Connection between covariant derivative and basis vectors.

Question

I read here, Curvilinear page 11, that $$\frac{\partial}{\partial x^i}e_j=\Gamma^k_{ij}e_k$$ where the $e_i$'s are basis vectors. There seems to be some connection, but when I calculate it, for example in polar coordinates, I don't get this. For example, $$\frac{\partial}{\partial\theta} \hat{r}=\hat{\theta}$$ but when I use the formula I wrote above, I get $\frac{1}{r}\hat{\theta}$, close, but no cigar. Is the page wrong, or have I messed up?