Reflective Abstraction in Advanced Mathematical Thinking

By Ed Dubinsky. 2002

828 citations recorded by the source corpus.

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Citations in the corpus

Levels of students’ “conception” of fractions (2012, background): “Structural operation and the action–process–object models (Sfard, 1991; Dubinsky, 1991) indicated that the process conception precedes the object conception; “a process solidifies into object, into a static structure” (Sfard, 1991, p.”
Distinguishing Two Stages of Mathematics Conceptual Learning (2004, methodology): “ Dubinsky (1991, 1994) and Sfard (1991) reported that many students do not accomplish an object or reified level of understanding.”
Children’s Use of Variables and Variable Notation to Represent Their Algebraic Ideas (2015, background): “…term mathematical object in the way that others have (e.g., Ayers, Davis, Dubinsky, & Lewin, 1988; Breidenbach, Dubinsky, Hawks, & Nichols, 1992; Dubinsky, 1991a, 1991b; Dubinsky & Harel, 1992; Sfard, 2000), alluding to the fact that the learner is able to perform an action on an expression (in…”
Explicating the Role of Mathematical Tasks in Conceptual Learning: An Elaboration of the Hypothetical Learning Trajectory (2004, result): “Consistent with the work of others (e.g., Dubinsky, 1991; Piaget, 1954; Pirie & Kieren, 1994; Sfard, 1991), our empirical work has led to distinctions among stages of conceptual learning.”
Doctoral students’ use of examples in evaluating and proving conjectures (2008, background): “Dubinsky 1991 ). This use of the term example is relatively narrow; Watson and Mason (2002, p.378) include it in their definition of the term as “examples of appropriate objects which satisfy certain conditions,” but also list “examples of classes, examples illustrating concepts, worked examples dem”
A Taxonomy for Categorizing Generalizations: Generalizing Actions and Reflection Generalizations (2007, background): “…or properties (Carpenter & Franke, 2001; English & Warren, 1995; Lee, 1996), (b) the extension or expansion of one’s range of reasoning beyond the case or cases considered (Dubinsky, 1991; Harel & Tall, 1991), and (c) the identification of commonalities across cases (Dreyfus, 1991; Kaput, 1999).”
Predications of the Limit Concept: An Application of Repertory Grids (2001, background): “Dubinsky (1991) notes that his approach is complementary to the two approaches discussed above (i.”
Undergraduate students’ performance and confidence in procedural and conceptual mathematics (2005, background): “Recognising the same distinction of components, Dubinsky et al (Dubinsky, 1991; Thomas and Holton, 2003) introduce the idea of processes being encapsulated as objects.”
The fundamental cycle of concept construction underlying various theoretical frameworks (2005, neutral): “Process-Object-Schema) in which actions are interiorised as processes, then thought of as objects within a wider schema (Dubinsky, 1991).”
Adding structure to the transition process to advanced mathematical activity (2010, background): “Gray and Tall [25,26]) introduced the idea of a procept and Dubinsky’s [27] APOS theory is well known.”
Generalizing-Promoting Actions: how Classroom Collaborations Can Support Students Mathematical Generalizations (2011, background): “Researchers historically have approached the notion of generalizing from a number of perspectives, viewing it as identifying commonalities (Dreyfus, 1991), extending one’s range of reasoning beyond the immediate case at hand (Dubinsky, 1991; Harel & Tall, 1991), or shifting from individual cases to ”
A Framework for Research and Curriculum Development in Undergraduate Mathematics Educationjpeg (2009, methodology): “The initial development of this theory and its relationship to Piaget is described in some detail in [6].”
Student learning of basis, span and linear independence in linear algebra (2010, background): “According to APOS theory [5,6] when an individual reflects on actions to the point where she/he establishes conscious control over them, can examine them mentally, such as describing or reflecting on their steps, reversing them, or examining only the inputs and outputs, then the action has become a ”
Understanding Primes: The Role of Representation. (2004, background): “Treating mathematical concepts as objects supports construction of corresponding mental objects in the mind of students (Dubinsky, 1991; Sfard, 1991).”
Handbook of research on the psychology of mathematics education: past, present and future (2006, background): “…are the various theoretical process-object distinctions that have been developed with respect to the construction of meaning in mathematics (e.g., Dubinsky, 1991; Sfard, 1991; Gray & Tall, 1994), as well as the recent models of abstraction (e.g., Hershkowitz, Schwarz & Dreyfus, 2001; Schwarz,…”