Advertisement

ZDM

, Volume 40, Issue 1, pp 3–22 | Cite as

Early algebra and mathematical generalization

  • David W. Carraher
  • Mara V. Martinez
  • Analúcia D. Schliemann
Original article

Abstract

We examine issues that arise in students’ making of generalizations about geometrical figures as they are introduced to linear functions. We focus on the concepts of patterns, function, and generalization in mathematics education in examining how 15 third grade students (9 years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra. Many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence. If this instructional route is pursued, educators need to recognize how students’ conceptualizations of functions depart from the closed form expressions ultimately aimed for. Even more fundamentally, it is important to nurture a transition from empirical generalizations, based on conjectures regarding cases at hand, to theoretical generalizations that follow from operations on explicit statements about mathematical relations.

Keywords

Young Student Algebraic Expression Mathematical Generalization Separate Table Dinner Table 
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.

Notes

Acknowledgments

This study was developed as part of the National Science Foundation supported project “Algebra in Early Mathematics” (NSF-ROLE grant 0310171). We thank Barbara Brizuela, Anne Goodrow, Darrell Earnest, Susanna Lara-Roth, Camille Burnett, and Gabrielle Cayton for their contributions to the work reported here. We thank also Judah Schwartz for many important insights regarding the role of functions in early mathematics.

