I read a novel as a teen about a girl who was trained by her scientist father to break 4th dimension. I remember isolation tanks and chess games were involved. The father really pushed his daughter through the experiments. There may have been a phrase like breaking through the wall or elevator... I read this in the 1980's. Yes, she did break through!
The Universe Between by Alan Nourse? It's in two parts - in the second part her son has visited the other dimension many times, and can function in it. From the Google Books description:
Bob Benedict is one of the few scientists able to make contact with the invisible, dangerous world of The Thresholders and return - sane! For years he has tried to transport - and receive - matter by transmitting it through the mysterious, parallel Threshold. At first his efforts met only with failure and madness. But now The Thresholders have risen in fury. Somehow Bob Benedict must make one more trip into that land of peril and pacify them before they succeed in hurling his planet - piece by piece - into the oblivion of infinity.