  In this paper, a specific family of $f(R)$ models that can produce the
$\Lambda$CDM background expansion history is constrained by using the currently
available geometric and dynamic probes. The scale dependence of the growth rate
$f(z,k)$ in this specific family of $f(R)$ model is shown. Therefore to
eliminate the scale dependence of $f\sigma_8(z)$ in theory, which usually is
defined as the product of $f(z,k)$ and $\sigma_8(z)$, we define
$f\sigma_8(z)=d\sigma_8(z)/d\ln a$ which is obviously scale independent and
reproduces the conventional definition in the standard $\Lambda$CDM cosmology.
In doing so, under the assumption that future probes having the same best fit
values as the current ten data points of $f\sigma_8(z)$, even having $20\%$
error bars enlarged, we find a preliminary constraint
$f_{R0}=-2.58_{-0.58}^{+2.14}\times 10^{-6}$ in $1\sigma$ regions. This
indicates the great potential that redshift space distortions have in
constraining modified gravity theories. We also discuss the nonlinear matter
power spectrum based on different halo fit models.
