Educational Studies in Mathematics

, Volume 96, Issue 1, pp 17–32 | Cite as

Students’ development of structure sense for the distributive law

  • Alexander Schüler-MeyerEmail author


After being introduced to the distributive law in meaningful contexts, students need to extend its scope of application to unfamiliar expressions. In this article, a process model for the development of structure sense is developed. Building on this model, this article reports on a design research project in which exercise tasks support students in developing their structure sense for the distributive law by means of structural mappings and guiding examples. A design experiment was conducted in six groups, each consisting of two eighth graders. Two contrasting cases are qualitatively analyzed and compared in terms of the development of the students’ structure sense for the distributive law. Theoretically, this article provides a development model for structure sense. Empirically, six characteristics of these development processes are reconstructed. Under certain conditions that are discussed in the end, the exercise tasks can help students to develop their structure sense for the distributive law.


Algebra Design research Distributive law Structure sense 



I would like to thank Susanne Prediger and the anonymous reviewers for their valuable and constructive feedback on previous versions of this paper.


  1. Banerjee, R., & Subramaniam, K. (2012). Evolution of a teaching approach for beginning algebra. Educational Studies in Mathematics, 80(3), 351–367.CrossRefGoogle Scholar
  2. Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In H. Moraová, M. Krátká, & A. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 126–154). Prague: Charles University.Google Scholar
  3. Bjuland, R. (2012). The mediating role of a teacher’s use of semiotic resources in pupils’ early algebraic reasoning. ZDM – The International Journal on Mathematics Education, 44(5), 665–675.CrossRefGoogle Scholar
  4. Brousseau, G. (1997). Theory of didactical situations in mathematics. Didactique des Mathématiques, 1970–1990. New York: Kluwer Academic.Google Scholar
  5. Capraro, M. M., & Joffrion, H. (2006). Algebraic equations: Can middle school students meaningfully translate from words to mathematical symbols? Reading Psychology, 27, 147–164.CrossRefGoogle Scholar
  6. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.CrossRefGoogle Scholar
  7. Drijvers, P., Goddijn, A., & Kindt, M. (2011). Algebra education: Exploring topics and themes. In P. Drijvers (Ed.), Secondary algebra education. Revisiting topics and themes and exploring the unknown (pp. 5–26). Sense: Rotterdam.CrossRefGoogle Scholar
  8. English, L. D., & Sharry, P. V. (1996). Analogical reasoning and the development of algebraic abstraction. Educational Studies in Mathematics, 30(2), 135–157.CrossRefGoogle Scholar
  9. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht: Reidel.Google Scholar
  10. Hoch, M., & Dreyfus, T. (2005). Students’ difficulties with applying a familiar formula in an unfamiliar context. In H. L. Chick & J. L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (vol. 3, pp. 145–152). Melbourne: PME.Google Scholar
  11. Jungwirth, H. (2003). Interpretative Forschung in der Mathematikdidaktik – Ein Überblick für Irrgäste, Teilzieher und Standvögel [Interpretative research in mathematics education – An overview for vagrants, partial migrants and sedentary birds]. ZDM Zentralblatt für Didaktik der Mathematik, 35(5), 189–200.Google Scholar
  12. Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick, & M. Kendal (Eds.), The future of the teaching and learning of algebra. The 12th ICMI study (pp. 21–33). Dordrecht: Springer.CrossRefGoogle Scholar
  13. Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics Education, 20(3), 274.CrossRefGoogle Scholar
  14. Küchemann, D. (1981). Algebra. In K. Hart, M. L. Brown, D. E. Kuchemann, D. Kerslake, G. Ruddock, & M. McCartney (Eds.), Children’s understanding of mathematics: 11–16 (pp. 102–119). London: Murray.Google Scholar
  15. Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40(2), 173–196.CrossRefGoogle Scholar
