  We analyze in some detail a recently proposed transfer matrix mean field
approximation which yields the exact critical point for several two dimensional
nearest neighbor Ising models. For the square lattice model we show explicitly
that this approximation yields not only the exact critical point, but also the
exact boundary magnetization of a semi--infinite Ising model, independent of
the size of the strips used. Then we develop a new mean field renormalization
group strategy based on this approximation and make connections with finite
size scaling. Applying our strategy to the quadratic Ising and three--state
Potts models we obtain results for the critical exponents which are in
excellent agreement with the exact ones. In this way we also clarify some
advantages and limitations of the mean field renormalization group approach.
