The training method of a Culdee is called the Quaternary which in Greek is Tetraktys, and in Gaelic is Cethrair. In English we say, Path of the Four, or Fourfold path.

It is a fourfold program which addresses the development of the Eikasia, Pistis, Dianoia, and Noetic levels of human consciousness.

The Quaternary is by analogy a temple which has one spire, two pillars, three steps and four gates.
Its symbol is the Pythagorean Tetraktys.
The spire, or aim, is Theosis and its symbol is the Monad, or point within the circle.
The pillars are initiation and revelation and their symbols are John the Baptist and John the Evangelist, because the professions of these two saints obviously allude to those two pillars.
The steps are Purgative, Illuminative and Unitive and their symbols are Faith, Hope and Charity.
The gates are, History, Ethics, Physics and Theology and their symbols are Earth, Water, Air and Fire. (Which are in turn symbolized by, the Lia Fáil, the Dagda's Cauldron, the Claíomh Solais, and the Spear Luin.)
The Gates, or elements, of the Quaternary are described by John Scotus Eriugena in his homily called Voice of the Eagle as, "a certain intelligible world, constituted of its four parts, its four elements. Whose earth, as it were, in the midst, at the lowest point, like a center, is history. Surrounding it, like the waters, is the abyss of moral understanding, that the Greeks are wont to call Ethike. And in this intelligible world, around these two, as it were, lower parts, that I have called history and ethics, floats what I call natural knowledge or knowledge of nature, that the Greeks called physike. Rolled around, outside and beyond all, is the celestial and burning fire of the empyrean heaven, that sublime contemplation that the Greeks named "theology," beyond which no intelligence passes."

We begin training in the Purgative training phase.

The Purgative training phase has one primary aim, and three parts.
The primary aim is to develop Justice within ourselves, (also called, purification of the heart).
The three parts are, Initiation, History, and Ethike.

In the Purgative training phase we begin with an initiation ceremony which is designed to empower the student to internalize the subject matter of the training phase.
We follow the initiation with the the study of mundane and divine history.
Mundane history is studied to help the student come to see the transient and impermanent nature of the material world, worldly things, and worldly affairs. The study of mundane history also helps the student learn the relative uselessness of the study of history for spiritual development. For example the student comes to see that giving the historicity, for instance, of the Bible, exaggerated importance does nothing to develop the student spiritually.
Mundane history's uselessness in spiritual development is due to mundane history being nothing but a subjective reflection of a impermanent, relativistic and subjective material world.
Simultaneously we study mythology, or divine history, in an effort to show the student that it is in the Myths that we find those useful allegories that, when internalized, truly can improve us, ethically, intellectually and spiritually.
To explain it another way, Mundane History shows the student the vain and impermenant nature of all temporal activity. Divine History, or the Myths, teaches us useful lessons though example, symbol and allegory, that pertain to all four levels of our spiritual experience.

Finally we continue the Purgative phase of training with the study and internalization of Ethike.

The study of Ethike, or Ethics, teaches us through careful training in, Reason and the Four Cardinal Virtues, to live and act in natural harmony with the Universe, our fellow creatures, and the Divine.
With concentrated training in rational ethics we gain peace of mind and generate in ourselves what Pythagoras called Universal Friendship.
This is primarily accomplished by training in attention to our thoughts, and coming to see the material world for what it truly is, what we are relative to the material world, and what we are relative to God.
For guidance in this training phase we rely on the aforementioned Myths, writings like the Pythagorean Golden Verses, (which mirrors the biblical Decalogue), Stoic methods of impression recognition, John Cassian's conferences, the Alphabet of Devotion, the sayings of the Jesus and other apostles, and any other useful training aids we can find.

The second training phase is the Illuminative Training Phase.