Crank Nicolson scheme approximating $\frac{\partial V}{\partial t}$

I am looking at Crank Nicolson scheme for solving the Black-Scholes equation. I have that $V$ is an option price. When I approximate $\frac{\partial V}{\partial t}$ I get some weird result that is not what I have in the notes.

So I am supposed to arrive at the same approximation given by both implicit and explicit schemes, namely: $\frac{\partial V}{\partial t} \approx \frac{V_n^m - V_n^{m-1}}{\delta t}$. Where $V_n^m = V(n\delta S, m \delta t)$.

So here are my steps for Crank-Nicolson method:

We let: $S \rightarrow S+\delta S$ and $t \rightarrow t+\frac{1}{2}\delta t$

Then: $V(S,t+\frac{1}{2}\delta t) = V(S,t) + \frac{\partial V}{\partial t}(\frac{1}{2}\delta t) + O(\delta t^2)$

Therefore:

$$\frac{\partial V}{\partial t} \approx \frac{2(V_n^{m+1/2} - V_n^m)}{\delta t}$$

And making the above a backward difference:

$$\frac{\partial V}{\partial t} \approx \frac{2(V_n^{m-1/2}-V_n^{m-1})}{\delta t}$$

Which is not the result indicated in the notes. How come I get a different result. What are the mistakes that I have done above? I am only interested in the mistakes above and how to correct them to arrive at the correct result.

$$V(S,t+\frac{1}{2} \delta t ) = V(S,t-\frac{1}{2} \delta t) + \frac{\partial V}{\partial t} \delta t + O(\delta t^2)$$