References

  1. Aleksandrov, A. D. (1989). A general view of mathematics. In A. Aleksandrov, A. Kolmogorov, & M. Lavrent’ev (Eds.), Mathematics, its content, methods, and meaning (pp. 1–64). Cambridge: The M.I.T. Press.Google Scholar
  2. Balacheff, N. (1987). Processus de preuves et situations de validation. Educational Studies in Mathematics, 18(2), 147–176.CrossRefGoogle Scholar
  3. Bastable, V., & Schifter, D. (2007). Classroom stories: examples of elementary students engaged in early algebra. In J. Kaput, D. W. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 165–184). Mahwah: Erlbaum.Google Scholar
  4. Bereiter, C. (1985). Toward a solution of the learning paradox. Review of Educational Research, 55(2), 201–226.CrossRefGoogle Scholar
  5. Blanton, M., & Kaput, J. (2000). Generalizing and progressively formalizing in a third grade mathematics classroom: conversations about even and odd numbers. In M. Fernández (Ed.), Proceedings of the 20th annual meeting of the psychology of mathematics education, North American chapter (p. 115). Columbus: ERIC Clearinghouse (ED446945).Google Scholar
  6. Bourke, S., & Stacey, K. (1988). Assessing problem solving in mathematics: some variables related to student performance. Australian Educational Researcher, 15, 77–83.Google Scholar
  7. Brizuela, B. M., & Schliemann, A. D. (2004). Ten-year-old students solving linear equations. For the Learning of Mathematics, 24(2), 33–40.Google Scholar
  8. Carpenter, T., & Franke, M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), The future of the teaching and learning of algebra. Proceedings of the 12th ICMI Study Conference (Vol. 1, pp. 155–162). The University of Melbourne, Australia.Google Scholar
  9. Carpenter, T., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades [Electronic Version]. Research Report 00–2. Madison, WI: National Center for Improving Student Learning and Achievement in Mathematics and Science.Google Scholar
  10. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: integrating arithmetic and algebra in elementary school. Portsmouth: Heinemann.Google Scholar
  11. Carraher, D. W., & Earnest, D. (2003). Guess My Rule Revisited. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 173–180). Honolulu: University of Hawaii.Google Scholar
  12. Carraher, D. W. & Schliemann, A. D. (2002a). Is everyday mathematics truly relevant to mathematics education? In J. Moshkovich, & M. Brenner (Eds.) Everyday mathematics. Monographs of the journal for research in mathematics education (Vol. 11, pp. 131–153). Reston: National Council of Teachers of Mathematics.Google Scholar
  13. Carraher, D. W., & Schliemann, A. D. (2002b). Modeling reasoning. In K. Gravemeijer, R. Lehrer, B. Oers, L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 295–304). The Netherlands: Kluwer.Google Scholar
  14. Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. Lester (Ed.), Handbook of research in mathematics education (pp. 669–705). Greenwich: Information Age Publishing.Google Scholar
  15. Carraher, D. W., Schliemann, A. D., & Brizuela, B. (2000). Early algebra, early arithmetic: treating operations as functions. In Plenary address at the 22nd meeting of the psychology of mathematics education, North American Chapter, Tucson, AZ (October) (available in CD-Rom).Google Scholar
  16. Carraher, D. W., Schliemann, A. D., & Brizuela, B. (2005). Treating operations as functions. In D. Carraher & R. Nemirovsky (Eds.), Monographs of the journal for research in mathematics education, XIII, CD-Rom Only Issue.Google Scholar
  17. Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2007). Early algebra is not the same as algebra early. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades. (pp. 235–272). Mahwah: Erlbaum.Google Scholar
  18. Cuoco, A. (1990). Investigations in Algebra: An Approach to Using Logo. Cambridge, MA, The MIT Press.Google Scholar
  19. Davis, R. B. (1967a). Mathematics teaching, with special reference to epistemological problems. Athens: College of Education, University of Georgia.Google Scholar
  20. Davis, R. B. (1967b). Exploration in mathematics: a text for teachers. Reading: Addison-Wesley.Google Scholar
  21. Davis, R. B. (1985). ICME-5 report: algebraic thinking in the early grades. Journal of Children’s Mathematical Behavior, 4, 198–208.Google Scholar
  22. Davis, R. B. (1989). Theoretical considerations: research studies in how humans think about algebra. In S. Wagner & C. Kieran (Eds.), Research agenda for mathematics education (Vol. 4, pp. 266–274). Hillsdale: Lawrence Erlbaum & National Council of Teachers of Mathematics.Google Scholar
  23. Davydov, V. (1990). Types of generalization in instruction (Soviet Studies in Mathematics Education, Vol. 2). Reston, VA: NCTM.Google Scholar
  24. Davydov, V. V. (1991). Psychological abilities of primary school children in learning mathematics (Soviet Studies in Mathematics Education, Vol. 6). Reston: National Council of Teachers of Mathematics.Google Scholar
  25. Dorfler, W. (1991). Forms and means of generalization in mathematics. In A. Bell (Ed.), Mathematical knowledge: it’s growth through teaching (pp. 63–85). The Netherlands: Kluwer.Google Scholar
  26. Dougherty, B. (2007). Measure up: a quantitative view of early algebra. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades. Mahwah: Lawrence Erlbaum.Google Scholar
  27. Duckworth, E. R. (1979). Either we’re too early and they can’t learn it or we’re too late and they know it already: the dilemma of ‘applying Piaget’. Harvard Educational Review, 49(3), 297–312.Google Scholar
  28. Filloy, E., & Rojano, T. (1989). Solving equations: the transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.Google Scholar
  29. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.Google Scholar
  30. Fridman, L. M. (1991). Features of introducing the concept of concrete numbers in the primary grades. In V. V. Davydov (Ed.), Psychological abilities of primary school children in learning mathematics. Soviet studies in mathematics education (Vol. 6, pp. 148–180). Reston: NCTM.Google Scholar
  31. Hargreaves. M., Threlfall, J., Frobisher, L., Shorrocks-Taylor, D. (1999). Children’s Strategies with Linear and Quadratic Sequences. In Orton, A. (Ed.), Pattern in the Teaching and Learning of Mathematics. Cassell, London.Google Scholar
  32. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operational structures. New York: Basic Books.Google Scholar
  33. Kaput, J. (1995). Transforming algebra from an engine of inequity to an engine of mathematical power by “algebrafying” the K-12 curriculum. Paper presented at the Annual Meeting of the National Council of Teachers of Mathematics, Boston, MA.Google Scholar
