Généralisation des séquences de motifs par des élèves des premières années
limites et possibilités
DOI :
https://doi.org/10.23925/1983-3156.2026.v28.e73217Mots-clés :
Généralisation algébrique, Séquence de motifs, Inconnue, École primaireRésumé
L'étude a examiné l'influence du type de séquence de motifs et de la position de l'inconnu sur la résolution de tâches nécessitant une généralisation algébrique. Les élèves du CP au CM2 devaient résoudre un problème de séquence avec un motif répétitif et un autre avec un motif non répétitif. Chaque problème comportait trois questions relatives à la position de l'inconnu : immédiate, proche et lointaine. Dans tous les niveaux, les performances étaient meilleures dans la séquence de motifs répétitifs que dans la séquence de motifs non répétitifs, et ce, lorsque l'inconnu était immédiate, suivie de la proche, puis de la lointaine. Le plus grand défi, même en CM2, était de résoudre des problèmes de motifs non répétitifs lorsque l'inconnu était lointain. Les aspects du développement de la pensée algébrique liés à la généralisation ont été identifiés, et les implications pédagogiques et les questions pour des recherches ultérieures sur la généralisation algébrique et l'identification de motifs ont été soulignées.
Téléchargements
Références
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.
Téléchargements
Publiée
Comment citer
Numéro
Rubrique
Licence
© João Alberto Da Silva, Alina Spinillo, Vania Finholdt Angelo Leite, Vinicius Carvalho Beck 2026
Ce travail est disponible sous licence Creative Commons Attribution - Pas d’Utilisation Commerciale 4.0 International.
Les auteurs qui publient dans EMP acceptent les conditions suivantes:
- Attribution — Vous devez créditer l'Œuvre, intégrer un lien vers la licence et indiquer si des modifications ont été effectuées à l'Oeuvre. Vous devez indiquer ces informations par tous les moyens raisonnables, sans toutefois suggérer que l'Offrant vous soutient ou soutient la façon dont vous avez utilisé son Oeuvre.
- Pas d’Utilisation Commerciale — Vous n'êtes pas autorisé à faire un usage commercial de cette Oeuvre, tout ou partie du matériel la composant.
- Pas de modifications — Dans le cas où vous effectuez un remix, que vous transformez, ou créez à partir du matériel composant l'Oeuvre originale, vous n'êtes pas autorisé à distribuer ou mettre à disposition l'Oeuvre modifiée.
