Why are my solutions to the Christoffel symbols of polar coordinates recipricoled?

Question

I just gave myself some practice with Christoffel symbols, using the polar coordinates formula of $[x, y] = [r\cos\theta,\,r\sin\theta]$. I then calculated the Christoffel symbols, and I got six correct ($\Gamma^r_{\theta\theta}$ was $-r$ and the others were zero) but the two that are giving me trouble are $\Gamma^\theta_{r\theta}$ and $\Gamma^\theta_{\theta r}$, for both of which I got an answer of $r$, when the video I was watching told me it should have been $1/r$. Why is this? The equation I was using for Christoffel symbol $\Gamma^k_{ij}$ was$$\frac{\partial e_i}{\partial x^j}\cdot e^k$$ Which I interpreted to be the dot product of the second derivative of $[r\cos\theta,\,r\sinθ\theta]$ with respect to $i$ and $j$ and the derivative with respect to $k$ (because I am under the impression that basis vectors are derivatives, and you may correct me if I'm wrong). Here is a link to the video and to the equation I used. I just can't see how the answer can be $1/r$.