Building data analysis skills is part of modern elementary school curricula. Recent research has explored how to facilitate children's understanding of visual data representations through completion exercises which highlight links between concrete and abstract mappings. This approach scaffolds visualization activities by presenting a target visualization to children. But how can we engage children in more free-form visual data mapping exercises that are driven by their own mapping ideas? How can we scaffold a creative exploration of visualization techniques and mapping possibilities? We present Construct-A-Vis, a tablet-based tool designed to explore the feasibility of free-form and constructive visualization activities with elementary school children. Construct-A-Vis provides adjustable levels of scaffolding visual mapping processes. It can be used by children individually or as part of collaborative activities. Findings from a study with elementary school children using Construct-A-Vis individually and in pairs highlight the potential of this free-form constructive approach, as visible in children's diverse visualization outcomes and their critical engagement with the data and mapping processes. Based on our study findings we contribute insights into the design of free-form visualization tools for children, including the role of tool-based scaffolding mechanisms and shared interactions to guide visualization activities with children.