The Trivium consists of the three verbal arts of grammar, dialectic (or logic), and rhetoric.  Grammar is emphasized in the early years of a student's education and provides the necessary foundation.  Dialectic requires critical thinking and builds on that foundation as students reason with what they learn.  Rhetoric involves applying the use of knowledge and reason in engaging, teaching, or persuading others.

For example, a grammar student might learn to locate countries on a map.  A dialectic student would consider how and why those borders were established.  A rhetoric student might debate the moral grounds on which those decisions were made and their implications for regional stability.

The Trivium might well be compared with the Bible's emphasis on knowledge, understanding, and wisdom.  These are essential tools of learning that focus not merely on what to think, but how to think; tools that may be effectively applied in life to any subject that a lifelong student may encounter.

The Quadrivium consists of the four mathematical arts including arithmetic, geometry, music, and astronomy.  Arithmetic is the art (or skill) that involves the properties of numbers and magnitude.  Geometry deals with the properties of shapes or magnitude at rest.  Music (not merely as we typically understand it) is the art of ratios, proportions, and harmony.  Astronomy involves the classical art of shapes or magnitude in time or motion (the movement of the stars, for example).

It has been said that the Trivium is about knowing who you are while the Quadrivium is about knowing where you are.  In that sense, the Quadrivium provides a student with knowledge and understanding of the world around them and serves as a gateway through which students may take up the four sciences.

In his letter to the church at Rome, Paul asserts that God's invisible attributes are clearly visible and may be understood through the things that He has made.  Through His creation, all mankind may understand something of His eternal power and Divine nature (Romans 1:20).  But although the heavens declare the glory of God and the firmament shows His handiwork, many refuse to glorify God and will be left without excuse before the bar of His justice.  For those students who apply themselves to a classical education involving the Quadrivium, it is a means through which they learn to glorify God and be thankful. 

The Natural Sciences are the sciences of the physical order such as biology, chemistry, and physics.  Physics involves understanding forces that bring about physical change, chemistry deals with the elements of which physical things consist, and biology deals with being and change within and among living things.

The Human Sciences are the sciences of moral order involving human behavior and the soul.  The human sciences will include studies in politics, ethics, history, art, and aesthetics to name a few.  The purpose of the human sciences is to pursue understanding of human behavior with the aim of cultivating virtue.

The Philosophical Sciences are the sciences of metaphysics and epistemology.  The aim of the philosophical sciences is to understand the causes and limits of human knowledge.

The Theological Sciences are the sciences concerned with knowledge of the First Cause, or God Himself.  All other sciences and subjects point to and culminate in theological understanding and should serve to order knowledge in relationship to our Creator.  The theological sciences acknowledge the limits of general revelation and point to God Himself as the source and provider of special revelation.