How did farmer's go about tilling fields and planting before the invention of the plow? From what I understand, it's a fairly recent invention.
Pliny has an entire section on types of ploughs in his Natural History, written in the first century AD. The plough is also an essential piece of imagery in the story (semi-mythical) of Cinncinatus (born in the 6th century):
From Livy, Bk 3
The one hope of Rome, L. Quinctius, used to cultivate a four-acre field on the other side of the Tiber, just opposite the place where the dockyard and arsenal are now situated; it bears the name of the " Quinctian Meadows." There he was found by the deputation from the senate either digging out a ditch or ploughing, at all events, as is generally agreed, intent on his husbandry.
So how "old school" are you meaning? Or are you talking about a specific technique?
While the plough has been around for a long time (over 5,000 years), there are places where other tools were used into recent times, or are still used today. These tools were also used before the plough in places where the plough replaced them. The alternative tools include digging sticks (straight sticks), hoes/spades, and a variety of specialised spade-like tools usually called "foot ploughs". Where animal traction (or modern mechanical traction) was not available, the plough was not used. Where land unsuited to animal traction (e.g., too steep, too small, inaccessible) or otherwise usuited to ploughing (e.g., too stony) was cultivated, digging sticks, hoes, and foot ploughs were used by people who knew and used animal-drawn ploughs.
Digging sticks have been used recently/are used currently in North America/Mesoamerica, New Guinea, The Horn of Africa, sub-Saharan Africa, and Polynesia.
Hoes have been used recently/are used currently in sub-Saharan Africa.
Foot ploughs have been used recently in Scotland, South America, New Zealand (Maori), and Japan (Ainu).
Grigg, D. B., Agricultural Systems of the World: An Evolutionary Approach, Cambridge University Press, 1974 has a concise global coverage of such tools.