The role of the teacher in supporting imagery in understanding integers

Bu çalışma, yedinci sınıf matematik öğrencilerinin tamsayı kavramı ve işlemlerini daha iyi anlayabilmeleri için yapılan bir araştırmanın sonuçlarını sunmaktadır. Çalışmada özellikle uzman bir öğretmenin öğrencilerin yeni matematiksel kavramları anlamalarında geriye dönük olarak kullanabilecekleri mantıklı imgelemeyi geliştirmelerindeki rolü araştırılmıştır. Çalışmada Toulmin tartışma modeli öğrencilerin oluşturdukları imgelerin tüm sınıf tarafından kabul edilip ortak olarak kullanılıp kullanılmadığını analiz etmek amacıyla kullanılmıştır. Sonuçlar, öğretmenin kullanmış olduğu yöntemlerin öğrencilerin tamsayı problemlerini anlaması ve doğru çözmesinde olduğu kadar, fikirlerini iletmede de etkili olan imgelemelerin gelişiminde önemli bir rol oynadığını göstermiştir.

Tamsayıların anlaşılmasında öğretmenin imgelemeyi desteklemedeki rolü

This paper presents the results of a design experiment conducted in a 7th grade mathematics classroom aimed at improving students’ understanding of integer concepts and operations. The study particularly focuses on an expert teacher’s role in helping students develop meaningful imagery which students can use as a foundation to fold back and rely on as they engage in further mathematical activities. Toulmin’s model of argumentation is used as an analytical tool to document when an image becomes taken-as-shared by the classroom community. The results suggest that the practices of the teacher played an important role in students’ development of various images in understanding and solving integer problems meaningfully as well as communicating their ideas effectively.

PDF

___

Battista, M. T. (1983). A complete model for operations on integers. Arithmetic Teacher, 30(9), 26-31.
Cobb, P., & McClain K. (2001). An approach for supporting teachers’ learning in social context. In F. Lin & T. J. Cooney (Eds.), Making sense of mathematics teacher education (pp. 27-231). Netherlands: Kluwer Academic Publishers.
Confrey, J. (1990): What constructivism implies for teaching. In: R. B. Davis, C. A. Maher & N.Noddings (Eds.), Constructivist views on the teaching and learning of mathematics (pp. 107–122). Reston, VA: National Council of Teachers of Mathematics.
Dirks, M (1984). The integer abacus. Arithmetic Teacher, 31(7), 50-54.
diSessa, A. A., & Sherin, B. L. (2000). Meta-representation: An introduction. Journal of Mathematical Behavior, 19(4), 385–398.
Glaser, B., & Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine.
Gravemeijer, K. (1994). Developing realistic mathematics education. Utrecht, Netherlands: CD-Press.
Gallardo, A., & Romero, M. (1999). Identification of difficulties in addition and subtraction of integers in the number line. In F. Hitt, & M. Santos (Eds.), Proceedings of the Twenty-first International Conference for the Psychology of Mathematics Education (Vol. I. pp. 275–282). North American Chapter, Mexico.
Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 49, 171–192.
Hayes, R. (1999). Teaching Negative Number Using Integer Tiles, 22nd Annual Conference of the Mathematics Education Research Group of Australasia (MERGA), University of Adelaide, Adelaide, SA.
Hayes, M. (2002). Elementary preservice teachers’ struggles to define inquiry-based science teaching. Journal of Science Teacher Education, 13, 147-165.
Janvier, C. (1985). Comparison of models aimed at teaching signed integers. Proceedings of the Nineth Meeting of the PME. State University of Utrecht, The. Netherlands, 135-140.
Kinach, B. M. (2002). A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: toward a model of effective practice. Teaching and Teacher Education, 18(1), 51-71.
Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work. Educational Psychologist, 41(2), 75-86.
Linchevski, L., & Williams, J. D. (1999). Using intuition from everyday life in “filling” the gap in children’s extension of their number concept to include the negative numbers. Educational Studies in Mathematics, 39, 131–147.
Leinhardt, G., & Smith, D. (1985). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 77(3), 247-271.
Lytle, P. (1994). Investigation of a Model Based on the Neutralization of Opposites to Teach Integer Addition and Subtraction. In Proceedings of the 18th International Group for the Psychology of Mathematics Education, Vol. 3, pp. 192-199) Concordia University, West Montreal, Canada.
Maccini, P., & Ruhl, K. L. (2000). Effects of a graduated instructional sequence on the algebraic subtraction of integers by secondary students with learning disabilities. Education and Treatment of Children, 23, 465–489.
Mukhopadhyay, S., Resnick, L. B., & Schauble, L. (1990). Social sense-making in mathematics;Children’s ideas of negative numbers. In G. Booker, P. Cobb, & T. N. de Mendicuti (Eds.), Proceedings of the 14th international conference for the Psychology in Mathematics Education, Vol. 3, pp. 281-288, Oaxtepec, Mexico: Conference Committee.
McClain, K., & Cobb, P. (1998). The role of imagery and discourse in supporting students’ mathematical development. In M. Lampert & M. L. Blunk (Eds.), Talking mathematics in school: Studies of teaching and learning (pp. 56–81). New York: Cambridge University Press.
Pinto, M. M. F, & Tall, D. O. (1999). Student constructions of formal theory: giving and extracting meaning. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of PME, Haifa, Israel,4, 65–73.
Pirie, S., & Kieren, T. (1989). A recursive theory of mathematical understanding. For the Learning of Mathematics, 9(3), 7–11.
Presmeg, N.C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23, 595–610.
Rasmussen, C., & Stephan, M. (2008). A Methodology for Documenting Collective Activity. In A. E. Kelly, R. A. Lesh & J. Y. Baek (Eds.), Handbook of design research in methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 195-215). New York and London: Routledge.
Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129-184.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.
Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. Journal of Mathematical Behavior, 21, 459–490.
Stephan, M. (2003). Reconceptualizing Linear Measurement Studies: The Development of Three Monograph Themes. In M. Stephan, J. S. Bowers, P. Cobb & K. P. E. Gravemeijer (Eds.),Supporting students’ development of measuring conceptions: Analyzing students’ learning in social context (Journal for Research in Mathematics Education, Monograph number 12, pp.17-35). Reston, VA: National Council of Teachers of Mathematics.
Stephan, M. (2009). What are you worth? Mathematics Teaching in Middle School, 15(1), 16-23.
Thompson, P. W., & Dreyfus, T. (1988). Integers as transformations. Journal for Research in Mathematics Education, 19(2), 115-133.
Thompson, P. W. (1992). Notations, conventions, and constraints: Contributions to effective uses of concrete materials in elementary mathematics. Journal for Research in Mathematics Education, 23(2), 123-147.
Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.), Theories of mathematical learning (267–285). Mahwah, NJ: Erlbaum.
Vlassis, J. (2001). Solving equations with negatives or crossing the formalizing gap. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the twenty-fifth international conference for the psychology of mathematics education (Vol. 4, pp. 375–382). Utrecht, Netherlands.