Growing patterns invention by primary education students

References

• Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241. https://doi.org/10.1023/A:1024312321077
   Google Scholar
• Ayala-Altamirano, C., & Molina, M. (2021). Fourth-graders’ justifications in early algebra tasks involving a functional relationship. Educational Studies in Mathematics, 107(2), 359–382. https://doi.org/10.1007/s10649-021-10036-1
   Web of Science ®Google Scholar
• Baumanns, L., & Rott, B. (2022). The process of problem posing: development of a descriptive phase model of problem posing. Educational Studies in Mathematics, 110, 251–269. https://doi.org/10.1007/s10649-021-10136-y
   Web of Science ®Google Scholar
• Blanton, M. L. (2017). Algebraic reasoning in Grades 3–5. In M. Battista (Ed.), Reasoning and sense making in grades 3–5 (pp. 67–102). NCTM.
   Google Scholar
• Blanton, M. L., & Kaput, J. J. (2011). Functional thinking as a route into algebra in the elementary grades. In J. Cai & E. Knuth (Eds.), Early algebraization, advances in mathematics education: A global dialogue from multiple perspective (pp. 5–23). Springer. https://doi.org/10.1007/978-3-642-17735-4_2
   Google Scholar
• Cai, J., & Hwang, S. (2020). Learning to teach through mathematical problem posing: Theoretical considerations, methodology, and directions for future research. International Journal of Educational Research, 102, 101391. https://doi.org/10.1016/j.ijer.2019.01.001
   Web of Science ®Google Scholar
• Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. Educational Studies in Mathematics, 83(1), 57–69. https://doi.org/10.1007/s10649-012-9429-3
   Web of Science ®Google Scholar
• Cañadas, M. C., Molina, M., & del Río, A. (2018). Meanings given to algebraic symbolism in problem-posing. Educational Studies in Mathematics, 98(1), 19–37. https://doi.org/10.1007/s10649-017-9797-9
   Web of Science ®Google Scholar
• Castro-Rodriguez, E., & Castro, E. (2016). Pensamiento lógico-matemático. In E. Castro & E. Castro (Eds.), Enseñanza y aprendizaje de las matemáticas en educación infantil (pp. 87–108). Pirámide.
   Google Scholar
• Dörfler, W. (2008). En route from patterns to algebra: Comments and reflections. ZDM, 40(1), 143–160. https://doi.org/10.1007/s11858-007-0071-y
   Google Scholar
• Fernández-Millán, E., & Molina, M. (2017). Secondary students’ implicit conceptual knowledge of algebraic symbolism. An exploratory study through problem posing. International Electronic Journal of Mathematics Education, 12(3), 799–826. https://doi.org/10.29333/iejme/649
   Google Scholar
• Fujita, T., & Yamamoto, S. (2011). The development of children’s understanding of mathematical patterns through mathematical activities. Research in Mathematics Education, 13(3), 249–267. https://doi.org/10.1080/14794802.2011.624730
   Google Scholar
• Hale, M. W. (2005). Using the standards. Algebra, Grade 4. Frank Schaffer Publications.
   Google Scholar
• Hunter, J., & Miller, J. (2022). The use of cultural contexts for patterning tasks: Supporting young diverse students to identify structures and generalise. ZDM – Mathematics Education, 54(6), 1349–1362. https://doi.org/10.1007/s11858-022-01386-y
   Google Scholar
• Lüken, M. M. (2012). Young children’s structure sense. Journal Für Mathematik-Didaktik, 33(2), 263–285. https://doi.org/10.1007/s13138-012-0036-8
   Google Scholar
• McGarvey, L. M. (2012). What is a pattern? Criteria used by teachers and young children. Mathematical Thinking and Learning, 14(4), 310–337. https://doi.org/10.1080/10986065.2012.717380
   Web of Science ®Google Scholar
• Ministerio de Educación y Formación Profesional. (2022). Real Decreto 157/2022 de 01 de marzo, por el que se establece la ordenación y enseñanzas mínimas de la Educación Primaria [Royal Decree 157/2022 of March 01, which establishes the basic curriculum of primary education]. BOE, 52, 24386-24504.
   Google Scholar
• Morales, R., Cañadas, M. C., & Castro, E. (2017). Creation and continuation of patterns by two 6-7 year-old students in sequences task. PNA. Revista de Investigación en Didáctica de la Matemática, 11(4), 233–252. https://doi.org/10.30827/pna.v11i4.6241
   Google Scholar
• Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49. https://doi.org/10.1007/BF03217544
   Google Scholar
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. NGA & CCSSO.
