Assessing knowledge of mathematical equivalence: A construct-modeling approach.
Abstract
Knowledge of mathematical equivalence, the principle that 2 sides of an equation represent the same value, is a foundational concept in algebra, and this knowledge develops throughout elementary and middle school. Using a construct-modeling approach, we developed an assessment of equivalence knowledge. Second through sixth graders (N = 175) completed the assessment on 2 occasions, 2 weeks apart. Evidence supported the reliability and validity of the assessment along a number of dimensions, and the relative difficulty of items was consistent with the predictions from our construct map. By Grade 5, most students held a basic relational view of equivalence and were beginning to compare the 2 sides of an equation. This study provides insights into the order in which students typically learn different aspects of equivalence knowledge. It also illustrates a powerful but underutilized approach to measurement development that is particularly useful for developing measures meant to detect changes in knowledge over time or after intervention
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In a study by Rittle-Johnson et al. (2011), students' understanding of the equal sign progressed steadily from operational to relational.
“In one study conducted to systematically evaluate second grade to sixth grade students' knowledge of mathematical equivalence (Rittle-Johnson, Matthews, Taylor & McEldoon, 2011), the researchers found that students give meaning to the equal sign in a constant progress, from operational to relational.”
From Relational thinking: The bridge between arithmetic and algebra (2017) · cites this paper as Rittle-Johnson, Matthews, Taylor & McEldoon, 2011
That study found that most fifth-grade students considered operations on both sides of the equation and were aware of the equal sign's relational meaning in addition to its operational meaning.
“The study demonstrated that especially most of the fifth grade students took the operations in both sides of the equation and that they were aware of the relational meaning of the equal sign besides its operational meaning.”
From Relational thinking: The bridge between arithmetic and algebra (2017) · cites this paper as Rittle-Johnson et al., 2011
The fifth-grade class has a vital place for the development of the relational meaning of the equal sign.
“This result is parallel to other research results which demonstrate that the fifth-grade class has a vital place for the development of the relational meaning of the equal sign (Rittle-Johnson et al., 2011).”
From Relational thinking: The bridge between arithmetic and algebra (2017) · cites this paper as Rittle-Johnson et al., 2011
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“This result is parallel to other research results which demonstrate that the fifth-grade class has a vital place for the development of the relational meaning of the equal sign (Rittle-Johnson et al., 2011).”
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“For example, they often exhibit misconceptions about the meaning of the equal sign, viewing it as an operator signal that means “adds up to” or “get the answer” (e.g., Matthews, Rittle-Johnson, McEldoon, & Taylor, 2012; McNeil & Alibali, 2005; Rittle-Johnson et al., 2011).”
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Teaching linear equations: Case studies from Finland, Flanders and Hungary
Andrews, P., & Sayers, J. (2012) · The Journal of Mathematical Behavior · doi:10.1016/j.jmathb.2012.07.002 · Cited by 28 · see citing works
“There is evidence that it is a function of children's cognitive maturity (Baroody and Ginsberg, 1983; Rittle-Johnson et al., 2011) and occurs at different ages in different students (Alibali et al., 2007).”
BackgroundSeparating Cognitive and Content Domains in Mathematical Competence
Harks, B., Klieme, E., Hartig, J., & Leiß, D. (2014) · Educational Assessment · doi:10.1080/10627197.2014.964114 · Cited by 25 · see citing works
Inducing mental set constrains procedural flexibility and conceptual understanding in mathematics
DeCaro, M. S. (2016) · Memory & Cognition · doi:10.3758/s13421-016-0614-y · Cited by 24 · see citing works
Lee, J., Chan, J. Y., Botelho, A. F., & Ottmar, E. (2022) · Educational Technology Research and Development · doi:10.1007/s11423-022-10138-4 · Cited by 27 · see citing works
“Students’ understanding of mathematical equivalence was measured with six items selected from two established measures (Rittle-Johnson et al., 2011; Star et al., 2015) at pretest and posttest.”
