From “You have to have three numbers and a plus sign” to “It’s the exact same thing”: K–1 students learn to think relationally about equations

Stephens, Ana C.; Torres, Ranza Veltri; Sung, Yewon; Strachota, Susanne; Gardiner, Angela Murphy; Blanton, Maria; Stroud, Rena; Knuth, Eric

doi:10.1016/j.jmathb.2021.100871

Cited by 7 publications

(5 citation statements)

References 27 publications

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“…The concept of mathematical equivalence, specifically the symbolic representation of equivalence relation using sign "=", is the accepted basis of algebraic thinking (Carpenter et al, 2003;National Governors & Stephens et al, 2021). Equivalence expressed by sign "=" is a relationship between two mathematical expressions, on the basis of which we can claim that the expressions have the same value, or that the expressions represent the same mathematical object (Kieran & Martínez-Hernández, 2022).…”

Section: Theoretical Basismentioning

confidence: 99%

Developing the concept of task substitution and transformation by defining own equivalences

Tirpáková,

Gonda,

Wiegerová

et al. 2023

Educ Stud Math

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The presented article is dedicated to a new way of teaching substitution in algebra. In order to effectively master the subject matter, it is necessary for students to perceive the equal sign equivalently, to learn to manipulate expressions as objects, and to perceive and use transformations based on defining their own equivalences. According to the results of several researches, these changes do not occur automatically, and the neglect of their development leads to students’ insufficient adoption of substitution. The submitted contribution presents a new way of teaching substitution, the stages of which support the gradual development of the necessary competences of students, so that substitution becomes part of their computing apparatus. The effectiveness of the mentioned method of teaching substitution was also verified experimentally. By conducting a pedagogical experiment, it was confirmed that the application of the substitution teaching method developed by us led to more frequent use of substitution by students from the experimental group (47 students) compared to students from the control group (82 students) who learned substitution in the usual way. It emerged from the interview with experimental group students that they considered the proposed method suitable and that it encouraged them to learn substitution in depth.

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Section: Theoretical Basismentioning

confidence: 99%

Developing the concept of task substitution and transformation by defining own equivalences

Tirpáková,

Gonda,

Wiegerová

et al. 2023

Educ Stud Math

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“…Students who hold an operational view tend to solve missing value equations by performing the given operations to the left of the equal sign or by performing all the given operations (e.g., for 8 + 4 = __ + 5, writing 12 or 17 in the blank; Carpenter et al, 2003 ). They also may refuse to endorse nonstandard equations, for example, rejecting 8 = 8 because it lacks an operation, or rejecting 9 = 5 + 4 because it is “backwards” (Falkner et al, 1999 ; Stephens et al, 2021 , Stephens et al, 2022 ).…”

Section: Students’ Conceptions Of Mathematical Equivalencementioning

confidence: 99%

“…The concept of mathematical equivalence—specifically, the use of the equal sign to symbolically represent an equivalence relation—is widely accepted as foundational to algebraic thinking (Baroody & Ginsburg, 1983 ; Carpenter et al, 2003 ; NCTM, 2000 ; NGA & CCSSO, 2010 ; Stephens et al, 2021 ). Deep understanding of the equal sign is connected to early algebraic competence in the elementary grades (e.g., Byrd et al, 2015 ; Carpenter et al, 2003 ; Hornburg et al, 2022 ; Matthews & Fuchs, 2020 ) and to equation-solving success in the middle grades (Fyfe et al, 2018 ; Knuth et al, 2006 ).…”

Section: Introductionmentioning

confidence: 99%

“…Deep understanding of the equal sign is connected to early algebraic competence in the elementary grades (e.g., Byrd et al, 2015 ; Carpenter et al, 2003 ; Hornburg et al, 2022 ; Matthews & Fuchs, 2020 ) and to equation-solving success in the middle grades (Fyfe et al, 2018 ; Knuth et al, 2006 ). Further, when students are provided with problem-solving experiences that highlight the equal sign's meaning, they grow in their understanding (e.g., Blanton et al, 2019 ; Stephens et al, 2021 ).…”

Section: Introductionmentioning

confidence: 99%

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Is a substitute the same? Learning from lessons centering different relational conceptions of the equal sign

Donovan

Stephens

Alapala

et al. 2022

ZDM Mathematics Education

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Understanding of the equal sign is associated with early algebraic competence in the elementary grades and equation-solving success in middle school. Thus, it is important to find ways to build foundational understanding of the equal sign as a relational symbol. Past work promoted a conception of the equal sign as meaning “the same as”. However, recent work highlights another dimension of relational understanding—a substitutive conception, which emphasizes the idea that an expression can be substituted for another equivalent one. This work suggests a substitutive conception may support algebra performance above and beyond a sameness conception alone. In this paper, we share a subset of results from an online intervention designed to foster a relational understanding of the equal sign among fourth and fifth graders ( n = 146). We compare lessons focused on a sameness conception alone and a dual sameness and substitutive conception to each other, and we compare both to a control condition. The lessons influenced students’ likelihood of producing and endorsing sameness and substitutive definitions of the equal sign. However, the impact of the lessons on students’ approaches to missing value equations was less clear. We discuss possible interpretations, and we argue that further research is needed to explore the roles of sameness and substitutive views of the equal sign in supporting structural approaches to algebraic equation solving.

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“…When the relevant literature is examined, studies show that students involved in an early algebra intervention can develop algebraic thinking starting from kindergarten (e.g., Blanton et al, 2015, Carraher et al, 2008Stephens, Veltri Torres, et al, 2021). For example, Blanton et al (2015)'s comprehensive early algebra intervention results with 3rd-grade students showed that although no significant difference was observed between the intervention and nonintervention groups in the pre-test, the intervention students in the post-test were significantly more successful than the nonintervention students.…”

Section: Introductionmentioning

confidence: 99%

The Effects of an Early Algebra Intervention on Third-Grade Students’ Algebraic Thinking Skills

İŞLER BAYKAL,

ÖZTÜRK TAVŞAN,

GÜZELLER

et al. 2023

Cukurova University Faculty of Education Journal

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The importance of early algebra has been emphasized in international literature, and it has been discussed in many studies that students who are involved in a comprehensive intervention improve their algebraic thinking skills. The aim of this study was to test the effects of an early algebra intervention on the algebraic thinking skills of 3rd-grade students. The 3rd-grade intervention and control groups were included in the study, and both groups were given pre-test and post-test. In the findings, although there was no statistical difference between the 3rd-grade intervention and control group performances in the pre-test, a statistically significant difference was found in the post-test. Analysis of the students' strategies revealed that the students in the intervention group used algebraic thinking strategies more in the post-test than the control group in three big ideas, which are equality, expressions, equations and inequalities, generalized arithmetic, and functional thinking.

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From “You have to have three numbers and a plus sign” to “It’s the exact same thing”: K–1 students learn to think relationally about equations

Cited by 7 publications

References 27 publications

Developing the concept of task substitution and transformation by defining own equivalences

Developing the concept of task substitution and transformation by defining own equivalences

Is a substitute the same? Learning from lessons centering different relational conceptions of the equal sign

The Effects of an Early Algebra Intervention on Third-Grade Students’ Algebraic Thinking Skills

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