This study aims to understand the instructional interactions that foster students’ generalizing. By analysing video-recorded lessons from 13 different Grade 3 classrooms in which an early algebra intervention was implemented, activities related to generalizing and students’ generalizing activities were identified, and the nature of the relationships between these activities were explored. Namely, I consider the activities that potentially encourage students’ generalizing and the generalizing that students themselves do, and describe ways in which these activities interact. The findings have both practical and theoretical implications; in the former case, they provide insight into the design of instruction to foster generalizing, and in the latter case, they contribute to an ongoing conversation about students’ mathematics learning.
Get full access to this article
View all access options for this article.
References
1.
BeckerJ. R.RiveraF. D. (2006). Sixth graders’ figural and numerical strategies for generalizing patterns in algebra (1). In AlatorreS.CortinaJ. L.SáizM.MéndezA. (Eds.), Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 95–101). Mérida: Universidad Pedagógica Nacional.
2.
BlantonM.IslerI.StephensA.GardinerA.KnuthE.DemersL.StrachotaS. (2017). Longitudinal growth in children’s algebraic thinking: Results of an early algebra intervention across grades 3–5. Paper presented at the National Council of Teachers of Mathematics Research Conference, San Antonio, TX.
3.
BlantonM.StephensA.KnuthE.GardinerA.IslerI.KimJ. (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, 39–87.
4.
BreiteigT.GrevholmB. (2006). The transition from arithmetic to algebra: To reason, explain, argue, generalize and justify. Proceedings of the 30th Conference of the international group for the psychology of mathematics education (Vol. 2, pp. 225–232). Prague: Charles University.
5.
BrittM. S.IrwinK. C. (2008). Algebraic thinking with and without algebraic representation: A three-year longitudinal study. ZDM, 40, 39–53.
6.
BrizuelaB. M.BlantonM.SawreyK.Newman-OwensA.GardinerA. (2015). Children’s use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17, 1–30.
7.
CarraherD. W.SchliemannA. D. (2007). Early algebra and algebraic reasoning. In LesterF. K.Jr. (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669–705). Charlotte, NC: Information Age Publishing.
8.
CooperT.WarrenE. (2011). Year 2 to Year 6 students’ ability to generalize: Models, representations and theory for teaching and learning. In CaiJ.KnuthE. (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187–214). Heidelberg: Springer.
9.
DekkerR.Elshout-MohrM. (2004). Teacher interventions aimed at mathematical level raising during collaborative learning. Educational Studies in Mathematics, 56, 39–65.
10.
DörflerW. (2008). En route from patterns to algebra: Comments and reflections. ZDM Mathematics Education, 40, 143–160.
11.
EllisA. B. (2005). Justification as a support for generalizing: Students’ reasoning with linear relationships. In LloydG. M.WilsonM. R.WilkinsJ. L. M.BehmS. L. (Eds.), Proceedings of the 27th annual meeting of the North American chapter of the international group for the psychology of mathematics education (CD-Rom). Eugene, OR: All Academic.
12.
EllisA. B. (2007). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations. The Journal of the Learning Sciences, 16, 221–262.
13.
EllisA. B. (2011). Generalizing-promoting actions. Journal in Research in Mathematics Education, 42, 308–345.
14.
EllisA. B.GrinsteadP. (2008). Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations. The Journal of Mathematical Behavior, 27, 277–296.
15.
EnglishL.WarrenE. (1995). General reasoning processes and elementary algebraic understanding: Implications for instruction. Focus on Learning Problems in Mathematics, 17(4), 1–19.
16.
HerbstP.ChazanD. (2012). On the instructional triangle and sources of justification for actions in mathematics teaching. ZDM, 44, 601–612.
17.
KaputJ. (1999). Teaching and learning a new algebra with understanding. In FennemaE.RombergT. (Eds.), Mathematics classrooms that promote understanding (pp. 133–155). Mahwah, NJ: Erlbaum.
18.
KaputJ. J.BlantonM. (2001). Algebrafying the elementary mathematics experience. Part 1: Transforming task structures. In ChickH.StaceyK.VincentJ.VincentJ. (Eds.), Proceedings of the 12th ICMI Study Conference: The Future of the Teaching and Learning of Algebra (pp. 344–351). Melbourne: The University of Melbourne.
19.
KieranC. (1996). The changing face of school algebra. In AlsinaC.AlvarezJ.HodgsonB.LabordeC.PerezA. (Eds.), 8th International Congress on Mathematical Education: Selected lectures (pp. 271–290). Sevilla: S.A.E.M. Thales.
20.
KieranC. (2004). The core of algebra: Reflections on its main activities. In The future of the teaching and learning of algebra the 12 th ICMI study (pp. 21–33). Dordrecht: Springer.
21.
KieranC. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In LesterF. K.Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Greenwich, CT: Information Age Publishing.
22.
KieranC.PangJ.SchifterD.NgS. F. (2016). Early algebra: Research into its nature, its learning, its teaching (pp. 3–32). Dordrecht: Springer.
23.
KnuthE.SlaughterM.ChoppinJ.SutherlandJ. (2002). Mapping the conceptual terrain of middle school students’ competencies in justifying and proving. In MewbornD.SztajnP.WhiteD.WiegelH.BryantR.NoonyK. (Eds.), Proceedings of the twenty-fourth annual meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 4, pp. 1693–1700). Columbus, OH: Educational Resources Information Center.
LanninJ. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7, 231–258.
26.
LeeL.WheelerD. (1987). Algebraic thinking in high school students: Their conceptions of generalization and justification(Research Report). Montreal: Concordia University, Department of Mathematics.
27.
LobatoJ. E. (2003). How design experiments can inform a rethinking of transfer and vice versa. American Educational Research Journal, 32, 17–20.
28.
MasonJ. (1996). Expressing generality and roots of algebra. In Approaches to algebra (pp. 65–86). Dordrecht: Springer.
29.
MasonJ. (2008). Making use of children’s powers to produce algebraic thinking. In KaputJ.CarraherD.BlantonM. (Eds.), Algebra in the early grades (pp. 57-94). Mahwah, NJ: Lawrence Erlbaum/Taylor & Francis Group & National Council of Teachers of Mathematics.
30.
WarrenE.CooperT. J. (2005). Introducing functional thinking in Year 2: A case study of early algebra teaching. Contemporary Issues in Early Childhood, 6, 150–162.
