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
Overview
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
Citations
Cites
The magical number seven, plus or minus two: Some limits on our capacity for processing information.
Miller, G. A. (1956) · Psychological Review · doi:10.1037/h0043158 · Cited by 17,812
The magical number seven, plus or minus two: Some limits on our capacity for processing information.
Miller, G. A. (1994) · Psychological Review · doi:10.1037/0033-295x.101.2.343 · Cited by 17,559
Paivio, A. (2013) · Psychology Press eBooks · doi:10.4324/9781315798868 · Cited by 4,184
Scribner, S., & Cole, M. (1981) · doi:10.4159/harvard.9780674433014 · Cited by 3,447
Mezei, Á., & Paivio, A. (1972) · Leonardo · doi:10.2307/1572599 · Cited by 3,265
Anderson, J. R. (2013) · Psychology Press eBooks · doi:10.4324/9780203780954 · Cited by 2,180
The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics
Penrose, R., & Gardner, M. (1989) · doi:10.1093/oso/9780198519737.001.0001 · Cited by 2,007
The Child's Conception of Number
Peel, E. A., & Piaget, J. (1953) · British Journal of Educational Studies · doi:10.2307/3118763 · Cited by 1,753
Relational Understanding and Instrumental Understanding
Skemp, R. R. (1978) · The Arithmetic Teacher · doi:10.5951/at.26.3.0009 · Cited by 1,316
Piaget, J. (1970) · Columbia University Press eBooks · doi:10.7312/piag91272 · Cited by 1,115
Flagged first, then by citations
Research works printed but not linked: 13
- Brownell, W. A.: 1935, ?Psychological considerations in learning and teaching arithmetic?, in The Teaching of Arithmetic: Tenth Yearbook of the NCTM, Columbia University Press, New York.
- Clements, K.: 1981, ?Visual imagery and school mathematics?, For the Learning of Mathematics 2(2), 33?38.
- Dörfler, W.: 1987, ?Empirical investigation of the construction of cognitive schemata from actions?, in Proceedings of the Eleventh International Conference of PME, Vol. III, pp. 3?9.
- Dörfler, W.: 1989, ?Protocols of actions as a cognitive tool for knowledge construction?, in Proceedings of the Thirteenth International Conference of PME, Paris, Vol. 1, pp. 212?9.
- Eisenberg, T. and Dreyfus, T. (eds.): 1989, Visualization in the mathematics curriculum, Special issue of Focus on Learning Problems in Mathematics 11(1&2).
- Kaput, J. J.: 1979, ?Mathematics and learning: Roots of epistemological status?, in Lochhead, J. and Clement, J. (eds.), Cognitive Process Instruction, Franklin Institute Press.
- Otte, M.: 1984, ?Komplementarität?, Dialektik 8, 60?75.
- Poincaré, H.: 1952, Science and Method, Dover Publications, New York.
- Sfard, A.: 1987, ?Two conceptions of mathematical notions: operational and structural?, in Proceedings of the Eleventh International Conference of PME, Montreal, Vol. 3, pp. 162?9.
- Sfard, A.: 1988, ?Operational vs structural method of teaching mathematics: A case study?, in Proceedings of the Twelfth International Conference of PME, Hungary, pp. 560?7.
- Sfard, A.: 1989, ?Transition from operational to structural conception: the notion of function revisited?, in Proceedings of the Thirteenth International Conference of PME, Paris, Vol. 3, pp. 151?8.
- Sinclair, H. and Sinclair, A.: 1986, ?Children's mastery of written numerals and the construction of basic number concepts?, in Hiebert, J. (ed.), Conceptual and Procedural Knowledge: The Case of Mathematics, Erlbaum, Hillsdale, N.J.
- Thompson, P. W.: 1985, ?Experience, problem solving, and learning mathematics: considerations in developing mathematical curricula?, in E. A.Silver (ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, Erlbaum, Hillsdale, N.J.
Cited by
Sorted by citations
Items 1,401–1,500 of 1,847
Molly and Equations in A[2]: A Case Study of Apprehending Structure.
