Pattern Generalization Processing of Elementary Students: Cognitive Factors Affecting the Development of Exact Mathematical Structures
F. D. Rivera 1  
More details
Hide details
Department of Mathematics & Statistics, San Jose State University, USA
F. D. Rivera   

Department of Mathematics & Statistics San Jose State University, USA 408 (924) 5170
Publish date: 2018-06-25
EURASIA J. Math., Sci Tech. Ed 2018;14(9):em1586
The fundamental aim in this article is to elucidate cognitive factors that influence the development of mathematical structures and incipient generalizations in elementary school children on the basis of their work on patterns, including how they use various representational forms such as gestures, words, and arithmetical symbols to convey their expressions of generality. We describe approximate and exact pattern generalizations and three cognitive factors that mutually influence the emergence of mathematical structures, namely, competence with number relationships, competence with shape similarity, and competence with figural property construction, discernment, and justification. We also highlight various representational modes that elementary students use to capture their emergent structures and incipient generalizations, grade-appropriate use and understanding of variables via the notions of intuited and tacit variables, and ways in which their structural incipient generalizations support their early understanding of functions.
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. https://doi.org/10.1111/j.0963....
Bhatt, R., & Quinn, P. (2011). How does learning impact development in infancy? The case of perceptual organization. Infancy, 16(1), 2-38. https://doi.org/10.1111/j.1532....
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.
Blanton, M., Brizuela, B., Gardiner, A., Sawrey, K., & Newman-Owens, A. (2015). A learning trajectory in 6-year-olds’ thinking about generalizing functional relationships. Journal for Research in Mathematics Education, 46(5), 511-558. https://doi.org/10.5951/jresem....
Cai, J., Ng, S.F., & Moyer, J. (2011). Developing students’ algebraic thinking in earlier grades: Lessons from China and Singapore. In J. Cai and E. Knuth (eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 25-42). New York: Springer. https://doi.org/10.1007/978-3-....
Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
Carraher, D., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematical generalization. ZDM, 40, 3-22. https://doi.org/10.1007/s11858....
Carraher, D., Schliemann, A., & Brizuela, B. (1999). Bringing out the algebraic character of arithmetic. Paper presented at the 1999 AERA Meeting, Montreal, Canada. Available at http://www.earlyalgebra.terc.e....
Carraher, D., Schliemann, A., Brizuela, B., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115.
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 Press. https://doi.org/10.1016/B978-0....
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. https://doi.org/10.1037/0096-3....
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 Verlag. https://doi.org/10.1007/978-3-....
Deacon, T. (1997). The symbolic species: The co-evolution of language and the brain. New York: W. W. Norton & Company.
Dehaene, S. (1997). The number sense. New York, NY: Oxford University Press.
Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana, & V. Villani (Eds.), Perspectives in the teaching of geometry for the 21st century (pp. 29–83). Boston: Kluwer.
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.
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 Press. https://doi.org/10.1016/B978-0....
Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants’ manual search. Developmental Science, 6, 568-584. https://doi.org/10.1111/1467-7....
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. https://doi.org/10.1007/s10649....
Goldstone, R., Son, J., & Byrge, L. (2011). Early perceptual learning. Infancy, 16(1), 45-51. https://doi.org/10.1111/j.1532....
Hill, C., & Bennett, D. (2008). The perception of size and shape. Philosophical Issues, 18, 294-315. https://doi.org/10.1111/j.1533....
Kay, D. (2001). College geometry: A discovery approach. Boston, MA: Addison Wesley Longman, Inc.
Kvasz, L. (2006). The history of algebra and the development of the form of its language. Philosophia Mathematica, 14, 287-317. https://doi.org/10.1093/philma....
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. https://doi.org/10.1016/j.cogn....
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. https://doi.org/10.1007/978-94....
Lipton, J., & Spelke, E. (2005). Preschool children master the logic of number word meanings. Cognition, 1-10.
Luck, S., & Vogel, E. (1997). The capacity of visual working memory for features and conjunctions. Nature, 390, 279-281. https://doi.org/10.1038/36846.
Moss, J., & London McNab, S. (2011). An approach to geometric and numeric patterning that fosters second grade students’ reasoning and generalizing about functions and co-variation. In J. Cai & E. Knuth (eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 277-302). New York: Springer. https://doi.org/10.1007/978-3-....
Mulligan, J., Prescott, A., & Mitchelmore, M. (2003). Taking a closer look at young students’ visual imagery. Australian Primary Mathematics, 8(4), 175-197.
Pinel, P., Dehaene, S., Riviere, D., & Le Bihan, D. (2001). Modulation of parietal activation by semantic distance in a number comparison task. Neuroimage, 14, 1013-1026. https://doi.org/10.1006/nimg.2....
Pizlo, Z., Sawada, T., Li, Y., Kropatsch, W., & Steinman, R. (2010). New approach to the perception of 3D shape based on veridicality, complexity, symmetry, and volume. Vision Research, 50, 1-11. https://doi.org/10.1016/j.visr....
Pothos, E., & Ward, R. (2000). Symmetry, repetition, and figural goodness: An investigation of the weight of evidence theory. Cognition, 75, 65-78. https://doi.org/10.1016/S0010-....
Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2-7.
Rivera, F. (2010). Second grade students’ preinstructional competence in patterning activity. In M. Pinto & T. Kawasaki (eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 81-88). Belo Horizante, Brazil: PME. https://doi.org/10.1007/978-94....
Rivera, F. (2011). Toward a visually-oriented school mathematics curriculum: Research, theory, practice, and issues (Mathematics Education Library Series 49). New York, NY: Springer.
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.
Schweitzer, K. (2006). Teacher as researcher: Research as a partnership. In S. Smith & M. Smith (eds.), Teachers engaged in research: Inquiry into mathematics classrooms, grades pre-k-2 (pp. 69-94). Greenwich, CT: Information Age Publishing.
Schyns, P., Goldstone, R., & Thibaut, J.-P. (1998). The development of features in object concepts. Behavioral and Brain Sciences, 21, 1-54. https://doi.org/10.1017/S01405....
Stavy, R., & Babai, R. (2008). Complexity of shapes and quantitative reasoning in geometry. Mind, Brain, and Education, 2(4), 170-176. https://doi.org/10.1111/j.1751....
Tanisli, D., & Özdas, A. (2009). The strategies of using generalizing patterns among primary school 5th grade students. Educational Sciences: Theory & Practice, 9(3), 1485-1497.
Taylor-Cox, J. (2003). Algebra in the early years? Young Children, 58(1), 15-21.
Triadafillidis, T. (1995). Circumventing visual limitations in teaching the geometry of shapes. Educational Studies in Mathematics, 15, 151-159. https://doi.org/10.1007/BF0127....
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 34th Conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 241-248). Belo Horizante, Brazil: PME.
Walkowiak, T. (2014). Elementary and middle school students’ analyses of pictorial growth patterns. Journal of Mathematical Behavior, 56-71. https://doi.org/10.1016/j.jmat....
Wallis, G., & Bülthoff, H. (1999). Learning to recognize objects. Trends in Cognitive Sciences, 3(1), 22-31. https://doi.org/10.1016/S1364-....
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.
Whitin, P., & Whitin, D. (2011). Mathematical pattern hunters. Young Children, 84-90.
Wilkie, K. (2014). Learning to like algebra through looking. Australian Primary Classroom, 24-33.
Wilkie, K., & Clarke, D. (2016). Developing students’ functional thinking in algebra through different visualizations of a growing pattern’s structure. Mathematics Education Research Journal, 28, 223-243. https://doi.org/10.1007/s13394....