Developing Algebraic Reasoning Through Variation in the United States

• In Event: 29.082 - Roundtable Session 9
  In Roundtable Session: 29.082-4 - Teaching and Learning Mathematics Through Variations: An International Perspective

Abstract

Algebraic reasoning involves building general mathematical relationships and expressing those relationships in increasingly sophisticated ways (Ontario Ministry of Education, 2005; Soares, Blanton, & Kaput, 2005; Warren & Cooper, 2008). This includes reasoning with generalizations of mathematical relationships (Blanton, 2008), which can be accomplished by varying a single task parameter (Blanton, 2008; Blanton & Kaput, 2003, 2005; Ontario Ministry of Education, n.d.; Soares et al., 2006). Varying a "parameter allows you to build a task that looks for a functional relationship between two quantities" (Blanton, 2008, p. 58) and "can shift the focus from arithmetic thinking to algebraic thinking" (Ontario Ministry of Education, n.d., p. 19). This emphasis on varying a parameter suggests that applying a theory of variation to the design of instruction may be an important means for engaging students in algebraic reasoning.
According to Marton, Runesson, and Tsui’s (2004) theory of variation, learning is a process in which students acquire a particular capability or way of seeing and experiencing. In order to see something in a certain way, students must discern critical features of an object. We designed a four-task lesson sequence that aimed to support the development of algebraic reasoning. The intended direct objects of learning during this task sequence were for students to be able to generalize a linear pattern given a series of geometric figures, give the generalization as an expression involving one variable (i.e., an + b where a and b are integers), and justify the generalization based on the geometric pattern. This paper presents the case in which the series of tasks whose development was informed by a theory of variation were enacted in a sixth grade classroom.
In the first lesson, a pattern involving corrals provided a starting point for the discussion of linear functions. Within this lesson, only the number of corrals was varied, bringing awareness to the relationship between an input and an output in a linear function. The subsequent lessons proceeded to vary one feature of linear functions at a time so as to bring attention to the characteristics of the parts of a linear function. The second lesson focused on a new pattern in which the number in each group varied from the previous corral pattern. The third lesson presented a third pattern in which the constant was varied. In order to adhere to variation theory, the explored pattern for each day was of the same style, therefore allowing this aspect of the discussion to remain invariant. In addition, only the position in each sequence (or input) was varied within the main activity in each lesson. By keeping these portions invariant, the lessons drew attention to the varied feature, allowing students to separate these features.
Data included lesson transcripts and student-generated artifacts from classwork and homework. Analyses revealed gains in understanding of the concepts of linear functions. From a theoretical perspective, careful analysis of the intended, enacted, and lived objects of learning found in this task sequence provides a clear picture of teaching through variation in the U.S.

Authors

• Angela T Barlow, Middle Tennessee State University

• Alyson Lischka, Middle Tennessee State University