Öğrencilerin Silindirin Hacmi Konusunda Geliştirdikleri Matematiksel Fikirler: Sınıf İçi Argümantasyon Modeli

Year 2022, , 36 - 64, 21.04.2022

Şule Şahin Doğruer , Didem Akyüz

https://doi.org/10.33711/yyuefd.1063106

Abstract

Bu çalışmanın amacı, silindirin hacmi konusu kapsamında sekizinci sınıf öğrencilerinin geliştirdikleri matematiksel fikirleri saptamak ve bunun için uygulanan içeriğin etkililiğini sekizinci sınıf matematik dersinde test etmektir. Bu bağlamda, bir varsayıma dayalı öğrenme yörüngesinin rehberliği ile bir öğretim dizisi kullanılmıştır. Konu olarak silindirin hacmi belirlenmiştir. Süreç iki hafta boyunca devam etmiştir. Sınıf içi argümantasyon, dinamik geometri yazılımı ve günlük yaşam örnekleri sınıf etkinliklerini desteklemiştir. Verilerin analizi için Krummheuer’in (2015) argümantasyon modeli kullanılmıştır. Analiz sonucunda dört matematiksel fikir elde edilmiştir; (a) hacim üçüncü boyut ile ilgilidir, (b) hacim bir cismin içini doldurmaktır, (c) hacim hesabı yükseklik, genişlik ve uzunluk kavramlarını içerir, (d) hacim taban alanı ile yüksekliğin çarpımıdır.

Keywords

matematiksel fikirler, argümantasyon, silindir, tasarım tabanlı çalışma, varsayıma dayalı öğrenme yörüngesi

Mathematical Ideas Developed by Students on Volume of Cylinder: A Classroom Argumentation Model

Abstract

The aim of this study is to determine the mathematical ideas developed by eighth-grade students in the context of the volume of the cylinder and to test the effectiveness of the applied content in the eighth-grade mathematics course. In this context, an instructional sequence with the guidance of a hypothetical learning trajectory was used. The volume of the cylinder was the subject. The process continued for two weeks. Classroom argumentations, dynamic geometry software, and daily life examples supported instructional activities. The argumentation model of Krummheuer (2015) was used to analyze classroom argumentations. The analysis of the data collected from the study yielded four mathematical ideas; (a) the volume is related to the third dimension; (b) the volume is to fill the solid; (c) the volume calculation includes the concepts of height, width, and length; (d) volume equals to the multiplication of base area and height.

Keywords

mathematical ideas, argumantation, cylinder, design-based study, Hypothetical learning trajectory

