Students’ Reasoning With Reversible Multiplicative Relationships

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On the idea of Learning Trajectories: Promises and Pitfalls (2011, background): “…in mathematics has an ample research base that illustrates, in some cases meticulously, how children’s thinking about fractions could progress (Behr, Harel, Post, & Lesh, 1992; Davydov & Tsvetkovich, 1991; Empson & Levi, 2011; Hackenberg, 2010; Steffe & Olive, 2010; Streefland, 1991; Tzur, 1999).”
Relationships between Students' Fractional Knowledge and Equation Writing. (2015, background): “Researchers have found that conceiving of fractions as multipliers on known quantities is a significant challenge for students (e.g., Behr et al., 1993; Hackenberg, 2010; Lamon, 2007).”
Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge? (2014, background): “…2002, English and Halford 1995, Grégoire and Meert 2005, Kilpatrick et al. 2001), we ensured that the test addressed all five sub-constructs as distinguished in literature (Charalambous and Pitta-Pantazi 2007, Hackenberg 2010, Kieren 1993, Kilpatrick et al. 2001, Lamon 1999, Moseley et al. 2007).”