The use of fully online (FO) mathematics teaching has been increasing worldwide. Despite claims and findings that mathematics is more challenging to teach FO than face-to-face (F2F), we know little about FO mathematics teaching. In this paper, we address this gap by working to elucidate the differences between teaching in the FO and F2F modalities. We do this by examining FO and F2F teaching from the perspective of Communities of Practice (Wenger, Social learning systems and communities of practice, 2010) by comparing and contrasting current FO practices (or "ways of doing") in the "general" undergraduate education community "with" current F2F practices in the undergraduate "mathematics" community. We identify six key differences between the two paradigms, which we recast to spotlight areas for technological and pedagogical development.