The Use of Variables in a Patterning Activity: Counting Dots
The present paper examines a patterning activity that was organised within a teaching experiment in order to analyse the different uses of variables by secondary school students. The activity presented in the paper can be categorised as a pictorial/geometric linear pattern. We adopted a student-oriented perspective for our analysis, in order to grasp how students perceive their own generalising actions. The analysis of our data led us to two broad categories for variable use, according to whether the variable is viewed as a generalised number or not. Our results also show that students sometimes treat the variable as closely linked to a referred object, as a superfluous entity or as a constant. Finally, the notion of equivalence, which is an important step towards understanding variables, proved difficult for our students to grasp.
Cobb, P., & Steffe, L. P. (1983). The constructivist researcher as teacher and model builder. Journal for
Research in Mathematics Education, 14(2), 83–94.
Dörfler, W. (2008). En route from patterns to algebra: Comments and reflections. ZDM Mathematics
Education, 40(1), 143–160.
Dretske, F. (1990). Seeing, believing, and knowing. In D. Osherson, S. M. Kosslyn, & J. Hollerback (Eds.),
Visual cognition and action: An invitation to cognitive science (pp. 129–148). Cambridge, MA: MIT Press.
Ellis, A. B. (2007a). Connections between generalizing and justifying: Students reasoning with linear
relationships. Journal for Research in Mathematics Education, 38(3), 194–229.
Ellis, A. B. (2007b). A taxonomy for categorizing generalizations: Generalizing actions and reflection
generalizations. Journal of the Learning Sciences, 16(2), 221–262.
Ellis, A. B. (2011). Generalizing-promoting actions: How classroom collaborations can support
students’ mathematical generalizations. Journal for Research in Mathematics Education, 42(4), 308–345.
English, L., & Warren, E. (1995). General reasoning processes and elementary algebraic understanding:
Implications for instruction. Focus on Learning Problems in Mathematics, 17(4), 1–19.
English, L. D., & Warren, E. (1998). Introducing the variable through pattern exploration. The
Mathematics Teacher, 91(2), 166–171.
Kieran, C. (1989). A perspective on algebraic thinking. In G. Vergnaud, J. Rogalski, & M. Artigue
(Eds.), Proceedings of the 13th conference of the international group for the psychology of mathematics
education (pp. 163–171). Paris: PME.
Krygowska, A. Z. (1980). Zarys Dydaktyki Matematyki, tom II [Overview of didactics of mathematics, Vol. 2]. Warsaw: Wydawnictwa Szkolne i Pedagogogiczne.
Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (pp. 87–106). Dordrecht: Kluwer.
Lee, L., & Wheeler, D. (1987). Algebraic thinking in high school students: Their conceptions of generalization and justification (Research Report). Montreal, CA: Concordia University, Department of Mathematics.
Legutko, M., & Stańdo J. (2008). Jakie działania powinny podjąć polskie szkoły w świetle badań PISA? [What actions should be taken by Polish schools in the light of PISA exams?]. In H. Kąkol (Ed.), Prace Monograficzne z Dydaktyki Matematyki. Współczesne problemy nauczania matematyki 1 (pp. 19–34). Bielsko-Biała: Stowarzyszenie Nauczycieli Matematyki.
Lobato, J. (2003). How design experiments can inform a rethinking of transfer and vice versa. Educational Researcher, 32(1), 17–20.
Malara, N. (2012). Generalization processes in the teaching/learning of algebra: Students behaviours and teacher role. In B. Maj-Tatsis & K. Tatsis (Eds.), Generalization in mathematics at all educational levels (pp. 57–90). Rzeszów: Wydawnictwo Uniwersytetu Rzeszowskiego.
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (pp. 65–86). Dordrecht: Kluwer.
National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Orton, A., & Orton, J. (1994). Students’ perception and use of pattern and generalization. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th international conference for the psychology of mathematics education (Vol. III, pp. 407–414). Lisbon: PME Program Committee.
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, UK: Cassell.
Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. Cirtina, M. Sáiz, & A. Méndez (Eds.), Proceedings of the 28th international conference for the psychology of mathematics education, NA Chapter (Vol. I, pp. 2–21). Mexico: UPN.
Radford, L. (2011). Grade 2 students’ non-symbolic algebraic thinking. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 303–322). Heidelberg: Springer.
Reznic, T., & Tabach, M. (2002). Armon HaMathematica - Algebra Be Sviva Memuchshevet, Helek Gimel [The mathematical palace - Algebra with computers for grade seven, Part C]. Rehovot: Weizmann Institute of Science.
Rivera, F. (2010). Visual templates in pattern generalization activity. Educational Studies in Mathematics, 73(3), 297–328.
Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20(2), 147–164.
Strauss, A., & Corbin, J. (1990). Basics of qualitative research. Grounded theory procedures and techniques. Newbury Park, CA: Sage Publications.
Wilkie, K. J. (2016). Students’ use of variables and multiple representations in generalizing functional relationships prior to secondary school. Educational Studies in Mathematics, 93(3), 333–361.
Zaręba, L. (2012). Matematyczne uogólnianie. Możliwości uczniów i praktyka nauczania [Mathematical generalisation. Abilities of students and teaching practices]. Krakow: Wydawnictwo naukowe Uniwersytetu Pedagogicznego.
Zazkis, R., & Hazzan, O. (1999). Interviewing in mathematics education research: Choosing the questions. Journal of Mathematical Behavior, 17(4), 429–439.
Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49(3), 379–402.
In order to ensure both the widest dissemination and protection of material published in CEPS Journal, we ask Authors to transfer to the Publisher (Faculty of Education, University of Ljubljana) the rights of copyright in the Articles they contribute. This enables the Publisher to ensure protection against infringement.