Building metaphorical bridges: A teaching experiment on first grade students' relational understanding of equivalence

Children often interpret addition equivalence statements in terms of what is "done" operationally to obtain answers rather than as representations of relationships between quantities. The purpose of this study was to determine if eight first grade children could develop a mental model of those relationships through instruction in the semantics of equivalence, transitive inference, and metaphors. Based on the theory that difficulty with the relational meaning of equivalence may hinder the subsequent learning of arithmetic and algebra, this study is significant because it employs a potentially lasting metaphor for that meaning of equivalence, which may in turn be useful to students in later work.;The study used a teaching experiment design. Pan balances and one-inch wood blocks were used to represent the balance between the elements of addition statements. The experiment included hypotheses, systematic instruction, and an analysis of the relative effectiveness of the strategies.;Participants were initially screened to determine their approximate level of cognitive development. Case studies were used to examine data collected through audiotaped clinical interviews, videotapes, informal observations, and worksheets. Conceptual beliefs about equivalence were ascertained, and the mental and physical associations that the children made between the blocks on pan balances and the numerals in equations were studied. The ability of the children to use the pan balance as a metaphor for equivalence was also determined. Cognitive changes were analyzed using individual case studies and a collective case study, and were reported in a narrative style.;Results revealed that in the areas of semantics of equivalence and transitive inference, instruction was of greater benefit to children at the concrete operational level of cognitive development than to those at the preoperational level. Instruction in the use of metaphors was of little benefit to children in either group. After extensive instruction, a clear change from an operational interpretation to a relational interpretation of equivalence did not occur for most of the participants. Thus, while the instruction was beneficial for certain students, it is likely that cognitive maturation also plays a role in the development of this particular concept.