In a doppler shift why does a moving source show no increase in speed of the mechanical waves emitted?

I'll first ask the student to explain the difference between a moving source and moving observer, prompting them until they get something like this(asking them to draw a diagram if needed)- When the observer moves(towards source): frequency increases but the wavelength is the same, therefore the wavelengths appear faster to the observer. When the source moves(towards observer): frequency stays the same but the wavelength shortens.

Despite what common sence would have you believe about velocities being cumulative, the wave's speed depends on the properties of the medium, not the motion of the source. The speed for a classical wave is constant in any given medium when stationary relative to the medium.