Recursive and explicit rules: How can we build student algebraic understanding?

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“Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students (2008, methodology): “This method does not seek to monitor development over time as in previous studies, such as Krebs (2003), Lanin et al. (2006), Steele (2005), Warren et al. (2006).”
A Learning Progression for Elementary Students’ Functional Thinking (2017, methodology): “Each of recursive, covariation, and correspondence perspectives of functions are valued in the literature (e.g., Carlson & Oehrtman, 2005; Confrey & Smith, 1994, 1995; Lannin et al., 2006).”
Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns (2008, background): “Third, purposeful instruction through well-designed classroom teaching experiments could scaffold the development of closed forms of constructive generalizations in middle school children ( Lannin, Barker, & Townsend, 2006; Martino & Maher, 1999; Steele & Johanning, 2004).”
Visual templates in pattern generalization activity (2010, background): “On the basis of recent findings at the K-9 and preservice elementary undergraduate levels involving generalization in patterning activity, we still need further research in terms of how such coordination could take place more effectively in a systematic manner (Mulligan & Mitchelmore, 2009; Warren &”