Graded Pattern Generalization Processing of Elementary Students (Ages 6 Through 10 Years)

  • Ferdinand Rivera
Chapter

Abstract

In this chapter, we focus on pattern generalization studies that have been conducted with elementary school children from Grades 1 through 5 (ages 6 through 10 years) in different contexts. Our contribution to the current research based on elementary students’ understanding of patterns involves extrapolating the graded nature of their pattern generalization schemes on the basis of their constructed structures, incipient generalizations, and the use of various representational forms such as gestures, words, and arithmetical symbols in conveying their expressions of generality. The gradedness condition foregrounds the dynamic emergence of parallel types of pattern generalization processing that is sensitive to a complex of factors (cognitive, sociocultural, neural, constraints in curriculum content, nature and type of tasks, etc.), where progression is seen not in linear terms but as states that continually evolve based on more learning. In a graded pattern generalization processing view, there are no prescribed stages or fixed rules but only states of conceptual coalescences and coherent covariations that change with more experiences. The chapter addresses different aspects of pattern generalization processing that matter to elementary school children. We also explore approximate and exact pattern generalizations along three dimensions, namely: whole number knowledge, shape sensitivity, and figural competence. We further discuss the representational modes that elementary students oftentimes use to capture their emergent structures and incipient generalizations. These modes include gestural, pictorial, verbal, and numerical. In another section, we address grade-level appropriate use and understanding of variables via the notions of intuited and tacit variables. We close the section with an analysis of the relationship between elementary children’s structural incipient generalizations and the natural emergence of their understanding of functions.

