Jesus’ teachings, like those of his contemporaries, consisted entirely of what was then called the Oral Torah. Contrary to popular belief, the written Torah was virtually inaccessible to all except a subset of the Temple Priests (essentially the High Priest and his direct acolytes). Apart from the High Holidays when the scrolls were read to thousands gathered in and around the Temple, the temple Priests did not teach. They were consumed with the duties involved with the daily sacrifices.
The Oral Torah encompassed the written Torah and the oral teachings (mostly parables), legal rulings, and commentary that had been developed over the previous millennia. The Oral Torah was a vast body of knowledge whose teaching reflected the personal interpretations of the particular Master doing the teaching.
In Jesus’ time, two major schools of Oral Torah predominated. The Be’it Hillel and the Be’it Shammai (house of Hillel and house of Shammai, respectively). A Master would have taught one or the other of these schools. However, even though Jesus’ teachings were closely inline with Be’it Hillel, Jesus teaching of the laws of marriage and divorce were identical to those of the school of Shammai.
Arguably, the two most famous sages of the Hillel School, Gamaliel, the Elder and his grandson, R. Gamaliel exerted profound influence on Saul of Tarsus (aka St. Paul). Paul claimed to have been a talmid of R. Gamaliel. Finally, Gamaliel is encountered in Acts 5:34 where he saves Peter and the apostles from execution for apostacy at the hands of the Sanhedrin.
After the destruction of the Temple in 70 CE, the school of Shammai died out and 200 years later, the sages of the school of Hillel codified the Oral Torah and established the beginning of the Jewish Rabbinate that we know today.