Layers of generality and types of generalization in pattern activities
DOI:
https://doi.org/10.30827/pna.v4i2.6169Palabras clave:
Estratos de generalidad, Generalización, Objetivación, Pensamiento algebraico, Pensamiento aritmético, SemióticaResumen
Pattern generalization is considered one of the prominent routes for introducing students to algebra. However, not all generalizations are algebraic. In the use of pattern generalization as a route to algebra, we —teachers and educators— thus have to remain vigilant in order not to confound algebraic generalizations with other forms of dealing with the general. But how to distinguish between algebraic and non-algebraic generalizations? On epistemological and semiotic grounds, in this article I suggest a characterization of algebraic generalizations. This characterization helps to bring about a typology of algebraic and arithmetic generalizations. The typology is illustrated with classroom examples.
Niveles de generalidad y tipos de generalizaciones en actividades de patrones
Handle: http://hdl.handle.net/10481/3505
Nº de citas en WOS (2017): 13 (Citas de 2º orden, 7)
Nº de citas en SCOPUS (2017): 6 (Citas de 2º orden, 4)
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Biografía del autor/a
Luis Radford, Université Laurentienne, Canadá
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Arzarello, F., & Edwards, L. (2005). Gesture and the construction of mathematical meaning. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 123-154). Melbourne, Australia: University of Melbourne.
Arzarello, F., & Robutti, O. (2001). From body motion to algebra through graphing. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI Study Conference (Vol. 1, pp. 33-40). Melbourne, Australia: University of Melbourne.
Bakhtin, M. (1990). Art and Answerability. Austin, TX: University of Texas Press. DOI: https://doi.org/10.7560/704114
Barallobres, G. (2005). La validation intellectuelle dans l'enseignement introductif de l'algèbre. Recherche en Didactiques des Mathématiques, 24(2/3), 285-328.
Bardini, C., Radford, L., & Sabena, C. (2005). Struggling with variables, parameters, and indeterminate objects or how to go insane in mathematics. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 129-136). Melbourne, Australia: University of Melbourne.
Bartolini-Bussi, M. G. (1995). Analysis of classroom interaction discourse from a Vygotskian perspective. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 95-98). Pernambuco, Brazil: Universidade Federal de Pernambuco.
Bednarz, N., & Janvier, B. (1996). Emergence and development of algebra as a problem-solving tool: continuities and discontinuities with arithmetic. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra, perspectives for research and teaching (pp. 115-136). Dordrecht, The Netherlands: Kluwer. DOI: https://doi.org/10.1007/978-94-009-1732-3_8
Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 99-119). Dordrecht, The Netherlands: Kluwer. DOI: https://doi.org/10.1007/0-306-47223-6_6
Carraher, D., Schliemann, A., & Brizuela, B. (2001). Can young students operate on unknowns? In M. V. D. Heuvel-Panhuizen (Ed.), Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 130-140). Stellenbosch, South Africa: Freudenthal Institute.
Castro, E. (1995). Exploración de patrones numéricos mediante configuraciones puntuales. Granada, Spain: Mathema.
Duval, R. (2002). L'apprentissage de l'algèbre et le problème cognitif de la signification des objets. In J. P. Drouhard & M. Maurel (Eds.), Séminaire Franco-Italien de Didactique de l'Algèbre (Vol. XIII, pp. 67-94). Nice, France: IREM de Nice.
Filloy, E., & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19-25.
Gibson, J. J. (1966). The senses considered as perceptual systems. Boston, MA: Houghton Mifflin.
Goldin-Meadow, S. (2003). Hearing gesture. How our hands help us think. Cambridge, MA: The Belknap Press of Harvard University Press. DOI: https://doi.org/10.1037/e413812005-377
Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115-141. DOI: https://doi.org/10.2307/749505
Hegel, G. W. F. (1977). The phenomenology of spirit. Oxford, United Kingdom: Oxford University Press.
Hoopes, J. (Ed.). (1991). Peirce on signs. Chapel Hill: The University of North Carolina Press.
