Why can a block of ice at $0^\circ\rm C$ and water at $0^\circ \rm C$ coexist in an insulated bucket without changes either way in the amount of ice? Question

Why can a block of ice at $0^\circ \rm C$ and water at $0^\circ \rm C$ coexist in an insulated bucket without changes either way in the amount of ice?


Note
The textbook I read mentions nothing of state transformations of matter; so if this phenomenon can be explained without it, I'd appreciate it very much.

Reference
This question is posed in chapter 6 ("Dynamics of Heat") of "Understanding Physics" (Cassidy, Holton, Rutherford). The book is free to download (legally) here.
 A: Plain and simple: at any phase transition, both states (in this case, water and ice) are in thermodynamic equilibrium, meaning there will be no spontaneous transition from one to the other (none of them is preferred at that temperature), but you still need to transfer energy to turn from one to the other: you need to heat to turn ice into water and vice-versa. So the amount of ice in the mixture stays constant unless you add or remove energy.
A: It is indeed a very active scene under this picture. Frequent transformation from ICE to WATER and WATER to ICE. But the total amount cannot change since the temperature is kept the same.If you look at this thing at the microscope level, you should only see the vibrating molecules. Ice molecules are more tightly bound than the liquid water molecules. If it is indeed at zero degree, then there could be no amount of energy being transferred to this mix. What you actually see is that for a small volume (suppose this is the ice form) , the molecules are vibrating. If suddenly one molecule at the boundary of this small region absorbs a bit energy from the others around, then the chemical bounds around this molecule become looser. You may regard this process as the transformation of the form ICE into WATER. But, since the energy is from some other molecules', for these molecules losing energy they may change from WATER to ICE.  
A: The statement "insulated bucket" tells one that there is no heat transfer between the ice and water and the outside world.
This means that the overall average number of hydrogen bonds made between the water molecules cannot change as it takes energy to break a bond.
The statement that both the ice and the water are at the same temperature means that if they come together they will be in thermal equilibrium which means that there will be no net heat transfer between them.  On the molecular scale this means that on average when liquid water molecules collide with ice molecules there is no net transfer of kinetic energy between them.  
So you have ice with each water molecule having four hydrogen bonds and liquid with on average each water molecule having slightly less than four hydrogen bonds coexisting together.
There can be an interchange as to which hydrogen bonds are actually made or broken but overall the number stays the same.
The interesting thing is that you allowed some heat into the bucket then as long as there was ice present the temperature would stay at $0^\circ \rm C$ with some of the ice turning into water.
