There is a broad impetus across policy and institutional domains to expand public engagement and involvement with emerging technology research and innovation. Yet innovative theory, methods, and practices to critically explore algorithmic system controversies and democratic possibilities are still in nascent form. In this paper, we bring together thinking about public pedagogy, infrastructuring, and democratic design to introduce a novel "pedagogic encounters toolkit," where pedagogic encounters are defined as: opportunities for learning together about the emergence, relationality, and uncertainty of algorithmic system controversies. A framework with accompanying practices and examples is outlined for researchers, policymakers, and practitioners to creatively and critically explore these pedagogic encounters within specific contexts. We conclude with implications for future investigations attentive to making democratic designs in a datafied society. In doing so, this paper informs theoretical and methodological innovation to the study of algorithmic system controversies and democratising technology.