10.1016/j.jmathb.2017.08.003
  • 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.
  • Saldanha, L., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous covariation. In S B. Berenson, K. R. Dawkins, M. Blanton, W. N. Coulombe, J. Kolb, K. Norwood, & L. Stiff (Eds.), Proceedings of the twentieth annual meeting of the North American chapter of the International group for the psychology of mathematics education (Vol. I, pp. 298–303). Raleigh: North Carolina State University.
  • Stalvey, H. E., & Vidakovic, D. (2015). Students’ reasoning about relationships between variables in a real-world problem. The Journal of Mathematical Behavior, 40, 192–210. doi:10.1016/j.jmathb.2015.08.002
  • Swan, M. (1985). The language of functions and graphs. Shell Centre & Joint Matriculation Board, Nottingham.
  • Şen-Zeytun, A., Cetinkaya, B., & Erbas, A. K. (2010). Mathematics teachers’ covariational reasoning levels and predictions about students’ covariational reasoning. Educational Sciences: Theory and Practice, 10(3), 1601–1612.
  • Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld & J. J. Kaput (Eds.), Research in collegiate mathematics education 1 (pp. 21–44). Providence, RI: American Mathematical Society.
  • 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 (pp. 33–57). Laramie: University of Wyoming.
  • Thompson, P. W. (2013). In the absence of meaning. In K. Leatham (Ed.), Vital directions for research in mathematics education (pp. 57–93). New York, NY: Springer.
  • Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), First compendium for research in mathematics education (pp. 421–456). Reston, VA: NCTM.
  • 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. doi:10.1016/j. jmathb.2017.08.001
  • Van de Walle, J., Karp, K.S., & Bay-Williams J.M. (2019). Elementary and middle school mathematics: teaching developmentally (10th ed.) New York, NY: Pearson Education, Inc.
  • 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.
  • Wilkie, K. J. (2019). Investigating students’ attention to covariation features of their constructed graphs in a figural pattern generalisation context. International Journal of Science and Mathematics Education, 1–22. https://doi.org/10.1007/s10763-019-09955-6
  • Yemen-Karpuzcu, S., Ulusoy, F., & Işıksal-Bostan, M. (2017). Prospective middle school mathematics teachers’ covariational reasoning for interpreting dynamic events during peer interactions. International Journal of Science and Mathematics Education. 15(1), 89–108.
  • Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. Journal of Mathematical Behavior, 25, 1–17.

Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri

Year 2020, Volume: 8 Issue: 2, 462 - 488, 30.04.2020

Abstract

Bu araştırma, ilköğretim matematik öğretmeni adaylarının iki değişkenin eş zamanlı değişimini içeren dinamik fonksiyonel durumları yorumlarken değişkenler arasındaki ilişkiyi grafik temsilleri aracılığıyla nasıl ifade ettiklerini ortaya çıkarmayı amaçlamaktadır. Bütüncül bir durum çalışması olan bu çalışmanın katılımcıları, bir devlet üniversitesinde öğrenim gören 100 ilköğretim matematik öğretmeni adayıdır. Veriler, içine su ile doldurulan iki farklı özellikteki şişeye ait yükseklik-hacim grafiğinin çizilmesini gerektiren bir dinamik fonksiyonel durum etkinliği için yapılan yazılı açıklamalar, grafik çizimleri ve klinik görüşmeler yoluyla elde edilmiştir. Bulgular, sadece altı öğretmen adayının grafik temsillerinin her iki durum için de doğru olduğunu göstermiştir. Grafik temsillerinde tespit edilen belirgin hatalar ve eksiklikler şunlar olmuştur: (i) değişkenler arasındaki farklı doğrusal ilişkiler için eğimleri koordine edememe, (ii) değişkenler arasındaki doğrusal olmayan ilişkileri doğrusal temsil etme, (iii) bağımlı ve bağımsız değişkenlerin rollerini değiştirerek temsil etme, (iv) değişkenler arasındaki ilişkiyi artan yerine azalan şekilde temsil etme ve (v) değişkenler arasındaki ilişkiyi verilen dinamik fonksiyonel durumun gerektirdiğinden daha az veya fazla sayıda bölüm içerecek biçimde temsil etme.

