Notation for partial derivatives

Congratulations, you have met one of the worst ambiguities in mathematical notation!

Assume you have a function of two variables, $f \colon A \times B \to \mathbb{R}$, where $A$ and $B$ are subsets of $\mathbb{R}$. The notation $$\frac{\partial f}{\partial x}(x_0,y_0)$$ is commonly used to denote the value of the partial derivative of $f$ with respect to the first variable, evaluated at $(x_0,y_0)$. This is the cleanest use of the notation for partial derivatives.

Anyway, it sometimes happens to use some lazy piece of notation such as $$\frac{\partial f(x,g(x,y))}{\partial x}$$ to denote the partial derivative of the map $(x,y) \mapsto f(x,g(x,y))$. This is imcompatible (in general) with the interpretation of the same formula as

The derivative of $f$ with respect to the first variable, evaluated at the point $(x,g(x,y))$.

This is bad, but it seems we have to live with it. Why? Just spend a couple of minutes and think about the second interpretation. To be rigorous, we should have written $$ \frac{\partial}{\partial x} \left( f \circ \left( (x,y) \mapsto (x,g(x,y)) \right) \right) (x,y), $$ which is a true nightmare.

I think Siminore's answer is good. But I checked some textbooks just for curiosity.

1. "Advanced calculus" by Folland uses the notation like $\partial_x f(0, 0)$. Of course the meaning is the partial derivative of $f$ w.r.t. $x$ evaluated at $(0, 0)$. It does not use the notation $\partial f(0, 0) / \partial x$ extensively but there is a comment that you can use the notation.
2. "Advanced calculus" by Kaplan and "The way of analysis" by Strichartz also follow the same convention.
3. "Real mathematical analysis" by Pugh uses the notation $\partial f(0, 0) / \partial x$ and in some place it uses $\partial f(x, g(x)) / \partial x$ to denote the partial derivative of $f$ w.r.t. $x$ evaluated at $(x, g(x))$. It doesn't mean the derivative of the composite function.
4. I also checked "Principles of mathematical analysis" by Rudin. It looks like to avoid the notation $\partial f(a, b) / \partial x$. Instead it says "$\partial f / \partial x$ at $(a, b)$". However, actually I found only one such occurrence. There is not much use of the round notation.
5. "Mathematical methods for physicists" by Afken uses the notation $\partial f(x, 0)/\partial t$ to denote $\partial f(x, t)/\partial t |_{t = 0}$. It uses the two notations interchangeably.

Thus, the use case of your teacher is fairly common.