BackgroundMethodMEASURING CONCEPTUAL UNDERSTANDING: THE CASE OF FRACTIONS
Jones, I., Inglis, M., Gilmore, C. K., et al. (2013) · Loughborough University Institutional Repository (Loughborough University) · Cited by 20 · see citing works
Ralston, N. C., Li, M., & Taylor, C. L. (2018) · Educational Assessment · doi:10.1080/10627197.2018.1483191 · Cited by 17 · see citing works
The relationship between mental computation and relational thinking in the seventh grade
Kindrat, A. N., & Osana, H. P. (2018) · Fields Mathematics Education Journal · doi:10.1186/s40928-018-0011-4 · Cited by 17 · see citing works
Young Students’ Arithmetic-Algebraic Structure Sense: an Empirical Model and Profiles of Students
Pittalis, M. (2022) · International Journal of Science and Mathematics Education · doi:10.1007/s10763-022-10333-y · Cited by 16 · see citing works
“The four types of numerical equivalence tasks tried to capture a broad spectrum of numerical equivalence tasks that are typically used for young students (Rittle-Johnson et al., 2010; Subramaniam & Banerjee, 2011).”
MethodSchoen, R. C., Bray, W., Wolfe, C. B., Tazaz, A. M., & Nielsen, L. (2017) · The Elementary School Journal · doi:10.1086/692912 · Cited by 16 · see citing works
Elkjær, M., & Jankvist, U. T. (2021) · Mathematics · doi:10.3390/math9050557 · Cited by 15 · see citing works
Somasundram, P. (2021) · Eurasia Journal of Mathematics Science and Technology Education · doi:10.29333/ejmste/9612 · Cited by 14 · see citing works
“Algebraic thinking is the most prevalent form of formal algebra learning, and algebra accounts for most fields in mathematics (Blanton & Kaput, 2003; Jacobs et al., 2007; Rittle-Johnson et al., 2011).”
BackgroundŞimşek, E., Xenidou‐Dervou, I., Karadeniz, I., et al. (2019) · Journal of Numerical Cognition · doi:10.5964/jnc.v5i1.147 · Cited by 14 · see citing works
“Three items adapted from Rittle-Johnson et al. (2011) were designed to investigate students’ understanding of arithmetic equations.”
BackgroundMethodInfluentialDonovan, A. M., Stephens, A. C., Alapala, B., et al. (2022) · ZDM · doi:10.1007/s11858-022-01405-y · Cited by 13 · see citing works
“Many students instead hold an operational conception of the equal sign, interpreting the symbol as meaning the “the total” or “the answer” (e.g., Baroody & Ginsburg, 1983; Knuth et al., 2006; McNeil, 2008; McNeil & Alibali, 2005; Rittle-Johnson et al., 2011).”
BackgroundSelf-citationLee, J., Hornburg, C. B., Chan, J. Y., & Ottmar, E. (2022) · Journal of Numerical Cognition · doi:10.5964/jnc.8323 · Cited by 17 · see citing works
“Students’ prior algebra knowledge was measured with 11 items selected from two previously validated measures (Rittle-Johnson et al., 2011; Star et al., 2015).”
MethodChi, S., Wang, Z., & Liu, X. (2022) · Research in Science Education · doi:10.1007/s11165-022-10056-8 · Cited by 13 · see citing works
“Developing a Construct Map A construct map is a unidimensional latent variable (Wilson, 2005), depicting a hierarchical complexity of skills that respondents are expected to progress through for the target construct (Rittle-Johnson et al., 2011).”
