  Julius Whiston calculated the maximum size of an irredundant generating set
for $S_n$ and $A_n$ by examination of maximal subgroups. Using analogous
considerations, we will compute upper bounds to this value for the first two
Mathieu groups, $M_{11}$ and $M_{12}$. Computational results gave explicit
irredundant generating sets of $M_{11}$ and $M_{12}$ of size 5 and 6,
respectively. Together these give the full results that the maximum size of an
irredundant generating set for $M_{11}$ is 5 and for $M_{12}$ it is 6.
