Advanced Mathematical Thinking Processes

By T. Dreyfus. 2002

364 citations recorded by the source corpus.

Subfields: Applied Mathematics, Education

View Paper: https://www.semanticscholar.org/paper/e25b68703e92efb5a4006b7ae822afadc27552e1

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In Applied Mathematics

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Citations in the corpus

Mathematics Majors' Perceptions of Conviction, Validity, and Proof (2010, background): “First, advanced mathematics courses are typically taught in a “definition-theorem-proof” format (e.g., Dreyfus, 1991; Weber, 2004).”
Abstraction in mathematics learning (2007, background): “We note that his conceptualisation includes a formal world of reasoning from definitions and axioms, where abstraction may be very different again (Dreyfus, 1991).”
Abstraction and consolidation of the limit procept by means of instrumented schemes: the complementary role of three different frameworks (2008, background): “Dreyfus and Tsamir (2004)”
Grade 6 Students' Preinstructional Use of Equations To Describe and Represent Problem Situations. (2000, background): “The process of generalization is one in which the commonalties in the specific cases are identified (Dreyfus, 1991).”
Mathematical Giftedness, Problem Solving, and the Ability to Formulate Generalizations: The Problem-Solving Experiences of Four Gifted Students (2003, background): “It includes abstracting properties (Davis & Hersh, 1981), identifying commonalties (Dreyfus, 1991) and expanding domains of validity (Davydov, 1990; Dienes, 1961; Polya, 1954).”
Abstraction in Context: Epistemic Actions (2001, background): “A notable exception is a study conducted by Goodson-Espy (1998), who observed abstraction during problem solving.”
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 ”
Lectures in Advanced Mathematics: Why Students Might Not Understand What the Mathematics Professor Is Trying to Convey. (2016, background): “Perhaps the most common complaint is that the domination of definitions, theorems, and proofs in mathematics lectures leads the lecturer to pay scant attention to other important types of mathematical thinking (e.g., Boero, 2007; Davis & Hersh, 1981; Dennis & Confrey, 1996; Dreyfus, 1991; Thurston, ”
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).”
Mathematicians’ perspectives on their pedagogical practice with respect to proof (2012, background): “Similarly, Dreyfus [2] writes that the typical mathematics instructor teaches ‘almost exclusively the one very convenient and important aspect which has been described above, namely the polished formalism, which so often follows the sequence theorem-proof-application’ (p. 27).”
A dialectical approach to the formation of mathematical abstractions (2007, background): “Dreyfus (1991), for example, writes of generalisation — a component part of abstraction: " To generalize is to derive or induce from particulars, to identify commonalities " (p. 35).”
Culturally Responsive Teaching of Mathematics: Three Models from Linked Studies (2009, methodology): “persistence with mathematical thinking and the integration of mathematical thinking in classroom activities and curriculum design (Cuoco, Goldenberg, & Mark, 1996; Dreyfus, 1991; Krutetskii, 1976; Polya, 1962; Watson, 2002).”
“Rising to the challenge”: using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students (2008, methodology): “…generalizations: empirical generalization, which refers to the recognition of common features (in line with the identification of commonalities by Dreyfus 1991); and theoretical generalization, which has to do with a ‘‘system of action’’ in which invariants are identified, abstracted and…”
Connections Between Generalizing and Justifying: Students' Reasoning with Linear Relationships (2007, methodology): “Generalization as the creation of a rule is similar to Peirce’s (1878) notion of induction, but it has also been described as the identification of commonalities (Dreyfus, 1991; Kaput, 1999), which shares some features of traditional views of lateral transfer (Anderson, Corbett, Koedinger, & Pelleti”