  A new mathematical framework is formulated to derive the effective equations
of motion for the constrained quantum system which possesses an internal clock.
In the realm close to classical behavior, the quantum evolution is approximated
by a finite system of coupled but ordinary differential equations adhered to
the weakly imposed Hamiltonian constraint. For the simplified version of loop
quantum cosmology in the Bianchi I model with a free massless scalar filed, the
resulting effective equations of motion affirm the bouncing scenario predicted
by the previous studies: The big bang singularity is resolved and replaced by
the big bounces, which take place up to three times, once in each diagonal
direction, whenever the directional density approaches the critical value in
the regime of Planckian density. It is also revealed that back-reaction arises
from the quantum corrections and modifies the precise value of the directional
density at the bouncing epoch. Additionally, as an example of symmetry
reduction, we study isotropy emerging from the anisotropic Bianchi I model in
the context of effective equations of motion.
