The distributed nature of pattern generalization

Autores/as

  • Ferdinand Rivera San José State University, Estados Unidos

DOI:

https://doi.org/10.30827/pna.v9i3.6102

Palabras clave:

Estructuras matemáticas, Generalización de patrones, Pensamiento algebraico, Pensamiento matemático, Vista distribuida del procesamiento de la generalización de patrones

Resumen

Drawing on a review of recent work conducted in the area of pattern generalization (PG), this paper makes a case for a distributed view of PG, which basically situates processing ability in terms of convergences among several different factors that influence PG. Consequently, the distributed nature leads to different types of PG that depend on the nature of a given PG task and a host of cognitive, sociocultural, classroom-related, and unexplored factors. Individual learners draw on a complex net of parallel choices, where every choice depends on the strength of ongoing training and connections among factors, with some factors appearing to be more predictable than others.

La naturaleza distribuida de la generalización de patrones

Sobre la base de una revisión de trabajos recientes en el área de generalización de patrones (PG), este artículo aboga por una visión distribuida de PG, que básicamente sitúa la capacidad de procesamiento en términos de convergencias entre diferentes factores que influyen en PG. En consecuencia, la naturaleza distribuida conduce a diferentes tipos de PG que dependen de la naturaleza de una tarea PG dada y una serie de factores cognitivos, socioculturales, inexplorados y relacionadas con el aula. Alumnos individuales se basan en una compleja red de opciones paralelas, donde cada elección depende de la fortaleza de la formación continua y las conexiones entre los factores, con algunos factores más predecibles que otros.

Handle: http://hdl.handle.net/10481/34989

WOS-ESCI

Nº de citas en WOS (2017): 1 (Citas de 2º orden, 1)

Descargas

Citas

Ainley, J., Wilson, K., & Bills, L. (2003). Generalizing the context and generalizing the calculation. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th International Conference of the Psychology of Mathematics Education (PME) (Vol. 2, pp. 9-16). Honolulu, HI: PME.

Bastable, V., & Schifter, D. (2008). Classroom stories: Examples of elementary students engaged in early algebra. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 165-184). New York, NY: Erlbaum.

Britt, M., & Irwin, K. (2011). Algebraic thinking with and without algebraic representation: A pathway for learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 137-160). Dordrecht, The Netherlands: Springer.

Carraher, D., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematical generalization. ZDM, 40(1), 3-22.

Clements, D., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Erlbaum.

Cooper, T., & Warren, E. (2011). Years 2 to 6 students' ability to generalize: Models, representations, and theory for teaching and learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187-214). Dordrecht, The Netherlands: Springer.

Ellis, A. (2007). Connections between generalizing and justifying: Students' reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194-229.

Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 184-212). Westport, CT: Greenwoood Press.

Holland, J., Holyoak, K., Nisbett, R., & Thagard, P. (1986). Induction: Processes of inference, learning, and discovery. Cambridge, MA: The MIT Press.

Iwasaki, H., & Yamaguchi, T. (1997). The cognitive and symbolic analysis of the generalization process: The comparison of algebraic signs with geometric figures. In E. Pehkonnen (Ed.), Proceedings of the 21st Annual Conference of the Psychology of Mathematics Education (Vol. 3, pp. 105-113). Lahti, Finland: PME.

Küchemann, D. (2010). Using patterns generically to see structure. Pedagogies, 5(3), 233-250.

Lee, L., & Freiman, V. (2004). Tracking primary students' understanding of patterns. In D. McDougall & J. Ross (Eds.), Proceedings of the 26th Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PMENA) (Vol. 2, pp. 245-251). Toronto, Canada: PMENA.

MacGregor, M., & Stacey, K. (1992). A comparison of pattern-based and equation-solving approaches to algebra. In B. Southwell, K. Owens, & B. Perry (Eds.), Proceedings of fifteenth Annual Conference of the Mathematics Education Research Group of Australasia (pp. 362-371). Brisbane, Australia: Mathematics Education Research Group of Australasia.

Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structures for all. Mathematics Education Research Journal, 21(2), 10-32.

Mulligan, J. T., Prescott, A., & Mitchelmore, M. C. (2004). Children's development of structure in early mathematics. In M. Høines & A. Fuglestad (Eds.), Proceedings of the 28th annual conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 393-401). Bergen, Norway: Bergen University College.

