The (activity-)effect of manipulatives on algebraic generalizations: a constructivist teaching experiment

To understand the ways that manipulatives might support changes in students’ reasoning about algebraic generalizations, a constructivist teaching experiment was conducted with two sixth-grade students. The students interpreted numerical situations with units of one and could construct units of units in mental activity. Initially, the students’ reasoning about Cuisenaire® rods did not lead to changes in their algebraic generalizations, whereas their reasoning about linking cubes did lead to such changes. The students’ learning with linking cubes is explained by their enactment of physical operations with units of one on the linking cubes, which were consistent with their mental operations in numerical situations. Over time, and after learning to generalize with linking cubes, the students also began to attribute meaning to their physical operations with Cuisenaire® rods. Thus, instruction with manipulatives that reflected the student’s interpretation of numerical situations supported their construction of algebra as generalized arithmetic. Instructional implications are discussed.