The Development of Children's Understanding of Equality and Inequality Relationships in Numerical Symbolic Contexts

Article excerpt

A fundamental concept in arithmetic and algebra is the equivalence relationship represented by the '=' symbol. In order to understand equations such as 7+8 = 7+2+6 or 2x+2 = 3x-6, children have to understand that the term 'equals' as represented by the '=' symbol is used as a relational symbol. As a relational symbol, '=' denotes a relationship of quantitative sameness between the two members of an equality sentence. There is considerable evidence, however, that children, especially those in the age range 5-8 years, consider the equals sign to be an operator rather than a relational symbol (Behr, Erlwanger, & Nichols, 1980). As an operator symbol, '=' is interpreted as a command to perform an arithmetical operation. For example, children tend to view the equals sign in the sentence 5+3 = [] as indicating that the addition should be performed. As a first grader in Baroody and Ginsburg's (1983) study explained, 'equals' in a sentence such as 5+3 = [] "means to put another number that they would add up to" (p. 207).

Children also appear to acquire the notion that the operator is one-directional or left-to-right; they tend to insist that a "math problem" must precede the equals sign and that the answer to the problem must come after the equals sign (Denmark, 1976). Thus children generally tend to view sentences like 4 = 1+3 as "backwards" and sentences like 1+3 = 2+2 as incorrect. They tend to reject atypical equations such as 5 = 5 and rewrite the sentence to include an operation (e.g., 5+0 = 5, 5-0 = 5) (Behr, Erlwanger, & Nichols, 1976). In only a few interviews with children in grades 1-6, and rarely in the lower grades, did Behr et al. (1976) encounter a child who could explain the meaning of sentences such as 5 = 5 and 1+3 = 2+2 in terms of a relation.

As children progress through school, they are increasingly exposed to equations where the interpretation of the '=' symbol as an operator is meaningless (e.g., 2 + (3+7) = (2+3) + 7, 2 x (10+3) = 2 x 10 + 2 x 3,47 = [] tens + [] ones, 3x + 7 = 9) (Anderson, 1976). However, it appears that many students in higher grades, even at the college level, do not develop a concept of an equals sign as an indication of a relation. This appears to affect algebra learning and the learning of other mathematical subjects that require algebraic competence (Falkner, Levi, & Carpenter, 1999; Frazer, 1976; Herscovics & Kieran, 1980; Renwick, 1932; Weaver, 1973).

The finding that children tend to interpret the equals sign as an operator is a robust one (Baroody & Ginsburg, 1983; Behr et al., 1980; Denmark, 1976; Falkner et al., 1999; Kieran, 1981; Renwick, 1932). This phenomenon has not been fully explained, however. Children's failure to interpret the equals sign as a relational symbol may reflect a lack of requisite understandings about equality and inequality relationships between symbolically represented numerical amounts. Alternatively, children may have the requisite conceptual understandings but may fail to associate the understandings with the symbol that represents an equality relationship. It is also possible that both explanations are valid but for different age groups. The level of instructional support that is required to assist children in grades 1-7 in acquiring a relational interpretation appears to decrease with age (Anderson, 1976; Denmark, 1976; Falkner et al., 1999; Herscovics & Kieran, 1980; Saenz-Ludlow & Walgamuth, 1998). This change may reflect a lack of requisite conceptual understandings for younger children whereas the only role for instruction for older children is to help them associate their understandings about equality relationships with the equals sign. These explanations cannot be assessed, however, because prior research has not established what children of different ages know about equality relationships in the context of symbolically represented numerical amounts or the relationship between children's interpretation of the equals sign and their conceptual understandings about equality relationships. …