Simpson, A. P., & Stehlíková, N. (2004) · Proceedings of the ... PME Conference · Cited by 1
Actes du séminaire national de didactique des mathématiques 2012
Coppé, S., & Haspékian, M. (2013) · HAL · Cited by 1
On the equivalence relation in the students' concept image of equation
Attorps, I., & Tossavainen, T. (2009) · Cited by 1
Pratt, D. (2012) · Cited by 1
Prime Numbers are Infinitely Many: Four Proofs from History to Mathematics Education
Bagni, G. T. (2004) · Cited by 1
RESEARCH FOR THE LEARNING AND TEACHING OF MATHEMATICS: AN EMERGENT DESIGN
Mostert, I. (2007) · SUNScholar (Stellenbosch University) · Cited by 1
Pettersson, K. (2004) · Chalmers Research (Chalmers University of Technology) · Cited by 1
Psicología y Didáctica de las matemáticas
Gómez-Granell, C., & Fraile, J. (1993) · Journal for the Study of Education and Development Infancia y Aprendizaje · doi:10.1080/02103702.1993.10822375 · Cited by 2
Valcarce, M. C., & Vázquez, M. S. (2007) · Cited by 1
Image-based reasoning in geometry
Handscomb, K. (2005) · Summit (Simon Fraser University) · Cited by 1
The role of instructional representations on students' written representations and achievements
Sun, Y. (2006) · OakTrust (Texas A&M University Libraries) · Cited by 1
Philosophy of Mathematics for the Masses : Extending the scope of the philosophy of mathematics
Buijsman, S. (2016) · DiVA (Stockholm University) · Cited by 1
Exploring and describing the growth points of learners as they encounter functions in equation form.
Clark, R. (2014) · University of the Witwatersrand, Johannesburg Institutional Repository on DSpace (University of the Witwatersrand, Johannesburg) · Cited by 1
INTEGER OPERATIONS IN THE PRIMARY SCHOOL: A SEMIOTIC ANALYSIS OF A "FACTUAL GENERALIZATION"
Koukkoufis, A., & Williams, J. S. (2005) · Cited by 1
OKAZAKI, M. (2006) · Cited by 1
Process or object? Ways of solving mathematical problems using CAS
Gjone, G. (2004) · Teaching Mathematics and Computer Science · doi:10.5485/tmcs.2004.0052 · Cited by 1
“the idea of turning a process into an autonomous entity should emerge, and finally to see this new entity as an integrated, object-like whole must be acquired (Sfard, [5], p.”
BackgroundInfluentialColeman, J. L. (2008) · QSpace (Queen's University Library) · Cited by 1
Procesos infinitos inherentes a la integral definida
Cuida, A. (2016) · UVaDOC UVaDOC University of Valladolid Documentary Repository (University of Valladolid) · doi:10.35376/10324/22112
Vincent, B. (2016) · The Research Repository @ WVU (West Virginia University) · doi:10.33915/etd.6876 · Cited by 1
“The following quote by Sfard (1991) captures the meaning of object (or structural conception): “Seeing a mathematical entity as an object means being capable of referring to it as if it was a real thing- a static structure, existing somewhere in space and time.”
BackgroundMethodInfluentialThe equal sign: Teachers’ specialised content knowledge and Learners’ misconceptions.
Meyer, B. C. (2016) · Cited by 1
Constructive habituation as an educational approach to process-object reification in mathematics
Peterson, A. (2004) · doi:10.31390/gradschool_dissertations.3083 · Cited by 1
“The lower level reification and the higher level interiorization are prerequisites for each other (Sfard, 1991).”
BackgroundInfluentialChapter 5: Phenomenography: Exploring the Roots of Numeracy
Neuman, D. (1997) · Journal for Research in Mathematics Education Monograph · doi:10.2307/749947 · Cited by 2
Salihu, F. O. (2017) · Unisa Institutional Repository (University of South Africa) · Cited by 1
Ssebaggala, L. (2018) · ISU Red - Research and eData (Illinois State University) · doi:10.30707/etd2019.ssebaggala.l · Cited by 1
What’s So Hard About Algebra? A Grounded Theory Study of Adult Algebra Learners
Steele Reese EdD, M. (2007) · doi:10.22371/05.2007.015 · Cited by 1
Zuccarini, G., & Malgieri, M. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2108.06919 · Cited by 1
“However, their findings and claims open further questions to the research community: according to Singh and Marshman (2015), research shows that students in upper-level QM “have common difficulties independent of their background, teaching style, textbook, and institution” (p.”
InfluentialProviding College Level Calculus Students with Opportunities to Engage in Theoretical Thinking
Challita, D. (2013) · Spectrum Research Repository (Concordia University) · Cited by 1
A cognitive framework for knowledge in informatics
Holmboe, C. (1999) · ACM SIGCSE Bulletin · doi:10.1145/384267.305833 · Cited by 1
Actions, Processes, Objects, Schemas (APOS) in Mathematics Education
Dubinsky, E. (2020) · Springer eBooks · doi:10.1007/978-3-030-15789-0_3 · Cited by 1
Psychological Approaches in Mathematics Education
Schwarz, B. B. (2014) · Springer eBooks · doi:10.1007/978-94-007-4978-8_167 · Cited by 1
Middle school students’ beliefs about mathematics as a discipline
Østergaard, M. K. (2022) · doi:10.7146/aul.462 · Cited by 1
Dynamic visualization in animated mathematics videos I: a classification of didactic rolesPreprint
ten Voorde, A., Piroi, M., & Bos, R. (2023) · Research Square · doi:10.21203/rs.3.rs-3358135/v1 · Cited by 1
Social Constructivism: Its Role in Algebra Classrooms.
