Problems, perseverance, and mathematical residue

This paper reports on a qualitative study in which three third grade students were presented with a mathematical challenge related to the volume of a cuboid. The task required the construction of containers and the enumeration of the multi-link cubes held by each of the containers. The study videotaped and investigated the students working as a group through potentially seven different problem solving categories, and how they dealt with the student-generated mathematical dilemmas that surfaced during their exploration of the original problem. The tapes were examined for the problem solving actions that the students demonstrated, and an analysis of the students' counting strategies and solutions explored the connection between the children's spatial structuring and their use of numerical operations in enumerating 3-D rectangular arrays of cubes. The notions of perseverance and control are considered as they emerge during autonomous problem solving and the mathematical residue that results from the developing understanding of volume is discussed. © 2002 Kluwer Academic Publishers.