  16. Malle, G. (1993). Didaktische Probleme der elementaren Algebra [Didactical problems of elementary algebra]. Wiesbaden: Vieweg.Google Scholar
  17. MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19.CrossRefGoogle Scholar
  18. Mariotti, M. A., & Cerulli, M. (2001). Semiotic mediation for algebra teaching and learning. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of mathematics education (vol. 3, pp. 225–232). Utrecht: Utrecht University.Google Scholar
  19. Mason, J. (2008). Doing = construing and doing + discussing = learning: The importance of the structure of attention. In M. Niss (Ed.), Proceedings of ICME 10 (CD). IMFUFA: Roskilde.Google Scholar
  20. Mason, J., Graham, A., & Johnston-Wilder, S. (2005). Developing thinking in algebra. London: Open University.Google Scholar
  21. Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman & J. S. Brown (Eds.), Intelligent tutoring systems (pp. 25–50). London: Academic Press.Google Scholar
  22. Meyer, A. (2014). Students’ transformation of algebraic expressions as ‘recognizing basic structures’ and ‘giving relevance’. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (vol. 4, pp. 209–216). Vancouver: PME.Google Scholar
  23. Novotná, J., & Hoch, M. (2008). How structure sense for algebraic expressions or equations is related to structure sense for abstract algebra. Mathematics Education Research Journal, 20(2), 93–104.CrossRefGoogle Scholar
  24. Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes: An overview on achievements and challenges. ZDM Mathematics Education, 47(6), 877–891.CrossRefGoogle Scholar
  25. Prediger, S., & Zwetzschler, L. (2013). Topic-specific design research with a focus on learning processes: The case of understanding algebraic equivalence in grade 8. In T. Plomp & N. Nieveen (Eds.), Educational design research - part B illustrative cases (pp. 407–424). Enschede: SLO.Google Scholar
  26. Radford, L. (2006). Elements of a cultural theory of objectification. Revista Latinoamericana de Investigación en Matemática Educativa, Special Issue on Semiotics, Culture and Mathematical Thinking, 103–129. Retrieved from
  27. Radford, L. (2008). Iconicity and contraction: A semiotic investigation of forms of algebraic generalizations of patterns in different contexts. ZDM – The International Journal on Mathematics Education, 40(1), 83–96.CrossRefGoogle Scholar
  28. Radford, L., & Puig, L. (2007). Syntax and meaning as sensuous, visual, historical forms of algebraic thinking. Educational Studies in Mathematics, 66(2), 145–164.CrossRefGoogle Scholar
  29. Rezat, S., & Sträßer, R. (2012). From the didactical triangle to the socio-didactical tetrahedron: Artifacts as fundamental constituents of the didactical situation. ZDM – The International Journal on Mathematics Education, 44(5), 641–651.CrossRefGoogle Scholar
  30. Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574.CrossRefGoogle Scholar
  31. Rittle-Johnson, B., Star, J. R., & Durkin, K. (2009). The importance of prior knowledge when comparing examples: Influences on conceptual and procedural knowledge of equation solving. Journal of Educational Psychology, 101(4), 836–852.CrossRefGoogle Scholar
  32. Rüede, C. (2012). The structuring of an algebraic expression as the production of relations. Journal für Mathematik-Didaktik, 33(1), 113–141.CrossRefGoogle Scholar
  33. 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), 1–36.CrossRefGoogle Scholar
  34. Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50(1), 29–47.CrossRefGoogle Scholar
  35. van Oers, B. (2001). Educational forms of initiation in mathematical culture. Educational Studies in Mathematics, 46(1/3), 59–85.CrossRefGoogle Scholar
  36. Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.CrossRefGoogle Scholar
  37. Zolkower, B., & Shreyar, S. (2007). A teacher’s mediation of a thinking-aloud discussion in a 6th grade mathematics classroom. Educational Studies in Mathematics, 65(2), 177–202.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Institute for Development and Research in Mathematics EducationTU Dortmund UniversityDortmundGermany

Personalised recommendations