Dreyfus, Hershkowitz and Schwarz (2001) have used three observable nested elements
to study the process of abstraction (Recognising, Building-with and Constructing).
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Recognising and Building-with are nested within Constructing. These nested elements of
abstraction can be described in terms of the categories in my hierarchy. Recognising
includes the process of identifying a context in which a previously abstracted entity
applies (Comprehending) and recognising a previously abstracted entity can be applied to a
given context (Applying). Building-with includes using previously abstracted entities as
part of several different processes such as: applying a previously abstracted entity in a
known context (Applying), or in a new context (Analysing), or applying several previously
abstracted entities in a familiar order (Applying) or a different order (Analysing). The
relationship between the observable elements of abstraction and my categories of cognitive
activities suggests my term “hierarchy” should be replaced by nested categories of
increasingly intellectually complex activity. Constructing—mathematically insightful
behaviour—is a process of integrating previously abstracted entities to develop a new
mathematical insight (Synthesising & Evaluating). Constructing may occur in a more
expert-directed learning culture where suggestions, hints, and corrections are made or
constructing may be a more autonomous activity where hints are not provided but
questions are asked for the purpose of eliciting more complex thinking (Williams, 2000).