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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We exclude younger children who have neurological impairments in approximate numerical processing (Ansari, 2010; Dehaene & Cohen, 1997).
- 2.
Mathematically, the pattern stages in Fig. 5.11a represent dilations (i.e. there is a fixed central point of projection). There is no research that deals with pattern stages that appear as a sequence of similitudes (i.e. similar figures that involve both isometry and dilation). The squares in Fig. 5.11b represent congruent figures that by definition can be established by applying at least one isometric action (Kay, 2001).
- 3.
For Duval (1999), visualization is an “intrinsically semiotic” (i.e., neither mental nor physical) cognitive activity. He distinguishes between visual perception (vision), which is primitive, and visualization, which has both epistemological and synoptic functions. Vision primarily engenders direct access and intuition of objects, while visualization involves the construction of a (semiotic) representation (epistemological function). In any semiotic representation, “relations or, better, organization of relations between representational units” are noted, including and especially those that are not at “all that accessible to vision” (Duval, p. 13). Also, while vision initially apprehends objects and their totality, it is never a “complete apprehension” (ibid) unlike visualization that engenders discourse and deductive actions (synoptic function).
- 4.
Context of the clinical interviews in third grade: The interviews took place toward the end of the school year without any intervening teaching experiment on patterns. Lack of instructional time prevented the third-grade class from exploring patterning activity. However, the author worked with the third grade teacher in ensuring that the entire third grade mathematics curriculum fostered structural and multiplicative thinking within and across the strands (i.e., number sense, algebra, statistics, data analysis, and probability, and geometry and measurement) were given the option to use the blocks to reconstruct the stages, however, they were asked to draw all the extended stages on paper. Like in the previous year, individual students were clinically interviewed on the five figural patterning tasks shown in Fig. 5.24. In each task, they were asked to: (1) construct stages 4 and 5 based on their initial interpretation of the given stages; (2) verbally describe stage 10; (3) try to transform their verbal description in arithmetical form involving whole numbers and the operations of multiplication and addition; and (4) state the total number of objects in stage 100 by using the arithmetical formula they established in (3). Concrete blocks were provided throughout the interview and drawn pictures of the pattern stages were shown one by one. The students were given the option to use the blocks to reconstruct the stages, however, they were asked to draw all the extended stages on paper.
References
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.
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.
Bhatt, R., & Quinn, P. (2011). How does learning impact development in infancy? The case of perceptual organization. Infancy, 16(1), 2–38.
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.
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.
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.
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.
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.
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.
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. (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.
Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants’ manual search. Developmental Science, 6, 568–584.
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.
Goldstone, R., Son, J., & Byrge, L. (2011). Early perceptual learning. Infancy, 16(1), 45–51.
Heeffer, A. (2008). The emergence of symbolic algebra as a shift in predominant models. Foundations of Science, 13, 149–161.
Hill, C., & Bennett, D. (2008). The perception of size and shape. Philosophical Issues, 18, 294–315.
Katz, V. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics, 66, 185–201.
Kvasz, L. (2006). The history of algebra and the development of the form of its language. Philosophia Mathematica, 14, 287–317.
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.
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.
Lipton, J., & Spelke, E. (2005). Preschool children master the logic of number word meanings. Cognition, 20, 1–10.
Luck, S., & Vogel, E. (1997). The capacity of visual working memory for features and conjunctions. Nature, 390, 279–281.
Mulligan, J., Prescott, A., & Mitchelmore, M. (2003). Taking a closer look at young students’ visual imagery. Australian Primary Mathematics, 8(4), 175–197.
Pothos, E., & Ward, R. (2000). Symmetry, repetition, and figural goodness: An investigation of the weight of evidence theory. Cognition, 75, 65–78.
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.
Radford, L. (2010). The eye as a theoretician: Seeing structures in generalizing activities. For the Learning of Mathematics, 30(2), 2–7.
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.
Schyns, P., Goldstone, R., & Thibaut, J.-P. (1998). The development of features in object concepts. The Behavioral and Brain Sciences, 21, 1–54.
Stavy, R., & Babai, R. (2008). Complexity of shapes and quantitative reasoning in geometry. Mind, Brain, and Education, 2(4), 170–176.
Tanisli, D. (2011). Functional thinking ways in relation to linear function tables of elementary school students, 30(3), 206–223.
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.
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.
Wallis, G., & Bülthoff, H. (1999). Learning to recognize objects. Trends in Cognitive Sciences, 3(1), 22–31.
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.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Rivera, F. (2013). Graded Pattern Generalization Processing of Elementary Students (Ages 6 Through 10 Years). In: Teaching and Learning Patterns in School Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2712-0_5
Download citation
DOI: https://doi.org/10.1007/978-94-007-2712-0_5
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-2711-3
Online ISBN: 978-94-007-2712-0
eBook Packages: Humanities, Social Sciences and LawEducation (R0)