one period after another.

It is awkward, and not redolent to naive speech. In a very lame fashion, that's what it means, but it is execrable English. Period in the United States means a brief measure of time, usually an hour, within the structure of a school day without reference to the subject to be taught. "I have chemistry during third period, then English until the lunch break." The other common use of period to mean a fragment of time is an era in which a certain cultural, intellectual or political paradigm holds sway: "The inter war years, between 1919 and 1939, saw the rise of fascism, and so can be called the fascist period of European history."

Thanks, Setanta. But how should it be phrased?

One might write: "Shakyamuni Buddha gradually taught higher level concepts one step after another." One might also say ". . . lesson after another." It is also facile and silly to write thatt someone taught teachings.

Thanks, Setanta, for the suggestion and for pointing out that "teachings" is the wrong word.

I don't have trouble with the use of "higher level teachings."

From the Heart of the Buddha's Teachings:

"Thich Nhat Hanh introduces us to the core teachings of Buddhism and shows us that the Buddha's teachings are accessible and applicable to our daily lives . . .Covering such teachings as the Four Noble Truths, the Noble Eightfold Path, . . . etc. etc. ."

Within the teachings, the breakdown for many of the subjects are referred to as levels, factors, aggregates, powers, and links.