Abstract
This chapter provides an empirical account of the formation of pattern generalization among a group of middle school students who participated in a three-year longitudinal study. Using pre-and post-interviews and videos of intervening teaching experiments, we document shifts in students’ ability to pattern generalize from figural to numeric and then back to figural, including how and why they occurred and consequences. The following six findings are discussed in some detail: development of constructive and deconstructive generalizations at the middle school level; operations needed in developing a pattern generalization; factors affecting students’ ability to develop constructive generalizations; emergence of classroom mathematical practices on pattern generalization; middle school students’ justification of constructive standard generalizations, and; their justification of constructive nonstandard generalizations and deconstructive generalizations. The longitudinal study also highlights the conceptual significance of multiplicative thinking in pattern generalization and the important role of sociocultural mediation in fostering growth in generalization practices.
This is an updated and thoroughly revised version of an earlier article that appeared in ZDM 40(1), 65–82. DOI 10.1007/s11858-007-0062-z. The ZDM article reported on findings drawn from the first two years of the study. This chapter reports on various aspects of the three-year study. This work was supported by Grant #DRL 044845 from the National Science Foundation (NSF) awarded to F. D. Rivera. The opinions expressed are not necessarily those of NSF and, thus, no endorsement should be inferred. This chapter is dedicated to Douglas Owens, mathematics educator, in honor of his retirement.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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
Becker, J. R., & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 121–128). Melbourne, Australia: University of Melbourne.
Becker, J. R., & Rivera, F. (2006). Establishing and justifying algebraic generalization at the sixth grade level. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 465–472). Prague: Charles University.
Becker, J. R., & Rivera, F. (2007). Factors affecting seventh graders’ cognitive perceptions of patterns involving constructive and deconstructive generalization. In J. Woo, K. Park, H. Sew, & D. Seo (Eds.), Proceedings of the 31 st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 129–136). Seoul, Korea: The Korea Society of Educational Studies in Mathematics.
Becker, J. R., & Rivera, F. (2008). Nature and content of generalization of 7th- and 8th-graders on a task that involves free construction of patterns. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PMENA XXX (Vol. 4, pp. 201–208). Morelia, Mexico: Cinvestav-UMSNH.
Bishop, J. (2000). Linear geometric number patterns: Middle school students’ strategies. Mathematics Education Research Journal, 12(2), 107–126.
Davydov, V. (1990). Types of Generalization in Instruction: Logical and Psychological Problems in the Structuring of School Curricula. Reston, VA: National Council of Teachers of Mathematics (J. Teller, Trans.).
Dörfler, W. (1991). Forms and means of generalization in mathematics. In A. Bishop, S. Mellin-Olsen, & J. van Dormolen (Eds.), Mathematical Knowledge: Its Growth Through Teaching (pp. 63–85). Dordrecht, Netherlands: Kluwer Academic Publishers.
Dörfler, W. (2008). En route from patterns to algebra: Comments and reflections. ZDM, 40(1), 143–160.
Dretske, F. (1990). Seeing, believing, and knowing. In D. Osherson, S. M. Kosslyn, & J. Hollerback (Eds.), Figural Cognition and Action: An Invitation to Cognitive Science (pp. 129–148). Cambridge, Massachusetts: MIT Press.
Dreyfus, T. (1991). Advanced mathematical thinking process. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 25–41). Dordrecht, Netherlands: Kluwer.
Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives in the Teaching of Geometry for the 21 st Century (pp. 29–83). Boston: Kluwer.
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1&2), 103–131.
English, L., & Warren, E. (1998). Introducing the variable through pattern exploration. Mathematics Teacher, 91, 166–170.
Garcia-Cruz, J. A., & Martinón, A. (1997). Actions and invariant schemata in linear generalizing problems. In Proceedings of the 23 rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 161–168). Haifa: Israel Institute of Technology.
Gelman, R. (1993). A rational-constructivist account of early learning about numbers and objects. In D. Medin (Ed.), Learning and Motivation (Vol. 30) New York: Academic Press.
Gelman, R., & Williams, E. (1998). Enabling constraints for cognitive development and learning: Domain specificity and epigenesis. In W. Damon, D. Kuhn, & R. Siegler (Eds.), Handbook of Child Psychology: Vol. 2. Cognition, Perception, and Language (5th ed., pp. 575–630). New York: Wiley.
Gravemeijer, K. P. E., & Doorman, L. M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1–3), 111–129.
Hartnett, P., & Gelman, R. (1998). Early understandings of numbers: Paths or barriers to the construction of new understandings? Learning and Instruction: The Journal of the European Association for Research in Learning and Instruction, 8(4), 341–374.
Hershkovitz, R. (1998). About reasoning in geometry. In C. Mammana & V. Villani (Eds.), Perspectives on the Teaching of Geometry for the 21 st Century (pp. 29–37). Boston: Kluwer.
