Young children’s ability to use the balance strategy to solve for unknowns

Abstract

This article examines students' ability to use the balance model to solve for unknowns. A teaching experiment was conducted in four Year 3 classrooms. This experiment focused on exploring the application of the balance model as an analogue for representing equations and solving for m@,nowns. The teaching experiment promoted a shift by students towards viewing addition and subtraction equations in terms of equivalence, where the situation is viewed in a multi-directional way (i.e., balance). Initially the lessons were trialed by the researchers in two classrooms. The lessons were then implemented in a further two classrooms by two classroom teachers in conjunction with the researchers. Two weeks after the conclusion of the teaching experiment, a one-on-one interview was conducted with a random sample comprising twenty students with an average age of eight years. The interviews indicated that while the balance model did assist students reach solutions for finding unknowns, for some students further explicit teaching was required.

Teaching linear equations: Case studies from Finland, Flanders and Hungary

“…At a general level it is argued that they allow new concepts, entities and operations to become endowed with meaning (Filloy and Rojano, 1989;Da Rocha Falcão, 1995) and can facilitate the link between concrete and abstract thinking Williams, 1996, Brown et al, 1999). However, while concrete models may act as analogues for the intended abstractions (English and Sharry, 1996;Warren and Cooper, 2005), they may mask teachers" intended learning outcomes, necessitating appropriate teacher interventions to ensure appropriate abstraction (Filloy and Sutherland, 1996).…”

“…Warren and Cooper (2005), mindful of the inadequacies of the balance with respect to negative quantities, concluded, that it helped "children in approaching problems with unknowns, particularly solving problems with unknowns on both sides of the equal sign" (70). Furthermore, it not only provided a vocabulary with which children could articulate their thinking but also facilitated their seeing the equals sign as representing equivalence rather than an instruction to calculate.…”

“…They did this in various ways, although their common emphases on algebraic equations as the provocations for their expositions and their invocations of the balance were clearly important in this regard. Sami and Emese, in the manner of Vlassis (2002), introduced the balance only when it became clear that students were unable to solve an algebraic equation by intuitive methods; while Pauline alluded to the balance during her activation, which involved the solution of arithmetic equations, before moving onto her formal algebraic equations-based exposition in much the same way as Warren and Cooper (2005).…”

Polish teachers’ conceptions of and approaches to the teaching of linear equations to grade six students: an exploratory case study

“…Systematic attempts to evaluate the balance's efficacy have shown that it helps students to understand the principles of equations, solve non-arithmetical equations with understanding, particularly from the perspective of the need to do the same thing to both sides (Araya et al, 2010;Warren and Cooper, 2005). In addition, it facilitates students' acquisition of an appropriate vocabulary and a relational understanding of the equals sign (Vlassis, 2002).…”

Developing mathematics understanding and abstraction: The case of equivalence in the elementary years

“…They persistently interpret it as either a syntactic indicator (i.e., a symbol indicating where the answer should be written) or an operator sign (i.e., a stimulus to action or "to do something") (Behr, Erlwanger & Nicols, 1980;Carpenter, Franke, & Levi, 2003;Saenz-Ludlow & Walgamuth, 1998;Warren & Cooper, 2005). Instead it should be interpreted as quantitative "sameness" (i.e., both sides of an equation are the same and information can be obtained from either direction in a symmetrical fashion) (Kieran & Chalouh, 1992).…”