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Artículos

Vol. 21 No. 1 (2018): March

HOW PRIMARY EDUCATION STUDENTS SOLVE PATTERN GENERALIZATION PROBLEMS. A LEARNING JOURNEY

DOI
https://doi.org/10.12802/relime.18.2114
Submitted
November 4, 2022
Published
2018-03-10

Abstract

The objective of this research is to study the way in which Primary Education students solve pattern generalization problems. For this, the level of success, the strategies used and the progression of 106 students in 3rd, 4th, 5th and 6th years of Primary Education have been analyzed by solving a pattern generalization problem. The results obtained show that the most successful students start with additive strategies and then switch to functional strategies, that reversing the process presents a higher cognitive demand and that very few students are able to express the general rule algebraically. The analysis of student progression has allowed us to define descriptors of a learning trajectory that help diagnose students' understanding and describe their progress in terms of growth through ten levels of development.

References

  1. Battista, M. (2004). Applying cognition-based assessment to elementary school students’ development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6 (2), 185-204. doi: 10.1207/s15327833mtl0602_6
  2. Callejo, M. L., y Zapatera, A. (2014). Flexibilidad en la resolución de problemas de identificación de patrones lineales en estudiantes de educación secundaria. BOLEMA, 28 (48), 64-88. doi:10.1590/1980-4415v28n48a04
  3. Cañadas, M. C., Castro, E., y Castro, E. (2007). Patrones, generalización y estrategias inductivas de estudiantes de 3º y 4º de la ESO en el problema de las baldosas. En M. Camacho, P. Flores & P. Bolea (Eds.), Investigación en Educación Matemática XI (pp. 283-294). Tenerife: SEIEM.
  4. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.
  5. Carraher, D. W., & Schliemann, A. D. (2015). Powerful ideas in elementary school mathematics. In L. D. English y D. Kirshner (Eds.), Handbook of international research in mathematics education (3rd ed., pp. 191-218). New York: Taylor & Francis.
  6. Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM. Mathematics Education, 40, 3-22. doi: 10.1007/s11858-007-0067-7
  7. Clements, D. H., & Sarama, J. (2014). Learning and teaching early math: The learning trajectories approach. London: Routledge.
  8. Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6 (2), 81-89. doi: 10.1207/s15327833mtl0602_1 De Bock, D., Van Dooren, W.,
  9. Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors. Educational Studies in Mathematics, 50 (3), 311-334. doi: 10.1023/A:1021205413749
  10. English, L. D., & Warren, E. A. (1998). Introducing the variable through pattern exploration. Mathematics Teacher, 91 (2), 166-170.
  11. García Cruz, J. A. (1998). El proceso de generalización desarrollado por alumnos de secundaria en problemas de generalización lineal (tesis doctoral no publicada). Universidad de la Laguna, Santa Cruz de Tenerife, España.
  12. Gómez, P., y Lupiáñez, J. L. (2006). Trayectorias hipotéticas de aprendizaje en la formación inicial de profesores de matemáticas de secundaria. PNA, 1 (2), 79-98.
  13. Gravemeijer, K. (2004). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6 (2), 105-128. doi:10.1207/s15327833mtl0602_3
  14. Kaput, J. (2000). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. Dartmouth, MA: National Center for Improving Student Learning and Achievement in Mathematics and Science. doi: 10.17226/6286
  15. Kaput, J., & Blanton, M. (2001). Student achievement in algebraic thinking: A comparison of thirdgraders’ performance on a state fourth-grade assessment. In R. Speiser, C. Maher, & C. Walter (Eds.), Proceedings of the 23rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 99-107). Utah, USA: PME.
  16. 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, France: Laboratoire PSYDEE.
  17. Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. Cham: Springer. doi: 10.1007/978-3-319-32258-2
  18. Krems, J. F. (1995). Cognitive flexibility and complex problem solving. Complex problem solving: The European perspective, 201-218.
  19. Lee, L. (1996). An initiation into algebraic culture through generalization activities. In Approaches to algebra (pp. 87-106). Dordrecht: Springer. doi: 10.1007/978-94-009-1732-3_6
  20. Mason, J., Burton, L., y Stacey, K. (1992). Pensar matemáticamente. Barcelona: Editorial Labor.
