Aprehensión cognitiva en problemas de generalización de patrones visuales

  1. María Luz Callejo 1
  2. Ceneida Fernández 1
  3. Álvaro García-Reche 1
  1. 1 Universitat d'Alacant
    info

    Universitat d'Alacant

    Alicante, España

    ROR https://ror.org/05t8bcz72

Journal:
Journal for the Study of Education and Development, Infancia y Aprendizaje

ISSN: 0210-3702,  1578-4126

Year of publication: 2019

Volume: 42

Issue: 4

Pages: 805-828

Type: Article

DOI: 10.1080/02103702.2019.1652447 DIALNET GOOGLE SCHOLAR

More publications in: Journal for the Study of Education and Development, Infancia y Aprendizaje

Abstract

The objective of this study was to identify cognitive apprehensions used by fifth- and sixth-grade students (10–12-year-olds) when answering far generalization questions in two problems of visual pattern generalization. A total of 81 students solved two linear generalizing problems, presented in two different configurations, in a succession of figures (square tables or trapezoid tables). The results showed that students used different types of cognitive apprehensions to solve problems and that these apprehensions sometimes changed according to the configuration of the sequence of figures. This finding indicates that configurations could determine apprehensions used by students, which in some cases led to the emergence of algebraic thinking. In addition, difficulties in modifying apprehension and a lack of coordination between spatial and numerical structures could explain some students’ difficulties in far generalization.

€ View funding

Funding information

This research was carried out with the support of EDU2017-87411-R from MINECO/FEDER (Spain) and PROMETEO/2017/135 from Generalitat Valenciana (Spain). / Esta investigación ha recibido el apoyo de los proyectos EDU2017-87411-R de MINECO/FEDER (España) y PROMETEO/2017/135 de la Generalitat Valenciana (España) .

Funders

Bibliographic References

  • Blanton, M., Brizuela, B., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015).A learning trajectory in 6-years-olds’thinking about generalizing functional relation-ships.Journal for Research in Mathematics Education,46,511–558.
  • Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization. A global dialogue from multiple perspectives. Berlin: Springer-Verlag.
  • Callejo, M. L., & Zapatera, A. (2014). Secondary school students’ flexibility when solving problems of recognition of lineal patterns. Bolema, 24, 64–88.
  • Cañadas, M. C., Brizuela, B. M., & Blanton, M. (2016). Second graders articulating ideas about linear functional relationships. Journal of Mathematical Behavior, 41, 87–103.
  • Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.
  • Carraher, D. W., Martinez, M. V., & Schliemann, A. D. (2008). Early algebra and mathematical generalization. ZDM. Mathematics Education, 40, 3–22.
  • Clemente, F., Llinares, S., & Torregrosa, G. (2017). Visualización y razonamiento configural. Bolema, 31, 497–516.
  • Cooper, T. J., & Warren, E. (2008). Generalising mathematical structures in years 3-4: A case study of equivalence of expression. In O. Figueras, J. L. Cortina, S. Alatorre, & A. Sepulveda (Eds.). Proceedings of the 32nd conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 369–376). PME.
  • Cooper, T. J., & 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). Berlin: Springer-Verlag.
  • Duval, R. (1993). Geometrical pictures: Kinds of representation and specific processing. In R. Sutherland, & J. Mason (Eds.), Exploiting mental imagery with computers in mathematics education (pp. 142–157). Berlin: Springer.
  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.
  • García-Cruz, J. A. (1998). Levels of generalization in linear patterns. In A. Olivier, & K. Newstead (Eds.), Proceedings of the 22th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 329–336). Stellendosh: PME.
  • Godino, J. D., Aké, L. P., Gonzato, M., & Wilhelmi, M. R. (2014). Algebrization levels of school mathematics activity. Implication for primary school teacher education. Enseñanza de las Ciencias, 32(1), 199–219.
  • Kaput, J., & Blanton, M. (2001). Student achievement in algebraic thinking: A comparison of third-graders’ 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, p. 99–108). Columbus, OH: ERIC.
  • Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator, 18, 139–151.
  • Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra. Research into its nature, its learning, its teaching. ICME-13 Topical Surveys. Hamburg: Springer International Publishing AG Switzerland.
  • Lannin, J. K. (2005). Generalization and justification, the challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7, 231–258.
  • Llinares, S., & Clemente, F. (2014). Characteristics of pre-service primary school teachers’ configural reasoning. Mathematical Thinking and Learning, 16, 234–250.
  • Martinez, M., & Brizuela, B. M. (2006). A third grader’s way of thinking about linear function tables. The Journal of Mathematical Behavior, 25, 285–298.
  • Merino, E., Cañadas, M. C., & Molina, M. (2013). Representations and patterns used by fifth grade students in a generalization task. Edma 0-6: Educación Matemática en la Infancia, 2(1), 24–40.
  • Nilsson, P., & Juter, K. (2011). Flexibility and coordination among acts of visualization and analysis in a pattern generalization activity. The Journal of Mathematical Behavior, 30, 194–205.
  • Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 2–21). Mérida: PME.
  • 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, 83–96.
  • Radford, L. (2010). Elementary forms of algebraic thinking in young students. 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. 73–80). Belo Horizonte: PME.
  • 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: PME.
  • Radford, L. (2014). The progressive development of early embodied algebraic thinking. Mathematics Education Research Journal, 26, 257–277.
  • 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: PME.
  • Rivera, F. D., & Becker, J. S. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra. The Journal of Mathematical Behavior, 26, 140–155.
  • Rivera, F. D., & Becker, J. S. (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 pPerspectives (pp. 323–366). Berlin: Springer-Verlag.
  • Schliemann, A. D., Carraher, D. V., & Brizuela, B. M. (2007). Bringing out the algebraic character of arithmetic: From children’s ideas to classroom practices. Mahwah, NJ: Lawrence Erlbaum.
  • Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.
  • Stephens, A. C., Ellis, A. B., Blanton, M., & Brizuela, B. M. (2017). Algebraic thinking in the elementary and middle grades. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 386–420). Reston: NCTM.
  • Strauss, A., & Corbin, J. (1994). Grounded theory methodology: An overview. In N. K. Denzin, & Y. Lincoln (Eds.), Handbook of qualitative research (pp. 273–285). Thousand Oaks, CA: Sage.
  • Torregrosa, G. (2017). Coordination of cognitive processes in problem solving: Relationship between geometry and algebra. Avances de Investigación en Educación Matemática, 12, 1–17.
  • Vergel, R. (2015). How does algebraic thinking emerge? The case of factual algebraic thinking. Uno. Revista de Didáctica de las Matemáticas, 68, 9–17.
  • Warren, E. A., & Cooper, T. J. (2008). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67, 171–185.
  • Warren, E. A., Cooper, T. J., & Lamb, J. T. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning. The Journal of Mathematical Behavior, 25, 208–223.
Data source: Dialnet