Children's construction of a volume calculation algorithm for a rectangular prism with a dynamic virtual manipulative

We report on the process of children's construction of a volume calculation algorithm for rectangular prisms. We provided third- and fourth-grade students with traditional volume tasks and a dynamic virtual manipulative to support their three-dimensional reasoning and use of unit cubes to structure space. We investigated the challenges faced, evolving understandings, and supports for growth. We found four threads of understanding we called interpretation, structure, representation, and numeration, that interacted in complex ways as students constructed volume calculation algorithms. Across six patterns of growth, we found that these threads could have both positive and negative influences on one another. The representation thread tended to have the strongest dragging influence on coordinated understanding, and the structuring thread tended to produce the most enduring conceptualizations. We also found that feedback from the dynamic virtual manipulative and the interviewer played a critical role in overcoming challenges and reaching new understandings about volume.