A hypothetical learning trajectory on measurement and flexible arithmetic

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A hypothetical learning trajectory for conceptualizing matrices as linear transformations (2017, background): “One such research-based construct that directly informs instructional practice is that of a hypothetical learning trajectory [1,2,11,12].”
Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions (2007, background): “Instead, the model-of phase usually includes concrete tools, graphs, or diagrams, and only later do analytic expressions function prominently in the model-for phase (e.g., Bakker, 2004; Doorman, 2005; Gravemeijer, 1994, 1999; Gravemeijer et al., 2003; Gravemeijer et al., 2000; Gravemeijer & Doorman,”
Mathematics Education and Neurosciences: Relating Spatial Structures to the Development of Spatial Sense and Number Sense (2007, background): “The aim, then, is not necessarily to conclude that the series of activities teach the children about spatial structures, but more to come to an analysis about why the series of activities may have stimulated the children’s thinking (Gravemeijer et al., 2003).”