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Diagrammatic Reasoning as the Basis for Developing Concepts: A Semiotic Analysis of Students' Learning about Statistical Distribution

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Abstract

In recent years, semiotics has become an innovative theoretical framework in mathematics education. The purpose of this article is to show that semiotics can be used to explain learning as a process of experimenting with and communicating about one's own representations (in particular ‘diagrams') of mathematical problems. As a paradigmatic example, we apply a Peircean semiotic framework to answer the question of how students develop a notion of ‘distribution' in a statistics course by ‘diagrammatic reasoning' and by forming ‘hypostatic abstractions', that is by forming new mathematical objects which can be used as means for communication and further reasoning. Peirce's semiotic terminology is used as an alternative to concepts such as modeling, symbolizing, and reification. We will show that it is a precise instrument of analysis with regard to the complexity of learning and communicating in mathematics classrooms.

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References

  • Anderson, M., Sáenz-Ludlow, A., Zellweger, S. and Cifarelli, V.V. (eds.): 2003, Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing, Legas, Ottawa, Canada.

    Google Scholar 

  • Bakker, A.: 2004, Design Research in Statistics Education; On Symbolizing and Computer Tools, CD Beta, Utrecht, The Netherlands.

    Google Scholar 

  • Bakker, A. and Gravemeijer, K.P.E.: 2004, ‘Learning to reason about distribution’, in D. Ben-Zvi and J. Garfield (eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking, Kluwer Academic, Dordrecht, The Netherlands, pp. 147–168.

    Google Scholar 

  • Ben-Zvi, D. and Arcavi, A.: 2001, ‘Junior high school students' construction of global views of data and data representations’, Educational Studies in Mathematics 45, 35–65.

    Article  Google Scholar 

  • Cobb, P.: 1999, ‘Individual and collective mathematical development: The case of statistical data analysis’, Mathematical Thinking and Learning 1(1), 5–43.

    Google Scholar 

  • Cobb, P.: 2002, ‘Modeling, symbolizing, and tool use in statistical data analysis’, in K.P.E. Gravemeijer, R. Lehrer, B. van Oers and L. Verschaffel (eds.), Symbolizing, Modeling and Tool Use in Mathematics Education, Kluwer Academic, Dordrecht, The Netherlands, pp. 171–196.

    Google Scholar 

  • Cobb, P., Yackel, E. and McClain, K. (eds.): 2000, Symbolizing and Communicating in Mathematics Classrooms. Perspectives on Discourse, Tools, and Instructional Design, Erlbaum, Hillsdale, NJ.

    Google Scholar 

  • Dörfler, W.: 2000, ‘Means for meaning’, in P. Cobb, E. Yackel and K. McClain (eds.), Symbolizing and Communicating in Mathematics Classrooms. Perspectives on Discourse, Tools, and Instructional Design, Erlbaum, Hillsdale, NJ, pp. 99–131.

    Google Scholar 

  • Duval, R.: in press, ‘The cognitive analysis of problems of comprehension in the learning of mathematics’, in A. Sáenz-Ludlow and N. Presmeg (eds.), Special issue on Semiotic Perspectives on Epistemology and Teaching and Learning of Mathematics, Educational Studies in Mathematics 61(1–3).

  • Ernest, P.: in press, ‘A semiotic perspective of mathematical activity: The case of number’, in A. Sáenz-Ludlow and N. Presmeg (eds.), Special issue on Semiotic Perspectives on Epistemology and Teaching and Learning of Mathematics, Educational Studies in Mathematics 61(1–3).

  • Goldin, G.A. and Kaput, J.J.: 1996, ‘A joint perspective on the idea of representation in learning and doing mathematics’. In L.P. Steffe, P. Nesher, P. Cobb, G.A. Golding and B. Greer (eds.), Theories of Mathematical Learning, Erlbaum, Mahwah, NJ, pp. 397–430.

  • Gravemeijer, K., Cobb, P., Bowers, J. and Whitenack, J.: 2000, ‘Symbolizing, modeling and instructional design’, in P. Cobb, E. Yackel and K. McClain (eds.), Symbolizing and Communicating in Mathematics Classrooms, Erlbaum, Mahwah, NJ, pp. 225–273.

    Google Scholar 

  • Hancock, C., Kaput, J.J. and Goldsmith, L.T.: 1992, ‘Authentic enquiry with data: Critical barriers to classroom implementation’, Educational Psychologist 27(3), 337–364.

    Google Scholar 

  • Hitt, F. (ed.): 2002, Representations and Mathematics Visualization, Departamento de Matemática Educativa, Cinvestav-IPN, Mexico City, Mexico.

    Google Scholar 

  • Hoffmann, M.H.G. (ed.): 2000, ‘Lernen als Zeichenprozess’, Special IssueZeitschrift für Semiotik 22(1).

