Fishing in/on/at (the) sea

OK. One of my 11 year old Japanese students has asked me which is correct.

I go fishing in the sea/on the sea/at sea.

This is how I see it.

On the sea - to fish from a boat that is floating on the ocean.

In the sea - Confusing one this one as although the line and bait is in the sea, the person fishing is not.

At sea - This suggests to me that a boat is away from the shore.

At the sea - This could be correct for beach fishing but would probably be conisdered as 'at the seaSIDE'.

What would any wordsmiths out there in cyberland suggest as the correct answer? Please bear in mind that Japanese examiners are really 'anal' about the correct use of grammar!

PS - I've also explained that we have specific terms for sea fishing and fresh water fishing but the kid wants to know the answr to in/on/at.

In the sea: someone is actually IN the water. Not great for fishing.

On the sea: On a boat, not fishing from a dock or at the shore.

At sea: Out on the open ocean, not close to land. Associated with deep sea fishing.

English has so many of these little differences that mean so much. It must be a nightmare to learn it as a second language and master all of these microscopic things.

"I go fishing in the sea." = underwater; you'd rarely use this.

"I go fishing on the sea." = no meaning.

"I go fishing at sea." = out on a boat; usual usage.

"I go fishing by the sea." = you're on land; also usual.

Agree, Jim...