ABSTRACT: An important calculation is to determine the stable operations associated with the relatively new cohomology theory of connective topological modular forms (or tmf for short). A possible starting point includes understanding the spectrum tmf ^ tmf which has proven to be a formidable problem. Mahowald (1981) showed that bo ^ bo splits as a bo-module spectrum into summands involving integral Brown-Gitler spectra. In this talk, I will discuss a similar splitting of bo ^ tmf and address why such a splitting may serve as a nice warm-up example to understanding tmf ^ tmf. As an application, I will give an explicit description of the bo_*-module structure of bo_* (tmf). (see doi:10.1016/j.jpaa.2009.06.005 for the paper)