Let the rate of the current of the river be \( c \) mph. When Connie is traveling upstream, her effective speed is \( 6 - c \) mph, and when she is traveling downstream, her effective speed is \( 6 + c \) mph.
We know that the time it takes to travel a certain distance is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Given that she travels 4 miles upstream and 44 miles downstream, we can set up the following equation for the times taken:
\[ \text{Time upstream} = \text{Time downstream} \]
Substituting the distances and speeds, we have:
\[ \frac{4}{6 - c} = \frac{44}{6 + c} \]
Now we can cross-multiply to eliminate the fractions:
\[ 4(6 + c) = 44(6 - c) \]
Expanding both sides:
\[ 24 + 4c = 264 - 44c \]
Now, we will combine like terms:
\[ 4c + 44c = 264 - 24 \]
\[ 48c = 240 \]
Now, divide by 48:
\[ c = \frac{240}{48} = 5 \]
Thus, the rate of the current of the river is:
\[ \boxed{5} \text{ mph} \]