Degree (oriented) of $\bar{z}^m$

Question

Let $f: S^1 \to S^1$ defined by $f(z) = \bar{z}^m$. I need to calculate the degree ofthis map, where the degree is defined by $I(f, \{\omega\})$ - the oriented intersection number-, where $\omega$ is any element of $S^1$. I was thinking of considering elements of $S^1 \setminus \{1\}$ as elements from $(0, 2\pi)$, by the association $t \mapsto e^{it}$, and then solving for $f^{-1}(i)$ - each element of this set is not $1$. Then, I would locally calculate $df_t$ for each $e^{-itm} = i$ and check that $df_t$ is orientation reversing for each of this cases (at least 4 cases - 1 for each sector of the plane), and conclude that $deg(f) = -m$. Does this make sense? Is there a more straightforward way of solving this?