The focus of this study is on issues of inclusion within collaborative groups in the context of research into children’s algebraic thinking.

Inclusion within collaborative groups

Traditionally in many classrooms, mathematics was viewed as an individual pursuit, where children worked alone, using paper and pencil, to answer questions and complete tasks. When conducting this research study, I held the position that children’s developing understanding of mathematics is supported by the social interactions and opportunities to communicate thinking that occur when they work with others. Children’s mathematical reasoning is thus best supported by opportunities to work collaboratively (Mueller, Yankelewitz & Maher, 2012).

Mueller et al. (2012) point to teacher moves which facilitate collaboration between peers in group settings, including an emphasis on the value of contrasting opinions. During the group interviews of this study I regularly encouraged children to compare their work with each other, and to explore the differences. Mercer and Littleton (2007) identify rich group work discussion as ‘exploratory talk’, wherein children build upon each other’s thinking through questioning, discussion and justification. Mercer and Littleton advise that teachers support children in engaging in high quality discussions by modelling desired behaviours, for example, seeking multiple opinions, or answers to a question; asking ‘why’ when relevant; drawing out reasons for answers proffered.

In order for peer interactions to best support collaborative sense-making, it is supportive if children engage in exploratory talk, where they feel comfortable disagreeing with each other, and calling upon each other to justify their thinking (Mercer and Littleton, 2007). Boaler (2002) states that the skills inherent in group work, and the construction of an exploratory talk scenario requires a skill level that must be honed over time. She also discusses the disadvantage experienced by children from working class backgrounds, or some ethnic groups, due to the norms of discussion which would be prevalent in their home, and differ from the disputational approach required to support a robust mathematical discussion.

Algebraic thinking

Stromskag (2015) defines a shape pattern as a sequence of terms, composed of constituent parts, where some or all elements of such parts may be increasing, or decreasing, in quantity in systematic ways. While a limited number of terms of a shape pattern may be presented for consideration, the pattern is perceivable as extending until infinity. Generalisation of a phenomenon, such as constructing general terms for shape patterns, is considered by many to be a highly significant component of algebraic thinking and in particular supports children in reasoning algebraically about covariance and rates of change (Rivera & Becker, 2011; Carpenter & Levi, 2000; Kaput, Blanton & Moreno, 2008). In order to construct a general term for a shape pattern, children must “grasp a regularity” in the structure of the terms presented, and generalize this regularity to terms beyond their perceptual field (Radford, 2010, p.6). Generalisation is not mentioned within the current Irish Primary School Mathematics Curriculum, and shape patterns are not presented as learning activities, beyond simple repeating patterns explored with very young children (Government of Ireland, 1999). It is most probable that shape patterning tasks, and requests to generalise, would be completely novel to the children who participated in the task-based group interviews of this study.

Research question

The focus of this paper is whether exclusion was experienced during the task-based group interviews, given the theoretical framework applied in seeking to support optimal collaboration. Secondly, if exclusion was experienced, I sought to explore whether there were identifiable catalysts that contributed to a child’s exclusion.

The research involved presenting shape patterning tasks to children and asking them to describe and extend the patterns, and to consider near and far terms. Due to the potential novelty of the tasks for the children involved in this research, it was necessary to conduct the research within a setting which allowed children to tease out their understanding of the tasks presented. As mentioned above, I held a socio-constructivist position, believing that children’s mathematical reasoning is best supported by opportunities for them to discuss their thinking, and to work collaboratively. The children engaged with the shape patterning tasks in groups of three or four. In all, 47 children with a mean age of 9.83 years took part in the task-based group interviews. During the interviews I applied theory from research findings into my practice in facilitating the children's engagement with tasks. I sought to encourage, and support collaboration among peers as they worked together to solve tasks (Mueller et al., 2012). Specifically, I asked for many children’s ideas to questions, rather than taking one answer and I regularly encouraged children to compare their work with each other, and to explore the differences. I aimed to support an understanding of the value of different approaches, and how somebody else’s thinking, while different to my own, may support me in thinking more broadly or in adopting a different perspective. I aimed to facilitate child agency by encouraging the children to report to, and seek justification from, each other, rather than reporting to me (Mercer, 1995). Interviews were recorded, and remaining cognisant of the complex task involved in interpreting children’s expressions, I analysed the strategies that selected groups of children used to generalise from the patterns, drawing from the work of Lannin (2005) and Rivera and Becker (2011). Taking a phenomenological approach, I identified and analysed factors that contributed to children’s strategy choices, which included explicit, recursive, numerical and figural approaches, among others. In exploring the role played by group interactions, I analysed the extent to which some children felt themselves to be included in or external to the group work. Analysis involved hermeneutic analysis of children’s verbal utterances and gestures. I also analysed the consistency with which the children a) contributed to the group discussion, b) their contributions were responded to by others in the group.

