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This paper studies grade 7 (age 13) students' expressed understanding about the equal sign/notion of equivalence in order to investigate what aspects of the concept that could be seen as an established knowledge at lower secondary school/middle school. Using items from different instruments and combining these to a new one that covered a broad spectrum of procedural and conceptual knowledge, we collected data from 159 students. The different statistical tests showed that if focusing only on separate items, it could confirm that students could be seen either having operational or relational understanding of the equal sign. However, when taking all results into account using several analyses, students' understanding appear to be much more complex. Instead of a dichotomized view, students' expressed knowledge of mathematical equivalence should be seen as a continuum. [ABSTRACT FROM AUTHOR]

Research on mathematical equivalence, and to some extent, on number structure in early grade mathematics, has shown that most students at primary and secondary levels understand these concepts instrumentally rather than relationally. The study reported in this paper uses a design research approach to explore both mathematical equivalence and number structure in a primary pre-service teacher education programme at a university in South Africa and reports on an intervention strategy intended to culminate in developing design principles comprising guidelines for the effective teaching of number structure and mathematics equivalence in primary pre-service teacher education programmes. The mixed method study comprises 70 first-year primary pre-service teachers (PSTs) and a teacher educator. The PSTs wrote pre- and post-tests, and were interviewed following the pre-test. Skemp's notion of relational and instrumental understanding guided the design of the intervention and the analysis of the data. The results from the study indicates that PSTs' understanding of the notion of equivalence is similar to what has been reported for students in schools, and that there were some learning gains in the post-test results. Based on the findings from the study in general, some emergent design principles are proposed. [ABSTRACT FROM AUTHOR]

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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. [ABSTRACT FROM AUTHOR]

Ensuring students correctly understand the notion of equality is a fundamental problem in teaching mathematics. Is it possible to teach arithmetic in such a way that students do not misinterpret the equal sign "=" as indicating the result of an arithmetic operation and, consequently, viewing the arithmetic operation symbols (+, -, or x) as systematically meaning perform a computation? The present paper describes the implementation of an experimental arithmetic teaching program (called ACE) drawn up in conjunction with teachers and focusing on these issues. We assessed the program's impact via an experiment involving 1140 experimental group students and 1155 control group students, and using a pretest/posttest design. The experimental group students achieved higher composite arithmetic scores, combining all four subdomains (arithmetic writing, mental computation, word-problem solving, and estimation), than control group students. The effect size, computed using individuals' posttest scores adjusted for pretest scores was d = 0.559. The ACE program was effective in all four subdomains tested; however, it was particularly successful in the arithmetic writing subdomain, especially in writing equalities. The program's effect not only held one year later, it was cumulative, as the benefits produced by following the program in both first and second grade were almost double those of following the program in just one grade. [ABSTRACT FROM AUTHOR]

It is important for students to develop a relational understanding of the equal sign, but students often have simultaneous operational and relational conceptions (i.e., SOR conceptions). This case study carefully explored how a student's conception of the equal sign changed during a classroom teaching experiment and analyzed the possibilities and difficulties in developing the student's conception of the equal sign. At first, the student demonstrated her understanding of only those equations in the form a ± b = c by using the operational conception of the equal sign. Given an unfamiliar equation in the form a ± b = c ± d = e, the student revealed the relational conception but still evaluated an incorrect equation as correct using the operational conception. Her SOR conception did not change easily despite considerable classroom discussion. When the teacher explained the meaning of the equal sign and connected it with a pan balance, the student understood the meaning of quantitative sameness of the equal sign and showed a stable relational conception. Later, in equations of the form a ± b = c ± d, the student's conception progressed to the comparative relational conception. Given the progression, this study focused on the noticeable shifts in the student's conceptions of the equal sign, and this article discusses both theoretical and pedagogical implications regarding students' conceptions of the equal sign. [ABSTRACT FROM AUTHOR]

The purpose of this study was to investigate how the 8th graders interpreted the equal sign in the pan balance activities they created on Scratch. Five eighth graders (4 girls and 1 boy) in Turkey participated in this study. Case study was employed as a qualitative research method. Data was gathered through the Scratch files created by the students and the think-aloud protocol interviews with them. The questions used in the activities were prepared in accordance with the test created by Matthews et al. (2012). For data analysis, content and descriptive analysis methods were employed. Findings showed that participants interpreted the operational questions as operational and relational questions as relational through using Scratch. Another significant finding emerged in Scratch activities was the fact that in order for the code block to provide an equation, the pan balance was supposed to balance. Employing Scratch in the class teachings may provide variety in learning environments for teaching of equal sign. [ABSTRACT FROM AUTHOR]

