Constructing the Self-similarity Concept

Fractals describe many natural phenomena; their strong visual-figurative nature found its mathematical conceptualization in the concept of self-similarity. In the current study, we investigate how students construct (fully or partially) the self-similarity concept while recursively constructing the Sierpiński triangle, working in small group and whole class settings in an inquiry-based MA level mathematics education course. We follow shifts of knowledge from individuals to groups and/or to the whole class community during the process of constructing the self-similarity concept. Our theoretical and methodological approach is based on networking between Abstraction in Context and Documenting Collective Activity. We found that the knowledge constructing processes of different students varied, some thinking recursively about finite cases and others thinking more directly about the infinite case. Some students acted as knowledge agents, with shifts of knowledge occasionally occurring in chains. We also observed a tendency to report results of group discussions back to the plenum only partially and in a purified manner.