BackgroundNabie, M. J., Akayuure, P., & Sofo, S. S. (2013) · Cited by 13 · see citing works
The Developmental Progression of Early Algebraic Thinking of Elementary School Students
Sun, S., Sun, D., & Xu, T. (2023) · Journal of Intelligence · doi:10.3390/jintelligence11120222 · Cited by 25 · see citing works
College developmental math students’ knowledge of the equal sign
Fyfe, E. R., Matthews, P. G., & Amsel, E. A. (2020) · Educational Studies in Mathematics · doi:10.1007/s10649-020-09947-2 · Cited by 12 · see citing works
Self-citationStudents’ opposing conceptions of equations with two equal signs
Lee, J., & Jeongsuk, P. (2020) · Mathematical Thinking and Learning · doi:10.1080/10986065.2020.1777364 · Cited by 12 · see citing works
“This section clarifies the diversity of students’ conceptions of the equal sign captured by five previous studies (Baroody & Ginsburg, 1983; Jones et al., 2013; Molina & Ambrose, 2008; Rittle-Johnson et al., 2011; Stephens et al., 2013).”
BackgroundBajwa, N. P., & Perry, M. (2019) · Mathematical Thinking and Learning · doi:10.1080/10986065.2020.1700587 · Cited by 12 · see citing works
Relational equity and mathematics learning: Mutual construction during collaborative problem solving
Gutiérrez, J. F., Brown, S. A., & Alibali, M. W. (2018) · Journal of Numerical Cognition · doi:10.5964/jnc.v4i1.91 · Cited by 11 · see citing works
Gade, S. (2012) · Educational Action Research · doi:10.1080/09650792.2012.727644 · Cited by 11 · see citing works
“Discussion 1: the equality sign, mathematics education research Much research in mathematics education with respect to the equality sign (Knuth et al. 2006; Jones and Pratt 2007; McNeil et al. 2006; Rittle-Johnson et al. 2011) refers to a very early study by Baroody and Ginsburg (1983).”
BackgroundBisson, M., Gilmore, C. K., Inglis, M., & Jones, I. (2019) · Research in Mathematics Education · doi:10.1080/14794802.2019.1692060 · Cited by 11 · see citing works
Stephens, A. C., Torres, R. V., Sung, Y., Strachota, S., Gardiner, A. M., BLANTON, M. L., Stroud, R., & Knuth, E. (2021) · The Journal of Mathematical Behavior · doi:10.1016/j.jmathb.2021.100871 · Cited by 11 · see citing works
Use of Photomath Applications in Helping Improving Students’ Mathematical (Algebra) Achievement
Zain, I. N. M., Setambah, M. A. B., Othman, M. S., et al. (2023) · European Journal of Education and Pedagogy · doi:10.24018/ejedu.2023.4.2.601 · Cited by 11 · see citing works
Effects of Symbolic and Nonsymbolic Equal-Sign Intervention in Second-Grade Classrooms
Chow, J. C., & Wehby, J. H. (2019) · The Elementary School Journal · doi:10.1086/703086 · Cited by 11 · see citing works
“Because this operational misunderstanding of the equal sign is difficult to overcome (Rittle-Johnson et al., 2011), only a portion (about 30”
BackgroundLee, J., Jindal, A., Patki, S. N., Gurung, A., Norum, R., & Ottmar, E. (2023) · Interactive Learning Environments · doi:10.1080/10494820.2023.2212726 · Cited by 14 · see citing works
“Both pretest and posttest math knowledge scores were measured using 11 items selected from two validated measures (Rittle-Johnson et al., 2011; Star et al., 2015).”
Using graphical representations to enhance the quality-of-care for colorectal cancer patients
Chien, T., Chang, Y.-J., Wen, K.-S., et al. (2016) · European Journal of Cancer Care · doi:10.1111/ecc.12591 · Cited by 10 · see citing works
The Importance of Spatial Reasoning in Early Childhood Mathematics
Rich, K., & Brendefur, J. L. (2019) · IntechOpen eBooks · doi:10.5772/intechopen.81564 · Cited by 12 · see citing works
“Developing and applying the knowledge of relational thinking to solve mathematical equivalence problems will increase early algebraic understanding [41, 44, 46, 53].”