Telese, J. A. (1999) · Cited by 1
Hino, K. (1999) · Journal of Science Education in Japan · doi:10.14935/jssej.23.283 · Cited by 1
Prospective Zimbabwean "A" level mathematics teacher's knowledge of the concept of a function
Nyikahadzoyi, M. R. (2006) · University of the Western Cape Electronic Theses and Dissertations Repository (University of the Western Cape) · Cited by 1
Tall, D. O. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139565202.013
Graves, A. A. (2008) · DigitalCommons (California Polytechnic State University)
Tall, D. O. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139565202.003
Quand un prof rencontre un autre prof... pour faire des mathématiques
Cherix, P.-A., Conne, F., Daina, A., et al. (2011) · Recherches en didactiques · doi:10.3917/rdid.012.0007
Bråthen, A. R. (2014) · NORA - Norwegian Open Research Archives
A tarefa como instrumento de desenvolvimento da flexibilidade de cálculo
Serrazina, L., & Rodrigues, M. (2014)
Una comprensión de la comprensión en matemáticas
Gómez, P. (2024) · instname:Universidad de los Andes · doi:10.57784/1992/32349
Wang, Y. (2015) · Durham e-Theses (Durham University)
Issues and Concerns in Teaching Mathematics
Mason, J. H. (2002) · Elsevier eBooks · doi:10.1533/9780857099648.161
Totality as a Possible New Stage and Levels in APOS Theory
Arnon, I., Cottrill, J., Dubinsky, E., et al. (2013) · Springer eBooks · doi:10.1007/978-1-4614-7966-6_8
MATHEMATICS AS A CONSTRUCTIVE ENTERPRISE
Mason, J. H. (2015)
Baker, W., & Czarnocha, B. (2016) · SensePublishers eBooks · doi:10.1007/978-94-6300-549-4_32
Fernández-Plaza, J. A., Ruiz-Hidalgo, J. F., & Romero, L. R. (2012) · Institutional Repository of the University of Granada (University of Granada)
The impact of a technology-rich intervention on grade 7 students' skills in initial algebra
Jupri, A., Drijvers, P. H. M., & van den Heuvel‐Panhuizen, M. (2014) · Utrecht University Repository (Utrecht University)
Making Meaning of Creativity and Mathematics Teaching
Misfeldt, M. (2015) · VBN Forskningsportal (Aalborg Universitet)
Pedagogical scenario involving Aplusix educational software
Trgalová, J. (2009) · HAL
You Can Lead a Horse to Water…: Issues in Deepening Learning Through Deepening Teaching
Mason, J. H. (2012) · Advances in mathematics education · doi:10.1007/978-94-007-5696-0_22
Emmanuel, C. (2011) · ResearchSpace (University of KwaZulu-Natal)
On the difficulties of acquiring mathematical experience: Case rural educationPreprint
Booß–Bavnbek, B. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1312.2125
An Evaluation of Student Learning and Engagement in a Technology-Enhanced Algebra Unit on Slope
Beck, E. (2000) · University of North Texas Digital Library (University of North Texas) · doi:10.12794/metadc2658
Skolår 4 elevers uppfattning av likhetstecknet och läroböckernas framställning
Haddad, J., & Sponheimer, G. (2010) · Malmö University Publications (Malmö University)
Des pistes pour enseigner le scandale des nombres irrationnels
Petit, M. (2014) · Nouveaux cahiers de la recherche en éducation · doi:10.7202/1025763ar
Ospina, A. G. (2013) · Revista Científica · doi:10.14483/23448350.7081
Bergholm, M. (2014) · Linköping University Electronic Press eBooks · doi:10.3384/lic.diva-106577
高等学校における数学的記号の理解に関する研究(II) : 置き換えを事例とした数学的記号の教育的役割
福田, 幸. (2009) · 数学教育学研究 : 全国数学教育学会誌 · doi:10.24529/jasme.15.1_29
Listening in the Mathematics Working Relationship of Two High-School students
Forster, P. A. (2004) · Australasian Journal of Paramedicine
Modeling the co-development of strategic and conceptual knowledge in mathematical problem solving
Campbell, M. E. (2011) · ProQuest Demo Repository
Uddin, R. S. (2011) · ResearchSpace (University of KwaZulu-Natal)
Joseph, A. K., Adu, I. K., Ruth, A., et al. (2015) · Journals & Books Hosting (International Knowledge Sharing Platform)
A Problem-Based Schema Analysis in Algebra
Chinnappan, M. (2002) · Research Online (University of Wollongong)
Aspects of learning and understanding in multivariable calculus
Deserti, F. (2006) · DSpace@MIT (Massachusetts Institute of Technology)
Representaciones de funciones matemáticas de una variable
Vanegas, D., & Escalona, M. (2010) · Omnia
Tall, D. O. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139565202.022
Fractions as Subtraction: An Activity‐Oriented Perspective from Elementary Children
Wood, M. B., Olson, A. M., Freiberg, E. J., et al. (2013) · School Science and Mathematics · doi:10.1111/ssm.12040
Student Difficulties with Mathematics
Mason, J. H. (2002) · Elsevier eBooks · doi:10.1533/9780857099648.7
Tall, D. O. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139565202.024
The similarity between geometry and conceptual domains as spatial environments
Diezmann, C. M. (1996) · QUT ePrints (Queensland University of Technology)
Using Design Sketch to Teach Bubble Sort in High SchoolPreprint
Liu, C., Jiu, Y.-W., & Chen, J. J.-Y. (2009) · arXiv (Cornell University) · doi:10.48550/arxiv.0912.3955
“However, some people will never comprehend certain concepts; that is to say, for some people, the conceptualization mentioned above may not likely occur, while, for others, it does [6].”
