I don’t think so.!!!

she's wrong.
you do think.
but the way you think, ie your opinion does not align with the stated one, thus,
i don't think so,
is nearer in meaning i do not share that opinion.
the so here refers to the stated opinion.
i do not think so, in that way, i think in my way.

“I think not” seems to me to be saying that you do not think. It is the addition of so that gives the phrase meaning.

I think not is a term I'm used to hearing

A sort of formal way of saying "no"

Lol my own punctuation makes it look like I am saying people tell me I don't think..

So I see your point

I just think people who use it know what it means

"I think therefore I am"
I don't think therefore I am not?

Whenever my elders said "I think not!" they were very definitely refusing me permission to do or say something.

"I don't think so." could either mean that the speaker did not entirely agree with whoever they were talking to, or could be used by an adult speaking to a child in the sense of "We'll see." an infuriating phrase that children of my generation rightly took to me an "No."

But strictly speaking, "I don't think so." implies that you do not think that whatever was said was correct, but you do no definitely know that it was incorrect, whereas "I think not." used in the sense OP's mother did, infers that she knew for a fact that the expression opinion was wrong.

I suspect, but have no proof, that there may be a difference between Scottish and English use of these two expressions.

Sorry about the clerical errors - I read this through twice!

Would you believe it?

And still hit the space bar wrongly me an instead of mean and left out a t, thus writing no instead of not!

IMO ‘ I think not!’ is considerably more definite than ‘I think so’, esp. if followed by the exclamation mark, whether written or verbal.

I can’t see that ‘I think not’ has ever meant that the speaker doesn’t think. It’s been used in the ‘I don’t think so’ sense by native speakers for ages.
I could understand a non-native speaker getting the wrong idea, though.

My mother was Welsh., so perhaps that is a clue.

Whenever I said, I think not! to my pupils, it meant a firm NO!

Ilovecheese

"I think therefore I am"
I don't think therefore I am not?

Renee Descartes said “cogito ergo sum” or “I think therefore I am.” Other philosophers contend you only need the I think as the I exists to do the thinking. Same argument with I think not. There has to be an I to not think. Hope that’s clear.