  • Hoffmann, M.H.G.: 2003, Peirce's “Diagrammatic reasoning” as a solution of the learning paradox, in G. Debrock (ed.), Process Pragmatism: Essays on a Quiet Philosophical Revolution, Rodopi, Amsterdam, pp. 121–143.

    Google Scholar 

  • Hoffmann, M.H.G.: in press-a, ‘Commentary paper: What is a “semiotic perspective”?’, in A. Sáenz-Ludlow and N. Presmeg (eds.), Educational Studies in Mathematics, Special Issue on “Semiotic Perspectives on Epistemology and Teaching and Learning of Mathematics”, 61(1–3).

  • Hoffmann, M.H.G.: 2004, ‘How to get it. Diagrammatic reasoning as a tool of knowledge development and its pragmatic dimension’, Foundations of Science 9(3), 285–305.

    Article  Google Scholar 

  • Hoffmann, M.H.G.: 2005, ‘Signs as means for discoveries. Peirce and his concepts of “diagrammatic reasoning”, “theorematic deduction”, “hypostatic abstraction”, and “theoric transformation”’, in M.H.G. Hoffmann, J. Lenhard and F. Seeger (eds.), Activity and Sign – Grounding Mathematics Education, Springer, New York, pp. 45–56.

    Google Scholar 

  • Hull, K.: 1994, ‘Why hanker after logic? Mathematical imagination, creativity and perception in Peirce's systematic philosophy’, Transactions of the Charles S. Peirce Society 30, 271–295.

    Google Scholar 

  • Janvier, C. (ed.): 1987, Problems of Representation in the Teaching and Learning of Mathematics, Erlbaum, Hillsdale, NJ.

    Google Scholar 

  • Kadunz, G.: 2003, ‘Geometrielernen mit computergenerierten Repräsentationen. Zur Bedeutung von Modulen’, in M.H.G. Hoffmann (ed.), Mathematik Verstehen -Semiotische Perspektiven, Franzbecker, Hildesheim, Germany, pp. 95–118.

    Google Scholar 

  • Konold, C. and Higgins, T.: 2003, ‘Reasoning about data’, in J. Kilpatrick, W.G. Martin and D. Schifter (eds.), A Research Companion to Principles and Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, VA, pp. 193–215.

    Google Scholar 

  • Noss, R. and Hoyles, C.: 1996, Windows on Mathematical Meaning, Kluwer Academic, Dordrecht, The Netherlands.

    Google Scholar 

  • Nöth, W.: 2000, Handbuch der Semiotik, Metzler, Stuttgart und Weimar.

    Google Scholar 

  • Otte, M.: 1997, ‘Analysis and synthesis in mathematics from the perspective of Charles S. Peirce's philosophy’, in M. Otte and M. Panza (eds.), Analysis and Synthesis in Mathematics. History and Philosophy, Kluwer, London, pp. 327–364.

    Google Scholar 

  • Otte, M.: in press, ‘Mathematical epistemology from a semiotic point of view’, in A. Sáenz-Ludlow and N. Presmeg (eds.), Special issue on Semiotic Perspectives on Epistemology and Teaching and Learning of Mathematics, Educational Studies in Mathematics 61(1–3).

  • Peirce, C.S. (CP). Collected Papers of Charles Sanders Peirce, Harvard Univeristy Press, Cambridge, MA.

  • Peirce, C.S. (EP). The Essential Peirce. Selected Philosophical Writings, Vol. 1 (1867–1893), Vol. 2 (1893–1913), Indiana University Press, Bloomington and Indianapolis 1992 + 1998, IN.

  • Peirce, C.S. (NEM). The New Elements of Mathematics by Charles S. Peirce, Vol. I–IV (1976), Mouton/Humanities, The Hague-Paris/Atlantic Highlands, NJ.

  • Peirce, C.S. (SEM). Charles S. Peirce, Semiotische Schriften (Vol. I–III) (1986–1994), Suhrkamp, Frankfurt a.M.

  • Petrosino, A.J., Lehrer, R. and Schauble, L.: 2003, ‘Structuring error and experimental variation as distribution in the fourth grade’, Mathematical Thinking and Learning 5(2 and 3), 131–156.

    Google Scholar 

  • Posner, R., Robering, K. and Sebeok, T.A. (eds.): 1997 ff., Semiotik. Semiotics: Ein Handbuch zu den zeichentheoretischen Grundlagen von Natur und Kultur, Vol. 1–3, de Gruyter, New York.

  • Presmeg, N.: in press, ‘Semiotics and the “connections” standard: Significance of semiotics for teachers of mathematics’ in A. Sáenz-Ludlow and N. Presmeg (eds.), Special issue on Semiotic Perspectives on Epistemology and Teaching and Learning of Mathematics, Educational Studies in Mathematics 61(1–3).