  21. Molina, M. (2009). Una propuesta de cambio curricular: integración del pensamiento algebraico en Educación Primaria. PNA, 3 (3), 135-156.
  22. Mouhayar, R. R., & Jurdak, M. E. (2012). Teachers’ ability to identify and explain students’ actions in near and far figural pattern generalization tasks. Educational Studies in Mathematics, 82 (3), 379-396. doi: 10.1007/s10649-012-9434-6
  23. Orton, A., & Orton, J. (1994). Students’ perception and use pattern and generalization. In J. P. Ponte & J. F. Matos (Eds.), Proceedings of the 8thInternational Conference for the Psychology, of Mathematics Education (Vol. 3, pp. 407-414). Lisbon: University of Lisbon.
  24. Radford, L. (2008). Iconicity and contraction: a semiotic investigation of forms of algebraic generalizations of patterns in different contexts. The International Journal on Mathematics Education (ZDM), 40 (1), 83-96. doi: 10.1007/s11858-007-0061-0
  25. Radford, L. (2011). Embodiment, perception and symbols in the development of early algebraic thinking. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 17-24). Ankara, Turkey: PME.
  26. Rico, L. (1997). Consideraciones sobre el currículo de matemáticas para educación secundaria. En L. Rico (Coord.), La educación matemática en la enseñanza secundaria (pp. 15-38). Barcelona: Horsori.
  27. Rivera, F. D., & Becker, J. R. (2007). Abduction-induction (generalization) processes of elementary majors on figural patterns in algebra. Journal of Mathematical Behavior, 26 (2), 140-155. doi:10.1016/j.jmathb.2007.05.001
  28. Rivera, F. D., & Becker, J. R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40 (1), 65-82.doi: 10.1007/s11858-007-0062-z
  29. Rivera, F. D. (2010). Second grade students’ preinstructional competence in patterning activity. In M. F. Pinto, & T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81-88). Belo Horizonte, Brazil: PME. Schliemann, A. D., Carraher, D. W., Brizuela, B. M., Earnest, D., Goodrow, A., Lara–Roth, S., &
  30. Peled, E. (2003). Algebra in elementary school. In N. Pateman, G. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th PME and the 25th PME-NA joined conference (Vol. 4, pp. 127-134). Honolulu, HI: University of Hawaii.
  31. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26 (2), 114-145. doi: 10.2307/749205
  32. Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory. Mathematical Thinking and Learning, 6 (2), 91-104. doi:10.1207/s15327833mtl0602_2
  33. Socas, M. (2011). La enseñanza del Álgebra en la Educación Obligatoria. Aportaciones de las investigaciones. Números, 77, 5-34.
  34. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20 (2), 147-164. doi: 10.1007/BF00579460
  35. Steffe, L. P. (2004). On the construction of learning trajectories of children: The case of commensurable fractions. Mathematical Thinking and Learning, 6 (2), 129-162. doi: 10.1207/s15327833mtl0602_4
  36. Warren, E. (2005). Young children’s ability to generalise the pattern rule for growing patterns. 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. 305-312). Melbourne, Australia: PME.
  37. Warren, E., Trigueros, M., & Ursini, S. (2016). Research on the learning and teaching of algebra. In A. Gutiérrez, G. Leder, & P. Boero (Eds.), The Second Handbook of Research on the Psychology of Mathematics Education (pp. 73-108). Rotterdam, Netherlands: Sense Publishers. doi: 10.1007/978-94-6300-561-6_3
  38. Zapatera, A. (2015). La competencia mirar con sentido de estudiantes para maestro (EPM) analizando el proceso de generalización en alumnos de Educación Primaria (tesis doctoral no publicada). Universidad de Alicante, Alicante, España.
  39. Zapatera, A., y Callejo, M. L. (2013). Cómo interpretan los estudiantes para maestro el pensamiento matemático de los alumnos sobre el proceso de generalización. En A. Berciano, G. Gutiérrez, A. Estepa & N. Climent (Eds.), Investigación en Educación Matemática XVII (pp. 535-544). Bilbao, España: SEIEM.
  40. Zapatera, A., y Callejo, M. L. (2011). Nivel de éxito y flexibilidad en el uso de estrategias resolviendo problemas de generalización de pautas lineales. En M. Marín, G. Fernández, L. J. Blanco, & M. Palarea (Eds.), Investigación en Educación Matemática XV (pp. 351-360). Ciudad Real, España: SEIEM.
  41. Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: the tension between algebraic notation. Educational Studies in Mathematics, 49 (3), 379-402. doi: 10.1023/A:1020291317178

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