Versatile thinking and the learning of statistical concepts

Statistics was for a long time a domain where calculation dominated to the detriment of statistical thinking. In recent years, the latter concept has come much more to the fore, and is now being both researched and promoted in school and tertiary courses. In this study, we consider the application of the concept of flexible or versatile thinking to statistical inference, as a key attribute of statistical thinking. Whilst this versatility comprises process/object, visuo/analytic and representational versatility, we concentrate here on the last aspect, which includes the ability to work within a representation system (or semiotic register) and to transform seamlessly between the systems for given concepts, as well as to engage in procedural and conceptual interactions with specific representations. To exemplify the theoretical ideas, we consider two examples based on the concepts of relative comparison and sampling variability as cases where representational versatility may be crucial to understanding. We outline the qualitative thinking involved in representations of relative density and sample and population distributions, including mathematical models and their precursor, diagrammatic forms.