Actually, the notation is confusing even to professional mathmaticians. The notation used in differential geometry is the archetype of this phenomenon; it's sometimes incredibly hard to work out what the simple-looking notation means.

In fact, differential geometry has been described as the study of objects "invariant under a change of notation" playing on the lack of clarity in this field, along with the standard description of a tensor as an object "invariant under a change of coordinates".

See, for example, the introduction to this piece: http://www.math.jhu.edu/~yrubinst/RiemGeom/Thesis-pp-23-30-a...