This study aims to determine how prospective middle school mathematics teachers respond to students' errors in the questions about the equal sign. This study utilizes case study method. In this case study, hypothetical scenarios, involving three common error types related to the equal sign, have been prepared by using the possible examples of student work. Through these scenarios, one-to-one interviews were conducted with seven prospective middle school mathematics teachers. In line with the data obtained in these interviews, it was seen that the prospective teachers used seven different ways to respond to students' errors related to the equal sign: showing the error, showing the right solution, guiding to find the right answer, guiding to find the error, re-explaining the concept, in-depth research, and false intervention. In addition, it was determined that two prospective teachers intervened incorrectly by taking an approach that could support the thought that led to the error. In the light of the findings, it was seen that the prospective teachers had a limited understanding of the equal sign. This study suggests that mathematics educators should create appropriate learning opportunities to improve prospective teachers' understanding of the equal sign and their ability to respond to students' errors. [ABSTRACT FROM AUTHOR]

This research focuses on ways in which balance scales mediate students' relational understandings of the equal sign. Participants included 21 Kindergarten–Grade 2 students who took part in an early algebra classroom intervention focused in part on developing a relational understanding of the equal sign through the use of balance scales. Students participated in pre-, mid- and post-intervention interviews in which they were asked to evaluate true-false equations and solve open number sentences. Students often worked with balance scales while solving these tasks. Interview analyses revealed several categories of affordances of these tools for supporting students' productive thinking about equations. [ABSTRACT FROM AUTHOR]

Despite many studies on students' understanding of the equal sign, little attention has been paid to whole-class discourse on their conceptions of equations with two equal signs. This study analyzed the classroom discourse of 30 fourth graders in Korea who participated in a 5-week classroom teaching experiment. A series of discussions on specific equations with two equal signs showed that the students had subtle but significantly different conceptions of the equal sign, even though they often shared the common approach of saying "is the same as" when referring to the equal sign – these findings differ from previous studies. Classroom discourse exposed the simultaneous coexistence of various conceptions in a single student's mind, including both operational and relational conceptions of the equal sign. The study's findings argue for and demonstrate the value of learning opportunities to explore equations with two equal signs in a systematic and explicit way. [ABSTRACT FROM AUTHOR]

Elementary school students struggle in interpreting the equal sign as a symbol denoting equivalence. Although many have advocated using a pan-balance scale to help students develop this understanding, less is known about what features associated with this model support learning. To attempt to control and examine these features, the investigators developed a digital, pan-balance computer applet. This allowed for experimentally manipulating three relatively grounded instructional conditions (involving the core principle of making two sides the same; a balancing analogy; or both, along with a dynamic demonstration), compared to a relatively idealized control condition. Results indicated that the relatively more grounded conditions promoted a relational understanding of mathematical equivalence among 148 second- and third-grade students and further suggest that providing dynamic, grounded support may not be as optimal as less-enriched supports to promote students' learning. [ABSTRACT FROM AUTHOR]

Decades of research have documented young students' misinterpretations of the equal sign and the impediments these present for children's mathematical development. Much less is known about individual differences in adults' knowledge of the equal sign. We assessed 182 college students from developmental math courses and present analyses from a subset of items on a previously validated assessment of mathematical equality. College students exhibited variability in their interpretations of the equal sign, and these individual differences were related both to their solutions for open equations with variables and to their interpretations of an algebraic expression. Students who provided a relational interpretation of the equal sign were most successful on target algebra problems and students who only provided an operational interpretation of the equal sign were least successful. These results show that misinterpretations of the equal sign can persist well beyond elementary school and can provide a critical indicator of broader mathematical reasoning among college students. [ABSTRACT FROM AUTHOR]

Students in the elementary grades often experience difficulty setting up and solving word problems. Using an equation to represent the structure of the problem serves as an effective tool for solving word problems, but students may require specific pre-algebraic reasoning instruction about the equal sign as a relational symbol to set up and solve such equations successfully. We identified students with mathematics difficulty (n = 138) from a sample of 916 third-grade students. We randomly assigned students to a word-problem intervention with a pre-algebraic reasoning component, a word-problem intervention without pre-algebraic reasoning, or the business-as-usual. Students in the 2 active intervention conditions participated in 45 individual sessions and learned about 3 additive word-problem schemas. Students who received word-problem intervention with a pre-algebraic reasoning component demonstrated improved nonstandard equation solving, equal sign understanding, and word-problem solving compared to students in the other two conditions. [ABSTRACT FROM AUTHOR]

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation solving and equal-sign understanding of students with mathematics difficulty (MD). In the present study, second-grade students with MD (n = 51) were randomly assigned to standard equations tutoring, combined tutoring (standard and nonstandard equations), and no-tutoring control. Combined tutoring students demonstrated greater gains on equation-solving assessments and equal-sign tasks compared to the other two conditions. Standard tutoring students demonstrated improved skill on equation solving over control students, but combined tutoring students’ performance gains were significantly larger. Results indicate that exposure to and practice with nonstandard equations positively influence student understanding of the equal sign. [ABSTRACT FROM AUTHOR]