On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin
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“For example, the conceptualization of ratios of integers as numbers, rather than operations, can be accounted for as an ontological shift of ratios from the category of processes to the category of mathematical objects (Sfard, 1991).”
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“…have been many studies on individuals’ learning and understanding of advanced mathematical concepts (e.g., Davis and Vinner, 1986; Tall, 1991; Pinto and Tall, 1999) and other reports on the nature of the concepts themselves (e.g., Sfard, 1991; Lakoff and Nunez, 2000; Dubinsky and McDonald, 2001).”
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“Tall, 1994), reification ( Sfard, 1991 ), and encapsulation (Piaget, 1977; Dubinsky & Levin, 1986) has both idiosyncratic aspects and more-orless shared aspects.”
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“Others focused on the polarity between pointwise and shape-based understanding of graphs, a polarity that has been presented as an instance of the more general dialectic between process/ object which, according to several authors, underlies all mathematics learning (Dubinsky & Harel, 1992; Sfard, 1991).”
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“…by starting with objects (e.g., numbers), defining processes that manipulate and transform those objects (e.g., functions), and then recategorizing the processes as a new kind of object (e.g., vectors in a function space), creating new processes that act on these objects, and so on (Sfard, 1991).”
BackgroundThe emergence of objects from mathematical practices
Font, V., Godino, J. D., & Romero, J. G. (2012) · Educational Studies in Mathematics · doi:10.1007/s10649-012-9411-0 · Cited by 214 · see citing works
“The process of reification has also been studied in the area of mathematics education (see, for example, Asiala et al., 1996; Sfard, 1991).”
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“In so doing, they appear to pass over some of the conceptual and structural underpinnings of symbolic forms of representation (e.g., Kieran, 1992; Sfard, 1991).”
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“The first are descriptions of stages of concept development (cf. Pirie & Kieren, 1994; Sfard, 1991), and the second are pedagogical theories that specify pedagogical situations or stages such as the French theory of didactical situations (Brousseau, 1997) and the Dutch Realistic Mathematics Education (Gravemeijer, 199…”
BackgroundInfluentialLearning the Algebraic Method of Solving Problems
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“It is interesting to note that some studies adopt a perspective that links historic, epistemological and cognitive analyses (Sfard 1991; Sierpinska 1992).”
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“Various authors, including Piaget (1972), Dubinsky (1991) and Sfard (1991), theorise that the growth of human knowledge starts with actions (first on the environment), some of which become repeatable processes and are later conceived as objects in their own right to be manipulated on a higher level…”
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“Sfard (1991) found that both students and teachers often expect immediate rewards for teaching and learning efforts.”
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“Although it cannot be pursued here, gesture analysis may shed further light on the pervasive metaphor in which numbers, procedures and other mathematical 134 L.D. Edwards “entities” are treated as “objects” by students and mathematicians alike (cf., Radford 2002; Sfard 1991).”
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Sherin, B. L. (2001) · International Journal of Computers for Mathematical Learning · doi:10.1023/a:1011434026437 · Cited by 152 · see citing works
“…I have relied significantly on research into how students understand and solve elementary word problems (e.g., Carpenter and Moser, 1983; Fuson, 1992; Reed, 1998; Riley, Greeno and Heller, 1983) and understand mathematical equations (Kieran, 1992; Herscovics and Kieran, 1980; Sfard, 1987, 1991).”
BackgroundAlgebraic thinking and the generalization of patterns: a semiotic perspective
Radford, L., Laurentienne, U., & Peirce, C. S. (2015) · Cited by 148 · see citing works
The balance model: Hindrance or support for the solving of linear equations with one unknown
Vlassis, J. (2002) · Educational Studies in Mathematics · doi:10.1023/a:1020229023965 · Cited by 143 · see citing works
“On the other hand, students assimilate only very gradu- ally the concepts implied by the properties of equality and by the acquisition of a ‘structural’ view of expressions (Sfard, 1991).”
