Subject:
The direction which is attributed to an electric current is based on a convention. Before the true direction of electron flow was discovered the direction of the electric current had been defined in such a way that the current flows from the plus to the minus terminal of a power supply (in the external part of the circuit).
Deficiencies:
When asking for the direction of an electric current, one is asking for the orientation of a vector. The vector that characterizes the direction of the electric current is the current density vector, just as the energy flow density vector characterizes the flow direction of the energy, or the mass flow density vector tells us the direction of a mass flow. Now, the direction of the electric current density vector does not depend on a convention. It follows from the continuity equation for the electric charge, which relates the charge density ρ to the current density j :
∂ ρ
∂ t
+ div j
=
0
The equation tells us, that the charge density ρ decreases at a given place if the divergence of j is positive at the same place. In other words: The electric charge decreases in a small region if an electric current is flowing out of this region. This statement is analogue to the following: The amount of water in a container decreases if a water current is flowing out of the container.
We see that the orientation of the current density vector is defined, as soon as we have disposed of the sign of the electric charge. We could indeed redefine the direction of the electric current, but only by redefining the sign of the electric charge. If we want to keep the minus sign for electrons and the plus sign for protons, then there is no choice for the direction of the electric current.
Origin:
When it is claimed that the direction of the electric current is based on a convention, what is meant is not the direction of the current density vector j but the direction of motion of the mobile charge carriers, i.e the vector of the drift velocity v of the carriers. Both vector quantities are related by j = ρ · v .
It does not matter if positive charge carriers move in one direction or negative carriers in the other –   the current density direction is the same.
Since the direction of v is the same as that of the mass current density or the particle current density of the charge carriers, one can diagnose, that the charge current is mistaken for the mass or the particle current.
Disposal:
Distinguish thoroughly between the concepts charge and charge carrier.
Distinguish also between two directions: the flow direction of the electric

charge and the direction of motion of the charge carriers (or the direction of the mass current density vector). Whereas the electric charge flows (outside of the battery or power supply) from high to low potential, the charge carriers move in one or the other direction depending on the sign of their charge.
To make the distinction clear in the class room I carry out the following experiment: Pupils sitting in a row pass red and blue tokens to each one’s neighbor. We imagine that each red token is 10 euros worth, and each blue one minus 10 euros. Every pupil, except the two at the ends, owns one red and one blue token, i.e. his monetary property is zero. We now realize several money value transports from the leftmost to the rightmost pupil. A metronome is beating, and at each beat each pupil passes a token to his neighbor. The first transport is as follows: At each metronome beat each pupil - except the two pupils at the end of the row - passes a red token to his neighbor at the right. Thereby each pupil remains with his monetary value zero except the two at the ends: The leftmost gets poorer and poorer, the rightmost richer and richer. Next we realize the transport of monetary value from left to right in another way: On each metronome beat each pupil passes a blue token to his left neighbor. Again all the pupils of the chain remain with zero euros except those at the two ends, and again the one the left end gets poorer and the one at the right end richer. A third possibility for a value transport from left to right is that each pupil passes one red token to his neighbor at the right and simultaneously a blue one to his left neighbor.
In each of the three transports the monetary value goes from left to right, whereas the “value carriers”, i.e. the tokens, move in one or the other direction.
Friedrich Herrmann, Karlsruhe Institute of Technology