Three Balloons for Two Dollars: Developing Proportional Reasoning.

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Promoting middle school students’ proportional reasoning skills through an ongoing professional development programme for teachers (2016, background): “Additionally, in some items, second tier responses were designed to determine whether students used qualitative or quantitative reasoning (see Lamon, 1993; Langrall & Swafford, 2000; Lesh et al., 1988).”
Development and application of a two-tier diagnostic instrument to assess middle-years students’ proportional reasoning (2013, methodology): “Different problem types elicit different forms of reasoning (Langrell and Swafford 2000) and students experience varying degrees of difficulty with different problem types.”