BackgroundBlack Boxes in Workplace Mathematics
Williams, J. S., & Wake, G. (2006) · Educational Studies in Mathematics · doi:10.1007/s10649-006-9039-z · Cited by 143 · see citing works
The role of multiple solution tasks in developing knowledge and creativity in geometry
Levav-Waynberg, A., & Лейкин, Р. (2011) · The Journal of Mathematical Behavior · doi:10.1016/j.jmathb.2011.11.001 · Cited by 143 · see citing works
Pedagogical Content Tools: Integrating Student Reasoning and Mathematics in Instruction
Rasmussen, C. L., & Marrongelle, K. A. (2006) · Journal for Research in Mathematics Education · doi:10.2307/30034860 · Cited by 136 · see citing works
Aspects of the Nature and State of Research in Mathematics Education
Niss, M. (1999) · Educational Studies in Mathematics · doi:10.1023/a:1003715913784 · Cited by 135 · see citing works
The array representation and primary children’s understanding and reasoning in multiplication
Barmby, P., Harries, T., Higgins, S., et al. (2008) · Educational Studies in Mathematics · doi:10.1007/s10649-008-9145-1 · Cited by 132 · see citing works
“Sfard (1991) highlighted the role that representations (in particular visual representations) play in the way we develop mathematical concepts as learners, moving from an operational or process view of a concept to a structural view (e.g.”
BackgroundKnowledge of Algebra for Teaching: A Framework of Knowledge and Practices
McCrory, R., Floden, R. E., Ferrini‐Mundy, J., Reckase, M. D., & Senk, S. L. (2012) · Journal for Research in Mathematics Education · doi:10.5951/jresematheduc.43.5.0584 · Cited by 132 · see citing works
“This item is advanced not because it is difficult but because it draws on the ability to see functions as algebraic objects in their own right, objects subject to rules governing operations on real numbers (cf. Dubinsky & McDonald, 2001; Kieran, 1992; Sfard, 1991).”
BackgroundZandieh, M., & Rasmussen, C. L. (2010) · The Journal of Mathematical Behavior · doi:10.1016/j.jmathb.2010.01.001 · Cited by 130 · see citing works
Understanding the integral: Students’ symbolic forms
Jones, S. R. (2013) · The Journal of Mathematical Behavior · doi:10.1016/j.jmathb.2012.12.004 · Cited by 129 · see citing works
Linchevski, L., & Herscovics, N. (1996) · Educational Studies in Mathematics · doi:10.1007/bf00163752 · Cited by 129 · see citing works
Procedural and Conceptual Knowledge: Exploring the Gap Between Knowledge Type and Knowledge Quality
Star, J. R., & Stylianides, G. J. (2013) · Canadian Journal of Science Mathematics and Technology Education · doi:10.1080/14926156.2013.784828 · Cited by 129 · see citing works
“In fact, many mathematics educators (e.g., Gray & Tall, 1994; Sfard, 1991) have raised questions about whether conceptual and procedural knowledge are distinct types of knowledge at all.”
BackgroundNaturally occurring data as research instrument
Lister, R., Clear, T., Simon, Bouvier, D. J., Carter, P., Eckerdal, A., Jacková, J., Lopez, M., McCartney, R., Robbins, P., et al. (2010) · ACM SIGCSE Bulletin · doi:10.1145/1709424.1709460 · Cited by 127 · see citing works
“It has also been observed [29] that these cycles are related both internally and sequentially: the desire to use a concept as a concrete object may drive the development of the earlier stages, and the need for a particular primitive in a concept may drive the reification in a previous concept.”
