Advertisement

Theoretical Issues and Educational Strategies for Encouraging Teachers to Promote a Linguistic and Metacognitive Approach to Early Algebra

  • Annalisa Cusi
  • Nicolina A. Malara
  • Giancarlo Navarra
Part of the Advances in Mathematics Education book series (AME)

Abstract

After an overview of the studies which led to the rise of the study of early algebra, we sketch our vision of this disciplinary area and of its teaching from a linguistic and socio-constructive point of view. We take into account the teacher’s role in the socio-constructive teaching process and stress the importance of reflecting upon the teaching and learning processes in order to reshape the teacher’s ways of being in the classroom. We dwell upon the strategies enacted and describe the tools we have shaped: theoretical, for the enculturation of early algebra teachers, and methodological, which aim at promoting their awareness and control of their action. We conclude with some considerations about the value of the tools and modalities we have used, as well as on the factors which determine their efficacy.

Keywords

Natural Language Primary School Mathematics Teacher Teacher Training Problem Situation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ainley, J. (coord.) (2001). Research forum on early algebra. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Annual Conference of the International Group for the Psychology of Mathematics Education, Utrecht (The Netherlands) (Vol. 1, pp. 128–159). Google Scholar
  2. Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning. Journal of Mathematics Teacher Education, 9, 33–52. CrossRefGoogle Scholar
  3. Arcavi, A. (1994). Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics, 14(3), 24–35. Google Scholar
  4. Arzarello, F. (1991). Procedural and relational aspects of algebraic thinking. In F. Furinghetti (Ed.), Proceedings of the 15th Annual Conference of the International Group for the Psychology of Mathematics Education, Assisi (Italy) (Vol. 1, pp. 80–87). Google Scholar
  5. Arzarello, F., Bazzini, L., & Chiappini, G. (1993). Cognitive processes in algebraic thinking: Towards a theoretical framework. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F. L. Lin (Eds.), Proceedings of 17th International Conference on the Psychology of Mathematics Education, Tokio (Japan) (Vol. 1, pp. 138–145). Google Scholar
  6. Bell, A. (1976). A study of pupils’ proof explanations. Educational Studies in Mathematics, 7, 23–40. CrossRefGoogle Scholar
  7. Bell, A., Swan, M., Onslow, B., Pratt, K., & Purdy, D. (1985). Diagnostic teaching for long term learning (Report of ESRC Project HR8491/1). Shell Centre for Mathematical Education, University of Nottingham. Google Scholar
  8. Blanton, M., & Kaput, J. (2001). Algebrafying the elementary mathematics experience: Transforming task structures: Transforming practice on a district-wide scale. In E. Chick et al. (Eds.), Proceedings of the 12th ICMI Study ‘The Future of the Teaching and Learning of Algebra’, Melbourne (Australia) (Vol. 1, pp. 87–95). Google Scholar
  9. Blanton, M., & Kaput, J. (2002). Design principles for tasks that support algebraic thinking in elementary school classroom. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of 26th International Conference on the Psychology of Mathematics Education, Norwich (UK) (Vol. 2, pp. 105–112). Google Scholar
  10. Blanton, M. L., Stylianou, D. A., & David, M. M. (2003). The nature of scaffolding in undergraduate students’ transition to mathematical proof. In N. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of 27th International Conference on the Psychology of Mathematics Education, Honolulu (Hawaii) (Vol. 2, pp. 113–120). Google Scholar
  11. Borasi, R., Fonzi, J., Smith, C., & Rose, B. (1999). Beginning the process of rethinking mathematics instruction: A professional development program. Journal of Mathematics Teacher education, 2(1), 49–78. CrossRefGoogle Scholar