References

• Abi-El-Mona, I. & Abd-El-Khalick, F. (2011). Perceptions of the nature and goodness of argument among college students, science teachers and scientists. International Journal of Science Education, 33(4), 573-605. https://doi.org/10.1080/09500691003677889
• Adolphus, T. (2011). Problems of teaching and learning of geometry in secondary schools in Rivers State, Nigeria. International Journal of Emerging Sciences, 1(2), 143-152.
• Agyei, D. D., & Benning, I. (2015). Pre-service teachers’ use and perceptions of GeoGebra software as an instructional tool in teaching mathematics. Journal of Educational Development and Practice, 5(1), 14-30.
• Akyüz, D. (2016). Farklı öğretim yöntemleri ve sınıf seviyesine göre öğretmen adaylarının TPAB analizi. Turkish Journal of Computer and Mathematics Education (TURCOMAT), 7(1), 89-111.
• Alqahtani, M. M., & Powell, A. B. (2017). Mediational activities in a dynamic geometry environment and teachers’ specialized content knowledge. The Journal of Mathematical Behavior, 48, 77-94. https://doi.org/10.1016/j.jmathb.2017.08.004
• Anderson, T., & Shattuck, J. (2012). Design-based research: A decade of progress in education research. Educational researcher, 41(1), 16-25. https://doi.org/10.3102/0013189X11428813
• Andreasen, J. B. (2006). Classroom mathematical practices in a preservice elementary mathematics education course using an instructional sequence related to place value and operations. Unpublished Dissertation. University of Central Florida, Orlando.
• Asterhan, C. S., & Schwarz, B. B. (2007). The effects of monological and dialogical argumentation on concept learning in evolutionary theory. Journal of Educational Psychology, 99(3), 626.
• Asterhan, C. S., & Schwarz, B. B. (2016). Argumentation for learning: Well-trodden paths and unexplored territories. Educational Psychologist, 51(2), 164-187. https://doi.org/10.1080/00461520.2016.1155458
• Baki, A. (2001). Bilişim Teknolojisi Işığı Altında Matematik Eğitiminin Değerlendirilmesi, Milli Eğitim Dergisi, 149, 26-31.
• Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. Multiple perspectives on the teaching and learning of mathematics, 83-104.
• Barab, S., & Squire, K. (2004). Design-based research: Putting a stake in the ground. The journal of the learning sciences, 13(1), 1-14.
• Battista, M. T. (2007). The development of geometric and spatial thinking. Second handbook of research on mathematics teaching and learning, 2, 843-908.
• Bauersfeld, H., Krummheuer, G., & Voigt, J. (1988). Interactional theory of learning and teaching mathematics and related microethnographical studies. Foundations and methodology of the discipline of mathematics education, 174-188.
• Ben-Chaim, D., Lappan, G. & Houang, R. T. (1985). Visualizing rectangular solids made of small cubes: Analyzing and affecting students’ performance. Educational Studies in Mathematics, 16(4), 389-409.
• Boaler, J. (2016). Designing mathematics classes to promote equity and engagement. Journal of Mathematical Behavior, (41), 172-178. https://doi.org/10.1016/j.jmathb.2015.01.002
• Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and instruction, 17(1), 25-66.
• Clark-Wilson, A., & Hoyles, C. (2017). Dynamic digital technologies for dynamic mathematics: Implications for teachers’ knowledge and practice: Final report. London: UCL Institute of Education Press.
• Cobb, P. (2000). Conducting classroom teaching experiments in collaboration with teachers. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307–334). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
• Cobb, P. (2003). Investigating students' reasoning about linear measurement as a paradigm case of design research. In M. Stephan, J. Bowers, P. Cobb, & K. Gravemeijer (Eds.), Supporting students' development of measuring conceptions: Analyzing students' learning in social context. Reston, VA: National Council of Teachers of Mathematics.
• Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational psychologist, 31(3-4), 175-190.
• Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for research in mathematics education, 258-277.
• Cobb, P., Gravemeijer, K., Yackel, E., McClain, K. & Whitenack, J. (1997). Mathematizing and symbolizing: The emergence of chains of signification in one first-grade classroom. In D. Kirshner & J. A. Whitson (Eds.), Situated cognition theory: Social semiotic, and psychological perspectives (pp. 151–233). Mahwah: Lawrence Erlbaum Associates, Inc.
• Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in mathematical practices. Journal of the Learning Sciences, 10(1&2), 113-163.
• Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics education, 2-33.
• Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. Educational Studies in Mathematics, 86(3), 401-429.
• Cramer, J. (2011). Everyday argumentation and knowlegde construction in mathematical tasks. In Proceedings of the 7th Congress of the European Society for Research in Mathematics Education. Rzeszów, Poland: University of Rzeszów.
• Drijvers, R., Newton, P., & Osborne, J. (2000). Establishing the norms of scientific argumentation in classrooms. Science education, 84(3), 287-312.
• Duschl, R. & Osborne, J. (2002). Supporting argumentation discourse in science education. Studies in Science Education, 38, 39-72.
• Flores, A., Park, J., & Bernhardt, S. A. (2016). Learning Mathematics and Technology through Inquiry, Cooperation, and Communication: A Learning Trajectory for Future. Handbook of Research on Transforming Mathematics Teacher Education in the Digital Age, 324.
• Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). “You're going to want to find out which and prove it”: Collective argumentation in a mathematics classroom. Learning and instruction, 8(6), 527-548.
• Fraenkel, J. R., Wallen, N. E. & Hyun, H. (2012). How to design and evaluate research in education. McGraw-Hill.
• Fujita, T., Kondo, Y., Kumakura, H., & Kunimune, S. (2017). Students’ geometric thinking with cube representations: Assessment framework and empirical evidence. The Journal of Mathematical Behavior, 46, 96-111. https://doi.org/10.1016/j.jmathb.2017.03.003
• Fukawa-Connelly, T., & Silverman, J. (2015). The Development of Mathematical Argumentation in an Unmoderated, Asynchronous Multi-User Dynamic Geometry Environment. Contemporary Issues in Technology and Teacher Education, 15(4), 445-488.
• Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education. Monograph, 3, i-196.
• Ganesh, B., Wilhelm, J., & Sherrod, S. (2009). Development of a geometric spatial visualization tool. School Science and Mathematics, 109(8), 461-472.
• Giannakoulias, E., Mastorides, E., Potari, D., & Zachariades, T. (2010). Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims. The Journal of Mathematical Behavior, 29(3), 160-168.
• Graveimejer, K. & Cobb, P. (2013). Design research from a learning design perspective. Educational design research (pp. 73-112). Taylor & Francis.
• Gravemeijer, K., & van Eerde, D. (2009). Design research as a means for building a knowledge base for teachers and teaching in mathematics education. The elementary school journal, 109(5), 510-524.
• Güven, B., Kosa, T. (2008). The effect of dynamic geometry software on student mathematics teachers’ spatial visualization skills. The Turkish Online Journal of Educational Technology, 7(4), 100-107.
• Hannafin, R. D., Truxaw, M. P., Vermillion, J. R., & Liu, Y. (2008). Effects of spatial ability and instructional program on geometry achievement. The Journal of Educational Research, 101(3), 148-157.
• Hollebrands, K. F., Conner, A., & Smith, R. C. (2010). The nature of arguments provided by college geometry students with access to technology while solving problems. Journal for Research in Mathematics Education, 324-350.
• Jonassen, D., & Kim, B. (2010). Arguing to learn and learning to argue: Design justifications and guidelines. Educational Technology Research and Development, 58, 439-457.
• Kesan, C., & Caliskan, S. (2013). The Effect of Learning Geometry Topics of 7th Grade in Primary Education with Dynamic Geometer's Sketchpad Geometry Software to Success and Retention. Turkish Online Journal of Educational Technology-TOJET, 12(1), 131-138.
• Kosko, K. W., Rougee, A., & Herbst, P. (2014). What actions do teachers envision when asked to facilitate mathematical argumentation in the classroom? Mathematics Education Research Journal, 26(3), 459-476.
• Krummheuer, G. (2015). Methods for Reconstructing Processes of Argumentation and Participation in Primary Mathematics Classroom Interaction. In Approaches to Qualitative Research in Mathematics Education (pp. 51-74). Springer, Dordrecht.
• Latsi, M., & Kynigos, C. (2012). Experiencing 3D simulated space through different perspectives. In A. Jimoyiannis (Ed.), Research on e-Learning and ICT in Education: Technological, Pedagogical and Instructional Issues (pp. 183–196). Berlin: Springer.
• Marchis, I. (2012). Preservıce prımary school teachers' elementary geometry knowledge. Acta Didactica Napocensia, 5(2), 33.
• MEB (2018). Matematik Dersi Öğretim Programı. Ankara.
