I remember that I had to learn division in my childhood, I could handle all the other mathematical concepts that were presented until then but division was a real pain to learn, somehow the idea of $n$ divided by $m$ equals $p$ was devilishly to grasp.
Then my father tried to teach me, and then he reworded it:
$$m \text{ divided by }n$$
To:
$$ \text{ how many n's are in m?}$$
Somehow, the rewording made complete sense for me, I could understand what is division about. Until the present date, thinking in the first form still doesn't make sense to me, if I think:
$$4\div 2=2$$
$$6\div 2=3$$
I know the answer because I've memorized the set of numbers, but if I think:
$$1657\div 1254$$
The meaning of $\div$ gets completely obscure and I have to reword it to perform the desired operation. It happened with me, I guess the effect do exist and I believe it could be very important to learn about it and use it to learn advanced concepts in higher mathematics. I don't know if there are studies about it (the nearest thing I know about is Polya's work) - But I'm almost certain that there may be a thing or two.