  12. Booth, L. R. (1984). Algebra: Children’s Strategies and Errors. A Report of the Strategies and Errors in Secondary Mathematics Project. Windsor, Berks: NFER-Nelson. Google Scholar
  13. Cai, J., Lew, H. C., Morris, A., Moyer, J. C., Ng, S. F., & Schmittau, J. (2005). The development of students’ algebraic thinking in earlier grades: A cross-cultural comparative perspective. ZDM, 37(1), 5–15. CrossRefGoogle Scholar
  14. Carpenter, T., & Franke, M. L. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In E. Chick, K. Stacey, Jl. Vincent, & Jn. Vincent (Eds.), Proceedings of the 12th ICMI Study ‘The Future of the Teaching and Learning of Algebra’, Melbourne (Australia) (Vol. 1, pp. 155–162). Google Scholar
  15. Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking Mathematically. Integrating Arithmetic and Algebra in the Elementary School. Portsmouth, NH: Heinemann. Google Scholar
  16. Carraher, D., Brizuela, B., & Schliemann, A. (2000). Bringing out the algebraic character of arithmetic: Instantiating variables in addition and subtraction. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education, Hiroshima (Japan) (Vol. 2, pp. 145–152). Google Scholar
  17. Chick, E., Stacey, K., Vincent, Jl., & Vincent, Jn. (Eds.) (2001). Proceedings of the 12th ICMI Study ‘The Future of the Teaching and Learning of Algebra’. Melbourne, Australia: University of Melbourne. Google Scholar
  18. Cusi, A. (2008). An approach to proof in elementary number theory focused on representation and interpretation aspects: The teacher’s role. In B. Czarnocha (Ed.), Handbook of Mathematics Teaching Research (pp. 107–122). Rzeszów: Rzeszów University Press. Google Scholar
  19. Cusi, A., & Malara, N. A. (2008). Approaching early algebra: Teachers’ educational processes and classroom experiences. Quadrante, 16(1), 57–80. Google Scholar
  20. Da Rocha Falcão, J. T. (1995). A case study of algebraic scaffolding: From balance to algebraic notation. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th Annual Conference of the International Group for the Psychology of Mathematics Education, Recife (Brazil) (Vol. 2, pp. 66–73). Google Scholar
  21. Dougherty, B. (2001). Access to algebra: A process approach. In E. Chick, K. Stacey, Jl. Vincent, & Jn. Vincent (Eds.), Proceedings of the 12th ICMI Study ‘The Future of the Teaching and Learning of Algebra’, Melbourne (Australia) (Vol. 1, pp. 207–212). Google Scholar
  22. DFE (Department for Education) (1995). Mathematics in the National Curriculum. London: HMSO. Google Scholar
  23. Ferreri, S. (2006). Parole tra quantità e qualità. In I. Tempesta & M. Maggio (Eds.), Linguaggio, mente, parole, Dall’infanzia all’adolescenza, Collana GISCEL (pp. 19–25). Milano: Franco Angeli Editore. Google Scholar
  24. Filloy, E. (1990). PME algebra research. A working perspective. In G. Booker, P. Cobb, & T. N. Mendicuti (Eds.), Proceedings of the 14th Annual Conference of the International Group for the Psychology of Mathematics Education, Oaxtepex (Mexico) (Vol. 1, pp. 1–33). Google Scholar
  25. Filloy, E. (1991). Cognitive tendencies and abstraction processes in algebra learning. In F. Furinghetti (Ed.), Proceedings of the 15th Annual Conference of the International Group for the Psychology of Mathematics Education, Assisi (Italy) (Vol. 2, pp. 48–55). Google Scholar
  26. Gray, E., & Tall, D. (1993). Success and failure in mathematics: The flexible meaning of symbols as process and concept. Mathematics Teaching, 142, 6–10. Google Scholar
  27. Harper, E. (Ed.) (1987–88). NMP Mathematics for Secondary School. Essex, UK: Longman. Google Scholar
  28. Kaput, J. (1991). Notations and representations as mediators of constructive processes. In E. von Glasersfeld (Ed.), Radical Constructivism in Mathematics Education (pp. 53–74). The Netherlands: Kluwer Academic Publishers. Google Scholar
  29. Kaput, J. (1995). A research base supporting long term algebra reform. In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of the 17th Annual Meeting of PME-NA (Vol. 1, pp. 71–94). Columbus (OH): ERIC Clearinghouse for Science, Mathematics and Environmental Education. Google Scholar
