Abstract
The concept of scaffolding can be used to describe various types of adult guidance, in multiple settings, across different time scales. This article clarifies what we mean by scaffolding, considering several questions specifically for scaffolding in mathematics: What theoretical assumptions are framing scaffolding? What is being scaffolded? At what level is scaffolding implemented? What is the setting for scaffolding? And lastly, how can scaffolding manage the tension between providing appropriate calibrated support while also providing opportunities beyond learners’ current understandings? The paper describes how attention to mathematical practices can maintain a sociocultural theoretical framing for scaffolding and move scaffolding beyond procedural fluency. The paper first specifies the sociocultural theoretical assumptions framing the concept of scaffolding, with particular attention to mathematical practices. The paper provides three examples of scaffolding mathematical practices in two settings, individual and whole-class. Lastly, the paper considers how two teacher moves during scaffolding, proleptic questioning and revoicing, can serve to provide appropriate calibrated support while also creating opportunities beyond current proficiency.
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Notes
For a discussion of differences between procedural fluency and conceptual understanding, see Kilpatrick, Swafford, & Findell (2001).
The study also described how appropriation functioned through the focus of attention, meaning for utterances, and goals for these three tasks, showing how the learner actively transformed some goals.
Throughout the sessions, the tutor fostered executive control activities, such as revising and evaluating, crucial for competent problem solving in this domain (Brown et al. 1983; Schoenfeld, 1985). The tutor also provided corrections, proleptic instruction (Stone, 1993), and guiding questions to scaffold student goal setting. Initially, the tutor was the problem poser, goal setter, critic, and evaluator; he asked for and suggested plans and overtly engaged in goal setting, checking, and evaluation. As tutoring proceeded, the student assumed some of these herself, setting new problems, developing new goals, transforming incomplete or inappropriate goals to reflect more content knowledge, checking results, and evaluating solutions. Thus, the student appropriated many of the executive control activities first experienced in interaction.
For a discussion of object and process perspectives of functions, see Moschkovich, Schoenfeld, & Arcavi (1993). The game environment and the tutoring were designed, in part, to introduce students to the object and process perspectives for functions; these were novel for this student.
Julian uttered “parallela” (turn 4) with hesitation and his voice trailed off. It is impossible to tell whether he said “parallela” or “parallelas.”
Adapted from handout by Harold Asturias, available online at Understanding Language http://ell.stanford.edu.
From “Creating Equations 1” Mathematics Assessment Resource Service, University of Nottingham. Available at http://map.mathshell.org/materials/tasks.php?taskid=292#task292.
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Moschkovich, J.N. Scaffolding student participation in mathematical practices. ZDM Mathematics Education 47, 1067–1078 (2015). https://doi.org/10.1007/s11858-015-0730-3
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DOI: https://doi.org/10.1007/s11858-015-0730-3