We all hate homework

Let's construct the puzzle grid first:

$\begin{array} {c | c | c | c } & \text{Mathematics} & \text{Geography} & \text{History} \\ \hline \text{Ava} & & & \\ \hline \text{Elle} & & & \\ \hline \text{Hannah} & & & \\ \end{array}$

We are given that Ava is currently not working on mathematics and Elle is currently not working on history. We fill the corresponding cells with X's to represent this information, as shown in the following table:

$\begin{array} {c | c | c | c } & \text{Mathematics} & \text{Geography} & \text{History} \\ \hline \text{Ava} & \times & & \\ \hline \text{Elle} & & & \times \\ \hline \text{Hannah} & & & \\ \end{array}$

Now we have 3 cases to work on:

Hannah is either
(i) currently not working on mathematics, or
(ii) currently not working on geography, or
(iii) currently not working on history.

Suppose that (i) Hannah is currently not working on mathematics, then adding an X to the relevant cell gives the following table:

$\begin{array} {c | c | c | c } & \text{Mathematics} & \text{Geography} & \text{History} \\ \hline \text{Ava} & \times & & \\ \hline \text{Elle} & & & \times \\ \hline \text{Hannah} & \times & & \\ \end{array}$

With that, we will know that Elle must be working on mathematics, as shown in the following table:

$\begin{array} {c | c | c | c } & \text{Mathematics} & \text{Geography} & \text{History} \\ \hline \text{Ava} & \times & & \\ \hline \text{Elle} & \checkmark & \times & \times \\ \hline \text{Hannah} & \times & & \\ \end{array}$

Then since Ava knows what subject she is studying, it's possible to complete the table as such (depending on her own subject), as shown in the following 2 tables:

$\begin{array} {c | c | c | c } & \text{Mathematics} & \text{Geography} & \text{History} \\ \hline \text{Ava} & \times & \checkmark & \times \\ \hline \text{Elle} & \checkmark & \times & \times \\ \hline \text{Hannah} & \times & \times & \checkmark \\ \end{array}$

$\begin{array} {c | c | c | c } & \text{Mathematics} & \text{Geography} & \text{History} \\ \hline \text{Ava} & \times & \times & \checkmark \\ \hline \text{Elle} & \checkmark & \times & \times \\ \hline \text{Hannah} & \times & \checkmark & \times \\ \end{array}$

So (i) can be a possible solution. Thus it is indeed true that Ava (truthfully) made the claim that she knows what subject everyone is doing.

Similarly, we can show that (ii) and (iii) are possible claims too.

Because we have shown that it is possible to have at least 2 possible claims, we as the reader can't determine what subject that Hannah says she is not currently working on.