To avoid confusion, here are the definitions of the objects in this question:
1) Let $\gamma:S^1\to\mathbb{R}^2\setminus\{0\}$ a smooth loop. The winding number of $\gamma$ is the number of times $\gamma$ encircles $0$. Similarly, if $\gamma:S^1\to\mathbb{R}^2\setminus\{p\}$ for some $p\in\mathbb{R}^2$, the winding number of $\gamma$ around $p$ is the winding number of the translation $\gamma-p$. As we all know, the winding number can be defined rigorously by means of algebraic topology / two-variable calculus / complex variables.
2) Let $\gamma:S^1\to\mathbb{R}^2$ smooth with non-vanishing derivative. The turning number of $\gamma$ is the winding number of $\dot{\gamma}$.
My question is about the following
Claim: Let $\gamma:S^1\to\mathbb{R}^2$ be a simple loop with non-vanishing derivative. Then the turning number of $\gamma$ is either $1$ or $-1$. Furthermore, for any $p$ in the area enclosed by $\gamma$, the winding number of $\gamma$ around $p$ is equal to the turning number.
This claim seems to be rather intuitive, but I can't think of an elementary way to prove it. One way, I guess, is to use the fact that any such loop is isotopic either to the standard embedding $S^1\hookrightarrow\mathbb{R}^2$, or to an embedding with the same image but reversed orientation. For these two embeddings the turning number can be computed directly, and then the claim follows from invariance under isotopy. However, the fact that every such $\gamma$ is isotopic to the unit circle is not so elementary, and so neither is this proof.
Any other approach? Any insight regarding turning numbers of simple loops? Anything simpler than the above argument?
