Tracing Robert's use of representations to solve counting problems: A 16-year case study

•A 16-year longitudinal case study of Robert working on combinatorial tasks.•He discovered several isomorphic relations in the counting tasks.•He used his elementary counting ideas to build complex counting schemas.•He elaborated his representations to solve new problems.•All students should be encouraged to create their own representations.

This study describes how Robert used his external representations (formulae, notations, sketches, models, and figures) to solve progressively more challenging counting tasks over a 16-year period. In his explorations of counting tasks, Robert discovered intricate connections between solutions to problems that looked different on the surface. Using video data from Robert's problem solving, analyses of his solutions are presented that shed light on how he built new ideas from existing ideas and how he modified external representations to make new mathematical discoveries and provide justifications for his solutions.