The DOI matches its Crossref record; 45 of the 45 references it cites were found in a registry or an index.
On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin
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“Numerous researchers have investigated the nature of mathematical objects and tried to define them (e.g., Davis, 1983, 1984; Piaget, 1985; Gray & Tall, 1991, 1994; Dubinsky, 1991; Dubinsky & McDonald, 2001; Sfard, 1987, 1989, 1991, 1992; Tall et al., 2000).”
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“In particular, the use of a graphical interpretation tends to encourage a structural approach while a purely algebraic representation might lead to a computational interpretation (Sfard 1991).”
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“The theory of the three worlds emphasizes the construction of mental representations of concepts and has emerged from several theories on concept development, such as Sfard’s (1991) work on encapsulation of processes to objects and Piaget’s abstraction theories (Tall, 2004).”
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“The movement from the operational to the structural is on a continuum, gradually progressing from one to the other through a series of stages: interiorisation, condensation and reification (for details, see Sfard, 1991).”
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“this is the case of semi groups (obtained by removing one axiom), or ring theory (obtained by adding a number of axioms) [30, 33].”
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“Every solution of a pre-algebraic problem, like those in Appendix 2 lives dialectically in a double polarity (see [13], for a theoretical framework of this kind of ideas).”
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“Conversely, an entity as an object means “being capable of referring to it as if it was a real thing and to recognize the idea ‘at a glance’ and to manipulate it as a whole, without going into details” [75].”
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“1More broadly, reflection on and coordination of specific mathematical activity is often considered an important component of mathematical learning (e.g., Dubinsky, 1991; Gravemeijer, Cobb, Bowers, & Whitenack, 2000; Piaget, 2001; Sfard, 1991).”
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“The operational growth model proposed by Sfard (1991) was another cognitive growth theory that was also considered by the researcher for this study.”
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“Indeed, those ‘who are not prepared to actively struggle for meaning (for reification) would soon resign themselves t o never understanding mathematics’ ( Sfard, 1991, p. 33).”
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“In one of the most famous examples, Sfard (1991) argued that a new algebraic object could emerge from a process of already known calculation.”
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“Students have difficulty in understanding algebra and they usually tend to engage in algebra without realizing the real purposes and thinking context (Chazan, 1996; Sfard, 1991; Kieran, 1992).”
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“This vicious circle occurs because ‘the lower-level reification and the higher-level interiorisation are prerequisite for each other’ (Sfard, 1991, p. 31).”
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“A. Sfard1991OnThe Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides Of the Same Coin Educational Studies in Mathematics 22, 1-36.”
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“In the research, we drew on Zandieh’s theoretical framework for the derivative (1997; 2000), which draws on the notion of concept images (Tall & Vinner, 1981; Thompson, 1994), Sfard’s (1991) work on the process-object duality of mathematical concepts and work on concept representations (Goldin & Shteingold, 2001; Grav…”
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“The question is, what is a conception? Sfard (1991) described a conception as “the whole cluster of internal representations and associations evoked by the concept – the concept’s counterpart in the internal, subjective universe of human knowing” (vol 22, p.”
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“Second, we will explore the idea of cognitive duality in mathematical thinking (Sfard, 1991; Skemp, 1987; Wachsmuth, 1981), applying it to the word problem solving situation.”
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“In terms of instruction, the lack of mathematical tasks at schools make students hard to gain a solid understanding of function [11].”
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Metadata
- Authors
- Sfard, A.
- Affiliations
- Hebrew University of Jerusalem (IL)
- Corresponding author
- Sfard, Anna
- ORCID
- Sfard, Anna
- Venue
- Educational Studies in Mathematics
- Published
- 1991-02
- Volume
- 22
- Issue
- 1
- Pages
- 1-36
- ISSN
- 0013-1954 (ISSN-L)
- 1573-0816
- OpenAlex
- W2072267223
- Semantic Scholar
- 57d32a56079c7a2d6b9c89c668386a1e09441137
- Cited by
- 1,852
- References
- 44
- Citations by year
- 2026: 51 · 2025: 44 · 2024: 54 · 2023: 48 · 2022: 82 · 2021: 69 · 2020: 68 · 2019: 78 · 2018: 55 · 2017: 72 · 2016: 87 · 2015: 90 · 2014: 93 · 2013: 102 · 2012: 72
- FWCI
- 77.21
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Educational Studies in Mathematics (publishedVersion)
- Topics
- Social Sciences › Arts and Humanities › History and Philosophy of Science › History and Theory of Mathematics
- Social Sciences › Education › Mathematics Education and Pedagogy
- Social Sciences › Education › Mathematics Education and Teaching Techniques
- Keywords
- structural conception, reification, interiorization, condensation, concept formation, mathematics learning, problem solving
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
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