Keywords

Function Table Elementary Student Elementary School Child Pattern Generalization Stage Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Alvarez, G., & Cavanagh, P. (2004). The capacity of visual short-term memory is set both by visual information load and by number of objects. Psychological Science, 15(2), 106–111.CrossRefGoogle Scholar
  2. Ansari, D. (2010). Neurocognitive approaches to developmental disorders of numerical and mathematical cognition: The perils of neglecting development. Learning and Individual Differences, 20, 123–129.CrossRefGoogle Scholar
  3. Bhatt, R., & Quinn, P. (2011). How does learning impact development in infancy? The case of perceptual organization. Infancy, 16(1), 2–38.CrossRefGoogle Scholar
  4. Blanton, M., & Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. 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: PME.Google Scholar
  5. Cai, J., Ng, S. F., & Moyer, J. (2011). Developing students’ algebraic thinking in earlier grades: Lessons from China and Singapore. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 25–42). New York: Springer.CrossRefGoogle Scholar
  6. Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.Google Scholar
  7. Carraher, D., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematical generalization. ZDM, 40, 3–22.CrossRefGoogle Scholar
  8. Cavanagh, P., & He, S. (2011). Attention mechanisms for counting in stabilized and in dynamic displays. In S. Dehaene & E. Brannon (Eds.), Space, time, and number in the brain: Searching for the foundations of mathematical thought (pp. 23–35). New York: Academic.CrossRefGoogle Scholar
  9. Condry, K., & Spelke, E. (2008). The development of language and abstract concepts: The case of natural number. Journal of Experimental Psychology. General, 137(1), 22–38.CrossRefGoogle Scholar
  10. Cooper, T., & 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). Netherlands: Springer.CrossRefGoogle Scholar
  11. Deacon, T. (1997). The symbolic species: The co-evolution of language and the brain. New York: W. W. Norton & Company.Google Scholar
  12. Dehaene, S. (1997). The number sense. New York, NY: Oxford University Press.Google Scholar
  13. Duval, R. (1999). Representation, vision, and visualization: Cognitive functions in mathematical thinking. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st North American PME Conference (pp. 3–26). Cuernavaca, Morelos, Mexico: PMENA.Google Scholar
  14. Feigenson, L. (2011). Objects, sets, and ensembles. In S. Dehaene & E. Brannon (Eds.), Space, time, and number in the brain: Searching for the foundations of mathematical thought (pp. 13–22). New York: Academic.CrossRefGoogle Scholar
  15. Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants’ manual search. Developmental Science, 6, 568–584.CrossRefGoogle Scholar
  16. Gal, H., & Linchevski, L. (2010). To see or not to see: Analyzing difficulties in geometry from the perspective of visual perception. Educational Studies in Mathematics, 74, 163–183.CrossRefGoogle Scholar
  17. Goldstone, R., Son, J., & Byrge, L. (2011). Early perceptual learning. Infancy, 16(1), 45–51.CrossRefGoogle Scholar
  18. Heeffer, A. (2008). The emergence of symbolic algebra as a shift in predominant models. Foundations of Science, 13, 149–161.CrossRefGoogle Scholar
  19. Hill, C., & Bennett, D. (2008). The perception of size and shape. Philosophical Issues, 18, 294–315.CrossRefGoogle Scholar
  20. Katz, V. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics, 66, 185–201.CrossRefGoogle Scholar
  21. Kvasz, L. (2006). The history of algebra and the development of the form of its language. Philosophia Mathematica, 14, 287–317.CrossRefGoogle Scholar
  22. Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438.CrossRefGoogle Scholar
  23. Lee, L. (1996). An initiation into algebra culture through generalization activities. In C. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 87–106). Dordrecht, Netherlands: Kluwer.CrossRefGoogle Scholar
  24. Lipton, J., & Spelke, E. (2005). Preschool children master the logic of number word meanings. Cognition, 20, 1–10.Google Scholar
  25. Luck, S., & Vogel, E. (1997). The capacity of visual working memory for features and conjunctions. Nature, 390, 279–281.CrossRefGoogle Scholar
  26. Mulligan, J., Prescott, A., & Mitchelmore, M. (2003). Taking a closer look at young students’ visual imagery. Australian Primary Mathematics, 8(4), 175–197.Google Scholar
  27. Pothos, E., & Ward, R. (2000). Symmetry, repetition, and figural goodness: An investigation of the weight of evidence theory. Cognition, 75, 65–78.CrossRefGoogle Scholar
  28. Radford, L. (2003). Gestures, speech, and the sprouting of signs: A semiotic-cultural approach to students’ types of generalization. Mathematical Thinking and Learning, 5(1), 37–70.CrossRefGoogle Scholar
  29. Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7.Google Scholar
  30. Rivera, F. (2011). Toward a visually-oriented school mathematics curriculum: Research, theory, practice, and issues (Mathematics Education Library Series 49). New York, NY: Springer.CrossRefGoogle Scholar
  31. Schliemann, A., Carraher, D., & Brizuela, B. (2007). Bringing out the algebraic character of arithmetic: From children’s ideas to classroom practice. New York, NY: Erlbaum.Google Scholar
  32. Schyns, P., Goldstone, R., & Thibaut, J.-P. (1998). The development of features in object concepts. The Behavioral and Brain Sciences, 21, 1–54.Google Scholar
  33. Stavy, R., & Babai, R. (2008). Complexity of shapes and quantitative reasoning in geometry. Mind, Brain, and Education, 2(4), 170–176.CrossRefGoogle Scholar
  34. Tanisli, D. (2011). Functional thinking ways in relation to linear function tables of elementary school students, 30(3), 206–223.Google Scholar
  35. Taylor-Cox, J. (2003). Algebra in the early years? Young Children, 58(1), 15–21.Google Scholar
  36. Triadafillidis, T. (1995). Circumventing visual limitations in teaching the geometry of shapes. Educational Studies in Mathematics, 15, 151–159.Google Scholar
  37. Vale, I., & Pimentel, T. (2010). From figural growing patterns to generalization: A path to algebraic thinking. In M. Pinto & T. Kawasaki (Eds.), Proceedings of the 34 th conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 241–248). Belo Horizante, Brazil: PME.Google Scholar
  38. Wallis, G., & Bülthoff, H. (1999). Learning to recognize objects. Trends in Cognitive Sciences, 3(1), 22–31.CrossRefGoogle Scholar
  39. Warren, E., & Cooper, T. (2007). Repeating patterns and multiplicative thinking: Analysis of classroom interactions with 9-year-old students that support the transition from the known to the novel. Journal of Classroom Interaction, 41(2), 7–17.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Ferdinand Rivera
    • 1
  1. 1.Department of MathematicsSan Jose State UniversitySan JoseUSA

Personalised recommendations