References

  • Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. In J. J. Kaput, E. Dubinsky, & A. H. Schoenfeld (Eds.), Research in collegiate mathematics education, III. issues in mathematics education (Vol. 7, pp. 115–162). Washington, DC: American Mathematical Society.
  • 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.
  • Carlson, M., Larsen, S., & Lesh, R. (2003). Integrating models and modeling perspective with existing research and practice. In R. Lesh, & H. Doerr (Eds.), Beyond constructivism: A models and modeling perspective (pp. 465–478). Mahwah, NJ: Erlbaum.
  • Confrey, J., & Smith, E. (1995). Splitting, covariation and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86.
  • Cooney, T. J., Beckmann, S., Lloyd, G. M., Wilson, P. S., & Zbiek, R. 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.
  • Ellis, A. B., Ozgur, Z., Kulow, T., Williams, 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
  • Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1-3), 111–129.
  • Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate. Educational Studies in Mathematics, 81, 85–101.
  • Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Mathematical Thinking and Learning, 17(1), 64–90.
  • Johnson, H. L., McClintock, E., & Hornbein, P. (2017). Ferris wheels and filling bottles: A case of a student’s transfer of covariational reasoning across tasks with different backgrounds and features. ZDM–Mathematics Education, 49(6), 851–864. doi:10.1007/s11858-017-0866-4
  • 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. doi:10.1016/j.jmathb.2016.11.002
  • Kertil, M. (2020). Matematik öğretmen adaylarının kovaryasyonel düşünme becerileri: Dinamik animasyonlar nasıl etkiliyor?. Turkish Journal of Computer and Mathematics Education. Advance online publication. doi:10.16949/turkbilmat.652481.
  • Kertil, M., Erbas, A. K., & Cetinkaya, B. (2017). Pre-service elementary mathematics teachers’ ways of thinking about rate of change in the context of a modeling activity. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 8(1), 188–217.
  • Kertil, M., Erbas, A. K., & Cetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21(3), 207–233. doi:10.1080/10986065.2019.1576001
  • Koklu, O., & Jakubowski, E. (2010). From interpretations to graphical representations: A case study investigation of covariational reasoning. Eurasian Journal of Educational Research, 40, 151–170.
  • Milli Eğitim Bakanlığı [MEB] (2018). Ortaokul matematik dersi (5, 6, 7, ve 8. sınıflar) öğretim programı. Ankara, Turkey.
  • Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel, & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA Notes (Vol. 25, pp. 175–193). Washington, DC: Mathematical Association of America.
  • Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In E. Dubinsky, A. H. Shoenfeld & J. J. Kaput (Eds.), Research in collegiate mathematics education I (pp. 139–168). Providence, RI: American Mathematical Society.
  • 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. doi:10.1016/j.jmathb.2011.09.001.
  • Moore, K. C., & Thompson, P. W. (2015, February). Shape thinking and students’ graphing activity. In Proceedings of the 18th Meeting of the MAA Special Interest Group on Research in Undergraduate Mathematics Education (pp. 782–789). Pittsburgh, PA: RUME.
  • National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author. Paoletti, T., & Moore, K. C. (2017). The parametric nature of two students’ covariational reasoning. The Journal of Mathematical Behavior, 48, 137–151. doi:10.1016/j.jmathb.2017.08.003
  • 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.
  • Saldanha, L., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous covariation. In S B. Berenson, K. R. Dawkins, M. Blanton, W. N. Coulombe, J. Kolb, K. Norwood, & L. Stiff (Eds.), Proceedings of the twentieth annual meeting of the North American chapter of the International group for the psychology of mathematics education (Vol. I, pp. 298–303). Raleigh: North Carolina State University.
  • Stalvey, H. E., & Vidakovic, D. (2015). Students’ reasoning about relationships between variables in a real-world problem. The Journal of Mathematical Behavior, 40, 192–210. doi:10.1016/j.jmathb.2015.08.002
  • Swan, M. (1985). The language of functions and graphs. Shell Centre & Joint Matriculation Board, Nottingham.
  • Şen-Zeytun, A., Cetinkaya, B., & Erbas, A. K. (2010). Mathematics teachers’ covariational reasoning levels and predictions about students’ covariational reasoning. Educational Sciences: Theory and Practice, 10(3), 1601–1612.
  • Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld & J. J. Kaput (Eds.), Research in collegiate mathematics education 1 (pp. 21–44). Providence, RI: American Mathematical Society.
  • 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 (pp. 33–57). Laramie: University of Wyoming.
  • Thompson, P. W. (2013). In the absence of meaning. In K. Leatham (Ed.), Vital directions for research in mathematics education (pp. 57–93). New York, NY: Springer.
  • Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), First compendium for research in mathematics education (pp. 421–456). Reston, VA: NCTM.
  • 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. doi:10.1016/j. jmathb.2017.08.001
  • Van de Walle, J., Karp, K.S., & Bay-Williams J.M. (2019). Elementary and middle school mathematics: teaching developmentally (10th ed.) New York, NY: Pearson Education, Inc.
  • 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.
  • Wilkie, K. J. (2019). Investigating students’ attention to covariation features of their constructed graphs in a figural pattern generalisation context. International Journal of Science and Mathematics Education, 1–22. https://doi.org/10.1007/s10763-019-09955-6
  • Yemen-Karpuzcu, S., Ulusoy, F., & Işıksal-Bostan, M. (2017). Prospective middle school mathematics teachers’ covariational reasoning for interpreting dynamic events during peer interactions. International Journal of Science and Mathematics Education. 15(1), 89–108.
  • Zandieh, M. J., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. Journal of Mathematical Behavior, 25, 1–17.
There are 37 citations in total.

Details

Primary Language Turkish
Journal Section Articles
Authors

Fadime Ulusoy This is me

Publication Date April 30, 2020
Published in Issue Year 2020 Volume: 8 Issue: 2

Cite

APA Ulusoy, F. (2020). Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri. Eğitimde Nitel Araştırmalar Dergisi, 8(2), 462-488.
AMA Ulusoy F. Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri. Derginin Amacı ve Kapsamı. April 2020;8(2):462-488.
Chicago Ulusoy, Fadime. “Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri”. Eğitimde Nitel Araştırmalar Dergisi 8, no. 2 (April 2020): 462-88.
EndNote Ulusoy F (April 1, 2020) Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri. Eğitimde Nitel Araştırmalar Dergisi 8 2 462–488.
IEEE F. Ulusoy, “Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri”, Derginin Amacı ve Kapsamı, vol. 8, no. 2, pp. 462–488, 2020.
ISNAD Ulusoy, Fadime. “Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri”. Eğitimde Nitel Araştırmalar Dergisi 8/2 (April 2020), 462-488.
JAMA Ulusoy F. Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri. Derginin Amacı ve Kapsamı. 2020;8:462–488.
MLA Ulusoy, Fadime. “Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri”. Eğitimde Nitel Araştırmalar Dergisi, vol. 8, no. 2, 2020, pp. 462-88.
Vancouver Ulusoy F. Öğretmen Adaylarının İki Niceliğin Eş Zamanlı Değişimini İçeren Dinamik Fonksiyonel Durumlar İçin Oluşturdukları Grafik Temsilleri. Derginin Amacı ve Kapsamı. 2020;8(2):462-88.