  We obtain a recursive formula for the number of rational degree $d$ curves in
$\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an
$m$-fold singular point. The special case of counting curves with a triple
point was solved earlier by other authors. We obtain the formula by considering
a family version of Kontsevich's recursion formula, in contrast to the excess
intersection theoretic approach of others. A large number of low degree cases
have been worked out explicitly.