• McClintock, E., Jiang, Z., & July, R. (2002). Students' Development of Three-Dimensional Visualization in the Geometer's Sketchpad Environment. In: Proceedings of the Annual Meeting [of the] North American Chapter of the International Group for the Psychology of Mathematics Education (24th, Athens, GA, October 26-29, 2002). Volumes 1-4.
• Mueller, M. F. (2009). The co-construction of arguments by middle-school students. The Journal of Mathematical Behavior, 28(2-3), 138-149.
• Mueller, M., Yankelewitz, D., & Maher, C. (2014). Teachers promoting student mathematical reasoning. Investigations in Mathematics Learning, 7(2), 1-20.
• National Council of Teachers of Mathematics [NCTM] (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
• Ng, O. L. (2015). Coherent and diverging discourse in mathematical activities with dynamic geometry. MEDS-C 2015, 71.
• Ng, O., & Sinclair, N. (2015b). “Area without numbers”: using touchscreen dynamic geometry to reason about shape. Canadian Journal of Science, Mathematics and Technology Education, 15(1), 84–101.
• Osborne, J., Erduran, S., & Simon, S. (2004). Enhancing the quality of argumentation in school science. Journal of Research in Science Teaching, 41(10), 994-1020.
• Owens, K., & Highfield, K. (2015). Visuospatial reasoning in contexts with digital technology. Visuospatial reasoning (pp. 275-289). Springer, Cham.
• Pei, Weintrop & Wilensky (2018). Cultivating Computational Thinking Practices and Mathematical Habits of Mind in Lattice Land, Mathematical Thinking and Learning, 20(1), 75-89. https://:10.1080/10986065.2018.1403543
• Pittalis, M., & Constantinou, C. (2010). Types of reasoning in 3D geometry thinking and their relation with spatial ability. Educational Studies in Mathematics, 75, 191–212.
• Pittalis, M., Christou, C., & Pitta-Pantazi, D. (2012). Enhancing prospective teachers’ technological pedagogical content knowledge in 3D shapes’ nets. Conference Proceedings of the 4th International Conference on Education and New Learning Technologies, Barcelona, Spain.
• Plomp, T. (2013). Educational design research: An introduction. Educational design research, 11-50.
• Prusak, N., Hershkowitz, R., & Schwarz, B. B. (2012). From visual reasoning to logical necessity through argumentative design. Educational Studies in Mathematics, 79(1), 19-40. https://10.1007/s 10649-0 11 -9335-0
• Rasmussen, C., & Stephan, M. (2008). 10 A Methodology for Documenting Collective Activity. Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching, 195.
• Sack, J. J. (2013). Development of a top-view numeric coding teaching-learning trajectory within an elementary grades 3-D visualization design research project. The Journal of Mathematical Behavior, 32(2), 183-196.
• Sadler, T. D., & Fowler, S. R. (2006). A threshold model of content knowledge transfer for socio-scientific argumentation. Science Education, 90(6), 986-1004.
• Sinclair, N., & Moss, J. (2012). The more it changes, the more it becomes the same: The development of the routine of shape identification in dynamic geometry environment. International Journal of Educational Research, 51, 28-44. https://doi.org/10.1016/j.ijer.2011.12.009
• Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. The Journal of Mathematical Behavior, 21(4), 459-490.
• Stephan, M., Bowers, J., Cobb, P., & Gravemeijer, K. (2003). Supporting students' development of measuring conceptions: Analyzing students' learning in social context. Reston, VA: National Council of Teachers of Mathematics.
• Stephan, M., Cobb, P., & Gravemeijer, K. (2003). Chapter 5: Coordinating social and individual analyses: Learning as participation in mathematical practices. Journal for Research in Mathematics Education. Monograph, 12, 67-102.
• Wheeldon, D. A. (2008). Developing mathematical practices in a social context: An instructional sequence to support prospective elementary teachers’ learning of fractions. Unpublished doctoral thesis. University of Central Florida, Orlando.
• Wood, T., Williams, G., & McNeal, B. (2006). Children's mathematical thinking in different classroom cultures. Journal for Research in Mathematics Education, 222-255.
• Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. The Journal of Mathematical Behavior, 21(4), 423-440.
• Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for research in mathematics education, 458-477.
• Zembaul-Saul, C. (2005, April). Pre-service teachers’ understanding of teaching elementary school science argument. Paper presented at the Annual Meeting of the National Association for Research in Science Teaching, Dallas.