If you studied chapter 9, then you know that free electron bands are parabolic and the weak potential introduces gaps at the BZ boundary. If these gaps are large, the electrons will remain in the same band in extended BZ. If the band gap are small, the electrons may end up in a higher band if the electric field is strong enough. After all, the electrons will follow the parabolic band if there is no potential. You may also see it like this: When you accelerate an electron, it may get reflected if the Bragg condition is fulfilled (i.e. the wavelength is equal to a BZ boundary vector). If an electron is accelerated slowly, it will fulfill this condition approximately during a long time, so that many reflections take place. However, if the electron is accelerated rapidly, the time it fulfills the Bragg condition is very short, so that reflection becomes improbable. What is short and what is long depends not only on the strength of the accelerating field, but also on the strength of the crystal potential, which also determines the width of the gap.
So finally we understand, if the gap is large, and the band is full, the occupancy of the k-levels won't change on average, hence the substance is an isolator. If the band is not full or if the band gap is 0, the occupancy of the band will change when a field is applied and the substance will behave like a metal.
