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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]

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]