Think of resistivity like the 'density' of a material; just as cutting a piece of metal in half doesn't change its density (the mass per unit volume remains the same), changing the wire's size doesn't change the material's inherent resistance to charge flow.

The resistance of a conductor is given by $R = \rho \frac{l}{A}$. While resistance ($R$) is inversely proportional to the cross-sectional area ($A$), resistivity ($\rho$) is a constant for a given material at a specific temperature. Changing $A$ will change the resistance, but the resistivity remains unchanged as it is a characteristic property of the material itself.

Always distinguish between Resistance ($R$), which is an extrinsic property dependent on dimensions ($l, A$), and Resistivity ($\rho$), which is an intrinsic material property independent of geometry.