BackgroundComprensión de frecuencias asociadas a las tablas de contingencia por estudiantes de psicología
Cañadas, G. R., Bernabeu, B., & Gea, M. M. (2013) · UNIPLURIVERSIDAD
Christiansen, I. M. (2008) · Perspectives in Education
Yusof, N. (2026) · Warwick Research Archive Portal (University of Warwick) · doi:10.82444/warw.33487387
The Effects of The Content Enhancement Model in College Algebra
VanCleave, J. M. (2010) · KU ScholarWorks (The University of Kansas)
The WebLabs Project: Building New Formalisms for Mathematical and Scientific Ideas
Hoyles, C., Kahn, K., Mor, Y., et al. (2005)
The Historical Evolution of Mathematics
Tall, D. O. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139565202.015
Tall, D. O. (2013) · Cambridge University Press eBooks · doi:10.1017/cbo9781139565202.023
Maimane, J. R., & Rankhumise, M. P. (2014) · Mediterranean Journal of Social Sciences · doi:10.5901/mjss.2014.v5n15p434
“She is of the opinion that successful learning and problem-solving require the ability to move from one domain to the other (Sfard, 1991).”
BackgroundReconceptualization of Histo-Genetic Principle
Yoo, Y. J. (2013) · Journal for History of Mathematics · doi:10.14477/jhm.2013.26.5_6.389
“대상화 는 Piaget [27]로부터 자극을 받아 여러 견해들이 제시되었는데 그 중에서도 Dubinsky [10]의 APOS 이론과 Sfard [37]의 구상화 (reification) 이론이 대표적이다.”
Lee, J.-E. (2014) · Mathematics Education Research and Practice · doi:10.63311/jksmed.2014.18.1.55
Procepts and proceptual solutions in undergraduate mathematics
Naidoo, I., & Huntley, B. (2010) · Pythagoras · doi:10.4102/pythagoras.v0i72.56
STUDENTS' INTERPRETATIONS AND REPRESENTATIONS OF NUMBER
O’Reilly, D. (1999) · Advances in Mathematics Education · doi:10.1080/14794809909461544
Design and Validation of a Model based Diagnosis System
Jean-Daubias, S., Delozanne, É., Jacoboni, P., et al. (1999) · HAL
Suthisung, N., Inprasitha, M., & Sangaroon, K. (2012) · Creative Education · doi:10.4236/ce.2012.37177
“Sfard (1991) comprised of interiorization, condensation and reification. APOS of Dubinsky (1991) comprised of action, process, and object and expanded to schema according to Pegg and Tall (2005: p.”
MethodDomini, J. (2008) · American book review/The American book review · doi:10.1353/abr.2008.0145
Culture and knowing in childhood for mathematical understanding
Dixon, P. W. (2004) · Human Evolution · doi:10.1007/bf02438912
“The problem remains, however, to determine how, "the lower level reification and higher-level interiorization are prerequisite for each other." ( Sfard, 1991, p. 31 ).”
BackgroundBorsani, V., Lamela, C., Luna, J. P., et al. (2014) · Yupana · doi:10.14409/yu.v1i7.4260
DYNAMICAL ASPECTS OF MATHEMATICAL PROOF
Barnard, T. (2014)
Função: Concepções daquele que ensina e daquele que aprende
Pires, R. F., & da Silva, B. A. (2015) · Em Teia | Revista de Educação Matemática e Tecnológica Iberoamericana · doi:10.36397/emteia.v5i3.2194
“Segundo Sfard (1991), o exercício de conceber um objeto matemático pode colocar em jogo uma dualidade sobre a natureza das concepções matemáticas: a concepção estrutural e a operacional, que para ela, são duas faces de uma mesma moeda.”
Background
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 ›
Are you an author, or do you see a problem with this page? Request a review
