Types and Levels of Pattern Generalization

  • Ferdinand Rivera


In this chapter, we synthesize at least 20 years of research studies on pattern generalization that have been conducted with younger and older students in different parts of the globe. Central to pattern generalization are the inferential processes of abduction, induction, and deduction that we discussed in some detail in  Chaps. 1 and  2 and now take as given in this chapter. Here we explore the other equally important (and overlapping) dimensions of pattern generalization, namely: natures and sources of generalization; types of structures; ways of attending to structures; and modes of representing and understanding generalizations. In this chapter we remain consistent as before in articulating the complexity of pattern generalization due to differences in, and the simultaneous layering of, processes relevant to constructing, expressing, and justifying interpreted structures.


Individual Learner Direct Expression Pattern Generalization Structural Generalization Algebraic Thinking 
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.


  1. Ainley, J., Wilson, K., & Bills, L. (2003). Generalizing the context and generalizing the calculation. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th international conference of the Psychology of Mathematics Education (PME) (Vol. 2, pp. 9–16). Honolulu, Hawai’i: PME.Google Scholar
  2. Bastable, V., & Schifter, D. (2008). Classroom stories: Examples of elementary students engaged in early algebra. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 165–184). New York, NY: Erlbaum.Google Scholar
  3. Bishop, J. (2000). Linear geometric number patterns: Middle school students’ strategies. Mathematics Education Research Journal, 12(2), 107–126.CrossRefGoogle Scholar
  4. Britt, M., & Irwin, K. (2011). Algebraic thinking with and without algebraic representation: A pathway for learning. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 137–160). Netherlands: Springer.CrossRefGoogle Scholar
  5. Brown, J. R. (1997). Proofs and pictures. British Journal for the Philosophy of Science, 48, 161–180.CrossRefGoogle Scholar
  6. Cañadas, M., & Castro, E. (2007). A proposal of categorization for analyzing inductive reasoning. PNA, 1(2), 67–78.Google Scholar
  7. Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.Google Scholar
  8. Carraher, D., Martinez, M., & Schliemann, A. (2008). Early algebra and mathematical generalization. ZDM, 40, 3–22.CrossRefGoogle Scholar
  9. Chua, B., & Hoyles, C. (2010). Generalization and perceptual agility: How did teachers fare in a quadratic generalizing problem? Research in Mathematics Education, 12(1), 71–72.CrossRefGoogle Scholar
  10. Chua, B., & Hoyles, C. (2011). Secondary school students’ perception of best help generalizing strategies. In: Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (CERME). Rzeszow, Poland: CERME. Retrieved December 23, 2011, from
  11. Clements, D., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Erlbaum.Google Scholar
  12. 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
  13. Ellis, A. (2007). Connections between generalizing and justifying: Students’ reasoning with linear relationships. Journal for Research in Mathematics Education, 38(3), 194–229.Google Scholar
  14. Empson, S., & Levi, L. (2011). Extending children’s mathematics: Fractions and decimals. Portsmouth, NH: Heinemann.Google Scholar
  15. Garcia, M., Benitez, A., & Ruiz, E. (2010). Using multiple representations to make and verify conjectures. In P. Brosnan, D. Erchick, & L. Flevares (Eds.), Proceedings of the 32nd annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 6, pp. 270–278). Columbus, OH: Ohio State University.Google Scholar
  16. Garcia-Cruz, J., & Martinón, A. (1998). Levels of generalization in linear patterns. In A. Olivier & K. Newstead (Eds.), Proceedings of the 22 nd conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 2, pp. 329–336). Stellenbosch, South Africa: PME.Google Scholar
  17. Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 184–212). Westport, CT: Greenwoood Press.Google Scholar
  18. Heid, K., & Blume, G. (Eds.). (2008). Research on technology and the teaching and learning of mathematics. New York, NY: Information Age Publishing.Google Scholar
  19. Holland, J., Holyoak, K., Nisbett, R., & Thagard, P. (1986). Induction: Processes of inference, learning, and discovery. Cambridge, MA: MIT.Google Scholar
  20. 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). Finland: Lahti.Google Scholar
  21. Kaput, J., Carraher, D., & Blanton, M. (2008). Algebra in the early grades. New York, NY: Erlbaum.Google Scholar
  22. Katz, V. (2007). Stages in the history of algebra with implications for teaching. Educational Studies in Mathematics, 66, 185–201.CrossRefGoogle Scholar
  23. Küchemann, D. (2010). Using patterns generically to see structure. Pedagogies, 5(3), 233–250.Google Scholar
  24. 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
  25. Lee, L., & Freiman, V. (2004). Tracking primary students’ understanding of patterns. In D. McDougall & J. Ross (Eds.), Proceedings of the 26th annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (PMENA) (Vol. 2, pp. 245–251). Toronto, Canada: PMENA.Google Scholar
  26. Lin, F., Yang, K., & Chen, C. (2004). The features and relationships of reasoning, proving, and understanding proof in number patterns. International Journal of Science and Mathematics Education, 2, 227–256.CrossRefGoogle Scholar
  27. 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 15 th annual conference of the Mathematics Education Research Group of Australasia (MERGA) (pp. 362–371). Brisbane, Australia: MERGA.Google Scholar
  28. Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematical structures for all. Mathematics Education Research Journal, 21(2), 10–32.CrossRefGoogle Scholar