In this paper I focus on the involvement of two children within their group of four. Both of the children experienced challenge with the mathematics of the patterning tasks presented to the group. One child, Luigi, showed indications that he perceived himself to be included in the group, whereas Orla indicated, verbally and through her actions, that she perceived herself to be external to the thinking of the group. A preliminary inspection of the number of contributions each child made to the group discussion shows that Luigi made 173 comments during the hour long group interview, while Orla made 41. For comparison the other two children in the group contributed 113 and 109 comments each. Examining how Luigi and Orla interacted with the thinking of their peers, Orla did not seem to make the persistent attempt to understand the perspectives of her peers that Wells and Arauz (2006) explain is required for collaborators to achieve a state of intersubjectivity. In contrast, Luigi responded to many of the comments made by Emily or Wyatt in the group discussions, seeking to agree, or to communicate his alternative perspective. As mentioned above, Boaler (2002) draws attention to the relative challenge some working class children might encounter in engaging in collaborative group work. Equally, Simensen, Fuglestad and Vos (2015) show evidence of low attaining children experiencing exclusion. While both of these factors may have contributed to Orla’s experience of exclusion, the findings of this research suggest that a child’s disposition within the group, and in particular his/her inclination to work towards intersubjectivity may also impact upon his/her experience of inclusion or exclusion. Teaching interventions that focus on responding to and building upon the thinking of others may support children who are at risk of exclusion in collaborative group work.

Boaler, J. (2002). Learning from teaching: Exploring the relationship between reform curriculum and equity. Journal for Research in Mathematics Education, 33(4), pp. 239-258. Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades. Research Report, 00(2). Madison: NCISLA. Retrieved April 25, 2014 from http://ncisla.wceruw.org/publications/reports/RR-002.pdf. Government of Ireland. (1999). Irish primary school curriculum mathematics. Dublin: The Stationery Office. Kaput, J. J., Blanton, M. L., & Moreno, L. (2008). Algebra from a symbolization point of view. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades. (pp. 19-56). New York: Lawrence Erlbaum Ass. Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258. Mercer, N. & Littleton, K. (2007). Dialogue and the development of children’s thinking: A sociocultural approach. Cornwall, UK: TJ International Ltd. Mercer, N. (1995). The guided construction of knowledge. Clevedon, UK: Multilingual Matters Mueller, M., Yankelewitz, D., & Maher, C. (2012). A framework for analysing the collaborative construction of arguments and its interplay with agency. Educational Studies in Mathematics, 80, 369-387. Radford, L. (2010). Algebraic thinking from a cultural semiotic perspective. Research in Mathematics Education, 12(1), 1-19. Rivera, F., & Becker, J. R. (2011). Formation of pattern generalization involving linear figural patterns among middle school students: Results of a three-year study. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives. (pp. 323-366). Heidelberg: Springer. Simensen, A.M., Fuglestad, A.B., & Vos, P. (2015). How much communication space is there for a low achiever in a heterogeneous group? In K. Krainer & N. Vondrová, (Eds.), Proceedings from the Ninth Congress of the European Society for Research in Mathematics Education (pp. 467-473). Prague: ERME Strømskag, H. (2015). A pattern-based approach to elementary algebra. In K. Krainer & N. Vondrová, (Eds.), Proceedings from the Ninth Congress of the European Society for Research in Mathematics Education (pp. 474-480). Prague: ERME Wells, G., & Arauz, R. M. (2006). Dialogue in the classroom. The Journal of the Learning Sciences, 15(3), 379–428.