All of this has been enlightening regarding common misconceptions that
can confuse students, and possible ways out of the confusion.
If we start with the definition of acceleration as the change in the
VECTOR quantity velocity WRT time, I think we avoid the misconception.
A lot has been said using the term speed, so that is an addition to
the confusion, which is unnecessary.
If a=(delta v)/(delta t), the minus sign possiblilites become
apparent, provided you insist that change in a vector quantity is
always the 2nd value minus the 1st value.
If a is negative, it is because v2-v1 is negative.
Assuming (as in the discussion) that you have picked + velocity to be
associated with increasing distance along some sign line, v2-v1 may be
negative as a result of each permutation of the +/- choices which
yield a *difference* which is (-), I think students can be drawn to
pick all the possibilities and describe their physical realities. Karl