  29. Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33–49.CrossRefGoogle Scholar
  30. Mulligan, J., Prescott, A., & Mitchelmore, M. (2004). Children’s development of structure in early mathematics. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology in Mathematics Education. Bergen, Norway: Bergen University College.Google Scholar
  31. Nathan, M., & Kim, S. (2007). Pattern generalization with graphs and words: A cross-sectional and longitudinal analysis of middle school students’ representational fluency. Mathematical Thinking and Learning, 9(3), 193–219.CrossRefGoogle Scholar
  32. Norton, A., & Hackenberg, A. (2010). Continuing research on students’ fraction schemes. In L. Steffe & J. Olive (Eds.), Children’s fractional knowledge (pp. 341–352). New York, NY: Springer.CrossRefGoogle Scholar
  33. Noss, R., Healy, L., & Hoyles, C. (1997). The construction of mathematical meanings: Connecting the visual with the symbolic. Educational Studies in Mathematics, 33, 203–233.CrossRefGoogle Scholar
  34. Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66, 23–41.CrossRefGoogle Scholar
  35. Plaut, D., McClelland, J., Seindenberg, M., & Patterson, K. (1996). Understanding normal and impaired word reading: Computational principles in quasi-regular domains. Psychological Review, 103(1), 56–115.CrossRefGoogle Scholar
  36. Radford, L. (1999). The rhetoric of generalization: A cultural semiotic approach to students’ processes of symbolizing. In O. Zaslavsky (Ed.), Proceedings of the 23 rd conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 89–96). Technion-Israel Institute of Technology, Israel: PME.Google Scholar
  37. Radford, L. (2000). Students’ processes of symbolizing in algebra: A semiotic analysis of the production of signs in generalizing tasks. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education (PME) (Vol. 4, pp. 81–88). Hiroshima University, Japan: PME.Google Scholar
  38. Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. Cortina, M. Saiz, & A. Mendez (Eds.), Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME) (Vol. 1, pp. 2–21). Universidad Pedagogica Nacional, Mexico: PME.Google Scholar
  39. Rivera, F. (2007). Accounting for students’ schemes in the development of a graphical process for solving polynomial inequalities in instrumented activity. Educational Studies in Mathematics, 65(3), 281–307.CrossRefGoogle Scholar
  40. 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
  41. Rivera, F., & Becker, J. (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 perspectives (advances in mathematics education) (Vol. 2, pp. 323–366). New York: Springer.Google Scholar
  42. Samson, D. (2011). Capitalizing on inherent ambiguities in symbolic expressions of generality. Australian Mathematics Teacher, 67(1), 28–32.Google Scholar
  43. Samson, D., & Schäfer, M. (2009). An analysis of the influence of question design on learners’ approaches to number pattern generalization tasks. In M. Schäfer & C. McNamara (Eds.), Proceedings of the 17 th annual Meeting of the Southern African Association for Research in Mathematics, Science, and Technology Education (SAARMSTE) (Vol. 2, pp. 516–523). Grahamstown, South Africa: SAARMSTE.Google Scholar
  44. Samson, D., & Schäfer, M. (2011). Enactivism, figural apprehension, and knowledge objectification: An exploration of figural pattern generalization. For the Learning of Mathematics, 31(1), 37–43.Google Scholar
  45. 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
  46. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147–164.CrossRefGoogle Scholar
  47. 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.Google Scholar
  48. Steele, D., & Johanning, D. (2004). A schematic-thoeretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57, 65–90.CrossRefGoogle Scholar
  49. 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.CrossRefGoogle Scholar
  50. Tabach, M., Arcavi, A., & Hershkowitz, R. (2008). Transitions among different symbolic generalizations by algebra beginners in a computer intensive environment. Educational Studies in Mathematics, 69(1), 53–71.CrossRefGoogle Scholar
  51. Tanisli, D. (2011). Functional thinking ways in relation to linear function tables of elementary school students, 30(3), 206–223.Google Scholar
  52. 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
  53. van den Heuvel-Panhuizen, M. (Ed.). (2008). Children learn mathematics: A learning-teaching trajectory with intermediate attainment targets for calculation with whole numbers in primary school. Rotterdam, Netherlands: Sense.Google Scholar
  54. Watson, A. (2009). Thinking mathematically, disciplined noticing, and structures of attention. In S. Lerman & B. Davis (Eds.), Mathematical action & structures of noticing (pp. 211–222). Rotterdam, Netherlands: Sense Publishers.Google Scholar
  55. Wilson, K., Ainley, J., & Bills, L. (2005). Naming a column on a spreadsheet: Is it more algebraic? In D. Hewitt & A. Noyes (Eds.), Proceedings of the Sixth British Congress of Mathematics Education (pp. 184–191). Warwick, UK: BCME.Google Scholar
  56. Yerushalmy, M., & Maman, H. (1988). The Geometric Supposer as the basis for class discussion in geometry. Haifa, Israel: University of Haifa Laboratory of Computers for Learning.Google Scholar
  57. Yerushalmy, M. (1993). Generalization in geometry. In J. Schwartz, M. Yerushalmy, & B. Wilson (Eds.), The geometric supposer: What is it a case of? (pp. 57–84). Hillsdale, NJ: Erlbaum.Google Scholar
  58. Yevdokimov, O. (2008). Making generalizations in geometry: Students’ views on the process. 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 and PME.Google Scholar
  59. Zazkis, R., Liljedahl, P., & Chernoff, E. (2008). The role of examples in forming and refuting generalizations. ZDM, 40, 131–141.CrossRefGoogle 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