Nathan, M., & Kim, S. (2007). Pattern generalization with graphs and words: A cross-sectional and longitudinal analysis of middle school students' representational fluency. Mathematical Thinking and Learning, 9(3), 193-219.

National Research Council (2013). The mathematical sciences in 2025. Washington, DC: The National Academies Press.

Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33(2), 203-233.

Papic, M., Mulligan, J. T., & Mitchelmore, M. (2009). The growth of mathematica patterning strategies in preschool children. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd conference of the International Groupfor the Psychology of Mathematics Education (Vol. 4, pp. 329-336). Thessaloniki, Greece: PME.

Papic, M., Mulligan, J. T., & Mitchelmore, M. (2011). Assessing the development of presechoolers' mathematical patterning. Journal for Research in Mathematics Education, 42(3), 237-268.

Plaut, D., McClelland, J., Seindenberg, M., & Patterson, K. (1996). Understanding normal and impaired word reading: Computational principles in quasi-regular domains. Psychological Review, 103(1), 56-115.

Radford, L. (1999). The rhetoric of generalization: A cultural semiotic approach to students' processes of symbolizing. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 89-96). Haifa, Israel: PME.

Radford, L. (2000). Students' processes of symbolizing in algebra: A semiotic analysis of the production of signs in generalizing tasks. In T. Nakahara and M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 81-88). Hiroshima, Japan: PME.

Radford, L. (2001a). Signs and meanings in students' emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237-268.

Radford, L. (2001b). Factual, contextual, and symbolic generalizations in algebra. In M. V. D. Hueuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 81-88). Utrecht, The Netherlands: PME.

Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students' types of generalization. Mathematical Thinking and Learning, 5(1), 37-70.

Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. Cortina, M. Saiz, & A. Mendez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME) (Vol. 1, pp. 2-21). Mexico, DF: PME.

Rivera, F. (2010). Second grade students' preinstructional competence in patterning activity. In M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 81-88). Belo Horizante, Brazil: PME.

Rivera, F. (2013). Teaching and learning patterns in school mathematics: Psychological and pedagogical considerations. New York, NJ: Springer.

Rivera, F., & Becker, J. (2011). Formation of pattern generalization involving linear figural patterns among middle school students: Results of a three-year study. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (Advances in mathematics education Vol. 2) (pp. 323-366). New York, NY: Springer.

Samson, D. (2011). Capitalizing on inherent ambiguities in symbolic expressions of generality. Australian Mathematics Teacher, 67(1), 28-32.

Samson, D., & Schäfer, M. (2011). Enactivism, figural apprehension, and knowledge objectification: An exploration of figural pattern generalization. For the Learning of Mathematics, 31(1), 37-43.

Stacey‚ K.‚ & MacGregor‚ M. (2001). Curriculum reform and approaches to algebra. In R. Sutherland‚ T. Rojano‚ A. Bell‚ & R. Lins. (Eds.)‚ Perspectives on school algebra (pp. 141-154). Dordrecht‚ The Netherlands: Kluwer Academic.

Steele, D., & Johanning, D. (2004). A schematic-theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57(1), 65-90.

Vale, I., & Pimentel, T. (2010). From figural growing patterns to generalization: A path to algebraic thinking. In M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 241-248). Belo Horizante, Brazil: PME.

Watson, A. (2009). Thinking mathematically, disciplined noticing, and structures of attention. In S. Lerman & B. Davis (Eds.), Mathematical action & structures of noticing (pp. 211-222). Rotterdam, The Netherlands: Sense Publishers.

Yerushalmy, M. (1993). Generalization in geometry. In J. Schwartz, M. Yerushalmy, & B. Wilson (Eds.), The geometric supposer: What is it a case of? (pp. 57-84). Hillsdale, NJ: Erlbaum.

Yevdokimov, O. (2008). Making generalizations in geometry: Students' views on the process. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PMENA XXX (Vol. 4, pp. 193-200). Morelia, Mexico: Cinvestav-UMSNH and PME.

Zazkis, R., Liljedahl, P., & Chernoff, E. (2008). The role of examples in forming and refuting generalizations. ZDM, 40(1), 131-141.

Descargas

Publicado

2015-03-01