  Using a family of mock modular forms constructed by Zagier, we study the
coefficients of a mock modular form of weight $3/2$ on
$\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. These
coefficients are related to the smallest parts function of Andrews. As an
application, we reprove a theorem of Garvan regarding the properties of this
function modulo $\ell$. As another application, we show that congruences modulo
$\ell$ for the smallest parts function are rare in a precise sense.
