How to calculate the amount of substance of sodium that are in 10 mL of 30% sodium chloride?
You are on the right track and with a few assumptions I'd say your calculation is correct. The one of your teacher is not.
Let's back up a moment and understand what is going on.

First of all we should make sure, that we are asking the right question. As you have given it, it seems incomplete. It should probably rather be:


  What amount of substance of sodium (ions) can be found in 10 mL of a 30% aqueous solution of sodium chloride?


We have been given a mass fraction, which we need to convert to the amount of substance concentration first. The general formula for this is
$$w_i = \frac{\rho_i}{\rho} = \frac{c_i M_i}{\rho},$$
where $\rho$ is the density of the solution, $\rho_i$ is the mass concentration of component $i$, $M_i$ is the molar mass of component $i$, and $c_i$ is the amount of substance concentration of $i$.

In your calculation you made the assumption, that the density of a 30% sodium chloride solution is $\rho = 1~\mathrm{g/mL}$, which is not correct. You can look up a couple of values for brine solutions on wikipedia.
For the purpose of this example, let's assume $\rho(\ce{NaCl~30\%}) \approx 1.2~\mathrm{g/mL}$.

The molar mass of sodium chloride is $M(\ce{NaCl}) = 58.4~\mathrm{g/mol}$. Be careful with the units, as you used the incorrect ones.

The amount concentration is given through the formula
$$c_i = \frac{n_i}{V},$$
where $n_i$ is the amount of substance (not number of moles!) and $V$ is the volume of the solution.

You correctly identified that there is a one-to-one correlation between sodium ions and sodium chloride as expressed by
$$\ce{NaCl (s) -&gt;C[H2O] Na+ (aq) + Cl- (aq)}.$$
Therefore we can write
$$n(\ce{Na+}) = n(\ce{NaCl}).$$

Putting everything together we arrive at the following formula:
\begin{align}
n(\ce{Na+}) &amp;= \frac{w_{\ce{NaCl}}\cdot \rho(\ce{NaCl~30\%})}{M(\ce{NaCl})} \cdot V\\
            &amp;= \frac{0.3\cdot 1.2~\mathrm{g/mL}}{58.4~\mathrm{g/mol}}\cdot 10~\mathrm{mL}\\
            &amp;= 0.061~\mathrm{mol}
\end{align}

Basically you thought process was correct under the assumption of the inaccurate density. And it was off by $10^{-3}$ because you (and your teacher) used the wrong units.