As a former Aerodynamicist, I think it is important to re-iterate the points that Taylor G and Bookworm have made.
The Cl/AoA debate is an issue of definition. By convention in aerodynamics, the chord line of an aerofoil is defined for it's clean configuration with nothing "hanging out". As we begin to deploy trailing edge high lift devices, we are indeed modifying the characteristics of the aerofoil - the most significant change being the introduction of a greater camber. Commensurate with this, the aerofoil's true chord is also modified - but by convention, the reference chord remains that for the clean aerofoil. Thus, on a characteristic Cl/AoA plot, it appears that the aerofoil stalls at a lower AoA, but with a greater Clmax. In reality, the AoA at which the aerofoil stalls is usually greater than for a clean aerofoil. A secondary effect of this is very beneficial to use as pilots - the pitch angle at which we a achieve a given AoA is reduced.
The same principle is true for Fowler slotted flaps and similar devices. The effect of such devices on a Cl/AoA plot often look too good to be true compared to other trailing edge devices, in that they offer a relatively large increase in Clmax. However, you have to consider that Fowler slotted flaps are also significantly increasing the aerofoil wetted area, but this fact is not reflected in the calculations - the aerofoil reference area remains that for the clean aerofoil. The increased area is hence absorbed into the increased Clmax the devices offer, slightly misleading!
Of course, day to day, the theory offered by BEagle is all we really need to know. However, when your really trying to understand what is going on, such simplifications can be misleading.
Rgds
Cuban_8