jmc

The hour hand is $\frac{20}{60}=\frac{1}{3}$ of the way from the 3 o'clock to the 4 o'clock position, so it has moved $\frac{1}{3}\cdot \frac{1}{12}\cdot 360^{\circ }=10$ degrees past the 3 o'clock position. The minute hand is in the 4 o'clock position, so it has moved $\frac{1}{12}\cdot 360^{\circ }=30$ degrees past the 3 o'clock position. The angle between the hour hand and the minute hand is $30-10=\boxed{20}$ degrees.