  • Radford, L.: 2000, ‘Signs and meanings in students' emergent algebraic thinking. A semiotic analysis’, Educational Studies in Mathematics 42(3), 237–268.

    Article  Google Scholar 

  • Radford, L.: in press, ‘On meaning, subjectivity, and mathematical objects’, in A. Sáenz-Ludlow and N. Presmeg (eds.), Educational Studies in Mathematics, Special Issue on “The Teaching and Learning of Mathematics: Semiotic and Epistemological Perspectives”.

  • Roth, W.-M.: 2003, Toward an Anthropology of Graphing: Semiotic and Activity-Theoretic Perspectives, Kluwer, Dordrecht, The Netherlands.

    Google Scholar 

  • Roth, W.-M. and McGinn, M.K.: 1998, ‘Inscriptions: Towards a theory of representing as social practice’, Review of Educational Research 68(1), 35–59.

    Google Scholar 

  • Sáenz-Ludlow, A.: 2003, ‘Classroom discourse in mathematics as an evolving interpreting game’, in M. Anderson, A. Sáenz-Ludlow, S. Zellweger and V.V. Cifarelli (eds.), Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing, Legas, Ottawa, Canada, pp. 253–281.

    Google Scholar 

  • Sáenz-Ludlow, A.: in press, ‘Classroom interpreting games as mediators in the construction of meaning’, in A. Sáenz-Ludlow and N. Presmeg (eds.), Educational Studies in Mathematics, Special Issue on “The Teaching and Learning of Mathematics: Semiotic and Epistemological Perspectives”.

  • Sáenz-Ludlow, A. and Presmeg, N. (eds.): in press, Special Issue on “The Teaching and Learning of Mathematics: Semiotic and Epistemological Perspectives”.

  • Saussure, F.d.: 1966(1916), Course in General Linguistics (Cours de Linguistique Generale), W. Baskin, Trans., McGraw-Hill, New York.

  • Seeger, F.: 1998, ‘Representations in the mathematical classroom: Reflections and constructions’, in F. Seeger, J. Voigt and U. Waschescio (eds.), The Culture of the Mathematics Classroom, Cambridge University Press, Cambridge, UP, pp. 308–343.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Simon, M.A.: 1995, ‘Reconstructing mathematics pedagogy from a constructivist perspective’, Journal for Research in Mathematics Education 26, 114–145.

    Google Scholar 

  • Steinbring, H.: in press, ‘What makes a sign a mathematical sign? – An epistemological perspective on mathematical interaction’ in A. Sáenz-Ludlow and N. Presmeg (eds.), Educational Studies in Mathematics, Special Issue on “The Teaching and Learning of Mathematics: Semiotic and Epistemological Perspectives”.

  • Stjernfelt, F.: 2000, ‘Diagrams as centerpiece of a Peircean epistemology’, Transactions of the Charles S. Peirce Society 36, 357–384.

    Google Scholar 

  • Van Someren, M., Reimann, P., Bozhimen, H. and De Jong, T. (eds.): 1998, Learning with Multiple Representations, Elsevier Science, Oxford, UK.

    Google Scholar 

  • Walkerdine, V.: 1988, The Mastery of Reason: Cognitive Development and the Production of Rationality, Routledge, London.

    Google Scholar 

  • Whitson, J.A.: 1997, ‘Cognition as a semiosic process: From situated mediation to critical reflective transcendence’, in D. Kirshner and J.A. Whitson (eds.), Situated Cognition: Social, Semiotic, and Psychological Perspectives, Erlbaum, Mahwah, NJ, pp. 97–149.

    Google Scholar 

  • Whitson, J.A.: 2003, Thinking, Learning, Knowing, Doing, and Becoming: Semiotic and Pragmatic Bases for an Alternative to “Transfer”, in Paper presented at the AERA, Chicago.

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Author notes

  1. Arthur Bakker

    Present address: School of Mathematics, Science and Technology Institute of Education, University of London, 23-29 Emerald Street, London, WC1N 3QS, United Kingdom

  2. Michael H. G. Hoffmann

    Present address: School of Public Policy Georgia Institute of Technology, D.M. Smitth Building 685 Cherry Street, Atlanta, Georgia, 30332-0345, U.S.A.

Authors and Affiliations

  1. Freudenthal Institute, Utrecht University, Aidadreef 12, NL 3561 GE, Utrecht, The Netherlands

    Arthur Bakker

  2. University of Victoria, Canada

    Michael H. G. Hoffmann

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  1. Arthur Bakker
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Correspondence to Arthur Bakker or Michael H. G. Hoffmann.

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Bakker, A., Hoffmann, M.H.G. Diagrammatic Reasoning as the Basis for Developing Concepts: A Semiotic Analysis of Students' Learning about Statistical Distribution. Educ Stud Math 60, 333–358 (2005). https://doi.org/10.1007/s10649-005-5536-8

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