When I introduce distribututions to students for the first time I usually emphasize the difference between $f$ (locally integrable function) and $T_f$ (the corresponding distribution). For a couple of lectures I avoid to write $\partial^\alpha f$ and keep the full formalization (it's a bit tedious) $T_{\partial^\alpha f} = \partial^\alpha T_f$. Basic formula are proven with this notation and then, gradually I leave those precautions ...

When I introduce distributions to students for the first time I usually emphasize the difference between $f$ (locally integrable function) and $T_f$ (the corresponding distribution). For a couple of lectures I avoid to write $\partial^\alpha f$ and keep the full formalization (it's a bit tedious) $T_{\partial^\alpha f} = \partial^\alpha T_f$. Basic formula are proven with this notation and then, gradually I leave those precautions ...