Improving children's understanding of mathematical equivalence

A great majority of children in Canada and the United States from Grades 2-6 fail to solve equivalence problems (e.g., 2 + 4 + 5 = 3 + __) despite having the requisite addition and subtraction skills. The goal of the present study was to determine the relative influence of two variables, instructional focus (procedural or conceptual) and use of manipulatives (with or without), in helping children learn to solve equivalence problems and develop an appropriate understanding of the equal sign. Instruction was provided in four conditions consisting of the combination of these two variables. Students in Grade 2 (n = 122) and Grade 4 (n = 151) participated in four sessions designed to assess the effectiveness of four instructional methods for learning and retention. Session 1 included a pretest of equivalence problem solving and three indicators of understanding of the equal sign. In Sessions 2 and 3 instruction was provided in one of the four instructional conditions or a control condition. Students were tested for their skill at solving equivalence problems immediately following instruction and at the beginning of Session 3 to assess what they had retained from Session 2. In Session 4, one month later, children were re-tested on all of the tasks presented in Session 1 to assess whether instruction had a lasting effect. All four instructional groups outperformed the control group in solving equivalence problems, but differences among instructional groups were minimal. Performance on indicators of understanding, however, favoured students who received conceptually focused instruction. Preliminary evidence was found that children’s understanding of problem structure and attentional skill may be associated with the ability to benefit from instruction on equivalence problems. Children clustered into four groups based on their performance across tasks that are consistent with the view that children’s understanding of the equal sign develops gradually, beginning with learning the definition. These findings suggest that a relatively simple intervention can markedly improve student performance in the area of mathematical equivalence, and that these improvements can be maintained over a period of time and show some limited generality to other indicators that children understand equivalence.