The Role of Definitions in the Teaching and Learning of Mathematics

By Shlomo Vinner. 2002

541 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/be96b0e89da456a6941c4b8355ce9a1d9ea24e46

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Citations in the corpus

Proof in School Mathematics: Insights from Psychological Research into Students' Ability for Deductive Reasoning (2008, background): “, definitions, axioms) that underlie a mathematical argument or a proof (Ball & Bass, 2000; Mariotti & Fischbein, 1997; Stylianides, 2007a; Vinner, 1991; Zaslavsky & Shir, 2005).”
Eigenvalues and eigenvectors: embodied, symbolic and formal thinking (2011, background): “Hence, it is not surprising that many students have real difficulties with definitions of such concepts, which are considered to be fundamental as a starting point for concept formation and deductive reasoning in advanced mathematics (Vinner 1991; Zaslavsky and Shir 2005).”
AN EXPLORATION OF DEFINITION AND PROCEDURAL FLUENCY IN INTEGRAL CALCULUS (2006, background): “The Role of Definitions in the Teaching and Learning of Mathematics, by Shlomo Vinner [5] describ es the differences between a "concept definition" and a "concept image" and the implicati ons that the formation of”
Analysis of Errors and Misconceptions in the Learning of Calculus by Undergraduate Students. (2012, methodology): “Vinner (1991) suggests that definitions do not only help in concept formation but they also help learners to handle tasks involving mathematics concepts.”
Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving (2008, background): “Attention to issues of mathematical language implies attention to both how mathematical arguments are represented and how mathematical ideas are defined (see, e.g., Ball & Bass, 2000b; Mariotti & Fischbein, 1997; Vinner, 1991; Zaslavsky & Shir, 2005).”
Students’ images and their understanding of definitions of the limit of a sequence (2008, background): “Another misconception about limits arises from regarding the infinite process itself as the limit rather than seeing the limit as the result of the infinite process (Vinner 1991).”
Exemplifying definitions: a case of a square (2008, methodology): “Based on the works of well-known mathematicians (e.g., Khinchin 1968, Poincare 1909/1952, Solow 1984, and Vinner 1991) they pointed out logical principles that should be fulfilled in defining mathematical concepts: (a) defining is giving a name: the statement used as a definition presents the name…”
Development of angle concepts by progressive abstraction and generalisation (2000, background): “However, verbal definitions are ineffective in teaching new concepts – not only in mathematics (Miller and Gildea, 1987; Vinner, 1991).”
Students' Conceptions of a Mathematical Definition (2005, background): “When dealing with alternative definitions, we must take into account students’ concept image and personal concept definition of the defined concept (Vinner, 1991).”
Letting the Intuitive bear on the Formal; A Didactical Approach for the Understanding of the Limit of a Sequence (2001, background): “Research in Mathematics Education is very consistent in saying that most students have extreme difficulties to obtain a reliable and robust image of the limit concept, but that it is very hard to find solutions to this situation, see especially Vinner (1991).”
LEARNERS’ UNDERSTANDING OF THE DEFINITIONS AND HIERARCHICAL CLASSIFICATION OF QUADRILATERALS: TOWARDS A THEORETICAL FRAMING (2007, background): “…and of classification, given that a range of studies is indicating that learners have difficulties both with defining (see, for example, de Villiers, 1998; Vinner, 1991; Zaslavsky and Shir, 2005) and classifying (see, for instance, Currie and Pegg, 1998; de Villiers, 1994; Monaghan, 2000).”
International group for the psychology of mathematics education (2008, background): “Certainly it goes beyond the typical lecture or book presentation of one or two “standard” examples for each definition, thus arguably encouraging the idea that we can ask about the applicability of properties more widely, and discouraging reliance on “prototypical” concept images (Vinner, 1991).”
Intuitive nonexamples: the case of triangles (2008, background): “In mathematics education, the use of examples and nonexamples has largely been investigated in relation to the acquisition of geometrical concepts (Cohen and Carpenter 1980; Petty and Jansson 1987, Vinner 1991; Wilson 1986).”
An Inquiry-Oriented Approach to Span and Linear Independence: The Case of the Magic Carpet Ride Sequence (2012, background): “face in using formal definitions and reasons for these challenges [5, 26, 28].”