Iwasaki, H., & Yamaguchi, T. (1997). The cognitive and symbolic analysis of the generalization process: The comparison of algebraic signs with geometric figures. In E. Pehkonnen (Ed.), Proceedings of the 21st Annual Conference of the Psychology of Mathematics Education (Vol. 3, pp. 105–113). Lahti, Finland.
Knuth, E. (2002). Proof as a tool for learning mathematics. Mathematics Teacher, 95(7), 486–490.
Lannin, J. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258.
Lannin, J., Barker, D., & Townsend, B. (2006). Recursive and explicit rules: How can we build student algebraic understanding. Journal of Mathematical Behavior, 25, 299–317.
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 Academic Publishers.
Lobato, J., Ellis, A., & Muñoz, R. (2003). How “focusing phenomena” in the instructional environment support individual students’ generalizations. Mathematical Thinking and Learning, 5(1), 1–36.
MacGregor, M., & Stacey, K. (1992). A comparison of pattern-based and equation-solving approaches to algebra. In B. Southwell, K. Owens, & B. Perry (Eds.), Proceedings of the Fifteenth Annual Conference of the Mathematics Education Research Group of Austalasia (pp. 362–371). Brisbane, Australia: MERGA.
Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and generalization in mathematics: What research has taught us. Journal of Mathematical Behavior, 18(1), 53–78.
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra: Perspectives for Research and Teaching (pp. 65–86). Dordrecht, Netherlands: Kluwer Academic Publishers.
Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing Thinking in Algebra. London: The Open University.
Mathematics in Context Team (2006a). Building Formulas: Algebra. Chicago, IL: Encyclopaedia Brittanica, Inc.
Mathematics in Context Team (2006b). Expressions and Formulas: Mathematics in Context. Chicago, IL: Encyclopaedia Brittanica, Inc.
Metzger, W. (2006/1936). Laws of Seeing. Cambridge, MA: MIT Press.
Orton, A., & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Patterns in the Teaching and Learning of Mathematics (pp. 104–123). London: Cassell.
Orton, J., Orton, A., & Roper, T. (1999). Pictorial and practical contexts and the perception of pattern. In A. Orton (Ed.), Patterns in the Teaching and Learning of Mathematics (pp. 121–136). London: Cassell.
Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237–268.
Radford, L. (2001). Factual, contextual, and symbolic generalizations in algebra. In M. van den Hueuvel-Panhuizen (Ed.), Proceedings of the 2th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 81–88). Netherlands: Freudenthal Institute.
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. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. Cortina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 2–21). México: UPN.
Radford, L. (2008). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different context. ZDM, 40(1), 83–96.
Richland, L., Holyoak, K., & Stigler, J. (2004). Analogy use in eight-grade mathematics classrooms. Cognition and Instruction, 22(1), 37–60.
Rivera, F. (2010a). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297–328.
Rivera, F. (2010b). Toward a Visually-Oriented School Mathematics Curriculum: Research, Theory, Practice, and Issues. Mathematics Education Library, Vol. 49. New York, NY: Springer.
Rivera, F., & Becker, J. R. (2003). The effects of figural and numerical stages on the induction processes of preservice elementary teachers. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the Joint Meeting PME and PMENA (Vol. 4, pp. 63–70). Honolulu, HI: University of Hawaii.
Rivera, F., & Becker, J. (2008). Sociocultural intimations on the development of generalization among middle school learners: Results from a three-year study. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PMENA XXX (Vol. 4, pp. 193–200). Morelia, Mexico: Cinvestav-UMSNH.
Rivera, F., & Becker, J. (2009a). Developing generalized reasoning in patterning activity. Mathematics Teaching in the Middle School.
Rivera, F., & Becker, J. (2009b). Visual templates in pattern generalization. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33 rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 473–480). Thessaloniki, Greece: PME.
Sasman, M., Olivier, A., & Linchevski, L. (1999). Factors influencing students’ generalization thinking processes. In O. Zaslavsky (Ed.), Proceedings of the 23 rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 161–168). Haifa: Israel Institute of Technology.
Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.
Stacey, K., MacGregor M. (2001). Curriculum reform and approaches to algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on School Algebra (pp. 141–154). Dordrecht, Netherlands: Kluwer Academic Publishers.
Steele, D., & Johanning, D. (2004). A schematic-theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57, 65–90.
Swafford, J., & Langrall, C. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89–112.
Taplin, M. L., & Robertson, M. E. (1995). Spatial patterning: A pilot study of pattern formation and generalization. In Proceedings of the 19 th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 273–280), Recife, Brazil.
Treffers, A. (1987). Three Dimensions: A Model of Goal and Theory Description in Mathematics Education. Reidel: Dordrecht.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Rivera, F.D., Becker, J.R. (2011). Formation of Pattern Generalization Involving Linear Figural Patterns Among Middle School Students: Results of a Three-Year Study. In: Cai, J., Knuth, E. (eds) Early Algebraization. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-17735-4_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17734-7
Online ISBN: 978-3-642-17735-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)