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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]

In this paper, it is the first time ever to suggest that we study the model theory of all finite structures and to put the equal sign in the same situtation as the other relations. Using formulas of infinite lengths we obtain new theorems for the preservation of model extensions, submodels, model homomorphisms and inverse homomorphisms. These kinds of theorems were discussed in Chang and Keisler's Model Theory, systematically for general models, but Gurevich obtained some different theorems in this direction for finite models. In our paper the old theorems manage to survive in the finite model theory. There are some differences between into homomorphisms and onto homomorphisms in preservation theorems too. We also study reduced models and minimum models. The characterization sentence of a model is given, which derives a general result for any theory T to be equivalent to a set of existential-universal sentences. Some results about completeness and model completeness are also given. [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]

Classroom communication has been recognized as a process in which ideas become objects of reflection, discussion, and amendments affording the construction of private mathematical meanings that in the process become public and exposed to justification and validation. This paper describes an explanatory model named “interpreting games”, based on the semiotics of Charles Sanders Peirce, that accounts for the interdependence between thought and communication and the interpretation of signs in which teacher and students engage in mathematics classrooms. Interpreting games account both for the process of transformation (in the mind of the learner) of written marks into mathematical signs that stand for mathematical concepts and for the continuous and converging private construction of mathematical concepts. Teacher–student and student–student collaborative interactions establish a mathematical communication that shapes and is also shaped by the conceptual domains and the domains of intentions and interpretations of the participants. A teaching episode with third graders is analyzed as an example of a classroom interpreting game. [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]

This case study investigates, at three Swedish schools, primary school students' knowledge of the equal sign. The schools were chosen as representatives of schools whose students have different socioeconomic backgrounds. The data consist of Grade 3 and 6 students' responses to an assessment form based on Matthews et al. (JRME, 43(3):316–350, 2012). A comparison between the schools indicates that there are great differences in the students' knowledge of the equal sign depending on which school they attend. The Swedish students perform better than South Korean students responding to the same assessment form when it comes to knowing that the equal sign does not mean an operation ("the answer to a problem"). We find this interesting to discuss, as South Korea is one of the countries that performs the best on the TIMSS and PISA. [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]

What kind of mathematical knowledge do prospective teachers need for teaching and for understanding student thinking? And how can its construction be enhanced? This article contributes to the ongoing discussion on mathematics-for-teaching by investigating the case of understanding students’ perspectives on equations and equalities and on meanings of the equal sign. It is shown that diagnostic competence comprises didactically sensitive mathematical knowledge, especially about different meanings of mathematical objects. The theoretical claims are substantiated by a report on a teacher education course, which draws on the analysis of student thinking as an opportunity to construct didactically sensitive mathematical knowledge for teaching for pre-service middle-school mathematics teachers. [ABSTRACT FROM AUTHOR]

This study examined awareness of equivalence and relational thinking exhibited by 30 preservice elementary teachers in order to assess their initial preparedness to engage students in these two important aspects of early algebraic reasoning. Findings indicated that preservice teachers collectively demonstrated an awareness of relational thinking both in identifying opportunities offered by tasks to engage students in this thinking and in identifying this thinking in samples of student work. However, in proposing difficulties students might have with selected tasks, few participants demonstrated the understanding that many elementary school students hold misconceptions about the meaning of the equal sign. Implications of these findings for preservice and inservice teacher education are discussed. [ABSTRACT FROM AUTHOR]

Relational thinking involves understanding equivalence and numerical relationships. The present study examined the relational thinking of seventh graders before and after a 15-day mental mathematics intervention in the context of whole number arithmetic. Using two intact seventh-grade classes and a staggered treatment design, students were assessed at three time points on their (a) ability to solve equivalence problems, and (b) reasoning abilities about true-false number sentences. The results indicated that the students in the class that received the intervention first (the Intervention First group) improved their performance on both measures after the intervention, and a similar pattern was found for the second class (the Intervention Second group), indicating that each group improved immediately following the mental mathematics intervention. Students in the Intervention First group were able to maintain their scores on the test of equivalence problems 4 weeks after the conclusion of the intervention. As the results suggest a link between mental mathematics and relational thinking, it is recommended that mental mathematics play a more prominent role in the seventh-grade mathematics classroom. [ABSTRACT FROM AUTHOR]