BackgroundDifferences in grade 7 students’ understanding of the equal sign
Sumpter, L., & Löwenhielm, A. (2022) · Mathematical Thinking and Learning · doi:10.1080/10986065.2022.2058160 · Cited by 8 · see citing works
“Firstly, open and closed (multiple choice) questions were designed to measure students’ understanding of the equal sign on different levels as the examples in Rittle-Johnson et al. (2011) and Matthews et al. (2012).”
BackgroundMethodInfluentialViegut, A. A., Stephens, A. C., & Matthews, P. G. (2024) · Investigations in Mathematics Learning · doi:10.1080/19477503.2024.2307805 · Cited by 8 · see citing works
“Work like this has been done in other domains like equal sign knowledge (e.g., Rittle-Johnson et al., 2011; Fyfe et al., 2018), teacher’s mathematical knowledge for teaching (e.g., Hill et al., 2004; Hill et al., 2008), and different early mathematics learning trajectories (Clements, et al., 2008;…”
Self-citationHakkarainen, A. M., Cordier, R., Parsons, L., Yoon, S., Laine, A., Pirjo Aunio, A., & Speyer, R. (2023) · International Journal of Educational Research · doi:10.1016/j.ijer.2023.102172 · Cited by 8 · see citing works
Young students’ understanding of mathematical equivalence across different schools in South Africa
McAuliffe, S., Tambara, C. T., & Şimşek, E. (2020) · South African Journal of Childhood Education · doi:10.4102/sajce.v10i1.807 · Cited by 7 · see citing works
“Other studies showed similar results in the use of shortcuts to solve arithmetic questions which were low in school students (Fyfe et al. 2018; Rittle-Johnson et al. 2011).”
MethodResultA Validity Argument for a Brief Assessment of Mature Number Sense
Kirkland, P. K., Cheng, Y., & McNeil, N. M. (2024) · Journal for Research in Mathematics Education · doi:10.5951/jresematheduc-2022-0071 · Cited by 7 · see citing works
Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions
DeWolf, M., Son, J. Y., Bassok, M., et al. (2017) · Cognitive Science · doi:10.1111/cogs.12468 · Cited by 7 · see citing works
Somasundram, P., Zamri, S. N. A. S., & Leong, K. E. (2019) · The New Educational Review · doi:10.15804/tner.2019.55.1.08 · Cited by 7 · see citing works
“Elementary school children often perceive the equal sign operationally rather relationally (Alibali, Knuth, Hattikudur, McNeil, & Stephens, 2007; Rittle-Johnson et al., 2011).”
BackgroundMuñoz-Catalán, M. C., García, M. R., Prieto, N. J., et al. (2021) · Journal for the Study of Education and Development Infancia y Aprendizaje · doi:10.1080/02103702.2021.1946640 · Cited by 7 · see citing works
“This allows the initial introduction of the students to Level 4, the relational comparative level, according to Rittle-Johnson et al. (2011) (form of epistemological genesis, KPM).”
BackgroundInfluentialRingo, S. S., Samsudin, A., & Ramalis, T. R. (2020) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1467/1/012054 · Cited by 7 · see citing works
Children’s confidence using incorrect strategies on mathematical equivalence problems
Grenell, A., Nelson, L. J., Gardner, B., & Fyfe, E. R. (2022) · Cognitive Development · doi:10.1016/j.cogdev.2022.101167 · Cited by 6 · see citing works
How physics textbooks embed meaning in the equals sign
Alaee, D. Z., Sayre, E. C., Kornick, K., & Franklin, S. (2022) · American Journal of Physics · doi:10.1119/10.0009096 · Cited by 6 · see citing works
Mathematical equivalence assessment: Measurement invariance across six countries
Şimşek, E., Jones, I., Hunter, J., & Xenidou‐Dervou, I. (2021) · Studies In Educational Evaluation · doi:10.1016/j.stueduc.2021.101046 · Cited by 6 · see citing works
Progressions in young learners’ understandings of parity arguments
BLANTON, M. L., Gardiner, A. M., Ristroph, I., Stephens, A. C., Knuth, E., & Stroud, R. (2022) · Mathematical Thinking and Learning · doi:10.1080/10986065.2022.2053775 · Cited by 6 · see citing works
“…abundance of studies have emerged concerning levels in students’ relational thinking of the equal sign, so has our understanding of the types of tasks and productive ways to sequence tasks that align with progressions in students’ thinking (e.g., Matthews et al., 2012; Rittle-Johnson et al., 2011).”
