For whatever reasons, industrialization leads to lower population growth.
It isn't industrialization but affluence. The average upper middle class parents could easily afford one, possibly two children going to Yale, but not four or five. Raising children in a middle-to-upper-middle class lifestyle is prohibitively expensive. Those economic groups that have affluence self-regulate procreation simply to preserve their lifestyles for the most part.
What's needed to arrest global population growth is to provide education, engineering expertise, contraception, and economic assistance to developing nations so they can modernize their economies ASAP.
This will do nothing to curb the population. Economic growth typically doesn't happen in the third world primarily because those countries tend not to have stable legal systems. It is impossible to conduct business with someone when you can't enforce things like contract law. Even those of modest education are capable of self sustaining enterprise. Many corporations have started, and small businesses run by individuals who lack a formal education. The pattern is that those business concerns again rely on a sound framework (legal and financial) to act as a foundation for trade.
Arithmetic is being able to count up to twenty without taking off your shoes. -- Mickey Mouse