Stirring the pot has three effects:

1. It increases the convective heat transfer between the hot bottom of the pot and the water.
2. It increases the convective heat transfer between the cool sides of the pot and the water.
3. It increases the evaporation/cooling rate at the top of the pot.

Process 1 increases the heating rate and decreases the time to boil. Processes 2 and 3 decrease the heating rate (increase the cooling rate) and increase the time to boil. The question is, will the increased heating due to 1 be greater or smaller than the decreases due to 2 and 3?

At this point you have to do some estimating or calculating. The overall heat transfer coefficient on the bottom will be the reciprocal of the sum of reciprocals of the heat transfer coefficient from the flame to the pot, the conduction through the pot, and the heat transfer coefficient from the pot to the water. You have similar terms for the sides of the pot. The top of the pot is dominated by the mass transfer coefficient between the water and the air. In SI units (watts/square meter Kelvin) reasonable values for the stirred and unstirred cases are:

The overall coefficient on the base increases by about 30% while the overall coefficient on the sides increases by about 0.4%. The heat loss at the top shows about a 400% increase upon stirring, but the magnitude is still less than 10% of the heat coming in through the bottom. Overall, you get about a 20% reduction in the time to reach a boil. Is it worth stirring constantly to reduce the time from 10 minutes to 8 minutes? Only your arm can tell you that.