  We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all
have finite forward orbit. Thurston's rigidity theorem states that outside a
single well-understood family, there are finitely many such maps over
$\mathbb{C}$ of fixed degree and critical orbit length. We provide an algebraic
proof of this fact for tamely ramified maps for which at least one of the
critical points is periodic. We also produce wildly ramified counterexamples.
