Teaching 3x

Year: 1997

Author: Cooper, Tom, Williams, Anne

Type of paper: Abstract refereed

A common misunderstanding in early algebra is related to binary algebraic expressions such as 3x. This simple algebraic expression represents the generalisation that any number has been multiplied by 3. However, it is syntactically similar to the arithmetic notation for two digit numerals (e.g., 32) and different from the arithmetic notation for a particular number multiplied by 3 (e.g., 3 x 2). Thus, many students believe that 3x represents a 3 beside a variable and write 32 when asked to substitute 2 for x. This paper reports on the findings with respect to 3x from two teaching experiments to introduce algebra to grade 8 classes in a middle class state secondary school. The experiments related algebraic representations, concepts and principles to arithmetic representations, concepts and principles (see Boulton-Lewis, Cooper, Atweh, Pillay, Wilss, & Mutch, 1997) via the development of informal generalisations. Establishing the meaning for 3x was a major component of both experiments. The expression was considered in five ways: (i) modelled by cups and counters; (ii) developed from patterns (e.g., 3, 6, 9, 12), transformations or function machines (e.g., 2 ---x3--> 6) and relationships (e.g., 2-->6, 8-->23, 5-->15); (iii) considered as repeated addition x + x + x (see Linchevski & Herscovics, 1996); (iv) used in more complex expressions (e.g., 3x+2 and 3(x+2); and (v) extended to relations such as 3 x 2x = 6x. The classes were videotaped, written materials collected, and selected students interviewed. The paper outlines the teaching episodes relating to 3x, describes the students reactions to this instruction, and discusses the students understandings of 3x as an algebraic expression in relation to the five ways above. It explores the successful and unsuccessful teaching episodes and attempts to explain, for binary algebraic expressions of the type 3x, the relationship between instruction, prior knowledge and learning.