Høyrup, J. (2002). Lengths, widths, surfaces. A portrait of old Babylonian algebra and its kin. New York: Springer. DOI: https://doi.org/10.1007/978-1-4757-3685-4
Innis, R. E. (1985). Semiotics. An introductory anthology. Bloomington, IN: Indiana University Press.
Kant, I. (1929). Critique of pure reason (N. K. Smith, Trans.). New York, NY: St. Marin's Press. (Original work published in 1781 and 1787)
Kant, I. (1974). Logic. Indianapolis, IN: The Bobbs-Merrill Company.
Kaput, J., & Sims-Knight, J. (1983). Errors in translations to algebraic equations: roots and implications. Focus on Learning Problems in Mathematics, 5(3), 63-78.
Kendon, A. (2004). Gesture: Visible action as utterance. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511807572
Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317-326. DOI: https://doi.org/10.1007/BF00311062
Kieran, C. (1989). A perspective on algebraic thinking. In G Vernand, J., Rogalski, & M. Artigue (Eds.), Proceedings of the 13th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 163-171). Paris: Laboratoire PSYDEE.
Kita, S. (2003). Pointing. Where language, culture, and cognition meet. Mahwah, NJ: Lawrence Erlbaum. DOI: https://doi.org/10.4324/9781410607744
Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 87-106). Dordrecht, The Netherlands: Kluwer. DOI: https://doi.org/10.1007/978-94-009-1732-3_6
Lerman, S. (1996). Intersubjectivity in mathematics learning: a challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27(2), 133-150. DOI: https://doi.org/10.5951/jresematheduc.27.2.0133
Lins, R. (2001). The production of meaning for algebra: a perspective based on a theoretical model of semantic fields. In R. Sutherland, T. Rojano, A. Bell, and R. Lins, (Eds.), Perspectives on school algebra (pp. 37-60). Dordrecht, The Netherlands: Kluwer. DOI: https://doi.org/10.1007/0-306-47223-6_3
Love, E. (1986). What is algebra? Mathematics Teaching, 117, 48-50.
MacGregor, M., & Stacey, K. (1992). A comparison of pattern-based and equation-solving approaches to algebra. In B. Southwell, K. Owens, & B. Penny (Eds.), Proceedings of the 15th Annual Conference, Mathematics Education Research Group of Australasia (MERGA) (pp. 362-370). Sydney, Australia: MERGA.
MacGregor, M., & Stacey, K. (1995). The effect of different approaches to algebra on students' perception of functional relationships. Mathematics Education Research Journal, 7(1), 69-85. DOI: https://doi.org/10.1007/BF03217276
Martzloff, J. C. (1997). A history of Chinese mathematics. Berlin: Springer.
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 65-86). Dordrecht, The Netherlands: Kluwer. DOI: https://doi.org/10.1007/978-94-009-1732-3_5
Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Mathematical Behavior, 3(1), 93-166.
McNeill, D. (2000). Language and gesture. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511620850
Peirce, C. S. (1931-1958). CP = Collected Papers, vol. I-VIII. Cambridge, Mass: Harvard University Press.
Poincaré, H. (1968). La science et l'hypothèse. Paris: Flammarion.
Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press. DOI: https://doi.org/10.1515/9781400828678
Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205-235). Rotterdam, The Netherlands: Sense Publishers. DOI: https://doi.org/10.1163/9789087901127_009
Puig, L. (2004, July). History of algebraic ideas and research on educational algebra. Regular lecture presented at ICME-10, Copenhagen. Available at http://www.uv.es/puigl/
Radford, L. (2001). The historical origins of algebraic thinking. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives in school algebra (pp. 13-63). Dordrecht, The Netherlands: Kluwer. DOI: https://doi.org/10.1007/0-306-47223-6_2
Radford, L. (2002a). Algebra as tekhne. Artefacts, symbols and equations in the classroom. Mediterranean Journal for Research in Mathematics Education, 1(1), 31-56.
Radford, L. (2002b). On heroes and the collapse of narratives. A contribution to the study of symbolic thinking. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81-88). Norwich, United Kingdom: University of East Anglia.
Radford, L. (2002c). The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14-23.
