In order to overcome this disconnection even before variables are introduced, a design research study was conducted in Grade 5 to design and investigate an early algebra learning environment to establish stronger connections between different mental models and representations of equivalence of expressions. A good algebraic understanding involves connecting the transformation rules to other characterizations of equivalence of expressions (e.g., description equivalence that both expressions describe the same situation or figure). One typical challenge in algebra education is that many students justify the equivalence of expressions only by referring to transformation rules that they perceive as arbitrary without being able to justify these rules. Theoretical and instructional implications are also discussed. This study contributes a breadth of pre-algebraic strategies that students used to solve word problems that can be modeled by systems of linear equations in relation to their numerical reasoning. Additionally, students with less sophisticated numerical reasoning commonly used pre-algebraic strategies, whereas students with more sophisticated numerical reasoning more commonly utilized algebraic reasoning strategies. Results show that students’ stages of numerical reasoning were more closely related to their problem-solving strategies than their mathematics instruction. Students’ numerical reasoning was assessed using the framework, number sequences, and their problem-solving was observed on two word problems that could be modeled using systems of linear equations. 18 students in grades 6 through 9 participated in task-based semi-structured clinical interviews.
This qualitative research study investigates the ways middle-grades students’ numerical reasoning relates to their solutions to word problems that can be modeled by systems of linear equations. (2019) listed three typical errors that students with the operational conception make: (a) their answers to an equivalence problem (e.g., a þ b = _ þ d) are often the sum of the numbers on the left side (a þ b) or the sum of all numbers in the equation (a þ b þ d), (b) they have difficulty accepting a = a is a (correct) mathematical equation and (c) they struggle to understand the a = b þ c form and tend to transform it to the operations on the left (a þ b = c) format. McNeil, Hornburg, Brletic-Shipley, et al. Much research has shown that students' conceptions of the equals sign relate to their equation-solving performance and that the operational conception of the equals sign is detrimental to equation-solving performance (Byrd et al., 2015 Knuth et al., 2006 Matthews et al., 2012. It is well documented (e.g., Kieran, 1981 Knuth et al., 2006 Molina & Ambrose, 2008) that many students hold the operational conception, interpreting the equals sign as an operational symbol that needs to be followed by an answer and/or as a "do something" signal. 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. However, the impact of the lessons on students' approaches to missing value equations was less clear. The lessons influenced students' likelihood of producing and endorsing sameness and substitutive definitions of the equal sign. 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. 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). This work suggests a substitutive conception may support algebra performance above and beyond a sameness conception alone. 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. Past work promoted a conception of the equal sign as meaning "the same as". Thus, it is important to find ways to build foundational understanding of the equal sign as a relational symbol. Understanding of the equal sign is associated with early algebraic competence in the elementary grades and equation-solving success in middle school.
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