jmc

There are 12 hours on a clock, so each hour mark is $360^{\circ }/12=30^{\circ }$ from its neighbors. At 6:44, the minute hand points at minute 44, which is $\frac{4}{5}$ of the way from hour 8 to hour 9. Therefore, the minute hand is $\frac{4}{5}\cdot 30^{\circ }=24^{\circ }$ past hour 8. The hour hand is $\frac{44}{60}=\frac{11}{15}$ of the way from hour 6 to hour 7, so it is $\frac{11}{15}\cdot 30^{\circ }=22^{\circ }$ past hour 6. This means that the hour hand is $30^{\circ }-22^{\circ }=8^{\circ }$ from hour 7. Since hours 7 and 8 are $30^{\circ }$ apart, the total angle between the two hands is $8^{\circ }+30^{\circ }+24^{\circ }=\boxed{62^{\circ }}$.