There was a thread on this a while ago, but I got a search error trying to find it.
A poster actually ran an experiment. IIRC water boiled sooner (and not by much) only with a large pot that was full or nearly so.
Absolutely. And I cook pasta with the lid off, at a vigourous boil, to keep the pasta from settling at the bottom of the pan and sticking together.
The physics are sometimes obvious. The lid keeps the heat in.
It’s faster with the lid on. Plus, if you’re boiling a specific amount of water, you’ll have less escapage via steam.
Yes, salt your pasta water for flavor, not to change the temperature (and while we’re add it, stop adding oil to the water!). But lots of people add salt thinking it does change the temperature–is the lid thing similarly silly?
I ran it myself just now using a small pot and 250ml (1cup) of cold water. The pot could hold ~5 cups so I had a pretty big airspace.
- Turned the burner on the electric stove to max let it heat up for several minutes
- Ran the tap until the water temperature stabalized, set the pot and measuring cup underneath and allowed to overflow for a minute or two so everything started at the same temp.
- Took the pot out, dried it with a towel, measured 250 mls of water from the still flowing tap, poured it in the pot, put glass lid on pot, had it on the red hot burner in about 30 seconds.
- Time to boiling = 2min 14sec.
- Left burner on, dumped water, put pot and measuring cup back underneath flowing tap water for a couple minutes so we start at the same temp again.
- Repeat same procedure, except do not put lid on pot.
- Time to boiling = 2min 37 seconds.
Now I didn’t have a thermometer so the exact time to boiling might be out by a few seconds; I went by the rolling bubbles. Using my times the % difference is about 15.8. If you add and subtrac say 3 seconds to my top and bottom numbers you might get 20% difference. Either way you’re only talking about 20-30 seconds.
Wow!
You watched a pot and it boiled? You should submit that to the Nobel Committee. Pictures or it didn’t happen! 
Buy a kettle.
And your mother hadn’t done enough cooking during her lifetime to know from her own experience that this was false?
You should encourage your mom to trust in herself, and not be swayed just because someone has a degree in mathematics. There are too many educated idiots around, and too few experienced mothers.
Thanks for running this, that’s awesome!
Keep in mind, though, that using the lower bound of your experiment, it took at minimum 20 seconds longer for a single cup to boil. That would mean if we were trying to boil a gallon, it would take 320 seconds, or over three minutes longer. I’m not sure the effect is linear due to edge effects (the sides of the pot will be heating the water, too), but this number will begin to become very significant when we’re cooking for a group of people, brewing beer, or just about any commercial endeavor.
In other words, your results are far from insignificant.