BackgroundMethodInfluentialEpistemologies of Mathematics and of Mathematics Education
Sierpinska, A., & Lerman, S. (1996) · Springer eBooks · doi:10.1007/978-94-009-1465-0_26 · Cited by 125 · see citing works
Symbols and the bifurcation between procedural and conceptual thinking
Tall, D. O., Gray, E. M., Ali, M. B., et al. (2001) · Canadian Journal of Science Mathematics and Technology Education · doi:10.1080/14926150109556452 · Cited by 123 · see citing works
“The process of compression from process to concept has been described in similar ways by various authors (e.g. Sfard, 1991 ; Dubinsky, 1991).”
MethodHandheld technology for mathematics education: flashback into the future
Trouche, L., & Drijvers, P. H. M. (2010) · ZDM · doi:10.1007/s11858-010-0269-2 · Cited by 122 · see citing works
“This suggests that the use of HHT can promote object thinking and encapsulation of mathematical processes, which is seen as an important mathematical achievement (e.g. see Sfard, 1991).”
BackgroundSfard, A. (2000) · Journal for Research in Mathematics Education · doi:10.2307/749809 · Cited by 120 · see citing works
Self-citationProcedural embodiment and magic in linear equations
de Lima, R. N., & Tall, D. O. (2007) · Educational Studies in Mathematics · doi:10.1007/s10649-007-9086-0 · Cited by 119 · see citing works
“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:”
MethodDevelopment 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 · see citing works
Nemirovsky, R., Kelton, M. L., & Rhodehamel, B. (2013) · Journal for Research in Mathematics Education · doi:10.5951/jresematheduc.44.2.0372 · Cited by 119 · see citing works
“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…”
BackgroundDoorman, M., Drijvers, P. H. M., Gravemeijer, K. P. E., et al. (2012) · International Journal of Science and Mathematics Education · doi:10.1007/s10763-012-9329-0 · Cited by 116 · see citing works
“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).”
BackgroundInfluentialLayers of generality and types of generalization in pattern activities
Radford, L. (2010) · PNA · doi:10.30827/pna.v4i2.6169 · Cited by 115 · see citing works
“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”
BackgroundHow humans learn to think mathematically: exploring the three worlds of mathematics
(2014) · Choice Reviews Online · doi:10.5860/choice.51-5072 · Cited by 115 · see citing works
Nemirovsky, R., Rasmussen, C. L., Sweeney, G., et al. (2011) · Journal of the Learning Sciences · doi:10.1080/10508406.2011.611445 · Cited by 114 · see citing works
“Historically speaking, viewing a complex number as single entity, in part via the complex plane, represents a significant conceptual advance (Penrose, 2004; Sfard, 1991).”
BackgroundDooren, W. V., Verschaffel, L., & Onghena, P. (2002) · Journal for Research in Mathematics Education · doi:10.2307/4149957 · Cited by 114 · see citing works
“") 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).”
BackgroundMethodHaspékian, M. (2005) · International Journal of Computers for Mathematical Learning · doi:10.1007/s10758-005-0395-z · Cited by 114 · see citing works
“…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).”
BackgroundGeneralising 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 · see citing works
diSessa, A. A. (2018) · ICME-13 monographs · doi:10.1007/978-3-319-72170-5_5 · Cited by 115 · see citing works
Hazzan, O. (2003) · Computer Science Education · doi:10.1076/csed.13.2.95.14202 · Cited by 111 · see citing works
“(Sfard, 1991, p. 4).”
BackgroundMethodInfluentialAsking mathematical questions mathematically
Mason, J. H. (2000) · International Journal of Mathematical Education in Science and Technology · doi:10.1080/002073900287426 · Cited by 109 · see citing works
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 107 · see citing works
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 · see citing works
“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).”
BackgroundMethodInfluentialKieran, C., Jeongsuk, P., Schifter, D., & Ng, S. F. (2016) · ICME-13 topical surveys · doi:10.1007/978-3-319-32258-2 · Cited by 106 · see citing works
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 · see citing works
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 · see citing works
“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
- 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
- Physical Sciences › Mathematics › Theoretical Computer Science › History and Theory of Mathematics
- Physical Sciences › Mathematics › Applied Mathematics › 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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