Melting point of alkaline earth metal carbonates
Melting and decomposition are different things, as you guessed in your question.  

Melting:

Something can melt, and then re-solidify, undergoing a physical but not chemical change. 

A simple example would be ice, but it is also true for many chemicals. For example tert-butanol is a solid at room temperature (at least in a well air-conditioned lab), but melts readily at slightly elevated temperatures without any degradation. Dimethylsulfoxide is another example of a chemical which often solidifies in a cool lab, but can be melted back down with only the heat of your hand.

Decomposition: 

Decomposition is a different thing, whereby the material undergoes a chemical change (usually in addition to physical change). This is often seen in laboratory settings when trying to measure the melting point of molecules with high melting points, such that at the temperatures needed to melt the solid, side reactions are often observed, destroying the material. For this reason, in the literature, you will see melting points reported as  mp. (decomp.), i.e. the temperature at which your molecule decomposed rather than melted.