Learning to Solve Equations in Three Swedish-Speaking Classrooms in Finland



In this chapter, video recorded episodes occurring in three Swedish-speaking Grade 6 classrooms in Finland constitute the empirical material. The focus of the instruction is the introduction of basic principles of equation solving. The analysis focuses on the classroom interactions, and the role of the textbooks in co-determining the structure of the lessons and the challenges students encounter. Through interviews with teachers on how they account for their approaches to equation solving, it is possible to discern different emphases on how to teach equation solving procedures.


  1. Anthony, G., & Burgess, T. (2014). Solving linear equations: A balanced approach. In F. K. S. Leung, K. Park, D. Holton, & D. Clarke (Eds.), Algebra teaching around the world (pp. 17–37). Rotterdam: Sense.Google Scholar
  2. Asikainen, K., Fälden, H., Nyrhinen, K., Rokka, P., Vehmans, P., Törnroos, S., et al. (2007a). Min matematik 5 [My mathematics 5]. Helsingfors: Schildts Förlag.Google Scholar
  3. Asikainen, K., Fälden, H., Nyrhinen, K., Rokka, P., Vehmans, P., Törnroos, S., et al. (2007b). Min matematik 5. Lärarhandledning [My mathematics 5. Teacher’s study guide]. Helsingfors: Schildts Förlag.Google Scholar
  4. Asikainen, K., Fälden, H., Nyrhinen, K., Rokka, P., Vehmans, P., Törnroos, S., et al. (2008a). Min matematik 6 [My mathematics 6]. Helsingfors: Schildts Förlag.Google Scholar
  5. Asikainen, K., Fälden, H., Nyrhinen, K., Rokka, P., Vehmans, P., Törnroos, S., et al. (2008b). Min matematik 6. Lärarhandledning [My mathematics 6. Teacher’s study guide]. Helsingfors: Schildts Förlag.Google Scholar
  6. 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
  7. Black, L., Mendick, H., & Solomon, Y. (Eds.). (2009). Mathematical relationships in education. Identities and participation. London: Routledge.Google Scholar
  8. Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87–115.Google Scholar
  9. Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(4), 232–236.Google Scholar
  10. Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19–25.Google Scholar
  11. Herscovics, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra. Educational Studies in Mathematics, 27, 59–78.CrossRefGoogle Scholar
  12. Hiebert, J. (Ed.). (1986). Conceptual and procedural knowledge: The case of mathematics. Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  13. Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317–326.CrossRefGoogle Scholar
  14. Kieran, C. (2014). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. In K. Leatham (Ed.), Vital directions in mathematics education research (pp. 153–172). New York, NY: Springer.Google Scholar
  15. Knuth, E. J., Alibali, M. W., McNeil, N., Weinberg, A., & Stephens, A. C. (2011). Middle school students’ understanding of core algebraic concepts: Equivalence & variable. In J. Cai & E. Knuth (Eds.), Early algebraization. A global dialogue from multiple perspectives (pp. 259–276). Berlin: Springer.Google Scholar
  16. Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312.Google Scholar
  17. National Board of Education. (2004). The curriculum for basic education. Helsinki: National Board of Education.Google Scholar
  18. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.CrossRefGoogle Scholar
  19. Rittle-Johnson, B., & Star, J. R. (2009). Compared with what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving. Journal of Educational Psychology, 101(3), 529–544.CrossRefGoogle Scholar
  20. Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2013). Bringing out the algebraic character of arithmetic. London: Routledge.CrossRefGoogle Scholar
  21. Stacey, K., & MacGregor, M. (2000). Learning the algebraic method of solving problems. Journal of Mathematical Behavior, 18(2), 149–167.CrossRefGoogle Scholar
  22. Törnroos, J. (2001, June). Finnish mathematics textbooks in grades 5-7. Presentation at the Second Scandinavian Symposium on Research Methods in Science and Mathematics Education, Helsinki, Finland.Google Scholar
  23. Vieira, L., Gimenez, J., & Palhares, P. (2013). Relational understanding when introducing early algebra in Portuguese schools. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the Eight Congress of the European Society for Research in Mathematics Education (pp. 550–557). Manavgat, Side: Middle East Technical University, Turkey on behalf of the European Society for Research in Mathematics Education.Google Scholar
  24. 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.CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Education and Welfare StudiesÅbo Akademi UniversityVasaFinland
  2. 2.Faculty of EducationUniversity of LaplandRovaniemiFinland

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