How do I stop my pans from boiling over? Yesterday the wife and I decided to boil some potatoes in a normal non-stick pan. 

We added salt,  potatoes and boiling water.  We put this on an electric hob, with no lid at maximum heat (marked at 6, which is ~10 times hotter than our 5 setting.  From experience, the remaining settings decrease operating temperature linearly). When come to the boil we cover with a lid. 

Almost immediately they boil over,  spilling water across the hob. We remove the pan from heat, remove the lid, put the pan back on heat, turn down to setting 3, and wait.  Once at the lower temperature,  we replace lid. The pan boils over. 

I know the basic mechanism behind boiling over.  Some agent increases the surface tension of the bubbles, causing many bubbles to form and then boil over. I am guessing that it is the starch. I don't know what factor the lid plays,  as occasionally the pan boils over without the lid on. 

Is there anything I can do? Are the pans not clean enough? Not enough/too much salt?  Wrong type of salt? 

I've heard of the wooden spoon trick, but I actually want to tackle the root cause.  Plus most of my spoons are silicone, and the trick doesn't work with them. 

So what can I do to prevent boiling over?
You are doing precisely the opposite of 'normal' procedure, which is to put the lid on the pan until the water starts boiling, then remove the lid (either partially or completely) to prevent boiling over. A reduction in the hob temperature will also probably be necessary, and is in any case desirable - mercilessly boiling any vegetable is rarely a good thing.

With the lid off, heat is lost at the top of the pan at a rate which is usually sufficient to prevent boiling over. With the lid on, much of the heat is retained, which A) helps the water come up to a boil quicker and B) causes it to boil over much more easily.

See this answer for more information.