  Recently, the elephant random walk has attracted a lot of attentions. A wide
range of literature is available for the asymptotic behavior of the process,
such as the central limit theorems, functional limit theorems and the law of
iterated logarithm. However, there is not result concerning Wassertein-1
distance for the normal approximations.In this paper, we show that the
Wassertein-1 distance in the central limit theorem is totally different when a
memory parameter $p$ belongs to one of the three cases $0< p < 1/2,$ $1/2<
p<3/4$ and $p=3/4.$