  30. Kaput, J., & Blanton, M. (2001). Algebrafying the elementary mathematics experience: Transforming task structures. In E. Chick, K. Stacey, Jl. Vincent, & Jn. Vincent (Eds.), Proceedings of the 12th ICMI Study ‘The Future of the Teaching and Learning of Algebra’, Melbourne (Australia) (Vol. 1, pp. 344–353). Google Scholar
  31. Kaput, J., Carraher, D. W., & Blanton, M. L. (Eds.) (2007). Algebra in Early Grades. Mahwah (NJ): Lawrence Erlbaum Associates. Google Scholar
  32. Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. Wagner & K. Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra (pp. 33–56). Reston (Virginia): LEA. Google Scholar
  33. Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 390–419). New York: Macmillan. Google Scholar
  34. Kieran, C. (1998). The changing face of school algebra. In C. Alsina, J. Alvarez, B. Hodgson, C. Laborde, & A. Perez (Eds.), 8th International Congress on Mathematics Education: Selected Lectures (pp. 271–290). Seville, Spain: S.A.E.M. Thales. Google Scholar
  35. Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator (Singapore), 8(1), 139–151. Google Scholar
  36. Kuchemann, D. E. (1981). Algebra. In K. Hart (Ed.), Children Understanding Mathematics: 11–16 (pp. 102–119). London: Murray. Google Scholar
  37. Jaworski, B. (1998). Mathematics teacher research: Process, practice and the development of teaching. Journal of Mathematics Teacher Education, 1, 3–31. CrossRefGoogle Scholar
  38. Jaworski, B. (2003). Research practice into/influencing mathematics teaching and learning development: Towards a theoretical framework based on co-learning partnerships. Educational Studies in Mathematics, 54, 249–282. CrossRefGoogle Scholar
  39. Jaworski, B. (2004). Grappling with complexity: Co-learning in inquiry communities in mathematics teaching development. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Conference of the International Group for the Psychology of Mathematics Education, Bergen (Norway) (vol. 1, pp. 17–36). Google Scholar
  40. Lee, L., & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathematics, 20(1), 41–54. CrossRefGoogle Scholar
  41. Linchevski, L. (1995). Algebra with numbers and arithmetic with letters: A definition of pre-algebra. Journal of Mathematical Behaviour, 14, 113–120. CrossRefGoogle Scholar
  42. Lins, R. C. (1990). A framework for understanding what algebraic thinking is. In G. Booker, P. Cobb, & T. N. Mendicuti (Eds.), Proceedings of the 14th Annual Conference of the International Group for the Psychology of Mathematics Education, Oaxtepex (Mexico) (Vol. 2, pp. 93–100). Google Scholar
  43. Malara, N. A. (2003). Dialectics between theory and practice: Theoretical issues and aspects of practice from an early algebra project. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.), Proceedings of the Joint Meeting of PME and PME-NA, Honolulu (Hawaii) (Vol. 1, pp. 33–48). Google Scholar
  44. Malara, N. A. (2005). Leading in-service teachers to approach early algebra. In L. Santos et al. (Eds.), Mathematics Education: Paths and Crossroads (pp. 285–304). Lisbon: Etigrafe. Google Scholar
  45. Malara, N. A. (2008). Methods and tools to promote a socio-constructive approach to mathematics teaching in teachers. In B. Czarnocha (Ed.), Handbook of Mathematics Teaching Research (pp. 89–102). Rzeszów: Rzeszów University Press. Google Scholar
  46. Malara, N. A., & Iaderosa, R. (1999). Theory and practice: A case of fruitful relationship for the renewal of the teaching and learning of algebra. In F. Jaquet (Ed.), Proceedings of CIEAEM 50 (pp. 38–54). Neuchatel: Neuchatel University Press. Google Scholar
  47. Malara, N. A., & Navarra, G. (2001). “Brioshi” and other mediation tools employed in a teaching of arithmetic with the aim of approaching algebra as a language. In E. Chick, K. Stacey, Jl. Vincent, & Jn. Vincent (Eds.), Proceedings of the 12th ICMI Study ‘The Future of the Teaching and Learning of Algebra’, Melbourne (Australia) (pp. 412–419). Google Scholar
