Concepts associated with the equality symbol
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Students need to understand that the equal sign expresses a relation and balance between numbers, not a direction or the result of an operation.
“As the key to relational thinking, students are supposed to understand that the equal sign refers to the relation and balance between numbers (Carpenter, Franke & Levi, 2003) not to a direction (Kieran, 1981) or the result of an operation.”
From Relational thinking: The bridge between arithmetic and algebra (2017) · cites this paper as Kieran, 1981
Arithmetic is the foundation of algebra teaching despite the differences between arithmetic and algebra.
“Therefore, it is obvious that arithmetic constitutes the basis of algebra teaching despite the differences resulting from the natures of arithmetic and algebra (Herscovics & Linchevski, 1994;Kieran, 1981;Knuth, Alibali, McNeil, Weinmberg & Stephens, 2005;Knuth, Stephens, McNeil & Alibali, 2006).”
From Relational thinking: The bridge between arithmetic and algebra (2017) · cites this paper as Kieran, 1981
Cited by
Sorted by citations
Sfard, A. (1991) · Educational Studies in Mathematics · doi:10.1007/bf00302715 · Cited by 1,852 · see citing works
Blanton, M., Stephens, A., Knuth, E., Gardiner, A. M., Isler, I., & Kim, J.-S. (2015) · Journal for Research in Mathematics Education · doi:10.5951/jresematheduc.46.1.0039 · Cited by 249 · see citing works
“…students have with algebra—and that our intervention addressed—are based on conceptions that include an operational view of the equal sign (e.g., Kieran, 1981), a focus on particular (numerical) answers rather than generalized claims (Booth, 1984), a lack of recognition of fundamental…”
BackgroundAssessing knowledge of mathematical equivalence: A construct-modeling approach.
Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L. (2011) · Journal of Educational Psychology · doi:10.1037/a0021334 · Cited by 160 · see citing works
Middle school students' understanding of core algebraic concepts: Equivalence & Variable1
Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005) · Zentralblatt für Didaktik der Mathematik · doi:10.1007/bf02655899 · Cited by 105 · see citing works
Stephens, A. C. (2006) · Journal of Mathematics Teacher Education · doi:10.1007/s10857-006-9000-1 · Cited by 81 · see citing works
Kieran, C. (2022) · ZDM – Mathematics Education · doi:10.1007/s11858-022-01435-6 · Cited by 68 · see citing works
Self-citationRelational thinking: The bridge between arithmetic and algebra
Kızıltoprak, A. (2017) · lnternational Electronic Journal of Elementary Education · doi:10.26822/iejee.2017131893 · Cited by 38 · see citing works
Molina, M., & Mason, J. (2009) · Canadian Journal of Science, Mathematics and Technology Education · doi:10.1080/14926150903191885 · Cited by 15 · see citing works
- Authors
- Kieran, C.
- Affiliations
- Concordia University (CA)
- Corresponding author
- Kieran, Carolyn
- ORCID
- Kieran, Carolyn
- Venue
- Educational Studies in Mathematics
- Published
- 1981-08
- Volume
- 12
- Issue
- 3
- Pages
- 317-326
- ISSN
- 0013-1954 (ISSN-L)
- 1573-0816
- OpenAlex
- W1992274553
- Cited by
- 562
- References
- 0
- Citations by year
- 2026: 14 · 2025: 21 · 2024: 21 · 2023: 11 · 2022: 35 · 2021: 28 · 2020: 20 · 2019: 25 · 2018: 23 · 2017: 36 · 2016: 20 · 2015: 28 · 2014: 34 · 2013: 23 · 2012: 28
- FWCI
- 3.49
- Type
- journal article
- Language
- en
- Open access
- closed
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Educational Studies in Mathematics (publishedVersion)
- Topics
- Social Sciences › Education › Mathematics Education and Teaching Techniques
- Physical Sciences › Mathematics › Applied Mathematics › Mathematics Education and Pedagogy
- Physical Sciences › Mathematics › Theoretical Computer Science › History and Theory of Mathematics
- Keywords
- equals sign, equality, equivalence, non-equivalence, operator symbols, algebraic equations
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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