Generalización de secuencias de patrón por estudiantes de años iniciales
límites y posibilidades
DOI:
https://doi.org/10.23925/1983-3156.2026.v28.e73217Palabras clave:
Generalización algebraica, Secuencia de patrón, Incógnita, Educación PrimariaResumen
El estudio investigó la influencia del tipo de secuencia de patrón y de la localización de la incógnita en resolución de tareas que requieren la generalización algebraica. Estudiantes de 1º a 5º año de Educación Primaria resolvieron un problema de patrón repetitivo y otro de patrón no repetitivo. Cada problema incluía tres preguntas relativas a la posición de la incógnita: inmediata, próxima y distante. En todos los años escolares, el desempeño fue mejor con el patrón repetitivo que con el patrón no repetitivo, y cuando la incógnita era inmediata, seguida de la próxima y de la distante. El mayor desafío, aún para 5° año, consistió en la resolución de problemas de patrón no repetitivo cuando la incógnita era distante. Se identificaron aspectos del desarrollo del pensamiento algebraico relacionados con la generalización, señalándose implicaciones educativas y cuestiones a ser investigadas en investigaciones sobre la generalización algebraica y la identificación de patrones.
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Amit, M., & Neria, D. (2007). ‘‘Rising to the challenge’’: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students. ZDM Mathematics Education, 40, 11–129.
Blanton, M., Schifter, D., Inge, V., Lofgren, P., Willis, C., Davis, F., & Confrey, J. (2007). Early Algebra. In V.J. Katz (Ed.), Algebra: Gateway to a Technological Future (pp. 7-14). New York: The Mathematical Association of America.
Blanton, M., Stroud, R., Stephens, A., Gardiner, A. M., Stylianou, D. A., Knuth, E., ... & Strachota, S. (2019). Does early algebra matter? The effectiveness of an early algebra intervention in grades 3 to 5. American Educational Research Journal, 56(5), 1930-1972.
Blanton, M., & Kaput, J (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 5-23). Berlin: Springer.
Blanton, M., & Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446.
Blanton, M., Schifter, D., Inge, V., Lofgren, P., Willis, C., & Davis, F. (2007), ‘Early algebra’, In V.J. Katz (Ed.), Algebra: Gateway to a technological future (pp. 7–14). New York: Mathematical Association of America.
Blanton, M., & Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. J. Hoines, & A. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 135-142). Bergen, Norway: International Group for the Psychology of Mathematics Education.
Borralho, A., Cabrita, I., Palhares, P., & Vale, I. (2007). Os padrões no ensino e aprendizagem da álgebra. In I. Vale, T. Pimentel, A. Barbosa, L. Fonseca, L. Santos, & P. Canavarro (Orgs.), Números e Álgebra (pp. 193-211). Lisboa: SEM-SPCE.
Brasil. (2018). Ministério da Educação. Secretaria de Educação Básica. Base Nacional Comum Curricular: Educação Infantil e Ensino Fundamental. Brasília, DF: MEC/SEB.
Canavarro, A. P. (2007). O pensamento algébrico na aprendizagem da Matemática nos primeiros anos. Quadrante, 16(2), 81-118
Carraher, D., & Blanton, M. (2007). Algebra in the Early Grades. London: Routledge.
Carraher, D., & Earnest, D. (2003). Guess my rule revisited. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the Twenty-seventh International Conference for the Psychology of Mathematics Education (vol. 2, pp. 173–180). Honolulu, Hawaii: University of Hawaii.
Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115.
Carraher, D.W., Schliemann, A.D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In J. Kaput, D. Carraher, & M. Blanton (Eds.). Algebra in the early grades (pp. 235-272). New York: Routledge.
Carraher D.W., Martinez M., & Schliemann A. D. (2008). Early algebra and mathematical generalization. ZDM Mathematics Education, 40, 3–22. https://doi.org/10.1007/s11858-007-0067-7
Carraher, D. W., & Schliemann, A. D. (2018). Cultivating early algebraic thinking. In C. Kieran (Ed.). Teaching and learning algebraic thinking with 5 – 12 year-olds. The Global Evolution of an Emerging Field of Research and Practice. ICME-13 Monographs (pp. 107-138). Chaim, Switzerland: Springer.
Carraher, D., Schliemann, A.D., Brizuela, B., & Earnest, D. (2016). Arithmetic and algebra in early mathematics education. In E.A. Silver, & P.A. Kenney (Eds). More Lessons Learned from Research: Volume 2. Helping All Students Understand Important Ma-thematics (pp. 109-122). NCTM.
Carraher, D.W. & A.D. Schliemann (2016). Powerful ideas in elementary school mathematics. In L. English, & D. Kirshner (Eds.) Handbook of International Research in Mathema-tics Education (pp. 191-218). New York: Taylor & Francis.
Erdogan, F. & Ay, S. (2022). Generalization, algebraic thinking, and pattern: An overview. In Ş. Durukan (Ed.), Education & Science 2022-IV (pp. 85-102). EFE Academy Publishing.
Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema, & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Erlbaum.
Kaput, J. J. (2008). ‘What is algebra? What is algebraic reasoning?’, in J. J.Kaput, D. Carraher, & M.L. Blanton (Eds.), Algebra in the early grades, (pp. 235–272). New York: Routledge.
Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: background, overar-ching dimensions, and new directions. ZDM Mathematics Education, 54, 1131–1150.
Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.
Lian, L. H. L., & Yew, W. T. (2011). Developing Pre-algebraic Thinking in Generalizing Repea-ting Pattern Using SOLO Model. US-China Education Review A 6, 774-780.
Lins, R., & Kaput, J. (2004). The early development of algebraic reasoning: The current state of the field. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra (pp. 47-70). Norwell, MA: Kluwer.
Mason, J. (1996). Expressing generality and roots of algebra. In: N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65–86). Dordrecht, The Netherlands: Kluwer.
Mason, J. (2018). How early is too early for thinking algebraically? In C. Kieran (Ed.). Tea-ching and learning algebraic thinking with 5 - 12 year-olds. The Global Evolution of an Emerging Field of Research and Practice. ICME-13 Monographs (pp. 329-350). Chaim, Switzerland: Springer.
Mason, J, Graham, A. Pimm, D. & Gowar, N. (1985). Routs to/ roots to algebra. Milton Keynes, UK: Open University.
Merlini, V.L., Spinillo, A.G. & Magina, S.M.P. (2025). O papel da localização da incógnita na resolução de problemas de função por estudantes do Ensino Fundamental, REnCiMa, 16 (3), 1-22.
Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33-49.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Papic, M. M., Mulligan, J. T., & Mitchelmore, M. C. (2011). Assessing the development of preschoolers' mathematical patterning. Journal for Research in Mathematics Education, 42(3), 237-268.
Du Plessis, J. (2018). Early algebra: Repeating pattern and structural thinking at foundation phase, South African Journal of Childhood Education, 8(2), a578. https:// doi.org/10.4102/sajce. v8i2.578
Radford, L. (2012). On the development of early algebraic thinking. PNA- Pensamiento Numérico Avanzado, 6(4), 117-133.
Radford, L. (2010). Layers of generality and types of generalization in pattern activities. PNA- Pensamiento Numérico Avanzado, 4(2), 37-62,
Radford, L., & Peirce, C. S. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, North American Chapter (Vol. 1, pp. 2-21). Mérida, México: Universidad Pedagógica Nacional.
Rodrigues, M., & Serra, P. (2015). Generalizing repeating patterns: a study with children aged four. The Eurasia Proceedings of Educational & Social Sciences (EPESS), Volume 2, Pages 81-95.
Santana, R. C. C. L., & Magina, S.M.P. (2025). Um estudo comparativo entre o desempenho de estudantes dos 4º e 6º anos na resolução de tarefas de sequência de padrões. Educação Matemática Debate, 9, 1-14.
Schliemann, A. D., Carraher, D., & Brizuela, B. (2007). Bringing out the algebraic character of arithmetic: from children’s ideas to classroom practice. Mahwah, New Jersey (NJ): Lawrence Erlbaum.
Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20(2), 147-164. doi:10.1007/BF00579460
Threlfall, J. (1999). Repeating patterns in the primary years. In A. Orton (Ed.), Patterns in the teaching and learning of mathematics (pp. 18-30). London: Cassell.
Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford, & A. P. Schulte (Eds.), Ideas of algebra, K-12, 1988 Yearbook (pp. 8–19). Reston, VA: National Council of Teachers of Mathematics
Vale, I. (2013). Padrões em contextos figurativos: um caminho para a generalização em matemática. REVEMAT Revista Eletrônica de Educação Matemática, 8 (2), 64-81.
Vale, I., & Barbosa, A. (2019). Pensamento algébrico: contributo da visualização na construção da generalização. Educação Matemática Pesquisa, 21 (3), 398-418.
Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert, & M. Behr (Eds.). Research agenda in mathematics education. Number Concepts and Operations in the Middle Grades (pp. 141–161). Hillsdale, N.J.: Lawrence Erlbaum.
Vergnaud, G. (1994). Multiplicative conceptual field: what and why? In H. Gershon, & J. Confrey (Eds.). The development of multiplicative reasoning in the learning of mathematics (pp. 41-59). Albany, NY: University of New York Press.
Vergnaud, G. (1997). The nature of mathematical concepts. In T. Nunes, & P. Bryant (Orgs.). Learning and teaching mathematics: An international perspective (pp. 5-28). Hove: Psychology Press.
Warren, E. (2005). Patterns supporting the development of early algebraic thinking: Building connections. Research, Theory and Practice, 2, 759-766.
Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379-402.
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Derechos de autor 2026 João Alberto Da Silva, Alina Spinillo, Vania Finholdt Angelo Leite, Vinicius Carvalho Beck

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