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On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin

Sfard, Anna·1991·Educational Studies in Mathematics

DOI 10.1007/bf00302715

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Self-citation
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de Lima, R. N., & Tall, D. O. (2007) · Educational Studies in Mathematics · doi:10.1007/s10649-007-9086-0 · Cited by 119
“In this case, theories of process-object encapsulation (Dubinsky, 1991, Sfard, 1991, Gray & Tall, 1994) are not relevant to describe what the students are actually doing:”
Method
Development of angle concepts by progressive abstraction and generalisation
Mitchelmore, M. C., & White, P. J. (2000) · Educational Studies in Mathematics · doi:10.1023/a:1003927811079 · Cited by 119
Playing Mathematical Instruments: Emerging Perceptuomotor Integration With an Interactive Mathematics Exhibit
Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2013) · Journal for Research in Mathematics Education · doi:10.5951/jresematheduc.44.2.0372 · Cited by 132
“Researchers have examined learning about functions from multiple points of view, such as: 1) Identifying different ways of thinking about functions (Carlson, 1998; Confrey & Smith, 1995; Schoenfeld, Smith, & Arcavi, 1993; Sfard, 1991; Thompson, 1994a; Yerushalmy, 1997); 2) Documenting errors and misconceptions about f…”
Background
TOOL USE AND THE DEVELOPMENT OF THE FUNCTION CONCEPT: FROM REPEATED CALCULATIONS TO FUNCTIONAL THINKING
Doorman, M., Drijvers, P. H. M., Gravemeijer, K. P. E., Boon, P., & Reed, H. C. (2012) · International Journal of Science and Mathematics Education · doi:10.1007/s10763-012-9329-0 · Cited by 129
“In summary, we conclude that the learning arrangement with a computer tool helped students to overcome the difficulty of integrating operational and structural aspects of the concept function (Falcade et al., 2007; Sfard, 1991).”
BackgroundInfluential
Layers of generality and types of generalization in pattern activities
Radford, L. (2010) · PNA · doi:10.30827/pna.v4i2.6169 · Cited by 115
“A bit later Sfard (1991) and Gray and Tall (1994) called attention to the students’ difficulties in distinguishing between objects and processes, while Bednarz and Janvier”
Background
How humans learn to think mathematically: exploring the three worlds of mathematics
(2014) · Choice Reviews Online · doi:10.5860/choice.51-5072 · Cited by 114
When the Classroom Floor Becomes the Complex Plane: Addition and Multiplication as Ways of Bodily Navigation
Nemirovsky, R., Rasmussen, C. L., Sweeney, G., et al. (2011) · Journal of the Learning Sciences · doi:10.1080/10508406.2011.611445 · Cited by 114
“Historically speaking, viewing a complex number as single entity, in part via the complex plane, represents a significant conceptual advance (Penrose, 2004; Sfard, 1991).”
Background
The Impact of Preservice Teachers' Content Knowledge on Their Evaluation of Students' Strategies for Solving Arithmetic and Algebra Word Problems
Dooren, W. V., Verschaffel, L., & Onghena, P. (2002) · Journal for Research in Mathematics Education · doi:10.2307/4149957 · Cited by 120
“") For this scoring, we used an a priori classification scheme based on the work of several other researchers (Filloy & Sutherland, 1996; Hall et al., 1989; Kieran, 1992; Linchevski & Herscovics, 1996; Reed, 1999; Sfard, 1991; Stacey & MacGregor, 2000; Wolters, 1976).”
BackgroundMethod
An “Instrumental Approach” to Study the Integration of a Computer Tool Into Mathematics Teaching: the Case of Spreadsheets
Haspékian, M. (2005) · International Journal of Computers for Mathematical Learning · doi:10.1007/s10758-005-0395-z · Cited by 114
“…the difficulties of algebraic learning have already been extensively investigated and often related to different analyses: procedural/ structural duality (Sfard, 1991), semantic/syntactic difficulty (Drouhard, 1992), arithmetic/algebra relationships (Vergnaud et al., 1988; Vergnaud, 1989/90).”
Background
Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking
Warren, E., & Cooper, T. (2008) · Educational Studies in Mathematics · doi:10.1007/s10649-007-9092-2 · Cited by 159
A Friendly Introduction to “Knowledge in Pieces”: Modeling Types of Knowledge and Their Roles in Learning
diSessa, A. A. (2018) · ICME-13 monographs · doi:10.1007/978-3-319-72170-5_5 · Cited by 129
How Students Attempt to Reduce Abstraction in the Learning of Mathematics and in the Learning of Computer Science
Hazzan, O. (2003) · Computer Science Education · doi:10.1076/csed.13.2.95.14202 · Cited by 111
“(Sfard, 1991, p. 4).”
BackgroundMethodInfluential
Asking mathematical questions mathematically
Mason, J. H. (2000) · International Journal of Mathematical Education in Science and Technology · doi:10.1080/002073900287426 · Cited by 119
Research on calculus: what do we know and where do we need to go?
Rasmussen, C. L., Marrongelle, K. A., & Borba, M. C. (2014) · ZDM · doi:10.1007/s11858-014-0615-x · Cited by 127
Levels of students’ “conception” of fractions
Pantziara, M., & Philippou, G. (2011) · Educational Studies in Mathematics · doi:10.1007/s10649-011-9338-x · Cited by 105
“Researchers viewing that in the concept formation procedural/operational understanding precedes the conceptual/structural one, developed frameworks describing the process of concept formation (Dubinsky, 1991; Gray & Tall, 1994; Sfard, 1991).”
BackgroundMethodInfluential
Early Algebra
Kieran, C., Jeongsuk, P., Schifter, D., & Ng, S. F. (2016) · ICME-13 topical surveys · doi:10.1007/978-3-319-32258-2 · Cited by 191
Characterizing Practices Associated with Functions in Middle School Textbooks: An Empirical Approach
Mesa, V. (2004) · Educational Studies in Mathematics · doi:10.1023/b:educ.0000040409.63571.56 · Cited by 105
Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus
Lagrange, J.-B. (1999) · International Journal of Computers for Mathematical Learning · doi:10.1023/a:1009858714113 · Cited by 103
“There is no permanent necessity to consider first an ‘operative’ (Sfard, 1991) approach of concepts.”
Background

Authors: Sfard, A.
Affiliations: Hebrew University of Jerusalem (IL)
Corresponding author: Sfard, Anna
ORCID: Sfard, Anna
Venue: Educational Studies in Mathematics
Publisher: Springer Science and Business Media LLC
Published: 1991-02
Volume: 22
Issue: 1
Pages: 1-36
ISSN: 0013-1954 (ISSN-L)
1573-0816
DOI: 10.1007/bf00302715
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

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On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin

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