A close reading of the Beraisa both here and in Shabbos (150a) will reveal that the Gemara is not permitting us to teach our children a trade on Shabbos, but rather it is permitting us to speak - on Shabbos - with someone who teaches a trade, about the possibility of teaching our children. This is clear from the wording of the Beraisa: "[we are permitted on Shabbos] to make a Shiduch for a daughter to get married and for a son to learn Torah or to learn a trade." The action - making a Shiduch - is referring to the son as well.

The language of the Shulchan Aruch (OC 306:6) is more ambiguous, but it is equally clear that the Mechaber is only referring to arranging a teacher; immediately after stating this Halachah he qualifies it by saying that one is only allowed to speak of the possibility of hiring the teacher but not to actually hire him or to discuss a salary.

This interpretation of the Beraisa and the Shulchan Aruch is stated explicitly by the Aruch ha'Shulchan (op cit).

Also, with this understanding of the Beraisa we avoid a contradicition with another Halachah of the Shulchan Aruch. In OC 307:17 he writes in the name of the Rambam that it is forbidden to learn anything on Shabbos other than Torah. The Ramban disagrees and his opinion is brought by the Mechaber as a Yesh Omrim, and the Mishnah Berurah says that the Minhag is to be lenient like the Ramban, but it seems that the Mechaber himself follows the opinion of the Rambam. According to this Halachah it would be problematic to teach one's child a trade on Shabbos, and even to allow the child to study himself from books would not seem to be good Chinuch.

Kol Tuv,

Yonasan Sigler