Skip to main content
SpringerLink
Log in
Book cover

Encountering Algebra pp 1–11Cite as

School Algebra

  • Chapter
  • First Online:

Abstract

Algebra is a central element of the curriculum in mathematics. In most parts of the world, a significant proportion of lower secondary school curricula in mathematics is devoted to algebra. The chapter introduces the origins of school algebra as well as issues related to how teaching and learning have been organized. What constitutes important elements of algebra as a school subject is discussed in light of the extensive literature on algebra and early algebra. Learning obstacles related to variables and representations are frequently observed in the literature, and constitutes the background for the empirical research project, VIDEOMAT, reported in this volume.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    https://profkeithdevlin.org/2011/11/20/what-is-algebra/ (retrieved 15 Dec. 2017).

References

  • Balacheff, N. (2001). Symbolic arithmetic vs algebra the core of a didactical dilemma. Postscript. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (pp. 249–260). Dordrecht: Kluwer.

    Google Scholar 

  • Bednarz, N., Kieran, C., & Lee, L. (Eds.). (1996). Approaches to algebra: Perspectives for research and teaching (Vol. 18). Dordrecht: Kluwer.

    Google Scholar 

  • Blanton, M., Stephens, A., Knuth, E., Murphy Gardiner, A., Isler, I., & Kim, J.-S. (2015). The development of children’s algebraic thinking: The impact of a comprehensive early algebra intervention in third grade. Journal for Research in Mathematics Education, 46(1), 39–87.

    Article  Google Scholar 

  • Britt, M. S., & Irwin, K. C. (2011). Algebraic thinking with and without algebraic representation. In J. Cai & E. Knuth (Eds.), Early algebraization (pp. 137–159). Berlin: Springer.

    Chapter  Google Scholar 

  • Bush, S., & Karp, K. (2013). Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32(3), 613–632.

    Article  Google Scholar 

  • Cai, J., & Knuth, E. (Eds.). (2011). Early algebraization. A global dialogue from multiple perspectives. Berlin: Springer.

    Google Scholar 

  • Cai, J., & Moyer, J. (2008). Developing algebraic thinking in earlier grades: Some insights from international comparative studies. In 70th Yearbook Algebra and algebraic thinking in school mathematics (pp. 69–182). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Carraher, D., & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F. Lester (Ed.), Handbook of research in mathematics education (pp. 669–705). Greenwich, CT: Information Age Publishing.

    Google Scholar 

  • Carraher, D. W., Schliemann, A. D., & Schwartz, J. (2008). 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). New York, NY: Routledge.

    Google Scholar 

  • Collis, K. F. (1975). The development of formal reasoning. New South Wales: University of Newcastle.

    Google Scholar 

  • Da Rocha Falcão, J. (1995). A case study of algebraic scaffolding: From balance scale to algebraic notation. In L. Meira & D. Carraher (Eds.), 19th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 66–73). Recife: Universidade Federal de Pernambuco.

    Google Scholar 

  • Davydov, V. V., Gorbov, S. F., Mikulina, G. G., & Savaleva, O. V. (1999). Mathematics class 1 (J. Schmittau, Trans.). Binghamton, NY: State University of New York.

    Google Scholar 

  • Even, R. (1998). Factors involved in linking representations of functions. The Journal of Mathematical Behavior, 17(1), 105–121.

    Article  Google Scholar 

  • Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.

    Google Scholar 

  • Gallardo, A. (2001). Historical-epistemological analysis in mathematics education: Two works in didactics of algebra. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.), Perspectives on school algebra (Vol. 22, pp. 121–139). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Goldin, G., & Shteingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. Cuoco & F. Curcio (Eds.), The roles of representation in school mathematics (pp. 1–23). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59–78.

    Article  Google Scholar 

  • Hewitt, D. (2014). A symbolic dance: The interplay between movement, notation, and mathematics on a journey toward solving equations. Mathematical Thinking and Learning, 16(1), 1–31.

    Article  Google Scholar 

  • Hohenwarter, M., & Jones, K. (2007). BSRLM Geometry Working Group: Ways of linking geometry and algebra, the case of Geogebra. Proceedings of the British Society for Research into Learning Mathematics, 27(3), 126–131.

    Google Scholar 

  • Kaput, J. (1987). Representation systems and mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 18–26). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Kaput, J., Blanton, M., & Moreno, L. (2008). Algebra from a symbolization point of view. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 19–56). New York, NY: Lawrence Erlbaum.

    Google Scholar 

  • Kaput, J., Carraher, D., & Blanton, M. (Eds.). (2008). Algebra in the early grades. New York, NY: Routledge.

    Google Scholar 

  • Kendal, M., & Stacey, K. (2004). Algebra: A world of difference. In R. Stacey (Ed.), The future of the teaching and learning of algebra (pp. 329–346). Dordrecht: Kluwer.

    Google Scholar 

  • 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, NY: Macmillan.