BackgroundMachaba, M. F. (2017) · Eurasia Journal of Mathematics Science and Technology Education · doi:10.12973/ejmste/78017 · Cited by 6 · see citing works
Primary School Students’ Knowledge of the Equal Sign—the Swedish Case
Madej, L. (2021) · International Journal of Science and Mathematics Education · doi:10.1007/s10763-020-10144-z · Cited by 6 · see citing works
A neural network model of learning mathematical equivalence.
Mickey, K. W., & McClelland, J. L. (2014) · eScholarship (California Digital Library) · Cited by 6 · see citing works
Iannacchione, A., Ottmar, E., Ngo, V., Mason, C. A., Chan, J. Y., Smith, H., Drzewiecki, K. C., & Shaw, S. T. (2022) · Instructional Science · doi:10.1007/s11251-022-09604-6 · Cited by 7 · see citing works
“Students’ equivalence understanding was measured with six items selected from a previously validated measure of algebra knowledge (Rittle-Johnson et al., 2011; Star et al., 2015).”
MethodWilkie, K. J., & Hopkins, S. (2024) · The Journal of Mathematical Behavior · doi:10.1016/j.jmathb.2023.101121 · Cited by 5 · see citing works
Algebraic Content and Pedagogical Knowledge of Sixth Grade Mathematics Teachers
Naseer, M. S. (2016) · ProQuest LLC eBooks · Cited by 5 · see citing works
Enhancing Conceptual Knowledge in Early Algebra Through Scaffolding Diagrammatic Self-explanation
Nagashima, T., Bartel, A. N., Silla, E., Vest, N. A., Alibali, M. W., & Aleven, V. (2020) · doi:10.22318/icls2020.35 · Cited by 5 · see citing works
Student Conceptions of the Equal Sign
Ralston, N. C., & Li, M. (2022) · The Elementary School Journal · doi:10.1086/717999 · Cited by 4 · see citing works
“These items were all devised based on the literature review and were similar in nature to items described in the literature above (i.e., Carpenter et al., 2005; Jacobs et al., 2007; Mann, 2004; McNeil, 2007; McNeil & Alibali, 2005; Rittle-Johnson & Alibali, 1999; Rittle-Johnson et al., 2011).”
MethodŞimşek, E., Xenidou‐Dervou, I., Hunter, J., Dowens, M. G., Jeongsuk, P., Lee, Y., McNeil, N. M., Kirkland, P. K., & Jones, I. (2022) · Journal of Educational Psychology · doi:10.1037/edu0000747 · Cited by 4 · see citing works
Equational reasoning: A systematic review of the Cuisenaire–Gattegno approach
Benson, I., Marriott, N., & McCandliss, B. D. (2022) · Frontiers in Education · doi:10.3389/feduc.2022.902899 · Cited by 4 · see citing works
“Other researchers, working from first principles, have independently discovered many of Gattegno’s findings especially those relating to the central importance of early algebra, pattern making, and mathematical equivalence (Davydov, 1962; Kaput, 1995a,b; Healy et al., 2002; Schmittau and Morris, 2004; Carraher et al.,…”
BackgroundThe equals sign: The problem of early algebra learning and how to solve it
Milinković, N. S., Maričić, S., & Đokić, O. (2022) · Inovacije u nastavi · doi:10.5937/inovacije2203026m · Cited by 4 · see citing works
Maguire, T. J. (2012) · USF Scholarship Repository (University of San Francisco) · Cited by 4 · see citing works
Enhancing and Enacting Curricular Progressions in Elementary Mathematics
Land, T. J., & Drake, C. (2014) · Mathematical Thinking and Learning · doi:10.1080/10986065.2014.889502 · Cited by 4 · see citing works
Assessing children’s capacity for procedural flexibility with mental addition
Hopkins, S., Russo, J., & Downton, A. (2025) · Mathematics Education Research Journal · doi:10.1007/s13394-024-00513-0 · Cited by 3 · see citing works
“Moreover, number-based procedures rely on, and reinforce, the concept of mathematical equivalence : that is, knowing that numbers (measurements or expressions) can be represented in a number of ways that are equal and interchangeable (Kieran, 1981; McNeil et al., 2015; Rittle-Johnson et al., 2011).”