   Google Scholar
• Ontario Ministry of Education and Training. (2020). The Ontario curriculum grades 1–8: Mathematics. Ministry of Education.
   Google Scholar
• Orton, J., Orton, A., & Roper, T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 121–136). Continuum.
   Google Scholar
• Outhred, L., & y Mitchelmore, M. (2000). Young children's intuitive understanding of rectangular area measurement. En Journal for Research in Mathematics Education, 31(2), 168–190.
   Web of Science ®Google Scholar
• 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. https://doi.org/10.5951/jresematheduc.42.3.0237
   Web of Science ®Google Scholar
• Pasnak, R. (2017). Empirical studies of patterning. Psychology (Savannah, Ga ), 08(13), 2276–2293. https://doi.org/10.4236/psych.2017.813144
   Google Scholar
• Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective. Research in Mathematics Education, 12(1), 1–19. https://doi.org/10.1080/14794800903569741
   Google Scholar
• Radford, L. (2011). Embodiment, perception and symbols in the development of early algebraic thinking. In B. Ubuz (Ed.), 35th conference of the international group for the psychology of mathematics education developing mathematical thinking (Vol. 4, pp. 17–24). PME.
   Google Scholar
• Rico, L. (2009). Sobre las nociones de representación y comprensión en la investigaciónen educación matemática. PNA, 4(1), 1–14.
   Google Scholar
• Rittle-Johnson, B., Fyfe, E. R., Hofer, K. G., & Farran, D. C. (2017). Early math trajectories: Low-income children’s mathematics knowledge from ages 4 to 11. Child Development, 88(5), 1727–1742. https://doi.org/10.1111/cdev.12662
   PubMed Web of Science ®Google Scholar
• Rivera, F. (2013). Teaching and learning patterns in school mathematics: Psychological and pedagogical considerations. Springer.
   Google Scholar
• Rivera, F., & Rossi-Becker, J. (2016). Middle school students’ patterning performance on semi-free generalization tasks. The Journal of Mathematical Behavior, 43(1), 53–69. https://doi.org/10.1016/j.jmathb.2016.05.002
   Google Scholar
• Stoyanova, E. (2000). Empowering students’ problem solving via problem posing: The art of framing ‘good’ questions. Australian Mathematics Teacher, 56(1), 33–37.
   Google Scholar
• Torres, M. D., Moreno, A., Vergel, R., & Cañadas, M. C. (2023). The evolution from “I think it plus three” towards “I think it is always plus three.” transition from arithmetic generalization to algebraic generalization. International Journal of Science and Mathematics Education. https://doi.org/10.1007/s10763-023-10414-6
   Web of Science ®Google Scholar
• Ubilla, C., & Cerda, V. (2020). Texto del Estudiante Matemática 2^o básico. Santillana.
   Google Scholar
• Warren, E., & Cooper, T. J. (2008). Patterns that support early algebraic thinking in elementary school. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 113–126). NCTM.
   Google Scholar
• Wijns, N., Torbeyns, J., De Smedt, B., & Verschaffel, L. (2019). Young children’s patterning competencies and mathematical development: A review. In K. M. Robinson, H. P. Osana, & D. Kotsopoulos (Eds.), Mathematical learning and cognition in early childhood (pp. 139–161). Springer International Publishing. https://doi.org/10.1007/978-3-030-12895-1_9
   Google Scholar
• Wijns, N., Verschaffel, L., De Smedt, B., & Torbeyns, J. (2021). Associations between repeating patterning, growing patterning, and numerical ability: A longitudinal panel study in 4- to 6-year olds. Child Development, 92(4), 1354–1368. https://doi.org/10.1111/cdev.13490
   PubMed Web of Science ®Google Scholar
• Wilkie, K. J. (2022). Generalization of quadratic figural patterns: Shifts in student noticing. The Journal of Mathematical Behavior, 65, 100917. https://doi.org/10.1016/j.jmathb.2021.100917
   Web of Science ®Google Scholar
• Yeo, J. B. W. (2017). Development of a framework to characterise the openness of mathematical tasks. International Journal of Science and Mathematics Education, 15(1), 175–191. https://doi.org/10.1007/s10763-015-9675-9
   Web of Science ®Google Scholar