  48. Malara, N. A., & Navarra, G. (2003). ArAl Project: Arithmetic Pathways Towards Favouring Pre-Algebraic Thinking. Bologna: Pitagora. Google Scholar
  49. Malara, N. A., & Navarra, G. (2009). The analysis of classroom-based processes as a key task in teacher training for the approach to early algebra. In B. Clarke, B. Grevholm, & R. Millman (Eds.), Tasks in Primary Mathematics Teacher Education (pp. 235–262). Berlin: Springer. CrossRefGoogle Scholar
  50. Malara, N. A., & Zan, R. (2002). The Problematic relationship between theory and practice. In L. English (Ed.), Handbook of International Research in Mathematics Education (pp. 553–580). Mahwah (NJ): LEA. Google Scholar
  51. Malara, N. A., & Zan, R. (2008). The complex interplay between theory and practice: Reflections and examples. In L. English (Ed.), Handbook of International Research in Mathematics Education (2nd ed.) (pp. 539–564). New York: Routledge. Google Scholar
  52. Mason, J. (1998). Enabling teachers to be real teachers: Necessary levels of awareness and structure of attention. Journal of Mathematics Teacher Education, 1, 243–267. CrossRefGoogle Scholar
  53. Mason, J. (2002). Researching Your Own Practice: The Discipline of Noticing. London: The Falmer Press. Google Scholar
  54. Meira, L. (1990). Developing knowledge of functions through manipulation of a physical device. In G. Booker, P. Cobb, & T. N. Mendicuti (Eds.), Proceedings of the 14th Annual Conference of the International Group for the Psychology of Mathematics Education, Oaxtepex (Mexico) (Vol. 2, pp. 101–108). Google Scholar
  55. Menzel, B. (2001). Language conceptions of algebra are idiosyncratic. In E. Chick, K. Stacey, Jl. Vincent, & Jn. Vincent (Eds.), Proceedings of the 12th ICMI Study ‘The Future of the Teaching and Learning of Algebra’, Melbourne (Australia) (Vol. 2, pp. 446–453). Google Scholar
  56. NCTM (National Council of Teacher of Mathematics) (1998). The Nature and Role of Algebra in the K-14 Curriculum. Washington, DC: National Academy Press. Google Scholar
  57. NCTM (National Council of Teacher of Mathematics) (2000). Principles and Standards for School Mathematics. Reston (VA): NTCM. http://standars.NTCM.org. Google Scholar
  58. Ponte, J. P. (2004). Investigar a nossa própria prática: una stratégia de formaçáo e de construçáo de conhecimento professional. In E. Castro & E. De la Torre (Eds.), Investigación en educación matemática, University of Coruña (Spain) (pp. 61–84). Google Scholar
  59. Potari, D., & Jaworski, B. (2002). Tackling complexity in mathematics teaching development: Using the teaching triad as a tool for reflection analysis. Journal of Mathematics Teacher Education, 5(4), 351–380. CrossRefGoogle Scholar
  60. Radford, L. (2000). Signs and meanings in students’ emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237–268. CrossRefGoogle Scholar
  61. Schoenfeld, A. (1998). Toward a theory of teaching in context. Issues in Education, 4(1), 1–94. CrossRefGoogle Scholar
  62. Schwarz, B., Dreyfus, T., Hadas, N., & Hershkowitz, R. (2004). Teacher guidance of knowledge construction. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Conference of the International Group for Psychology of Mathematics Education, Bergen (Norway) (Vol. 4, pp. 169–176). Google Scholar
  63. Sherin, M. G. (2002). A balancing act: Developing a discourse community in a mathematics classroom. Journal of Mathematics Teacher Education, 5, 205–233. CrossRefGoogle Scholar
  64. Wood, T. (1999). Approaching teacher development: Practice into theory. In B. Jaworski, T. Wood, & S. Dawson (Eds.), Mathematics Teacher Education: Critical International Perspectives (pp. 163–179). London: The Falmer Press. Google Scholar
  65. Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477. CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Annalisa Cusi
    • 1
  • Nicolina A. Malara
    • 1
  • Giancarlo Navarra
    • 1
  1. 1.Mathematics DepartmentUniversity of Modena and Reggio EmiliaModenaItaly

Personalised recommendations