    Google Scholar 

  • Kieran, C. (1996). The changing face of school algebra. In C. Alsina, J. Alvarez, B. Hodgson, C. Laborde, & A. Pérez (Eds.), 8th International Congress on Mathematical Education: Selected lectures (pp. 271–290). Seville: S.A.E.M. Thales.

    Google Scholar 

  • Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Teacher, 8(1), 139–151.

    Google Scholar 

  • Kieran, K. (Ed.). (2018). Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice. New York, NY: Springer.

    Google Scholar 

  • Kieran, C., Pang, J. S., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. ICME 13 Topical Surveys. Hamburg: Springer Open.

    Book  Google Scholar 

  • Küchemann, D. (1978). Children’s understanding of numerical variables. Mathematics in School, 9, 23–26.

    Google Scholar 

  • Küchemann, D. (1981). Algebra. In K. Hart (Ed.), Children’s understanding of mathematics: 11-16 (pp. 102–119). London: John Murray.

    Google Scholar 

  • Lesh, R., & Landau, M. (1983). Acquisition of mathematics concepts and processes. New York, NY: Academic.

    Google Scholar 

  • Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  • Leung, F. K. S., Park, K., Holton, D., & Clarke, D. (Eds.). (2014). Algebra teaching around the world. Rotterdam: Sense Publishers.

    Google Scholar 

  • MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33(1), 1–19.

    Article  Google Scholar 

  • Mason, J. (1996). Expressing generality and the roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (pp. 65–86). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Ng, S. F. (2004). Developing algebraic thinking in the early grades: Case study of the Singapore primary mathematics curriculum. The Mathematics Educator, 8(1), 39–59.

    Google Scholar 

  • Philipp, R. A. (1992). The many uses of algebraic variables. The Mathematics Teacher, 85(7), 557–561.

    Google Scholar 

  • Radford, L. (2011). Embodiment, perception and symbols in the development of early algebraic thinking. In B. Ubuz (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 17–14). Ankara: PME.

    Google Scholar 

  • Radford, L. (2018). The emergence of symbolic algebraic thinking in primary school. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 1–23). New York: Springer.

    Google Scholar 

  • Rystedt, E., Kilhamn, C., & Helenius, O. (2016). What’s there in an n? Nordic Studies in Mathematics Education, 21(1), 5–26.

    Google Scholar 

  • Schmittau, J. (2011). The role of theoretical analysis in developing algebraic thinking: A Vygotskian perspective. In J. Cai & E. Knuth (Eds.), Early algebraization (pp. 71–81). Berlin: Springer.

    Chapter  Google Scholar 

  • Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.

    Article  Google Scholar 

  • Sutherland, R. (1987). A longitudinal study of the development of pupils’ algebraic thinking in a logo environment. Diss., University of London, Institute of Education.

    Google Scholar 

  • Sutherland, R. (1993). Connecting theory and practice: Results from the teaching of Logo. Educational Studies in Mathematics, 24(1), 95–113.

    Article  Google Scholar 

  • Treffers, A. (1987). Three dimensions: A model of goal and theory. In Description in mathematics instruction—The Wiskobas Project (p. 247). Dordrecht: Reidel.

    Google Scholar 

  • Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford & A. Shulte (Eds.), The ideas of algebra, K-12 (pp. 8–19). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Varadarajan, V. (1998). Algebra in ancient and modern times. Providence, RI: American Mathematical Society.

    Book  Google Scholar 

  • Vergnaud, G. (1998). A comprehensive theory of representation for mathematics education. The Journal of Mathematical Behavior, 17(2), 167–181.

    Article  Google Scholar 

  • Vlassis, J. (2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49, 341–359.

    Article  Google Scholar 

  • Watanabe, T. (2011). Shiki: A critical foundation for school algebra in Japanese elementary school mathematics. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 109–124). Berlin: Springer.

    Chapter  Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Gothenburg, Sweden

    Cecilia Kilhamn

  2. Faculty of Education and Welfare Studies, Åbo Akademi University, Vasa, Finland

    Ann-Sofi Röj-Lindberg & Ole Björkqvist

Authors
  1. Cecilia Kilhamn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ann-Sofi Röj-Lindberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ole Björkqvist
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cecilia Kilhamn .

Editor information

Editors and Affiliations

  1. Department of Pedagogical, Curricular and Professional Studies, University of Gothenburg, Gothenburg, Sweden

    Cecilia Kilhamn

  2. Department of Education, Communication and Learning, University of Gothenburg, Gothenburg, Sweden

    Roger Säljö

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kilhamn, C., Röj-Lindberg, AS., Björkqvist, O. (2019). School Algebra. In: Kilhamn, C., Säljö, R. (eds) Encountering Algebra. Springer, Cham. https://doi.org/10.1007/978-3-030-17577-1_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-17577-1_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-17576-4

  • Online ISBN: 978-3-030-17577-1

  • eBook Packages: EducationEducation (R0)

Publish with us

Policies and ethics

Search

Navigation