Radford, L. (2003). Gestures, speech and the sprouting of signs. Mathematical Thinking and Learning, 5(1), 37-70. DOI: https://doi.org/10.1207/S15327833MTL0501_02
Radford, L. (2005a). The semiotics of the schema. Kant, Piaget, and the calculator. In M. H. G. Hoffmann, J. Lenhard, & F. Seeger (Eds.), Activity and sign. Grounding mathematics education (pp. 137-152). New York: Springer.
Radford, L. (2005b). Why do gestures matter? Gestures as semiotic means of objectification. In H. Chick & J. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 143-145). Melbourne, Australia: University of Melbourne.
Radford, L. (2006). The anthropology of meaning. Educational Studies in Mathematics, 61(1-2), 39-65. DOI: https://doi.org/10.1007/s10649-006-7136-7
Radford, L. (2008). The ethics of being and knowing: towards a cultural theory of learning. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: epistemology, history, classroom, and culture (pp. 215-234). Rotterdam, The Netherlands: Sense Publishers. DOI: https://doi.org/10.1163/9789087905972_013
Radford, L., & Demers, S. (2004). Communication et apprentissage. Repères conceptuels et pratiques pour la salle de classe de mathématiques. Ottawa, Canada: Centre Franco-Ontarien des Ressources Pédagogiques.
Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the general: the multisemiotic dimension of students' algebraic activity. Journal for Research in Mathematics Education, 38(5), 507-530.
Radford, L., & Puig, L. (2007). Syntax and meaning as sensuous, visual, historical forms of algebraic thinking. Educational Studies in Mathematics, 66(2), 145-164. DOI: https://doi.org/10.1007/s10649-006-9024-6
Rivera, F. (2006). Sixth Graders' ability to generalize patterns in algebra: issues and insights. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 320). Prague, Czech Republic: Faculty of Education, Charles University in Prague.
Robutti, O. (2009). Space-time representations in young children: Thinking through gestures and graphs. In C. Andersen, N. Scheuer, M. Echeverría, & E. Teubal (Eds.), Representational systems and practices as learning tools in different fields of knowledge (pp. 59-75). Rotterdam, The Netherlands: Sense Publishers. DOI: https://doi.org/10.1163/9789087905286_005
Roth, M. W. (2001). Gestures: their role in teaching and learning. Review of Educational Research, 71(3), 365-392. DOI: https://doi.org/10.3102/00346543071003365
Roth, M.-W. (2006). On the relation of abstract and concrete and the contradiction of a double ascension: a dialectical approach to mathematical learning. (Submitted).
Sabena, C., Radford, L., & Bardini, C. (2005). Synchronizing gestures, words and actions in pattern generalizations. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 129-136). Melbourne, Australia: University of Melbourne.
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), 1-36. DOI: https://doi.org/10.1007/BF00302715
Ursini, S., & Trigueros, M. (2001). A model for the uses of variable in elementary algebra. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 327-334). Utrecht, The Netherlands: Freudenthal Institute.
Vergnaud, G. (1996). Au fond de l'apprentissage, la conceptualisation. In R. Noirfalise & M. J. PerrinGlorian (coord.), Actes de l'École d'Été de Didactique des Mathématiques (174-185). Clermont-Ferrand, France: IREM de Clermont-Ferrand.
Vygotsky, L. S. (1962). Thought and language. Cambridge: MIT Press. DOI: https://doi.org/10.1037/11193-000
Wagner, S., & Kieran, C. (Eds.). (1989). Research issues in the learning and teaching of algebra. Virginia, VI: Lawrence Erlbaum & NCTM.
Warren, E. (2006). Teacher actions that assist young students write generalizations in words and in symbols. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 377-384). Prague, Czech Republic: Faculty of Education, Charles University in Prague.
Wartofsky, M. (1979). Models, representation and the scientific understanding. Dordrecht, The Netherlands: D. Reidel. DOI: https://doi.org/10.1007/978-94-009-9357-0_10
You, H. (1994). Defining rhythm: aspects of an anthropology of rhythm. Culture, Medicine and Psychiatry, 18, 361-384. DOI: https://doi.org/10.1007/BF01379231