  34. Kaput, J. (1998a). Transforming algebra from an engine of inequity to an engine of mathematical power by ‘algebrafying’ the K-12 Curriculum. In National Council of Teachers of Mathematics and Mathematical Sciences Education Board Center for Science, Mathematics and Engineering Education, National Research Council (Sponsors). The Nature and Role of Algebra in the K-14 Curriculum (pp. 25–26). Washington: National Academies Press.Google Scholar
  35. Kaput, J. (1998b). Transforming algebra from an engine of inequity to an engine of mathematical power by ‘algebrafying’ the K-12 Curriculum. In The nature and role of algebra in the K-14 curriculum (pp. 25–26). Washington: National Council of Teachers of Mathematics and the Mathematical Sciences Education Board, National Research Council.Google Scholar
  36. Kaput, J., Carraher, D. W., & Blanton, M. (Eds.). (2007). Algebra in the early grades. Hillsdale/Reston: Erlbaum/The National Council of Teachers of Mathematics.Google Scholar
  37. Lee, L. (1996) An initiation into algebraic culture through generalization activities 87–106. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 87–106). Dordrecht: Kluwer.Google Scholar
  38. Martinez, M., & Brizuela, B. (2006). A third grader’s way of thinking about linear function tables. Journal of Mathematical Behavior, 25, 285–298.CrossRefGoogle Scholar
  39. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra (pp. 65–86). Dordrecht: Kluwer.Google Scholar
  40. Mill, J. S. (1965/1843). Mathematics and experience. In P. Edwards & A. Pap (Eds.), A modern introduction to philosophy: Readings from Classical and Contemporary Sources (pp. 624–637). New York: Collier-Macmillian.Google Scholar
  41. Moss, J., Beatty, R., McNab, S. L., & Eisenband, J. (2006). The potential of geometric sequences to foster young students’ ability to generalize in mathematics. Paper presented at the annual meeting of the American Educational Research Association, San Francisco (April).Google Scholar
  42. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston: National Council of Teachers of Mathematics.Google Scholar
  43. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: National Council of Teachers of Mathematics.Google Scholar
  44. Orton, A. (1999), (Ed.). Patterns in the Teaching and Learning of Mathematics. London: Cassell.Google Scholar
  45. Orton, A., & Orton J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Patterns in the Teaching and Learning of Mathematics (pp. 104–120). London: Cassell.Google Scholar
  46. Piaget, J. (1952). The Child's Conception of Number. London: Routledge.Google Scholar
  47. Piaget, J. (1978). Recherches sur la Généralisation (Etudes d’Epistémologie Génétique XXXVI). Paris, Presses Universitaires de France.Google Scholar
  48. Radford, L. (1996). Some Reflections on Teaching Algebra Through Generalization. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to Algebra: perspectives for research and teaching, (pp. 107–111). Dordrecht/Boston/London: Kluwer.Google Scholar
  49. RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: toward a strategic research and development program in mathematics education (No. 083303331X). Santa Monica: RAND.Google Scholar
  50. Rivera, F. D. (2006). Changing the face of arithmetic: teaching children algebra. Teaching Children Mathematics 6, 306–311.Google Scholar
  51. Schifter, D. (1999). Reasoning about operations: early algebraic thinking, grades K through 6. In L. Stiff, & F. Curio (Eds.), Mathematical reasoning, K-12: 1999 National Council of Teachers of mathematics yearbook (pp. 62–81). Reston: The National Council of Teachers of Mathematics.Google Scholar
  52. Schliemann, A. D., & Carraher, D. W. (2002). The evolution of mathematical understanding: everyday versus idealized reasoning. Developmental Review, 22(2), 242–266.CrossRefGoogle Scholar
  53. Schliemann, A. D., Carraher, D. W. & Brizuela, B. M. (2001). When tables become function tables. In M. v. d. Heuvel-Panhuizen (Ed.), Proceedings of the 25th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 145–152). Utrecht: Freudenthal Institute.Google Scholar
  54. Schliemann, A. D., Carraher, D. W., Brizuela, B. M., Earnest, D., Goodrow, A., Lara-Roth, S., et al. (2003). Algebra in elementary school. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), International conference for the psychology of mathematics education (Vol. 4, pp. 127–134). Honolulu: University of Hawaii.Google Scholar
  55. Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2007). Bringing out the algebraic character of arithmetic: from children’s ideas to classroom practice. Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  56. Schoenfeld, A. (1995). Report of Working Group 1. In C. B. Lacampagne (Ed.), The algebra initiative colloquium: Vol. 2: Working group papers (pp. 11–18). Washington: US Department of Education, OERI.Google Scholar
  57. Schwartz, J. L. (1996). Semantic aspects of quantity. Cambridge, MA, Harvard University, Department of Education: 40 pp.Google Scholar
  58. Schwartz, J. L. (1999). Can technology help us make the mathematics curriculum intellectually stimulating and socially responsible? International Journal of Computers for Mathematical Learning, 4(2/3), 99–119.CrossRefGoogle Scholar
  59. Schwartz, J., & Yerushalmy, M. (1992a). Getting students to function on and with algebra. In E. Dubinsky, & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 261–289). Washington, DC: Mathematical Association of America.Google Scholar
  60. Schwartz, J. L., & Yerushalmy, M. (1992b). The Function Supposer: Symbols and graphs. Pleasantville, NY: Sunburst Communications.Google Scholar
  61. Schwartz, J. L., & Yerushalmy, M. (1992c). The geometric supposer series. Pleasantville: Sunburst Communications.Google Scholar
  62. Smith, J., & Thompson, P. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 95–132). Mahwah: Erlbaum.Google Scholar
  63. Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20(2), 147–164.CrossRefGoogle Scholar
  64. Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford, & A. P. Schulte (Eds.), Ideas of algebra, K-12: 1988 Yearbook (pp. 8–19). Reston, VA: NCTM.Google Scholar
  65. Vergnaud, G. (1983). Multiplicative structures. In R. A. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 127–174). New York: Academic.Google Scholar
  66. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.CrossRefGoogle Scholar
  67. Weisstein, E. W. (1999). Arithmetic progression; closed-form solution; function; geometric sequence; generalized hypergeometric function; sequence [Electronic entries]. MathWorld—A Wolfram Web Resource from http://mathworld.wolfram.com.

Copyright information

© FIZ Karlsruhe 2007

Authors and Affiliations

  • David W. Carraher
    • 1
  • Mara V. Martinez
    • 2
  • Analúcia D. Schliemann
    • 2
  1. 1.TERCCambridgeUSA
  2. 2.Department of EducationTufts UniversityMedfordUSA

Personalised recommendations