I've been unable to find any comments directly from Roger Waters (writer of the music and lyrics of the song) on the meaning of those lines. Looking at the lyrics on Genius.com, the annotation for the first line says that if you read it as a double negative, it could mean two different things
- "We do need education"—A call for education
- "We don't need this type of education"—A stab at the education the children are getting
Personally, I'd be inclined to disagree with this assessment. I've always thought that it was just slang for "We don't need education", for a few reasons.
First, if you remember the second time they sing through the verse, it's just a bunch of kids singing it; "we don't need no" is absolutely something a school age working class kid with a Cockney accent in England in the 70s might say instead of "we need no" or "we don't need".
Second, either potential meaning of the double negative doesn't really make sense for the second line. "We need thoughts control"? "We don't need this type of thoughts control"? Neither of those make sense, at least to me. And if the second line is not a double negative, then why would the first one be? Furthermore, do children need "dark sarcasm in the classroom"? I don't think so either.
Third, the whole song is a rebellion against the school, and when "Another Brick In the Wall" plays in The Wall film, Pink has a day dream of the children chanting "we don't need no education", destroying the school building with sledgehammers and crowbars, setting it on fire, and dragging the teachers out kicking and screaming. To me, this doesn't say "we need education" or "we need a different type of education", it says "we don't need education at all". I don't know if the call for alternative education is even something a kid of that age would think of; school is the way that it is and Pink hates it.