Benefits of Practicing 4 = 2 + 2: Nontraditional Problem Formats Facilitate Children’s Understanding of Mathematical Equivalence

Abstract

This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional "operations on left side" format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving traditional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children's understanding of fundamental concepts.

Persistent and Pernicious Errors in Algebraic Problem Solving

“…Results from this study suggest that any classroom time devoted to improving students' conceptual understanding of the equals sign is likely well spent; fortunately, a variety of tools, materials, and instructional suggestions are already available for this purpose, including utilizing a balance scale (Vlassis, 2002) or introducing arithmetic sentences that are in unconventional forms, such as 7 = 2 + 5, etc. (McNeil, Fyfe, Petersen, Dunwiddie, & Brletic-Shipley, 2011).…”

Teaching the substitutive conception of the equals sign

“…As they move through secondary school many children gradually develop a sameness-relational conception of the equals sign in which "=" is seen as meaning "is the same as" and a wide variety of statement types are accepted McNeil et al 2011;Rittle-Johnson et al 2011). …”

“…Much research has investigated how children move from an operational understanding of the equals sign in their early years of schooling to a more relational understanding in later years (Baroody and Ginsburg 1983;Behr, Erlwanger and Nichols 1976;Denmark, Barco and Voran 1976;Kieran 1981;McNeil 2008;McNeil et al 2011;Rittle-Johnson et al 2011). Jones and Pratt (2012) argued that research to date has operationalised the relational understanding of the equals sign exclusively in terms of sameness.…”

Early algebraization: a global dialogue from multiple perspectives

“…However the traditional curriculum did not better prepare children for solving equations despite including substantially more equation solving practice. Analogously, McNeil et al (2011) found that emphasising a computational view of arithmetic equations hampered young children's relational understanding, but emphasising a relational view did not hamper computational proficiency. Nevertheless, even when learners display a conceptual understanding of algebra development is unlikely to be straightforward.…”

“…This is a disappointment in light of the research team's impressive contribution to theory on the development of algebraic concepts elsewhere (e.g. McNeil et al, 2011). Conversely, many chapters are explicitly theory-based, and many authors exemplify their theoretical approach with excerpts from curricula, resources, classroom episodes and interview transcripts.…”