Evolution of a teaching approach for beginning algebra

Abstract

The article reports aspects of the evolution of a teaching approach over repeated trials for beginning symbolic algebra. The teaching approach emphasized the structural similarity between arithmetic and algebraic expressions and aimed at supporting students in making a transition from arithmetic to beginning algebra. The study was conducted with grade 6 students over 2 years. Thirty-one students were followed for a year, and data were analysed as they participated in the three trials conducted that year. Analysis of students' written and interview responses as the approach evolved revealed the potential of the approach in creating meaning for symbolic transformations in the context of both arithmetic and algebra as well as making connections between arithmetic and symbolic algebra. Students by the end of the trials learnt to use their understanding of both procedures and a sense of structure of expressions to evaluate/simplify expressions and reason about equality/ equivalence of expressions both in the arithmetic and the algebraic contexts.

Examining Two Middle School Mathematics Teachers’ Knowledge for Teaching Manipulation of Algebraic Expressions during Lesson Planning and Instruction

“…On the other hand, Eda"s knowledge to choose which examples to start teaching addition and subtraction appeared efficiently by connecting the properties of arithmetical operations that the students had learned before. Eda started by connecting the properties of arithmetical operations, which was one of the commonly suggested approaches for connecting arithmetic with algebra (Banerjee & Subramaniam, 2012;Livneh & Linchevski, 2007;Subramaniam & Banerjee, 2004). Molina et al (2017) also pointed out that the lack of understanding of the relationships in arithmetic expressions can cause errors in structuring algebraic expressions.…”

“…In order to build the students" thinking ability, teachers can present the procedures of operations with algebraic expressions by connecting the arithmetic procedures based on the similarity of the structure of expressions. This improves the students" understanding of rules and procedures in operations, simplifications of algebraic expressions, and the equivalence of the expressions (Banerjee & Subramaniam, 2012). Besides, teachers should use strategies and tasks that would attract students' interest in combining pedagogical elements of social constructivism in teaching algebra (Prendergast & O"Donoghue, 2014).…”

“…It is important to support students" ability to think algebraically in the elementary grades in order to help middle and high school students from not experiencing difficulties in algebra (Cai, Ng, & Moyer, 2011). More specifically, the studies about learning algebra in the early grades show that students generally have difficulty in manipulating algebraic expressions (e.g., Banerjee & Subramaniam, 2012;Booth, 1988;Gunnarsson, Sönnerhed, & Hernell, 2016;Livneh & Linchevski, 2007;MacGregor & Stacey, 1997;Seng, 2010). Gallardo (2000) asserts that there is a didactic cut in the child"s thought in the transition from arithmetic to algebra and that students have a resistance to operating with unknowns in the transition to algebraic thinking.…”

Recombinant Enaction: Manipulatives Generate New Procedures in the Imagination, by Extending and Recombining Action Spaces

“…that are used to test the understanding of area. Restructuring is used widely by experts (Cross & Cross, 1998;Kothiyal, Murthy, & Chandrasekharan, 2016;Wankat & Oreovicz, 2015), and restructuring based on perception and simulation is a key element of solving equations, and possibly symbolic operations in general (Banerjee & Subramaniam, 2012;Landy & Goldstone, 2009;Landy et al, 2014). When systematically used across many situations, manipulatives may help in developing general restructuring capabilities, which are applicable across many problems.…”

The Literature Review of Algebra Learning: Focusing on the Contributions to Students’ Difficulties

“…As Sadovsky and Sessa (2005: p. 90) pointed out, "the 'object' of arithmetic in primary school is numbers, whereas elementary algebra focuses on relationships between quantities". It is also shown that students' prior exposure in computing binary operations does not prepare them very well to handle algebra (Banerjee & Subramaniam, 2012). For instance, students often apply procedures that have been employed in arithmetic context to simplify algebraic expressions and make the similar mistakes (Fischbein & Barash, 1993).…”