What is so Complicated in Developing Students’ Conception of the Equal Sign?
Lee, J., & Jeongsuk, P. (2022) · International Journal of Science and Mathematics Education · doi:10.1007/s10763-022-10248-8 · Cited by 3 · see citing works
“Lee & Pang (2021) categorized six types of students’ conceptions of the equal sign by synthesizing five representative studies dealing with students’ various concep‐ tions (i.e., Baroody & Ginsburg, 1983; Jones et al., 2013; Molina & Ambrose, 2008; Rittle‐Johnson et al., 2011; Stephens et al., 2013): rigid operational…”
BackgroundXu, C., Li, H., Burr, S. D. L., et al. (2023) · Journal of Cognition and Development · doi:10.1080/15248372.2023.2245507 · Cited by 3 · see citing works
Developing the concept of task substitution and transformation by defining own equivalences
Tirpáková, A., Gonda, D., Wiegerová, A., & Navrátilová, H. (2023) · Educational Studies in Mathematics · doi:10.1007/s10649-023-10264-7 · Cited by 3 · see citing works
“However, a lot of students hold an operational conception of equal sign, interpreting the sign as “total” or “answer” (e.g., Knuth et al., 2006; McNeil, 2008; Rittle-Johnson et al., 2011).”
BackgroundComputing in Differential Equations with Mathematical Thinking Approach among Engineering Students
Ismail, Z. B., Zeynivandnezhad, F., Mohammad, Y. B., et al. (2014) · doi:10.1109/latice.2014.39 · Cited by 3 · see citing works
“For example, explanations as to why procedures work or drawing a concept map [6].”
Background
- Authors
- Rittle-Johnson, B., Matthews, P. G., Taylor, R. S., & McEldoon, K. L.
- Affiliations
- Vanderbilt University (US)
- State University of New York (US)
- Venue
- Journal of Educational Psychology
- Published
- 2011-02
- Volume
- 103
- Issue
- 1
- Pages
- 85-104
- ISSN
- 0022-0663 (ISSN-L)
- 1939-2176
- OpenAlex
- W2099233120
- Cited by
- 160
- References
- 73
- Citations by year
- 2026: 4 · 2025: 9 · 2024: 11 · 2023: 13 · 2022: 21 · 2021: 15 · 2020: 7 · 2019: 10 · 2018: 13 · 2017: 8 · 2016: 8 · 2015: 4 · 2014: 15 · 2013: 9 · 2012: 13
- FWCI
- 33.24
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of Educational Psychology (publishedVersion)
- Funders
- National Science Foundation (DRL-0746565)
- US Department of Education (R305B040110; R305B080008)
- Topics
- Social Sciences › Education › Mathematics Education and Teaching Techniques
- Physical Sciences › Mathematics › Statistics and Probability › Cognitive and developmental aspects of mathematical skills
- Social Sciences › Decision Sciences › Management Science and Operations Research › Psychometric Methodologies and Testing
- Keywords
- mathematical equivalence, equation solving, assessment development, construct maps, algebraic thinking, elementary school students, junior high school students, reliability and validity, item difficulty
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